chap12 decision analysis

Chapter 12 - Decision Analysis 1

Chapter 12

Decision Analysis

Introduction to Management Science

8th Edition

by

Bernard W. Taylor III


Components of Decision Making

Decision Making without Probabilities

Decision Making with Probabilities

Decision Analysis with Additional Information

Chapter Topics


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


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


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


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


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


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


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


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


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


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


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


Table 12.9Payoff Table with Decisions, Given Perfect Information

Decision Making with ProbabilitiesEVPI Example (1 of 2)


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)


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)


Figure 12.1Decision Tree for Real Estate Investment Example



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.



Figure 12.2Decision Tree with Expected Value at Probability Nodes



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)


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



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


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pPpPPgPgPP

gPgPPPgP


Table 12.12Computation of Posterior Probabilities

Decision Analysis with Additional InformationComputing Posterior Probabilities with Tables


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.


Figure 12.5Decision Tree with

Posterior Probabilities




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


Figure 12.6Decision Tree Analysis



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


Table 12.13Payoff Table for Auto Insurance Example

Decision Analysis with Additional InformationUtility (1 of 2)


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)


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)



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.


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



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



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



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



Step 4 (part d): Develop a decision tree.



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



Step 6 (part f):

Decision tree analysis.


chap12 decision analysis

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