chap12 decision analysis
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Chapter 12 - Decision Analysis 1
Chapter 12
Decision Analysis
Introduction to Management Science
8th Edition
by
Bernard W. Taylor III
Chapter 12 - Decision Analysis 2
Components of Decision Making
Decision Making without Probabilities
Decision Making with Probabilities
Decision Analysis with Additional Information
Chapter Topics
Chapter 12 - Decision Analysis 3
Table 12.1Payoff Table
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.
Decision AnalysisComponents of Decision Making
Chapter 12 - Decision Analysis 4
Decision situation:
• Decision-Making Criteria: maximax, maximin, minimax, minimax regret, Hurwicz, and equal likelihood
Table 12.2Payoff Table for the Real Estate Investments
Decision AnalysisDecision Making without Probabilities
Chapter 12 - Decision Analysis 5
Table 12.3Payoff Table Illustrating a Maximax Decision
In the maximax criterion the decision maker selects the decision that will result in the maximum of maximum payoffs; an optimistic criterion.
Decision Making without ProbabilitiesMaximax Criterion
Chapter 12 - Decision Analysis 6
Table 12.4Payoff Table Illustrating a Maximin Decision
In the maximin criterion the decision maker selects the decision that will reflect the maximum of the minimum payoffs; a pessimistic criterion.
Decision Making without ProbabilitiesMaximin Criterion
Chapter 12 - Decision Analysis 7
Table 12.6 Regret Table Illustrating the Minimax Regret Decision
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.
Decision Making without ProbabilitiesMinimax Regret Criterion
Chapter 12 - Decision Analysis 8
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 Values
Apartment building $ 50,000(0.4) + 30,000(0.6) = 38,000
Office building $ 100,000(0.4) - 40,000(0.6) = 16,000
Warehouse $ 30,000(0.4) + 10,000(0.6) = 18,000
Decision Making without ProbabilitiesHurwicz Criterion
Chapter 12 - Decision Analysis 9
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 Values
Apartment building $50,000(0.5) + 30,000(0.5) = 40,000
Office building $100,000(0.5) - 40,000(0.5) = 30,000
Warehouse $30,000(0.5) + 10,000(0.5) = 20,000
Decision Making without ProbabilitiesEqual Likelihood Criterion
Chapter 12 - Decision Analysis 10
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
Decision Making without ProbabilitiesSummary of Criteria Results
Chapter 12 - Decision Analysis 11
Expected value is computed by multiplying each decision outcome under each state of nature by the probability of its occurrence.
EV(Apartment) = $50,000(0.6) + 30,000(0.4) = 42,000
EV(Office) = $100,000(0.6) - 40,000(0.4) = 44,000
EV(Warehouse) = $30,000(0.6) + 10,000(0.4) = 22,000
Table 12.7Payoff table with Probabilities for
States of Nature
Decision Making with ProbabilitiesExpected Value
Chapter 12 - Decision Analysis 12
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.
EOL (Apartment) = $50,000(0.6) + 0(0.4) = 30,000
EOL (Office) = $0(0.6) + 70,000(0.4) = 28,000
EOL (W/house) = $70,000(0.6) + 20,000(0.4) = 50,000
Table 12.8Regret (Opportunity Loss) Table with Probabilities for
States of Nature
Decision Making with ProbabilitiesExpected Opportunity Loss
Chapter 12 - Decision Analysis 13
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.
Decision Making with ProbabilitiesExpected Value of Perfect Information
Chapter 12 - Decision Analysis 14
Table 12.9Payoff Table with Decisions, Given Perfect Information
Decision Making with ProbabilitiesEVPI Example (1 of 2)
Chapter 12 - Decision Analysis 15
Decision with perfect information:
$100,000(0.60) + 30,000(0.40) = $72,000
Decision without perfect information:
EV(office) = $100,000(0.60) - 40,000(0.40) = $44,000
EVPI = $72,000 - 44,000 = $28,000
EOL(office) = $0(0.60) + 70,000(0.4) = $28,000
Decision Making with ProbabilitiesEVPI Example (2 of 2)
Chapter 12 - Decision Analysis 16
A decision tree is a diagram consisting of decision nodes, represented as (squares), probability nodes (circles), and decision alternatives (branches).
Table 12.10Payoff Table for Real Estate Investment Example
Decision Making with ProbabilitiesDecision Trees (1 of 4)
Chapter 12 - Decision Analysis 17
Figure 12.1Decision Tree for Real Estate Investment Example
Decision Making with ProbabilitiesDecision Trees (2 of 4)
Chapter 12 - Decision Analysis 18
The expected value is computed at each probability node:
EV(node 2) = 0.60($50,000) + 0.40(30,000) =
$42,000
EV(node 3) = 0.60($100,000) + 0.40(-40,000) =
$44,000
EV(node 4) = 0.60($30,000) + 0.40(10,000) =
$22,000
Branches with the greatest expected value are selected.
Decision Making with ProbabilitiesDecision Trees (3 of 4)
Chapter 12 - Decision Analysis 19
Figure 12.2Decision Tree with Expected Value at Probability Nodes
Decision Making with ProbabilitiesDecision Trees (4 of 4)
Chapter 12 - Decision Analysis 20
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 12.11Payoff Table for the Real Estate Investment Example
Decision Analysis with Additional InformationBayesian Analysis (1 of 3)
Chapter 12 - Decision Analysis 21
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(Pg) = 0.80 P(Ng) = 0.20
P(Pp) = 0.10 P(Np) = 0.90
Decision Analysis with Additional InformationBayesian Analysis (2 of 3)
Chapter 12 - Decision Analysis 22
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) = 0.60; P(p) = 0.40
Posteria probabilities by Bayes’ rule:
Posteria (revised) probabilities for decision:
P(gN) = 0.250 P(pP) = 0.077 P(pN) = 0.750
Decision Analysis with Additional InformationBayesian Analysis (3 of 3)
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)6.0)(8.0(
)()./()()./(
)()./()/(
pPpPPgPgPP
gPgPPPgP
Chapter 12 - Decision Analysis 23
Table 12.12Computation of Posterior Probabilities
Decision Analysis with Additional InformationComputing Posterior Probabilities with Tables
Chapter 12 - Decision Analysis 24
Decision Analysis with Additional InformationDecision 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.
Chapter 12 - Decision Analysis 25
Figure 12.5Decision Tree with
Posterior Probabilities
Decision Analysis with Additional InformationDecision Trees with Posterior Probabilities (2 of 4)
Chapter 12 - Decision Analysis 26
Decision Analysis with Additional InformationDecision Trees with Posterior Probabilities (3 of 4)
EV (apartment building) = $50,000(0.923) + 30,000(0.077)
= $48,460
EV (strategy) = $89,220(0.52) + 35,000(0.48) = $63,194
Chapter 12 - Decision Analysis 27
Figure 12.6Decision Tree Analysis
Decision Analysis with Additional InformationDecision Trees with Posterior Probabilities (4 of 4)
Chapter 12 - Decision Analysis 28
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
Decision Analysis with Additional InformationExpected Value of Sample Information
Chapter 12 - Decision Analysis 29
Table 12.13Payoff Table for Auto Insurance Example
Decision Analysis with Additional InformationUtility (1 of 2)
Chapter 12 - Decision Analysis 30
Expected Cost (insurance) = 0.992($500) + 0.008(500) = $500
Expected Cost(no insurance)= 0.992($0) + 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 low-probability disaster.
Risk takers take a chance for a bonanza on a very low-probability event in lieu of a sure thing.
Decision Analysis with Additional InformationUtility (2 of 2)
Chapter 12 - Decision Analysis 31
States of Nature
Decision Good Foreign
Competitive Conditions Poor Foreign Competitive
Conditions
Expand Maintain Status Quo Sell now
$ 800,000 1,300,000 320,000
$ 500,000 -150,000 320,000
Decision Analysis Example Problem Solution (1 of 9)
Chapter 12 - Decision Analysis 32
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 0.70 probability of good conditions, 0.30 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(Pg) = 0.70, P(Ng) = 0.30, P(Pp) = 20, P(Np) = 0.80, determine posteria probabilities using Bayes’ rule.
f. Perform a decision tree analysis using the posterior probability obtained in part e.
Chapter 12 - Decision Analysis 33
Step 1 (part a): Determine decisions without probabilities.
Maximax Decision: Maintain status quo
Decisions Maximum Payoffs
Expand $800,000Status quo 1,300,000 (maximum)Sell 320,000
Maximin Decision: Expand
Decisions Minimum Payoffs
Expand $500,000 (maximum)Status quo -150,000Sell 320,000
Decision Analysis Example Problem Solution (3 of 9)
Chapter 12 - Decision Analysis 34
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(0.3) + 500,000(0.7) = $590,000
Status quo $1,300,000(0.3) - 150,000(0.7) = $285,000
Sell $320,000(0.3) + 320,000(0.7) = $320,000
Decision Analysis Example Problem Solution (4 of 9)
Chapter 12 - Decision Analysis 35
Equal Likelihood Decision: Expand
Expand $800,000(0.5) + 500,000(0.5) = $650,000
Status quo $1,300,000(0.5) - 150,000(0.5) = $575,000
Sell $320,000(0.5) + 320,000(0.5) = $320,000
Step 2 (part b): Determine Decisions with EV and EOL.
Expected value decision: Maintain status quo
Expand $800,000(0.7) + 500,000(0.3) = $710,000
Status quo $1,300,000(0.7) - 150,000(0.3) = $865,000
Sell $320,000(0.7) + 320,000(0.3) = $320,000
Decision Analysis Example Problem Solution (5 of 9)
Chapter 12 - Decision Analysis 36
Expected opportunity loss decision: Maintain status quo
Expand $500,000(0.7) + 0(0.3) = $350,000
Status quo 0(0.7) + 650,000(0.3) = $195,000
Sell $980,000(0.7) + 180,000(0.3) = $740,000
Step 3 (part c): Compute EVPI.
EV given perfect information = 1,300,000(0.7) + 500,000(0.3)
= $1,060,000
EV without perfect information = $1,300,000(0.7) - 150,000(0.3)
= $865,000
EVPI = $1.060,000 - 865,000 = $195,000
Decision Analysis Example Problem Solution (6 of 9)
Chapter 12 - Decision Analysis 37
Step 4 (part d): Develop a decision tree.
Decision Analysis Example Problem Solution (7 of 9)
Chapter 12 - Decision Analysis 38
Step 5 (part e): Determine posterior probabilities.
P(gP) = P(Pg) P(g) / [P(Pg) P(g) + P(Pp) P(p)]
= (0.70)(0.70)/[(0.70)(0.70) + (0.20)(0.30)] = 0.891
P(pP) = 0.109
P(gN) = P(Ng) P(g) / [P(Ng) P(g) + P(Np) P(p)]
= (0.30)(0.70)/[(0.30)(0.70) + (0.80)(0.30)] = 0.467
P(pN) = 0.533
Decision Analysis Example Problem Solution (8 of 9)
Chapter 12 - Decision Analysis 39
Step 6 (part f):
Decision tree analysis.
Decision Analysis Example Problem Solution (9 of 9)
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