12-1 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Decision Analysis Chapter 12.

12-1Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Decision Analysis

Chapter 12


■ Components of Decision Making

■ Decision Making without Probabilities

■ Decision Making with Probabilities

■ Decision Analysis with Additional Information

■ Utility

Chapter Topics


Previous chapters used an assumption of certainty

with regards to problem parameters.

This chapter relaxes the certainty assumption

Two categories of decision situations:

Probabilities can be assigned to future

occurrences

Probabilities cannot be assigned to future

occurrences

Decision AnalysisOverview


Table 12.1 Payoff 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 AnalysisDecision Making Without Probabilities

Figure 12.1 Decision situation with real estate investment alternatives


Decision-Making Criteria

maximax maximin minimax

minimax regret Hurwicz equal likelihood

Decision AnalysisDecision Making without Probabilities

Table 12.2 Payoff table for the real estate investments


Table 12.3 Payoff 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 12.4 Payoff 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 12.5 Regret table

Regret is the difference between the payoff from the best decision and all other decision payoffs.

Example: under the Good Economic Conditions state of nature, the best payoff is $100,000. The manager’s regret for choosing the Warehouse alternative is $100,000-$30,000=$70,000

Decision Making without ProbabilitiesMinimax Regret Criterion

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Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Table 12.6 Regret table illustrating the minimax regret decision

The manager calculates regrets for all alternatives under each state of nature. Then the manager identifies the maximum regret for each alternative.

Finally, the manager attempts to avoid regret by selecting the decision alternative that minimizes the maximum regret.

Decision Making without ProbabilitiesMinimax Regret Criterion

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The Hurwicz criterion is a compromise between the maximax and maximin criteria.

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. Here, = 0.4.

Decision Making without ProbabilitiesHurwicz Criterion

Decision ValuesApartment building

$50,000(.4) + 30,000(.6) = 38,000

Office building $100,000(.4) - 40,000(.6) = 16,000

Warehouse $30,000(.4) + 10,000(.6) = 18,000

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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 Making without ProbabilitiesEqual Likelihood Criterion

Decision ValuesApartment building

$50,000(.5) + 30,000(.5) = 40,000

Office building $100,000(.5) - 40,000(.5) = 30,000

Warehouse $30,000(.5) + 10,000(.5) = 20,000

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■ 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

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Exhibit 12.1

Decision Making without ProbabilitiesSolution with QM for Windows (1 of 3)

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Exhibit 12.2


Equal likelihood weight

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Exhibit 12.3


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Decision Making without ProbabilitiesSolution with Excel

Exhibit 12.4

=MIN(C7,D7)

=MAX(E7,E9)=MAX(C18,D18)

=MAX(F7:F9)

=MAX(C7:C9)-C9

=C7*C25+D7*C26

=C7*0.5+D7*0.5

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Expected value is computed by multiplying each decision outcome under each state of nature by the probability of its occurrence.

EV(Apartment) = $50,000(.6) + 30,000(.4) = $42,000EV(Office) = $100,000(.6) - 40,000(.4) = $44,000EV(Warehouse) = $30,000(.6) + 10,000(.4) = $22,000

Table 12.7 Payoff table with probabilities for states of nature

Decision Making with ProbabilitiesExpected Value

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■ 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(.6) + 0(.4) = 30,000EOL(Office) = $0(.6) + 70,000(.4) = 28,000EOL(Warehouse) = $70,000(.6) + 20,000(.4) =

50,000

Table 12.8 Regret table with probabilities for states of nature

Decision Making with ProbabilitiesExpected Opportunity Loss

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Exhibit 12.5

Expected Value ProblemsSolution with QM for Windows

Expected values

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Exhibit 12.6

Expected Value ProblemsSolution with Excel and Excel QM (1 of 2)

Expected value for apartment building

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Expected Value ProblemsSolution with Excel and Excel QM (2 of 2)

Exhibit 12.7

Click on “Add-Ins” to access the “Excel QM” menu

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■ 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

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Table 12.9 Payoff table with decisions, given perfect information

Decision Making with ProbabilitiesEVPI Example (1 of 2)

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■ 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

Decision Making with ProbabilitiesEVPI Example (2 of 2)

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Exhibit 12.8

Decision Making with ProbabilitiesEVPI with QM for Windows

The expected value, given perfect information, in Cell F12

=MAX(E7:E9)

=F12-F11

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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

Decision Making with ProbabilitiesDecision Trees (1 of 4)

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Figure 12.2 Decision tree for real estate investment example


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■ 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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Figure 12.3 Decision tree with expected value at probability nodes


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Decision Making with ProbabilitiesDecision Trees with QM for Windows

Exhibit 12.9

Select node to add from

Number of branches from node 1

Add branches from node 1 to 2, 3, and 4

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Decision Making with ProbabilitiesDecision Trees with Excel and TreePlan (1 of 4)

Exhibit 12.10

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Exhibit 12.11

Decision Making with ProbabilitiesDecision Trees with Excel and TreePlan (2 of 4)To create another

branch, click “B5,” then the “Decision Tree” menu, and select “Add Branch”

Invoke TreePlan from the “Add Ins” menu

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Exhibit 12.12


Click on cell “F3,” then “Decision Tree”

Select “Change to Event Node” and add two new branches

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Exhibit 12.13

Add numerical dollar and probability values in these cells in column H

These cells contain decision tree formulas; do not type in these cells in columns E and I

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Exhibit 12.14

Sequential Decision Tree AnalysisSolution with QM for Windows

Cell A16 contains the expected value of $44,000

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Decision Making with ProbabilitiesSequential 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.

■ The next slide shows the real estate investment example modified to encompass a ten-year period in which several decisions must be made.

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Figure 12.4 Sequential decision tree


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■ Expected value of apartment building is:

$1,290,000-800,000 = $490,000

■ Expected value if land is purchased is:

$1,360,000-200,000 = $1,160,000

■ The decision is to purchase land; it has the highest net expected value of $1,160,000.

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Figure 12.5 Sequential decision tree with nodal expected values


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Exhibit 12.15

Sequential Decision Tree AnalysisSolution with Excel QM

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Exhibit 12.16

Sequential Decision Tree AnalysisSolution with TreePlan

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■ Bayesian analysis uses additional information to alter the marginal probability of the occurrence of an event.

■ In the real estate investment example, using the expected value criterion, the best decision was to purchase the office building with an expected value of $444,000, and EVPI of $28,000.

Table 12.11 Payoff table for real estate investment

Decision Analysis with Additional InformationBayesian Analysis (1 of 3)

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■ A conditional probability is the probability that an event will occur given that another event has already occurred.

■ An economic analyst provides additional information for the 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) = .80 P(NG) = .20

P(Pp) = .10 P(Np) = .90


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■ A posterior probability is the altered marginal probability of an event based on additional information.

■ Prior probabilities for good or poor economic conditions in the real estate decision:

P(g) = .60; P(p) = .40

■ Posterior probabilities by Bayes’ rule:

(gP) = P(PG)P(g)/[P(Pg)P(g) + P(Pp)P(p)]

= (.80)(.60)/[(.80)(.60) + (.10)(.40)] = .923

■ Posterior (revised) probabilities for decision:

P(gN) = .250 P(pP) = .077 P(pN) = .750


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Decision Analysis with Additional InformationDecision Trees with Posterior Probabilities (1 of 4)Decision trees with posterior probabilities differ

from earlier versions in that:

■Two new branches at the beginning of the tree represent report outcomes.

■Probabilities of each state of nature are posterior probabilities from Bayes’ rule.

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Figure 12.6 Decision tree with posterior probabilities

Decision Analysis with Additional InformationDecision Trees with Posterior Probabilities (2 of 4)

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Decision Analysis with Additional InformationDecision 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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Figure 12.7 Decision tree analysis for real estate investment

Decision Analysis with Additional InformationDecision Trees with Posterior Probabilities (4 of 4)

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Table 12.12 Computation of posterior probabilities

Decision Analysis with Additional InformationComputing Posterior Probabilities with Tables

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Decision Analysis with Additional Information Computing Posterior Probabilities with Excel

Exhibit 12.17

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■ 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

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Table 12.13 Payoff table for auto insurance example

Decision Analysis with Additional InformationUtility (1 of 2)

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Expected Cost (insurance) = .992($500) + .008(500) = $500

Expected Cost (no insurance) = .992($0) + .008(10,000) = $80

The 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)

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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 corporate raider contemplates the future of a recent acquisition. Three alternatives are being considered in two states of nature. The payoff table is below.

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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 the following, P(Pg) = .70, P(Ng) = .30,

P(Pp) = 20, P(Np) = .80, determine posterior probabilities using Bayes’ rule.

f. Perform a decision tree analysis using the posterior probability obtained in part e.

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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


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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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Equal Likelihood Decision: Expand

Expand $800,000(.5) + 500,000(.5) = $650,000

Status quo $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 $800,000(.7) + 500,000(.3) = $710,000

Status quo $1,300,000(.7) - 150,000(.3) = $865,000

Sell $320,000(.7) + 320,000(.3) = $320,000


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Expected opportunity loss decision: Maintain status quo

Expand $500,000(.7) + 0(.3) = $350,000

Status quo 0(.7) + 650,000(.3) = $195,000

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


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Step 5 (part e): Determine posterior probabilities.

P(gP) = P(Pg)P(g)/[P(Pg)P(g) + P(Pp)P(p)]

= (.70)(.70)/[(.70)(.70) + (.20)(.30)] = .891

P(pP) = .109

P(gN) = P(Ng)P(g)/[P(Ng)P(g) + P(Np)P(p)]

= (.30)(.70)/[(.30)(.70) + (.80)(.30)] = .467

P(pN) = .533


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Step 6 (part f): Decision tree analysis.


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12-1 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Decision Analysis Chapter 12.

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