2005 Thomson/South

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Transcript 2005 Thomson/South

EMGT 501 HW Solutions

12-9 a.

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Chapter 14 Decision Analysis       Problem Formulation Decision Making without Probabilities Decision Making with Probabilities Risk Analysis and Sensitivity Analysis Decision Analysis with Sample Information Computing Branch Probabilities © 2005 Thomson/South-Western Slide 6

Problem Formulation    A decision problem is characterized by decision alternatives, states of nature, and resulting payoffs.

The decision alternatives are the different possible strategies the decision maker can employ.

The states of nature refer to future events, not under the control of the decision maker, which may occur. States of nature should be defined so that they are mutually exclusive and collectively exhaustive.

Influence Diagrams      An influence diagram is a graphical device showing the relationships among the decisions, the chance events, and the consequences.

Squares or rectangles depict decision nodes.

Circles or ovals depict chance nodes.

Diamonds depict consequence nodes.

Lines or arcs connecting the nodes show the direction of influence.

Payoff Tables    The consequence resulting from a specific combination of a decision alternative and a state of nature is a payoff.

A table showing payoffs for all combinations of decision alternatives and states of nature is a payoff table.

Payoffs can be expressed in terms of profit, cost, time, distance or any other appropriate measure.

Decision Trees  A decision tree is a chronological representation of the decision problem.

The branches leaving each round node represent the different states of nature while the branches leaving each square node represent the different decision alternatives.

Decision Making without Probabilities  Three commonly used criteria for decision making when probability information regarding the likelihood of the states of nature is unavailable are: • the optimistic approach • the conservative approach • the minimax regret approach. © 2005 Thomson/South-Western Slide 12

Optimistic Approach    The optimistic approach would be used by an optimistic decision maker.

The decision with the largest possible payoff is chosen. If the payoff table was in terms of costs, the decision with the lowest cost would be chosen.

Conservative Approach    The conservative approach would be used by a conservative decision maker. For each decision the minimum payoff is listed and then the decision corresponding to the maximum of these minimum payoffs is selected. (Hence, the minimum possible payoff is maximized.) If the payoff was in terms of costs, the maximum costs would be determined for each decision and then the decision corresponding to the minimum of these maximum costs is selected. (Hence, the maximum possible cost is minimized.) © 2005 Thomson/South-Western Slide 14

Minimax Regret Approach     The minimax regret approach requires the construction of a regret table or an opportunity loss table. This is done by calculating for each state of nature the difference between each payoff and the largest payoff for that state of nature. Then, using this regret table, the maximum regret for each possible decision is listed. The decision chosen is the one corresponding to the minimum of the maximum regrets.

Example Consider the following problem with three decision alternatives and three states of nature with the following payoff table representing profits:

1 Decisions d 2

3 States of Nature

Example: Optimistic Approach An optimistic decision maker would use the optimistic (maximax) approach. We choose the decision that has the largest single value in the payoff table. Maximax decision Decision

Example: Optimistic Approach

1 2 3 4 5 6 7 8 9

 Formula Spreadsheet

A B

PAYOFF TABLE

Decision Alternative d1 d2 d3 Best Payoff

State of Nature s1 4 0 1 s2 4 3 5 s3 -2 -1 -3

=MAX(E5:E7)

Example: Optimistic Approach

1 2 3 4 5 6 7 8 9

 Solution Spreadsheet

A B

PAYOFF TABLE

Decision Alternative d1 d2 d3 Best Payoff

State of Nature s1 4 0 1 s2 4 3 5 s3 -2 -1 -3

Maximum Payoff 4 3 5

5 F

Recommended Decision

Example: Conservative Approach A conservative decision maker would use the conservative (maximin) approach. List the minimum payoff for each decision. Choose the decision with the maximum of these minimum payoffs.

Maximin decision Decision

Example: Conservative Approach  Formula Spreadsheet

1 2 3 4 5 6 7 8 9 A

PAYOFF TABLE

Decision Alternative d1 d2 d3

State of Nature s1 4 0 1 s2 4 3 5 s3 -2 -1 -3 Best Payoff

D E

=MAX(E5:E7)

Example: Conservative Approach

1 2 3 7 8 9 4 5 6

 Solution Spreadsheet

PAYOFF TABLE

B C

Decision Alternative d1 d2 d3 Best Payoff

State of Nature s1 4 0 1 s2 4 3 5 s3 -2 -1 -3

Minimum Payoff -2 -1 -3

-1 F

Recommended Decision

Example: Minimax Regret Approach For the minimax regret approach, first compute a regret table by subtracting each payoff in a column from the largest payoff in that column. In this example, in the first column subtract 4, 0, and 1 from 4; etc. The resulting regret table is:

d d d

1 2 3

Example: Minimax Regret Approach For each decision list the maximum regret. Choose the decision with the minimum of these values.

Minimax decision Decision

Example: Minimax Regret Approach  Formula Spreadsheet

1 2 A

PAYOFF TABLE Decision

B C

State of Nature

D 3 4 5 6 7 8

Altern.

d1 d2 d3 s1 4 0 1 s2 4 3 5 s3 -2 -1 -3

9 10 11 12 13 14

OPPORTUNITY LOSS TABLE Decision Altern.

s1 d1 d2 d3 State of Nature s2 Minimax Regret Value s3 =MAX($B$4:$B$6)-B4 =MAX($C$4:$C$6)-C4 =MAX($D$4:$D$6)-D4 =MAX($B$4:$B$6)-B5 =MAX($C$4:$C$6)-C5 =MAX($D$4:$D$6)-D5 =MAX($B$4:$B$6)-B6 =MAX($C$4:$C$6)-C6 =MAX($D$4:$D$6)-D6

E F

Example: Minimax Regret Approach  Solution Spreadsheet

PAYOFF TABLE Decision

B C

State of Nature

D 1 2 3 4 5 6

Alternative d1 d2 d3 s1 4 0 1 s2 4 3 5 s3 -2 -1 -3

7 8 9 10 11 12 13 14

Maximum Regret 1 4 3

1 F

Recommended Decision

Slide 26

Decision Making with Probabilities  Expected Value Approach • If probabilistic information regarding the states of nature is available, one may use the expected value (EV) approach. • Here the expected return for each decision is calculated by summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring. • The decision yielding the best expected return is chosen.

Expected Value of a Decision Alternative   The expected value of a decision alternative is the sum of weighted payoffs for the decision alternative.

The expected value (EV) of decision alternative d defined as:

d i i

) )

j j

 1

j j

where: N = the number of states of nature P(s

) = the probability of state of nature s

j V ij

= the payoff corresponding to decision alternative d

and state of nature s

Example: Burger Prince Burger Prince Restaurant is considering opening a new restaurant on Main Street. It has three different models, each with a different seating capacity. Burger Prince estimates that the average number of customers per hour will be 80, 100, or 120. The payoff table for the three models is on the next slide. © 2005 Thomson/South-Western Slide 29

Payoff Table Average Number of Customers Per Hour

Expected Value Approach Calculate the expected value for each decision. The decision tree on the next slide can assist in this calculation. Here d 1 , d 2 , d 3 represent the decision alternatives of models A, B, C, and s 1 , s 2 , s 3 represent the states of nature of 80, 100, and 120.

3 .4

Payoffs 10,000 15,000 14,000 8,000 18,000 12,000 6,000 16,000 21,000 Slide 32

Expected Value for Each Decision 1

1 EMV = .4(10,000) + .2(15,000) + .4(14,000) 2 = $12,600 Model A Model B

2 EMV = .4(8,000) + .2(18,000) + .4(12,000) 3 = $11,600 Model C

3 EMV = .4(6,000) + .2(16,000) + .4(21,000) 4 = $14,000 Choose the model with largest EV, Model C.

Expected Value Approach  Formula Spreadsheet

1 2 3 4 5 6 7 8 9 A

PAYOFF TABLE

Decision

State of Nature

D E

Expected

Recommended Alternative s1 = 80 s2 = 100 s3 = 120 Value Decision d1 = Model A 10,000 15,000 14,000 =$B$8*B5+$C$8*C5+$D$8*D5 =IF(E5=$E$9,A5,"") d2 = Model B 8,000 18,000 12,000 =$B$8*B6+$C$8*C6+$D$8*D6 =IF(E6=$E$9,A6,"") d3 = Model C 6,000 Probability 0.4

16,000 21,000 =$B$8*B7+$C$8*C7+$D$8*D7 =IF(E7=$E$9,A7,"") 0.2

0.4

Expected Value Approach  Solution Spreadsheet

1 2 3 4 5 6 7 8 9 A

PAYOFF TABLE

B C D

Decision State of Nature Alternative s1 = 80 s2 = 100 s3 = 120 d1 = Model A 10,000 15,000 14,000 d2 = Model B 8,000 d3 = Model C 6,000 Probability 0.4

18,000 12,000 16,000 21,000 0.2

0.4

Maximum Expected Value

Expected Value 12600 11600 14000

14000 F

Recommended Decision

d3 = Model C

Expected Value of Perfect Information    Frequently information is available which can improve the probability estimates for the states of nature. The expected value of perfect information (EVPI) is the increase in the expected profit that would result if one knew with certainty which state of nature would occur. The EVPI provides an upper bound on the expected value of any sample or survey information. © 2005 Thomson/South-Western Slide 36

Expected Value of Perfect Information  EVPI Calculation • Step 1: Determine the optimal return corresponding to each state of nature.

• Step 2: Compute the expected value of these optimal returns.

• Step 3: Subtract the EV of the optimal decision from the amount determined in step (2).

Expected Value of Perfect Information Calculate the expected value for the optimum payoff for each state of nature and subtract the EV of the optimal decision.

Expected Value of Perfect Information  Spreadsheet

1 2 3 10 11 12 4 5 6 7 8 9 A

PAYOFF TABLE

B C

Decision State of Nature Alternative s1 = 80 s2 = 100 s3 = 120 d1 = Model A 10,000 15,000 14,000 d2 = Model B 8,000 d3 = Model C 6,000 Probability 0.4

18,000 12,000 16,000 21,000 0.2

0.4

Maximum Expected Value Expected Maximum Payoff

10,000 18,000 21,000

Value 12600 11600 14000

14000

EVwPI

16000 F

Recommended Decision

d3 = Model C

EVPI

2000

Risk Analysis   Risk analysis helps the decision maker recognize the difference between: • • the expected value of a decision alternative, and the payoff that might actually occur The risk profile for a decision alternative shows the possible payoffs for the decision alternative along with their associated probabilities.

Risk Profile  Model C Decision Alternative .50

.40

.30

.20

.10

Sensitivity Analysis   Sensitivity analysis can be used to determine how changes to the following inputs affect the recommended decision alternative: • • probabilities for the states of nature values of the payoffs If a small change in the value of one of the inputs causes a change in the recommended decision alternative, extra effort and care should be taken in estimating the input value.

Bayes’ Theorem and Posterior Probabilities     Knowledge of sample (survey) information can be used to revise the probability estimates for the states of nature. Prior to obtaining this information, the probability estimates for the states of nature are called prior probabilities. With knowledge of conditional probabilities for the outcomes or indicators of the sample or survey information, these prior probabilities can be revised by employing Bayes' Theorem. The outcomes of this analysis are called posterior probabilities or branch probabilities for decision trees.

Computing Branch Probabilities  Branch (Posterior) Probabilities Calculation • Step 1: For each state of nature, multiply the prior probability by its conditional probability for the indicator -- this gives the joint probabilities for the states and indicator.

Computing Branch Probabilities  Branch (Posterior) Probabilities Calculation • Step 2: Sum these joint probabilities over all states -- this gives the marginal probability for the indicator.

• Step 3: For each state, divide its joint probability by the marginal probability for the indicator -- this gives the posterior probability distribution.

Expected Value of Sample Information  The expected value of sample information (EVSI) is the additional expected profit possible through knowledge of the sample or survey information. © 2005 Thomson/South-Western Slide 46

Expected Value of Sample Information  EVSI Calculation • Step 1: Determine the optimal decision and its expected return for the possible outcomes of the sample using the posterior probabilities for the states of nature. • Step 2: Compute the expected value of these optimal returns.

• Step 3: Subtract the EV of the optimal decision obtained without using the sample information from the amount determined in step (2).

Efficiency of Sample Information   Efficiency of sample information is the ratio of EVSI to EVPI. As the EVPI provides an upper bound for the EVSI, efficiency is always a number between 0 and 1.

Sample Information Burger Prince must decide whether or not to purchase a marketing survey from Stanton Marketing for $1,000. The results of the survey are "favorable" or "unfavorable". The conditional probabilities are: P(favorable | 80 customers per hour) = .2

P(favorable | 100 customers per hour) = .5 P(favorable | 120 customers per hour) = .9 Should Burger Prince have the survey performed by Stanton Marketing?

Influence Diagram

Decision Chance Consequence

Posterior Probabilities Favorable State Prior Conditional Joint Posterior 80 .4 .2 .08 .148

100 .2 .5 .10 .185

120 .4 .9 .36

.667

Total .54 1.000

P(favorable) = .54

Posterior Probabilities Unfavorable State Prior Conditional Joint Posterior 80 .4 .8 .32 .696

100 .2 .5 .10 .217

120 .4 .1 .04

.087

Total .46 1.000

P(unfavorable) = .46

Posterior Probabilities  Formula Spreadsheet

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A B

Market Research Favorable Prior State of Nature Probabilities s1 = 80 0.4

s2 = 100 s3 = 120 0.2

0.4

Conditional Probabilities 0.2

0.5

0.9

P(Favorable) = Market Research Unfavorable Prior State of Nature Probabilities s1 = 80 0.4

s2 = 100 s3 = 120 0.2

0.4

Conditional Probabilities 0.8

0.5

0.1

P(Unfavorable) =

Joint Probabilities =B4*C4 =B5*C5 =B6*C6 =SUM(D4:D6) Joint Probabilities =B12*C12 =B13*C13 =B14*C14 =SUM(D12:D14)

Posterior Probabilities  Solution Spreadsheet

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A

Market Research Favorable State of Nature Probabilities s1 = 80 0.4

s2 = 100 s3 = 120

Prior 0.2

0.4

Market Research Unfavorable Prior State of Nature Probabilities s1 = 80 0.4

s2 = 100 s3 = 120 0.2

0.4

Conditional Probabilities 0.2

0.5

0.9

P(Favorable) = Conditional Probabilities 0.8

0.5

0.1

Joint Probabilities 0.08

0.10

0.36

0.54

Joint Probabilities 0.32

0.10

0.04

0.46

Posterior Probabilities 0.148

0.185

0.667

Posterior Probabilities 0.696

0.217

0.087

Slide 54

Decision Tree  Top Half I 1 (.54) 2

1 (.148)

3 (.185) (.667)

3 (.148) (.185) (.667)

1 (.148)

2 (.185)

3 (.667) $10,000 $15,000 $14,000 $8,000 $18,000 $12,000 $6,000 $16,000 $21,000 Slide 55

Decision Tree  Bottom Half 1

2 (.46) 3

1 (.696)

3 (.217) (.087)

1 (.696)

2 (.217)

3 (.087)

1 (.696)

3 (.217) (.087) $10,000 $15,000 $14,000 $8,000 $18,000 $12,000 $6,000 $16,000 $21,000 Slide 56

Decision Tree 1

1 (.54) $17,855 2

d d d

1 2 3 4 5 6 EMV = .148(10,000) + .185(15,000) + .667(14,000) = $13,593 EMV = .148 (8,000) + .185(18,000) + .667(12,000) = $12,518 EMV = .148(6,000) + .185(16,000) +.667(21,000) =

$17,855

EMV = .696(10,000) + .217(15,000) +.087(14,000)=

$11,433

2 (.46) 3 $11,433

d d

Expected Value of Sample Information If the outcome of the survey is "favorable”, choose Model C. If it is “unfavorable”, choose Model A.

EVSI = .54($17,855) + .46($11,433) - $14,000 = $900.88 Since this is less than the cost of the survey, the survey should not be purchased.

Efficiency of Sample Information The efficiency of the survey: EVSI/EVPI = ($900.88)/($2000) = .4504

Bayes’ Decision Rule

: Using the best available estimates of the probabilities of the respective states of nature (currently the prior probabilities), calculate the expected value of the payoff for each of the possible actions. Choose the action with the maximum expected payoff.

Bayes’ theory

: State of Nature (

i =

1 ~

) P(S

): Prior Probability I

: Professional Information (Experiment)(

j =

1 ~

) P(I

P(I

| S

): Conditional Probability  S

) = P(S

 I

): Joint Probability P(S

P(S

| I

): Posterior Probability | I

Home Work 14-20 Due Day: Nov 7

2005 Thomson/South

Transcript 2005 Thomson/South

EMGT 501 HW Solutions

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Bayes’ Decision Rule

Bayes’ theory

Home Work 14-20 Due Day: Nov 7

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