Transcript Document

Introduction to Management Science
8th Edition
by
Bernard W. Taylor III
Chapter 3
Decision Analysis
Chapter 3 - Decision Analysis
1
Chapter Topics
Components of Decision Making
Decision Making without Probabilities
Decision Making with Probabilities
Decision Analysis with Additional Information
Utility
Chapter 3 - Decision Analysis
2
Decision Analysis
Components of Decision Making
A state of nature is an actual event that may occur in the
future.
A payoff table is a means of organizing a decision situation,
presenting the payoffs from different decisions given the
various states of nature.
Table 3.1
Payoff Table
Chapter 3 - Decision Analysis
3
Decision Analysis
Decision Making without Probabilities
Decision situation:
Table 3.2
Payoff Table for the Real Estate Investments
• Decision-Making Criteria: maximax, maximin, minimax
(minimal regret), Hurwicz, and equal likelihood
Chapter 3 - Decision Analysis
4
Decision Making without Probabilities
Maximax Criterion
In the maximax criterion the decision maker selects the
decision that will result in the maximum of maximum payoffs;
an optimistic criterion.
Table 3.3
Payoff Table Illustrating a Maximax Decision
Chapter 3 - Decision Analysis
5
Decision Making without Probabilities
Maximin Criterion
In the maximin criterion the decision maker selects the
decision that will reflect the maximum of the minimum (best
of the worst-case) payoffs; a pessimistic criterion.
conservative
Table 3.4
Payoff Table Illustrating a Maximin Decision
Chapter 3 - Decision Analysis
6
Decision Making without Probabilities
Minimax Regret Criterion
Regret is the difference between the payoff from the best
decision and all other decision payoffs.
The decision maker attempts to avoid regret by selecting the
decision alternative that minimizes the maximum regret.
Highest
payoff
Maximal
regrets
$ 50,000
$ 70,000
$ 70,000
$100,000
- $50,000
Table 3.6
Regret Table Illustrating the Minimax Regret Decision
Chapter 3 - Decision Analysis
7
Decision Making without Probabilities
Hurwicz Criterion
The Hurwicz criterion is a compromise between the maximax
(optimist) and maximin (conservative) criterion.
A coefficient of optimism, , is a measure
of the decision maker’s optimism.
The Hurwicz criterion multiplies the best payoff by  and the
worst payoff by (1- ), for each decision, and the best result
is selected.
Decision
Apartment building
Office building
Warehouse
Chapter 3 - Decision Analysis
 = 0.4
Values
$50,000(.4) + 30,000(.6) = 38,000
$100,000(.4) - 40,000(.6) = 16,000
$30,000(.4) + 10,000(.6) = 18,000
8
Decision Making without Probabilities
Equal Likelihood Criterion
The equal likelihood ( or Laplace) criterion multiplies the
decision payoff for each state of nature by an equal weight,
thus assuming that the states of nature are equally likely to
occur.
For 2 states of nature, the =.5 case of the Hurwicz method
In general, it is essentially different !
Decision
Apartment building
Office building
Warehouse
Chapter 3 - Decision Analysis
Values
$50,000(.5) + 30,000(.5) = 40,000
$100,000(.5) - 40,000(.5) = 30,000
$30,000(.5) + 10,000(.5) = 20,000
9
Decision Making without Probabilities
Summary of Criteria Results
A dominant decision is one that has a better payoff than
another decision under each state of nature.
The appropriate criterion is dependent on the “risk”
personality and philosophy of the decision maker.
Criterion
Decision (Purchase)
Maximax
Office building
Maximin
Apartment building
Minimax regret
Apartment building
Hurwicz
Apartment building
Equal likelihood
Apartment building
Chapter 3 - Decision Analysis
10
Decision Making without Probabilities
Solution with QM for Windows (1 of 3)
Exhibit 3.1
Chapter 3 - Decision Analysis
11
Decision Making without Probabilities
Solution with QM for Windows (2 of 3)
Exhibit 3.2
Chapter 3 - Decision Analysis
12
Decision Making without Probabilities
Solution with QM for Windows (3 of 3)
Exhibit 3.3
Chapter 3 - Decision Analysis
13
Decision Making with Probabilities
Expected Value
Expected value is computed by multiplying each decision
outcome under each state of nature by the probability of its
occurrence.
Table 3.7
Payoff table with
Probabilities for States
of Nature
EV(Apartment) =
EV(Office) =
$50,000(.6) + $30,000(.4) = $42,000
$100,000(.6) – $40,000(.4) = $44,000
EV(Warehouse) = $30,000(.6) + $10,000(.4) = $22,000
Chapter 3 - Decision Analysis
14
Decision Making with Probabilities
Expected Opportunity Loss
The expected opportunity loss is the expected value of the
regret for each decision.
The expected value and expected opportunity loss criterion
result in the same decision.
Table 3.8
Regret (Opportunity Loss) Table
with Probabilities for States of
Nature
EOL(Apartment) = $50,000(.6) + $0(.4) = $30,000
EOL(Office) = $0(.6) + $70,000(.4) = $28,000
EOL(Warehouse) = $70,000(.6) + $20,000(.4) = $50,000
Chapter 3 - Decision Analysis
15
Expected Value Problems
Solution with QM for Windows
Exhibit 3.4
Chapter 3 - Decision Analysis
16
Expected Value Problems
Solution with Excel and Excel QM (1 of 2)
Exhibit 3.5
Chapter 3 - Decision Analysis
17
Decision Making with Probabilities
Expected Value of Perfect Information
The expected value of perfect information (EVPI) is the
maximum amount a decision maker should pay for additional
information.
EVPI equals the expected value (with) given perfect
information (insider information, genie) minus the expected
value calculated without perfect information.
EVPI equals the expected opportunity loss (EOL) for the best
decision.
Chapter 3 - Decision Analysis
18
Decision Making with Probabilities
EVPI Example (1 of 2)
Table 3.9
Payoff Table with Decisions, Given Perfect Information
Chapter 3 - Decision Analysis
19
Decision Making with Probabilities
EVPI Example (2 of 2)
Decision with perfect (insider/genie) information:
$100,000(.60) + $30,000(.40) = $72,000
Decision without perfect information:
EV(office) = $100,000(.60) - $40,000(.40) = $44,000
EVPI = $72,000 - $44,000 = $28,000
EOL(office) = $0(.60) + $70,000(.4) = $28,000
The
“genie pick”
Chapter 3 - Decision Analysis
EV
$42,000
$44,000
$22,000
20
Expected Value Problems
Solution with Excel and Excel QM (2 of 2)
$100,000*0.6+$30,000*0.4 = $72,000
Exhibit 3.6
Chapter 3 - Decision Analysis
21
Decision Making with Probabilities
EVPI with QM for Windows
Exhibit 3.7
Chapter 3 - Decision Analysis
22
Decision Making with Probabilities
Decision Trees (1 of 4)
A decision tree is a diagram consisting of decision nodes
(represented as squares), probability nodes (circles), and
decision alternatives (branches).
Table 3.10
Payoff Table for Real Estate Investment Example
Chapter 3 - Decision Analysis
23
Decision Making with Probabilities
Decision Trees (2 of 4)
uncontrollable
controllable
Figure 3.1
Decision Tree for Real Estate Investment Example
Chapter 3 - Decision Analysis
24
Decision Making with Probabilities
Decision Trees (3 of 4)
The expected value is computed at each probability
(uncontrollable) node:
EV(node 2) = .60($50,000) + .40(30,000) = $42,000
EV(node 3) = .60($100,000) + .40(-40,000) = $44,000
EV(node 4) = .60($30,000) + .40(10,000) = $22,000
populating the decision tree from right to left.
The branch(es) with the greatest expected value are then
selected, starting from the left and progressing to the right.
Chapter 3 - Decision Analysis
25
Decision Making with Probabilities
Decision Trees (4 of 4)
Figure 3.2
Decision Tree with Expected Value at Probability Nodes
Chapter 3 - Decision Analysis
26
Decision Making with Probabilities
Decision Trees with QM for Windows
Exhibit 3.8
Chapter 3 - Decision Analysis
27
Decision Making with Probabilities
Decision Trees with Excel and TreePlan (1 of 4)
Exhibit 3.9
Chapter 3 - Decision Analysis
28
Decision Making with Probabilities
Decision Trees with Excel and TreePlan (2 of 4)
Exhibit 3.10
Chapter 3 - Decision Analysis
29
Decision Making with Probabilities
Decision Trees with Excel and TreePlan (3 of 4)
Exhibit 3.11
Chapter 3 - Decision Analysis
30
Decision Making with Probabilities
Decision Trees with Excel and TreePlan (4 of 4)
Exhibit 3.12
Chapter 3 - Decision Analysis
31
Decision Making with Probabilities
Sequential Decision Trees (1 of 4)
A sequential decision tree is used to illustrate a situation
requiring a series (a sequence) of decisions.
It is often chronological, and always logical in order.
Used where a payoff table, limited to a single decision,
cannot be used.
Real estate investment example modified to encompass a
ten-year period in which several decisions must be made:
Chapter 3 - Decision Analysis
32
Decision Making with Probabilities
Sequential Decision Trees (2 of 4)
The decision to be
made at [1] logically
depends on the decisions
(to be) made at [4] and [5].
Chapter 3 - Decision Analysis
Figure 3.3
Sequential Decision Tree
33
Decision Making with Probabilities
Sequential Decision Trees (3 of 4)
Figure 3.4
Sequential Decision Tree with Nodal Expected Values
Chapter 3 - Decision Analysis
34
Decision Making with Probabilities
Sequential Decision Trees (4 of 4)
Decision is to purchase land; highest net expected value
($1,160,000, at node [1] ).
Payoff of the decision is $1,160,000. (That’s the payoff that
this decision is expected to yield.)
Chapter 3 - Decision Analysis
35
Sequential Decision Tree Analysis
Solution with QM for Windows
Exhibit 3.13
Chapter 3 - Decision Analysis
36
Sequential Decision Tree Analysis
Solution with Excel and TreePlan
Exhibit 3.14
Chapter 3 - Decision Analysis
37
Decision Analysis with Additional Information
Bayesian Analysis (1 of 3)
Bayesian analysis uses additional information to alter the
marginal probability of the occurrence of an event.
In real estate investment example, using expected value
criterion, best decision was to purchase office building with
expected value of $44,000, and EVPI of $28,000.
Table 3.11
Payoff Table for the Real Estate Investment Example
Chapter 3 - Decision Analysis
38
Decision Analysis with Additional Information
Bayesian Analysis (2 of 3)
A conditional probability is the probability that an event will
occur given that another event has already occurred.
Economic analyst provides additional information for real
estate investment decision, forming conditional probabilities:
g = good economic conditions
p = poor economic conditions
P = positive economic report
as before…
new info
N = negative economic report
P(Pg) = .80
P(Ng) = .20
P(Pp) = .10
P(Np) = .90
Chapter 3 - Decision Analysis
new, given
39
Decision Analysis with Additional Information
Bayesian Analysis (3 of 3)
A posterior probability is the altered marginal probability of an
event based on additional information.
Prior probabilities for good or poor economic conditions in
real estate decision:
P(g) = .60; P(p) = .40
Posterior probabilities by Bayes’ rule:
P(gP) = P(Pg)P(g)/[P(Pg)P(g) + P(Pp)P(p)]
= (.80)(.60)/[(.80)(.60) + (.10)(.40)] = .923
Posterior (revised) probabilities for decision:
P(gN) = .250
Chapter 3 - Decision Analysis
P(pP) = .077
P(pN) = .750
40
Decision Analysis with Additional Information
Decision Trees with Posterior Probabilities (1 of 4)
Decision tree with posterior probabilities differ from earlier
versions (prior probabilities) in that:
Two (or more) new branches at beginning of tree
represent report/survey… outcomes.
Probabilities of each state of nature, thereafter, are
posterior probabilities from Bayes’ rule.
Bayes’ rule can be simplified, since P(A|B)P(B)=P(AB) is
the joint prob., and iP(ABi)=P(A) is the marginal prob. So:
P(Bk|A)=P(A|Bk)P(Bk)/[iP(A|Bi)P(Bi)] = P(ABk)/P(A),
much quicker, if the joint and marginal prob’s are known.
Chapter 3 - Decision Analysis
41
Decision Analysis with Additional Information
Decision Trees with Posterior Probabilities (2 of 4)
P(P|g)=.80
P(N|g)=.20
P(P|p)=.10
P(N|p)=.90
P(g)=.60
P(p)=.40
Figure 3.5
Decision Tree with
Posterior Probabilities
P(g|P)=.923
P(p|P)=.077
P(g|N)=.250
P(p|N)=.750
Chapter 3 - Decision Analysis
42
Decision Analysis with Additional Information
Decision Trees with Posterior Probabilities (3 of 4)
EV (apartment building) = $50,000(.923) + 30,000(.077)
= $48,460
EV (office building) = $100,000(.923) – 40,000(.077)
= $89,220
EV (warehouse) = $30,000(.923) + 10,000(.077)
= $28,460
Then do the same with the “Negative report” probabilities.
So, finally:
EV (whole strategy) = $89,220(.52) + 35,000(.48) = $63,194
“Positive report”
Chapter 3 - Decision Analysis
“Negative report”
43
Decision Analysis with Additional Information
Decision Trees with Posterior Probabilities (4 of 4)
Figure 3.6
Decision Tree Analysis
Chapter 3 - Decision Analysis
44
Decision Analysis with Additional Information
Computing Posterior Probabilities with Tables
Table 3.12
Computation of Posterior Probabilities
Indeed, this equals [ P(P|g)P(g)+P(P|p)P(p) ] = P(P&g) + P(P&p) = P(P) .
Chapter 3 - Decision Analysis
45
Decision Analysis with Additional Information
Expected Value of Sample Information
The expected value of sample information (EVSI) is the
difference between the expected value with and without
information:
For example problem, EVSI = $63,194 - 44,000 = $19,194
The efficiency of sample information is the ratio of the
expected value of sample information to the expected value
of perfect information:
efficiency = EVSI /EVPI = $19,194/ 28,000 = .68
Chapter 3 - Decision Analysis
46
Decision Analysis with Additional Information
Utility (1 of 2)
Table 3.13
Payoff Table for Auto Insurance Example
Cost
Chapter 3 - Decision Analysis
47
Decision Analysis with Additional Information
Utility (2 of 2)
Expected Cost (insurance) = .992($500) + .008(500) = $500
Expected Cost (no insurance) = .992($0) + .008(10,000) = $80
Decision should be “do not purchase insurance”, but people
almost always do purchase insurance.
Utility is a measure of personal satisfaction derived from
money.
Utiles are units of subjective measures of utility.
Risk averters (evaders) forgo a high expected value to avoid
a low-probability disaster.
Risk takers take a chance for a bonanza on a very lowprobability event in lieu of a sure thing.
Chapter 3 - Decision Analysis
48
Decision Analysis
Example Problem Solution (1 of 9)
States of Nature
Decisions
Good Foreign
Poor Foreign
Competitive Conditions Competitive Conditions
Expand
$800,000
$500,000
Maintain Status Quo
$1,300,00
–$150,000
Sell Now
$320,000
$320,000
Chapter 3 - Decision Analysis
49
Decision Analysis
Example Problem Solution (2 of 9)
a. Determine the best decision without probabilities using the
5 criteria of the chapter.
b. Determine best decision with probabilities assuming .70
probability of good conditions, .30 of poor conditions. Use
expected value and expected opportunity loss criteria.
c. Compute expected value of perfect information.
d. Develop a decision tree with expected value at the nodes.
e. Given following, P(Pg) = .70, P(Ng) = .30, P(Pp) = .20,
P(Np) = .80, determine posterior probabilities using Bayes’
rule.
f. Perform a decision tree analysis using the posterior
probability obtained in part e.
Chapter 3 - Decision Analysis
50
Decision Analysis
Example Problem Solution (3 of 9)
Step 1 (part a): Determine decisions without probabilities.
Maximax (Optimist) Decision: Maintain status quo
Decisions
maximum Payoffs
Expand
Status quo
Sell
$800,000
1,300,000 (Maximum)
320,000
Maximin (Conservative) Decision: Expand
Decisions
minimum Payoffs
Expand
Status quo
Sell
$500,000 (Maximum)
-150,000
320,000
Chapter 3 - Decision Analysis
51
Decision Analysis
Example Problem Solution (4 of 9)
Minimax (Optimal) Regret Decision: Expand
Decisions
maximum Regrets
Expand
$500,000 (Minimum)
Status quo
650,000
Sell
980,000
Hurwicz ( = .3) Decision: Expand
Expand
$800,000(.3) + 500,000(.7) = $590,000
Status quo
$1,300,000(.3) - 150,000(.7) = $285,000
Sell
$320,000(.3) + 320,000(.7) = $320,000
Chapter 3 - Decision Analysis
52
Decision Analysis
Example Problem Solution (5 of 9)
Equal Likelihood (Laplace) Decision: Expand
Expand
Status quo
Sell
$800,000(.5) + 500,000(.5) = $650,000
$1,300,000(.5) - 150,000(.5) = $575,000
$320,000(.5) + 320,000(.5) = $320,000
Step 2 (part b): Determine Decisions with EV and EOL.
Expected value decision: Maintain status quo
Expand
Status quo
Sell
Chapter 3 - Decision Analysis
$800,000(.7) + 500,000(.3) = $710,000
$1,300,000(.7) - 150,000(.3) = $865,000
$320,000(.7) + 320,000(.3) = $320,000
53
Decision Analysis
Example Problem Solution (6 of 9)
Expected opportunity loss decision: Maintain status quo
Expand
Status quo
Sell
$500,000(.7) + 0(.3)
= $350,000
0(.7) + 650,000(.3) = $195,000
$980,000(.7) + 180,000(.3) = $740,000
Step 3 (part c): Compute EVPI.
EV given perfect information
= 1,300,000(.7) + 500,000(.3) = $1,060,000
EV without perfect information
= $1,300,000(.7) - 150,000(.3) = $865,000
EVPI = $1,060,000 - 865,000 = $195,000
Chapter 3 - Decision Analysis
54
Decision Analysis
Example Problem Solution (7 of 9)
Step 4 (part d): Develop a decision tree.
Chapter 3 - Decision Analysis
55
Decision Analysis
Example Problem Solution (8 of 9)
Step 5 (part e): Determine posterior probabilities.
P(gP) = P(Pg)P(g)/[P(Pg)P(g) + P(Pp)P(p)]
= (.70)(.70)/[(.70)(.70) + (.20)(.30)] = .891
P(pP) = .109
P(gN) = P(Ng)P(g)/[P(Ng)P(g) + P(Np)P(p)]
= (.30)(.70)/[(.30)(.70) + (.80)(.30)] = .467
P(pN) = .533
Chapter 3 - Decision Analysis
56
Decision Analysis
Example Problem Solution (9 of 9)
Step 6 (part f): Decision tree analysis.
Without the report, maintain status
quo, based on the expected payoff
value $865,000.
With the report, the payoff may be
expected to be even $1,141,950.
Thus, the opportunity loss is
$1,141,950 – $865,000 = $276,950.
Therefore, no more than
$276,950 should be paid
to obtain such a report. (EVPI)
Chapter 3 - Decision Analysis
57
Chapter 3 - Decision Analysis
58