Chapter 3 Decision Analysis

Chapter 3 - Decision Analysis 1

Chapter 3

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

Utility

Chapter Topics


Table 3.1Payoff 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 (minimal regret), Hurwicz, and equal likelihood

Table 3.2Payoff Table for the Real Estate Investments

Decision AnalysisDecision Making without Probabilities


Table 3.3Payoff 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 3.4Payoff Table Illustrating a Maximin Decision

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.

Decision Making without ProbabilitiesMaximin Criterion

conservative


Table 3.6 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

Maximal

regrets

$ 50,000

$ 70,000

$ 70,000

Maximal

regrets

$ 50,000

$ 70,000

$ 70,000

Highestpayoff

$100,000- $50,000


The Hurwicz criterion is a compromise between the maximax (optimist) and maximin (conservative) criterion.

A coefficient of optimism, , is a measureof 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(.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

Decision Making without ProbabilitiesHurwicz Criterion

= 0.4


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 methodIn general, it is essentially different !

Decision Values

Apartment 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

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


Exhibit 3.1

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


Exhibit 3.2



Exhibit 3.3



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) = $42,000

EV(Office) = $100,000(.6) – $40,000(.4) = $44,000

EV(Warehouse) = $30,000(.6) + $10,000(.4) = $22,000

Table 3.7Payoff 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(.6) + $0(.4) = $30,000

EOL(Office) = $0(.6) + $70,000(.4) = $28,000

EOL(Warehouse) = $70,000(.6) + $20,000(.4) = $50,000

Table 3.8Regret (Opportunity Loss) Table

with Probabilities for States of Nature

Decision Making with ProbabilitiesExpected Opportunity Loss


Exhibit 3.4

Expected Value ProblemsSolution with QM for Windows


Exhibit 3.5

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


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.

Decision Making with ProbabilitiesExpected Value of Perfect Information


Table 3.9Payoff Table with Decisions, Given Perfect Information

Decision Making with ProbabilitiesEVPI Example (1 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

Decision Making with ProbabilitiesEVPI Example (2 of 2)

EV

$42,000

$44,000

$22,000

The“genie pick”


Exhibit 3.6

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

$100,000*0.6+$30,000*0.4 = $72,000


Exhibit 3.7

Decision Making with ProbabilitiesEVPI with QM for Windows


A decision tree is a diagram consisting of decision nodes (represented as squares), probability nodes (circles), and decision alternatives (branches).

Table 3.10Payoff Table for Real Estate Investment Example

Decision Making with ProbabilitiesDecision Trees (1 of 4)


Figure 3.1Decision Tree for Real Estate Investment Example


controllable

uncontrollable


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.



Figure 3.2Decision Tree with Expected Value at Probability Nodes



Exhibit 3.8

Decision Making with ProbabilitiesDecision Trees with QM for Windows


Exhibit 3.9

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


Exhibit 3.10



Exhibit 3.11



Exhibit 3.12



Decision Making with ProbabilitiesSequential 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:


Figure 3.3Sequential Decision Tree


The decision to bemade at [1] logicallydepends on the decisions(to be) made at [4] and [5].


Figure 3.4Sequential Decision Tree with Nodal Expected Values




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.)


Exhibit 3.13

Sequential Decision Tree AnalysisSolution with QM for Windows


Exhibit 3.14

Sequential Decision Tree AnalysisSolution with Excel and TreePlan


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 3.11Payoff 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) = .80 P(Ng) = .20

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


as before…

new info

new, given


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(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



Decision Analysis with Additional InformationDecision 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.


Figure 3.5Decision Tree with

Posterior Probabilities


P(P|g)=.80

P(N|g)=.20

P(P|p)=.10

P(N|p)=.90

P(g)=.60

P(p)=.40

P(g|P)=.923

P(p|P)=.077

P(g|N)=.250

P(p|N)=.750



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” “Negative report”


Figure 3.6Decision Tree Analysis



Table 3.12Computation of Posterior Probabilities

Decision Analysis with Additional InformationComputing Posterior Probabilities with Tables

Indeed, this equals [ P(P|g)P(g)+P(P|p)P(p) ] = P(P&g) + P(P&p) = P(P) .


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 3.13Payoff Table for Auto Insurance Example

Decision Analysis with Additional InformationUtility (1 of 2)

Cost


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 low-probability event in lieu of a sure thing.

Decision Analysis with Additional InformationUtility (2 of 2)


Decision Analysis Example Problem Solution (1 of 9)

DecisionsGood ForeignCompetitive Conditions

Poor ForeignCompetitive Conditions

Expand $800,000 $500,000

Maintain Status Quo $1,300,00 –$150,000

Sell Now $320,000 $320,000

States of Nature



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(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.


Step 1 (part a): Determine decisions without probabilities.

Maximax (Optimist) Decision: Maintain status quo

Decisions maximum Payoffs

Expand $800,000Status quo 1,300,000 (Maximum)Sell 320,000

Maximin (Conservative) Decision: Expand

Decisions minimum Payoffs

Expand $500,000 (Maximum)Status quo -150,000Sell 320,000



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



Equal Likelihood (Laplace) 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



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



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)]

= (.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



Step 6 (part f): Decision tree analysis.


Without the report, maintain statusquo, based on the expected payoff

value $865,000.

With the report, the payoff may beexpected 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 paidto obtain such a report. (EVPI)


Chapter 3 Decision Analysis

Documents

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decision payoffs

best decision

decision valuesapartment

decision analysisintroduction

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decision makers optimism

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