Slide 1

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

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A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

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A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

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A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
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A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
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A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

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A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
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A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

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A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
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A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

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A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

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Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

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A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
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Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

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A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

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A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

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A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

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A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

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A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

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A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
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A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

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A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

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A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

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A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

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A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 2

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 3

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 4

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

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A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

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A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 5

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

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A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

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Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

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A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

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A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

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A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
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A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 6

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

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A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 7

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 8

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 9

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

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A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

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A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

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A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 10

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

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Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

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Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

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A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

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Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

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Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

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Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

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A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
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Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 11

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

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A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

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A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 12

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 13

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 14

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

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A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

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A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

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A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

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A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

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A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 15

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

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Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

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Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

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Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

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Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

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Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 16

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

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A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 17

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 18

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 19

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

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A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

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A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

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A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

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A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 20

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

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Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

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A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

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A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
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Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 21

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

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A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

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A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

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A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
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A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 22

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

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A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 23

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 24

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 25

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

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Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

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Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

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Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

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A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
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Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 26

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 27

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

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A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 28

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

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A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

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A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

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A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
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A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 29

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

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A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

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A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

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A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
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A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

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A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

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A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
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A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 30

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

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Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 31

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

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A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

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A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

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A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
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A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 32

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Slide 33

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

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A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

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A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
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A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 34

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

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A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

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A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

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A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

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A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 35

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

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Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 36

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 37

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Slide 38

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

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A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

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A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
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A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 39

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

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A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

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A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 40

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

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Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

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Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 41

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 42

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

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A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

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A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

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Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

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A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Slide 43

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

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A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

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A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

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A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

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A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
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A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 44

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

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A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

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A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
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A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

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A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

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A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 45

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 46

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 47

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

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A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Slide 48

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

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A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

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A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 49

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

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A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 50

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 51

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

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A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 52

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

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A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Slide 53

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 54

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

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A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 55

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
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A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 56

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

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A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

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A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

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A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

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A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

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A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

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A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 57

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

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A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Slide 58

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

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A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

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A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

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A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

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A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

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Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 59

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

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A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 60

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 61

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

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A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
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A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

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A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

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A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

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A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

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A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 62

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

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A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

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A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

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A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Slide 63

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

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A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 64

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 65

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

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A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
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A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 66

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

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A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
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A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

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A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
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A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
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A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
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A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
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A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

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A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

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A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
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A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

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A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
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A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

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A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

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A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

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A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

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A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

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A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

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A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
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A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
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A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

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A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 52

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 53

Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 54

Application A.6

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A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 56

Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 58

Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 60

Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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A – 61

Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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A – 63

Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 64

Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 65

Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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A – 66

Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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A – 67

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 68


Slide 67

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
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Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
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Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

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Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
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Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

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Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

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Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

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Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

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Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

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Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Application A.6

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 55

Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 57

Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 59

Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 62

Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

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Slide 68

A

Decision Making

PowerPoint Slides
by Copyright
Jeff Heyl
© 2010 Pearson Education, Inc. Publishing as Prentice Hall.

For Operations Management, 9e by
Krajewski/Ritzman/Malhotra
© 2010 Pearson Education
A–1

Break-Even Analysis
 Evaluating Services or Products
 Is

the predicted sales volume of the service or
product sufficient to break even (neither
earning a profit nor sustaining a loss)?

 How

low must the variable cost per unit be to
break even, based on current prices and sales
forecasts?

 How

low must the fixed cost be to break even?

 How

do price levels affect the break-even
quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–2

Break-Even Analysis
 Break-even analysis is based on the assumption
that all costs related to the production of a specific
service or product can be divided into two
categories: variable costs and fixed costs
 Variable cost, c, is the portion of the total cost that
varies directly with volume of output

 If Q = the number of customers served or units
produced per year, total variable cost = cQ
 Fixed cost, F, is the portion of the total cost that
remains constant regardless of changes in levels
of output

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–3

Break-Even Analysis
 So
Total cost = F + cQ
Total revenue = pQ
 By setting revenue equal to total cost

pQ = F + cQ
F
Q=
p-c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–4

Finding the Break-Even Quantity
EXAMPLE A.1
A hospital is considering a new procedure to be offered at $200
per patient. The fixed cost per year would be $100,000, with
total variable costs of $100 per patient. What is the break-even
quantity for this service? Use both algebraic and graphic
approaches to get the answer.
SOLUTION
The formula for the break-even quantity yields

100,000
F
=
Q=
= 1,000 patients
p-c
200 – 100

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–5

Finding the Break-Even Quantity
 To solve graphically we plot two lines: one for costs and one
for revenues
 Begin by calculating costs and revenues for two different
output levels
 The following table shows the results for Q = 0 and Q = 2,000
Quantity
(patients) (Q)

Total Annual Cost ($)
(100,000 + 100Q)

Total Annual Revenue ($)
(200Q)

0

100,000

0

2,000

300,000

400,000

 Draw the cost line through points (0, 100,000) and (2,000,
300,000)
 Draw the revenue line through (0, 0) and (2,000, 400,000)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–6

Finding the Break-Even Quantity

Dollars (in thousands)

400 –

300 –

(2000, 400)

Profits

Total annual revenues

(2000, 300)
Total annual costs

200 –

Break-even quantity

100 –
Loss

0–

Fixed costs
|

|

|

|

500

1000

1500

2000

 The two lines
intersect at
1,000 patients,
the break-even
quantity

Patients (Q)
FIGURE A.1 – Graphic Approach to Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–7

Application A.1
The Denver Zoo must decide whether to move twin polar bears
to Sea World or build a special exhibit for them and the zoo.
The expected increase in attendance is 200,000 patrons. The
data are:
Revenues per Patron for Exhibit
Gate receipts
$4
Concessions
$5
Licensed apparel
$15
Estimated Fixed Costs
Exhibit construction
Salaries
Food

$2,400,000
$220,000
$30,000

 Is the
predicted
increase in
attendance
sufficient to
break even?

Estimated Variable Costs per Person
Concessions
$2
Licensed apparel
$9
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–8

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

7–

Cost and revenue
(millions of dollars)

6–
5–
4–
3–
2–
1–
|

0–

|

|

|

|

|

50

100

150

200

250

Q (thousands of patrons)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A–9

Application A.1
Q
0
250,000

TR = pQ
$0
$6,000,000

TC = F + cQ
$2,650,000
$5,400,000

Where
p = 4 + 5 + 15 = $24
F = 2,400,000 + 220,000 + 30,000
= $2,650,000
c = 2 + 9 = $11

Algebraic solution of Denver Zoo problem
pQ = F + cQ
24Q = 2,650,000 + 11Q
13Q = 2,650,000
Q = 203,846

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 10

Sensitivity Analysis
EXAMPLE A.2
If the most pessimistic sales forecast for the proposed service
in Figure A.1 were 1,500 patients, what would be the
procedure’s total contribution to profit and overhead per year?
SOLUTION

The graph shows that even the pessimistic forecast lies above
the break-even volume, which is encouraging. The product’s
total contribution, found by subtracting total costs from total
revenues, is
pQ – (F + cQ) = 200(1,500) – [100,000 + 100(1,500)]
= $50,000

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A – 11

Evaluating Processes
 Choices must be made between two
processes or between an internal process
and buying services or materials on the
outside
 We assume that the decision does not
affect revenues
 The analyst finds the quantity for which the
total costs for two alternatives are equal

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 12

Evaluating Processes
 Let Fb equal the fixed cost (per year) of the buy
option, Fm equal the fixed cost of the make option,
cb equal the variable cost (per unit) of the buy
option, and cm equal the variable cost of the make
option
 The total cost to buy is Fb + cbQ and the total cost
to make is Fm + cmQ
 To find the break-even quantity, we set the two
cost functions equal and solve for Q:

Fb + cbQ = Fm + cmQ
Fm – Fb
Q= c –c
b
m
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 13

Break-Even Analysis for
Make-or-Buy Decisions
EXAMPLE A.3
 A fast-food restaurant featuring hamburgers is adding salads
to the menu
 The price to the customer will be the same
 Fixed costs are estimated at $12,000 and variable costs
totaling $1.50 per salad
 Preassembled salads could be purchased from a local
supplier at $2.00 per salad
 Preassembled salads would require additional refrigeration
with an annual fixed cost of $2,400
 Expected demand is 25,000 salads per year
 What is the break-even quantity?

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 14

Break-Even Analysis for
Make-or-Buy Decisions
SOLUTION
The formula for the break-even quantity yields the following:

Fm – Fb
Q= c –c
b
m
=

12,000 – 2,400
= 19,200 salads
2.0 – 1.5

Figure A.2 shows the solution from OM Explorer’s BreakEven Analysis Solver. The break-even quantity is 19,200
salads. As the 25,000-salad sales forecast exceeds this
amount, the make option is preferred. Only if the restaurant
expected to sell fewer than 19,200 salads would the buy
option be better.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 15

Break-Even Analysis for
Make-or-Buy Decisions

FIGURE A.2 – Break-Even Analysis
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.
Solver of OM Explorer for Example A.3

A – 16

Application A.2
At what volume should the Denver Zoo be indifferent
between buying special sweatshirts from a supplier or have
zoo employees make them?
Buy

Make

Fixed costs

$0

$300,000

Variable costs

$9

$7

$300,000 – $0
Fm – Fb
Q= c –c =
= $150,000
$9
–
$7
b
m

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 17

Preference Matrix
 A Preference Matrix is a table that allows you to
rate an alternative according to several
performance criteria
 The criteria can be scored on any scale as long as
the same scale is applied to all the alternatives
being compared
 Each score is weighted according to its perceived
importance, with the total weights typically
equaling 100
 The total score is the sum of the weighted scores
(weight × score) for all the criteria and compared
against scores for alternatives
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A – 18

Evaluating an Alternative
EXAMPLE A.4
The following table shows the performance criteria, weights,
and scores (1 = worst, 10 = best) for a new thermal storage air
conditioner. If management wants to introduce just one new
product and the highest total score of any of the other product
ideas is 800, should the firm pursue making the air conditioner?
Performance Criterion

Weighted Score (A  B)

Weight (A)

Score (B)

Market potential

30

8

240

Unit profit margin

20

10

200

Operations compatibility

20

6

120

Competitive advantage

15

10

150

Investment requirements

10

2

20

5

4

20

Project risk

Weighted score =

750

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 19

Evaluating an Alternative
SOLUTION
Because the sum of the weighted scores is 750, it falls short of
the score of 800 for another product. This result is confirmed by
the output from OM Explorer’s Preference Matrix Solver in
Figure A.3.

Total

750

FIGURE A.3 – Preference Matrix Solver for Example A.4
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 20

Application A.3
Performance Criterion

Weight

Score

Market potential

10

5

Unit profit margin

30

8

Operations compatibility

20

10

Competitive advantage

25

7

Investment requirements

10

3

5

4

Project risk

Weighted Score

Total weighted score =

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 21

Application A.3
Performance Criterion

Weight

Score

Weighted Score

Market potential

10

5

50

Unit profit margin

30

8

240

Operations compatibility

20

10

200

Competitive advantage

25

7

175

Investment requirements

10

3

30

5

4

20

Project risk

Total weighted score = 715

 Repeat this process for each alternative – pick the one
with the largest weighted score

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 22

Criticisms
 Requires the manager to state criterion
weights before examining alternatives, but
they may not know in advance what is
important and what is not
 Allows one very low score to be overridden
by high scores on other factors
 May cause managers to ignore important
signals

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 23

Decision Theory
 Decision theory is a general approach to decision
making when the outcomes associated with
alternatives are in doubt
 A manager makes choices using the following
process:
1. List a reasonable number of feasible alternatives
2. List the events (states of nature)
3. Calculate the payoff table showing the payoff for each
alternative in each event
4. Estimate the probability of occurrence for each event
5. Select the decision rule to evaluate the alternatives

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A – 24

Decision Making Under Certainty
 The simplest solution is when the
manager knows which event will occur
 Here the decision rule is to pick the
alternative with the best payoff for the
known event

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A – 25

Decisions Under Certainty
EXAMPLE A.5
 A manager is deciding whether to build a small or a large
facility

 Much depends on the future demand
 Demand may be small or large
 Payoffs for each alternative are known with certainty
 What is the best choice if future demand will be low?
Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 26

Decisions Under Certainty
SOLUTION
 The best choice is the one with the highest payoff
 For low future demand, the company should build a small
facility and enjoy a payoff of $200,000
 Under these conditions, the larger facility has a payoff of
only $160,000

Possible Future Demand
Alternative

Low

High

Small facility

200

270

Large facility

160

800

0

0

Do nothing
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 27

Decision Making Under Uncertainty
 Can list the possible events but can not
estimate the probabilities
1. Maximin: The best of the worst, a pessimistic
approach
2. Maximax: The best of the best, an optimistic
approach

3. Laplace: The alternative with the best weighted
payoff assuming equal probabilities
4. Minimax Regret: Minimizing your regret (also
pessimistic)

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A – 28

Decisions Under Uncertainty
EXAMPLE A.6
Reconsider the payoff matrix in Example A.5. What is the best
alternative for each decision rule?
SOLUTION
a. Maximin. An alternative’s worst payoff is the lowest number
in its row of the payoff matrix, because the payoffs are
profits. The worst payoffs ($000) are
Alternative

Worst Payoff

Small facility

200

Large facility

160

The best of these worst numbers is $200,000, so the
pessimist would build a small facility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 29

Decisions Under Uncertainty
b. Maximax. An alternative’s best payoff ($000) is the highest
number in its row of the payoff matrix, or

Alternative

Best Payoff

Small facility

270

Large facility

800

The best of these best numbers is $800,000, so the optimist
would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 30

Decisions Under Uncertainty
c. Laplace. With two events, we assign each a probability of
0.5. Thus, the weighted payoffs ($000) are

Alternative

Weighted Payoff

Small facility

0.5(200) + 0.5(270) = 235

Large facility

0.5(160) + 0.5(800) = 480

The best of these weighted payoffs is $480,000, so the
realist would build a large facility

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 31

Decisions Under Uncertainty
d. Minimax Regret. If demand turns out to be low, the best
alternative is a small facility and its regret is 0 (or 200 –
200). If a large facility is built when demand turns out to be
low, the regret is 40 (or 200 – 160).
Regret
Alternative

Low Demand

High Demand

Maximum
Regret

Small facility

200 – 200 = 0

800 – 270 = 530

530

Large facility

200 – 160 = 40

800 – 800 = 0

40

The column on the right shows the worst regret for each
alternative. To minimize the maximum regret, pick a large
facility. The biggest regret is associated with having only a
small facility and high demand.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 32

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 33

Application A.4
Fletcher (a realist), Cooper (a pessimist), and Wainwright (an
optimist) are joint owners in a company. They must decide
whether to make Arrows, Barrels, or Wagons. The government
is about to issue a policy and recommendation on pioneer
travel that depends on whether certain treaties are obtained.
The policy is expected to affect demand for the products;
however it is impossible at this time to assess the probability of
these policy “events.” The following data are available:
Payoffs (Profits)
Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 34

Application A.4
 Which product would be favored by Fletcher (realist)?
 Which product would be favored by Cooper (pessimist)?
 Which product would be favored by Wainwright (optimist)?
 What is the minimax regret solution?
Payoffs (Profits)
Alternative

 The short answers
are:

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

 Fletcher (realist – Laplace) would choose arrows
 Cooper (pessimist – Maximin) would choose barrels
 Wainwright (optimist – Maximax) would choose wagons
 The Minimax Regret solution is arrows
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 35

Decisions Under Risk
 The manager can list the possible events
and estimate their probabilities
 The manager has less information than
decision making under certainty, but more
information than with decision making
under uncertainty
 The expected value rule is widely used
 This rule is similar to the Laplace decision
rule, except that the events are no longer
assumed to be equally likely
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 36

Decisions Under Risk
EXAMPLE A.7
Reconsider the payoff matrix in Example A.5. For the expected
value decision rule, which is the best alternative if the
probability of small demand is estimated to be 0.4 and the
probability of large demand is estimated to be 0.6?
SOLUTION
The expected value for each
alternative is as follows:

Alternative

Possible Future
Demand
Alternative

Small

Large

Small facility

200

270

Large facility

160

800

Expected Value

Small facility

0.4(200) + 0.6(270) = 242

Large facility

0.4(160) + 0.6(800) = 544

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 37

Application A.5
For Fletcher, Cooper, and Wainwright, find the best decision
using the expected value rule. The probabilities for the events
are given below.
Payoffs (Profits)

What alternative has
the best expected
results?

Alternative

Land Routes
No treaty

Land Routes
Treaty

Sea Routes
Only

Arrows

$840,000

$440,000

$190,000

Barrels

$370,000

$220,000

$670,000

Wagons

$25,000

$1,150,000

($25,000)

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Arrows

(0.50)(840,000)

+

(0.30)(440,000) + (0.20)(190,000)

= 590,000

Barrels

(0.50)(370,000)

+

(0.30)(220,000) + (0.20)(670,000)

= 385,000

Wagons

(0.50)(25,000)

+

(0.30)(1,150,000) +

Sea routes,
Only (0.20)

(0.20)(-25,000)

Expected
Value

= 352,500

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 38

Decision Trees
 Are schematic models of available
alternatives and possible consequences
 Are useful with probabilistic events and
sequential decisions
 Square nodes represent decisions
 Circular nodes represent events

 Events leaving a chance node are
collectively exhaustive

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 39

Decision Trees
 Conditional payoffs for each possible
alternative-event combination shown at
the end of each combination
 Draw the decision tree from left to right
 Calculate expected payoff to solve the
decision tree from right to left

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 40

Decision Trees
E1 & Probability
E2 & Probability
E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Alternative 3

1
1st
decision

2

Alternative 4
Alternative 5

Possible
2nd decision

E2 & Probability
= Event node

E3 & Probability

Payoff 1
Payoff 2
Payoff 3

Payoff 1
Payoff 2

= Decision node
Ei = Event i
P(Ei) = Probability of event i

FIGURE A.4 – A Decision Tree Model

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 41

Decision Trees
 After drawing a decision tree, we solve it
by working from right to left, calculating
the expected payoff for each of its
possible paths
1. For an event node, we multiply the payoff of
each event branch by the event’s probability
and add these products to get the event
node’s expected payoff
2. For a decision node, we pick the alternative
that has the best expected payoff

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 42

Decision Trees
 Various software is available for drawing
and/or analyzing decision trees, including


PowerPoint can be used to draw decision
trees, but does not have the capability to
analyze the decision tree



POM with Windows



SmartDraw



PrecisionTree decision analysis from Palisade
Corporation



TreePlan

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 43

Analyzing a Decision Tree
EXAMPLE A.8
 A retailer will build a small or a large facility at a new location
 Demand can be either small or large, with probabilities
estimated to be 0.4 and 0.6, respectively
 For a small facility and high demand, not expanding will have
a payoff of $223,000 and a payoff of $270,000 with expansion
 For a small facility and low demand the payoff is $200,000
 For a large facility and low demand, doing nothing has a
payoff of $40,000
 The response to advertising may be either modest or sizable,
with their probabilities estimated to be 0.3 and 0.7,
respectively
 For a modest response the payoff is $20,000 and $220,000 if
the response is sizable
 For a large facility and high demand the payoff is $800,000
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 44

Analyzing a Decision Tree
SOLUTION

The decision tree in Figure A.5 shows the event probability
and the payoff for each of the seven alternative-event
combinations. The first decision is whether to build a small or
a large facility. Its node is shown first, to the left, because it is
the decision the retailer must make now. The second decision
node is reached only if a small facility is built and demand
turns out to be high. Finally, the third decision point is
reached only if the retailer builds a large facility and demand
turns out to be low.

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 45

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise
Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 46

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

0.3 x $20 = $6

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$6 + $154 = $160

Sizable response [0.7]

$220

0.7 x $220 = $154
High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 47

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand
$270

1

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 48

Analyzing a Decision Tree
Low demand [0.4]
$200

Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 49

Analyzing a Decision Tree
Low demand [0.4]
$200

x 0.4 = $80

$80 + $162 = $242
Don’t expand
$223

2

Expand

$270
1

$270

x 0.6 = $162

Do nothing
$40

3

Modest response [0.3]

$20

Advertise

$160
$160

Sizable response [0.7]

$220

High demand [0.6]
$800
FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 50

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

3

$20

Advertise

$160
0.4 x $160 = $64

$544

Modest response [0.3]

$160

High demand [0.6]
$800

Sizable response [0.7]

$220

x 0.6 = $480

FIGURE A.5 – Decision Tree for Retailer (in $000)

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

A – 51

Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

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Analyzing a Decision Tree
Low demand [0.4]
$200

$242
Don’t expand
$223

2

Expand

$270
1

$270
Do nothing
$40

$544

3

$20

Advertise

$160
$160

$544

Modest response [0.3]

Sizable response [0.7]

$220

High demand [0.6]
$800

FIGURE A.5 – Decision Tree for Retailer (in $000)

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Application A.6
a. Draw the decision tree for the Fletcher, Cooper, and
Wainwright Application A.5 problem
b. What is the expected payoff for the best alternative in the
decision tree below?

Alternative

Land routes,
No Treaty
(0.50)

Land Routes,
Treaty Only
(0.30)

Sea routes,
Only (0.20)

Arrows

840,000

440,000

190,000

Barrels

370,000

220,000

670,000

Wagons

25,000

1,150,000

-25,000

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Application A.6

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Solved Problem 1
 A small manufacturing business has patented a new device
for washing dishes and cleaning dirty kitchen sinks
 The owner wants reasonable assurance of success
 Variable costs are estimated at $7 per unit produced and
sold
 Fixed costs are about $56,000 per year
a. If the selling price is set at $25, how many units must be
produced and sold to break even? Use both algebraic
and graphic approaches.
b. Forecasted sales for the first year are 10,000 units if the
price is reduced to $15. With this pricing strategy, what
would be the product’s total contribution to profits in the
first year?

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Solved Problem 1
SOLUTION
a. Beginning with the algebraic approach, we get
56,000
F
=
Q=
p–c
25 – 7
= 3,111 units
Using the graphic approach, shown in Figure A.6, we first
draw two lines:
Total revenue = 25Q
Total cost = 56,000 + 7Q
The two lines intersect at Q = 3,111 units, the break-even
quantity
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Solved Problem 1
b. Total profit contribution = Total revenue – Total cost
= pQ – (F + cQ)
= 15(10,000) – [56,000 + 7(10,000)]
= $24,000

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Solved Problem 1
250 –

Dollars (in thousands)

200 –
Total revenues
150 –

Break-even
quantity

100 –
$77.7

Total costs

50 –

3.1
0–
FIGURE A.6

|

|

|

|

|

|

|

|

1

2

3

4

5

6

7

8

Units (in thousands)

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Solved Problem 2
Binford Tool Company is screening three new product idea: A,
B, and C. Resource constraints allow only one of them to be
commercialized. The performance criteria and ratings, on a
scale of 1 (worst) to 10 (best), are shown in the following table.
The Binford managers give equal weights to the performance
criteria. Which is the best alternative, as indicated by the
preference matrix method?
Rating
Performance Criteria

Product A

Product B

Product C

1. Demand uncertainty and project risk

3

9

2

2. Similarity to present products

7

8

6

10

4

8

4. Compatibility with current
manufacturing process

4

7

6

5. Competitive Strategy

4

6

5

3. Expected return on investment (ROI)

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Solved Problem 2
Rating

SOLUTION

Performance Criteria

Each of the five criteria
receives a weight of 1/5
or 0.20

Product A

Product B

Product C

1. Uncertainty and risk

3

9

2

2. Similarity

7

8

6

10

4

8

4. Compatibility

4

7

6

5. Strategy

4

6

5

3. ROI

Product

Calculation

Total Score

A

(0.20 × 3) + (0.20 × 7) + (0.20 × 10) +
(0.20 × 4) + (0.20 × 4)

= 5.6

B

(0.20 × 9) + (0.20 × 8) + (0.20 × 4) +
(0.20 × 7) + (0.20 × 6)

= 6.8

C

(0.20 × 2) + (0.20 × 6) + (0.20 × 8) +
(0.20 × 6) + (0.20 × 5)

= 5.4

The best choice is product B as Products A and C are well
behind in terms of total weighted score
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Solved Problem 3
Adele Weiss manages the campus flower shop. Flowers must
be ordered three days in advance from her supplier in Mexico.
Although Valentine’s Day is fast approaching, sales are almost
entirely last-minute, impulse purchases. Advance sales are so
small that Weiss has no way to estimate the probability of low
(25 dozen), medium (60 dozen), or high (130 dozen) demand for
red roses on the big day. She buys roses for $15 per dozen and
sells them for $40 per dozen. Construct a payoff table. Which
decision is indicated by each of the following decision criteria?
a. Maximin
b. Maximax
c. Laplace
d. Minimax regret

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Solved Problem 3
SOLUTION
The payoff table for this problem is
Demand for Red Roses
Alternative

Low
(25 dozen)

Medium
(60 dozen)

High
(130 dozen)

Order 25 dozen

$625

$625

$625

Order 60 dozen

$100

$1,500

$1,500

($950)

$450

$3,250

$0

$0

$0

Order 130 dozen
Do nothing

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Solved Problem 3
a. Under the maximin criteria, Weiss should order 25 dozen,
because if demand is low, Weiss’s profits are $625, the best
of the worst payoffs.
b. Under the maximax criteria, Weiss should order 130 dozen.
The greatest possible payoff, $3,250, is associated with the
largest order.
c. Under the Laplace criteria, Weiss should order 60 dozen.
Equally weighted payoffs for ordering 25, 60, and 130 dozen
are about $625, $1,033, and $917, respectively.
d. Under the minimax regret criteria, Weiss should order 130
dozen. The maximum regret of ordering 25 dozen occurs if
demand is high: $3,250 – $625 = $2,625. The maximum regret
of ordering 60 dozen occurs if demand is high: $3,250 –
$1,500 = $1,750. The maximum regret of ordering 130 dozen
occurs if demand is low: $625 – (–$950) = $1,575.
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Solved Problem 4
White Valley Ski Resort is planning the ski lift operation for its
new ski resort and wants to determine if one or two lifts will be
necessary
Each lift can accommodate 250 people per day and skiing
occurs 7 days per week in the 14-week season and lift tickets
cost $20 per customer per day
The table below shows all the costs and probabilities for each
alternative and condition
Should the resort purchase one lift or two?
Alternatives

Conditions

One lift

Two lifts

Utilization

Installation

Operation

Bad times (0.3)

0.9

$50,000

$200,000

Normal times (0.5)

1.0

$50,000

$200,000

Good times (0.2)

1.0

$50,000

$200,000

Bad times (0.3)

0.9

$90,000

$200,000

Normal times (0.5)

1.0

$90,000

$400,000

Good times (0.2)

1.0

$90,000

$400,000

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Solved Problem 4
SOLUTION
The decision tree is shown in Figure A.7. The payoff ($000) for
each alternative-event branch is shown in the following table.
The total revenues from one lift operating at 100 percent
capacity are $490,000 (or 250 customers × 98 days ×
$20/customer-day).
Payoff Calculation (Revenue – Cost)

Alternatives

Economic Conditions

One lift

Bad times

0.9(490) – (50 + 200) = 191

Normal times

1.0(490) – (50 + 200) = 240

Good times

1.0(490) – (50 + 200) = 240

Bad times

0.9(490) – (90 + 200) = 151

Normal times

1.5(490) – (90 + 400) = 245

Good times

1.9(490) – (90 + 400) = 441

Two lifts

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Solved Problem 4
Bad times [0.3]

0.3(191) + 0.5(240) +
0.2(240) = 225.3
Normal times [0.5]

$191

$240

One lift
$225.3
$256.0

Good times [0.2]
Bad times [0.3]

Two lifts

Normal times [0.5]
$256.0

0.3(151) + 0.5(245) +
0.2(441) = 256.0

Good times [0.2]

$240
$151

$245

$441

FIGURE A.7
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