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Introduction to Management Science
8th Edition
by
Bernard W. Taylor III
Chapter 12
Decision Analysis
Chapter 12 - Decision Analysis
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Chapter Topics
Components of Decision Making
Decision Making without Probabilities
Decision Making with Probabilities
Decision Analysis with Additional Information
Utility
Chapter 12 - Decision Analysis
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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 12.1
Payoff Table
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Decision Analysis
Decision Making without Probabilities
Decision situation:
Table 12.2
Payoff Table for the Real Estate Investments
• Decision-Making Criteria: maximax, maximin, minimax,
minimax regret, Hurwicz, and equal likelihood
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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 12.3
Payoff Table Illustrating a Maximax Decision
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Decision Making without Probabilities
Maximin Criterion
In the maximin criterion the decision maker selects the
decision that will reflect the maximum of the minimum
payoffs; a pessimistic criterion.
Table 12.4
Payoff Table Illustrating a Maximin Decision
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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.
Table 12.6
Regret Table Illustrating the Minimax Regret Decision
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Decision Making without Probabilities
Hurwicz Criterion
The Hurwicz criterion is a compromise between the maximax
and maximin 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 12 - Decision Analysis
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
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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.
Decision
Apartment building
Office building
Warehouse
Chapter 12 - 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
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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
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Decision Making without Probabilities
Solution with QM for Windows (1 of 3)
Exhibit 12.1
Chapter 12 - Decision Analysis
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Decision Making without Probabilities
Solution with QM for Windows (2 of 3)
Exhibit 12.2
Chapter 12 - Decision Analysis
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Decision Making without Probabilities
Solution with QM for Windows (3 of 3)
Exhibit 12.3
Chapter 12 - Decision Analysis
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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 12.7
Payoff table with
Probabilities for States
of Nature
EV(Apartment) = $50,000(.6) + 30,000(.4) = 42,000
EV(Office) = $100,000(.6) - 40,000(.4) = 44,000
EV(Warehouse) = $30,000(.6) + 10,000(.4) = 22,000
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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 12.8
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
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Expected Value Problems
Solution with QM for Windows
Exhibit 12.4
Chapter 12 - Decision Analysis
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Expected Value Problems
Solution with Excel and Excel QM (1 of 2)
Exhibit 12.5
Chapter 12 - Decision Analysis
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Expected Value Problems
Solution with Excel and Excel QM (2 of 2)
Exhibit 12.6
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Decision Making with Probabilities
Expected Value of Perfect Information
The expected value of perfect information (EVPI) is the
maximum amount a decision maker would pay for additional
information.
EVPI equals the expected value given perfect information
minus the expected value without perfect information.
EVPI equals the expected opportunity loss (EOL) for the best
decision.
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Decision Making with Probabilities
EVPI Example (1 of 2)
Table 12.9
Payoff Table with Decisions, Given Perfect Information
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Decision Making with Probabilities
EVPI Example (2 of 2)
Decision with perfect 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
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Decision Making with Probabilities
EVPI with QM for Windows
Exhibit 12.7
Chapter 12 - Decision Analysis
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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 12.10
Payoff Table for Real Estate Investment Example
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Decision Making with Probabilities
Decision Trees (2 of 4)
Figure 12.1
Decision Tree for Real Estate Investment Example
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Decision Making with Probabilities
Decision Trees (3 of 4)
The expected value is computed at each probability 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
Branches with the greatest expected value are selected.
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Decision Making with Probabilities
Decision Trees (4 of 4)
Figure 12.2
Decision Tree with Expected Value at Probability Nodes
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Decision Making with Probabilities
Decision Trees with QM for Windows
Exhibit 12.8
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Decision Making with Probabilities
Decision Trees with Excel and TreePlan (1 of 4)
Exhibit 12.9
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Decision Making with Probabilities
Decision Trees with Excel and TreePlan (2 of 4)
Exhibit 12.10
Chapter 12 - Decision Analysis
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Decision Making with Probabilities
Decision Trees with Excel and TreePlan (3 of 4)
Exhibit 12.11
Chapter 12 - Decision Analysis
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Decision Making with Probabilities
Decision Trees with Excel and TreePlan (4 of 4)
Exhibit 12.12
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Decision Making with Probabilities
Sequential Decision Trees (1 of 4)
A sequential decision tree is used to illustrate a situation
requiring a series of decisions.
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:
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Decision Making with Probabilities
Sequential Decision Trees (2 of 4)
Figure 12.3
Sequential Decision Tree
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Decision Making with Probabilities
Sequential Decision Trees (3 of 4)
Decision is to purchase land; highest net expected value
($1,160,000).
Payoff of the decision is $1,160,000.
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Decision Making with Probabilities
Sequential Decision Trees (4 of 4)
Figure 12.4
Sequential Decision Tree with Nodal Expected Values
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Sequential Decision Tree Analysis
Solution with QM for Windows
Exhibit 12.13
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Sequential Decision Tree Analysis
Solution with Excel and TreePlan
Exhibit 12.14
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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 $444,000, and EVPI of $28,000.
Table 12.11
Payoff Table for the Real Estate Investment Example
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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
N = negative economic report
P(Pg) = .80
P(NG) = .20
P(Pp) = .10
P(Np) = .90
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Decision Analysis with Additional Information
Bayesian Analysis (3 of 3)
A posteria 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
Posteria probabilities by Bayes’ rule:
(gP) = P(PG)P(g)/[P(Pg)P(g) + P(Pp)P(p)]
= (.80)(.60)/[(.80)(.60) + (.10)(.40)] = .923
Posteria (revised) probabilities for decision:
P(gN) = .250
Chapter 12 - Decision Analysis
P(pP) = .077
P(pN) = .750
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Decision Analysis with Additional Information
Decision Trees with Posterior Probabilities (1 of 4)
Decision tree with posterior probabilities differ from earlier
versions in that:
Two new branches at beginning of tree represent report
outcomes.
Probabilities of each state of nature are posterior
probabilities from Bayes’ rule.
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Decision Analysis with Additional Information
Decision Trees with Posterior Probabilities (2 of 4)
Figure 12.5
Decision Tree with
Posterior Probabilities
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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 (strategy) = $89,220(.52) + 35,000(.48) = $63,194
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Decision Analysis with Additional Information
Decision Trees with Posterior Probabilities (4 of 4)
Figure 12.6
Decision Tree Analysis
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Decision Analysis with Additional Information
Computing Posterior Probabilities with Tables
Table 12.12
Computation of Posterior Probabilities
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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
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Decision Analysis with Additional Information
Utility (1 of 2)
Table 12.13
Payoff Table for Auto Insurance Example
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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 forgo a high expected value to avoid a lowprobability disaster.
Risk takers take a chance for a bonanza on a very lowprobability event in lieu of a sure thing.
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Decision Analysis
Example Problem Solution (1 of 9)
States of Nature
Decision
Expand
Maintain Status Quo
Sell now
Chapter 12 - Decision Analysis
Good Foreign
Competitive Conditions
$ 800,000
1,300,000
320,000
Poor Foreign Competitive
Conditions
$ 500,000
-150,000
320,000
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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 posteria probabilities using Bayes’
rule.
f. Perform a decision tree analysis using the posterior
probability obtained in part e.
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Decision Analysis
Example Problem Solution (3 of 9)
Step 1 (part a): Determine decisions without probabilities.
Maximax Decision: Maintain status quo
Decisions
Maximum Payoffs
Expand
Status quo
Sell
$800,000
1,300,000 (maximum)
320,000
Maximin Decision: Expand
Decisions
Minimum Payoffs
Expand
Status quo
Sell
$500,000 (maximum)
-150,000
320,000
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Decision Analysis
Example Problem Solution (4 of 9)
Minimax 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
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Decision Analysis
Example Problem Solution (5 of 9)
Equal Likelihood Decision: Expand
Expand
Status quo
$800,000(.5) + 500,000(.5) = $650,000
$1,300,000(.5) - 150,000(.5) = $575,000
Sell
$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
$800,000(.7) + 500,000(.3) = $710,000
$1,300,000(.7) - 150,000(.3) = $865,000
Sell
Chapter 12 - Decision Analysis
$320,000(.7) + 320,000(.3) = $320,000
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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
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Decision Analysis
Example Problem Solution (7 of 9)
Step 4 (part d): Develop a decision tree.
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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
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Decision Analysis
Example Problem Solution (9 of 9)
Step 6 (part f): Decision tree analysis.
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