11

ChapterChapter 88Decision AnalysisDecision Analysis

Problem FormulationProblem Formulation Decision Making without ProbabilitiesDecision Making without Probabilities Decision Making with ProbabilitiesDecision Making with Probabilities Risk Analysis and Sensitivity AnalysisRisk Analysis and Sensitivity Analysis Decision Analysis with Sample InformationDecision Analysis with Sample Information Computing Branch ProbabilitiesComputing Branch Probabilities

Problem FormulationProblem Formulation

A decision problem is characterized by decisionA decision problem is characterized by decisionalternatives, states of nature, and resulting payoffs.alternatives, states of nature, and resulting payoffs.

TheThe decision alternativesdecision alternatives are the different possibleare the different possiblestrategies the decision maker can employ.strategies the decision maker can employ.

TheThe states of naturestates of nature refer to future events, notrefer to future events, notunder the control of the decision maker, whichunder the control of the decision maker, whichmay occur. States of nature should be defined somay occur. States of nature should be defined sothat they are mutually exclusive and collectivelythat they are mutually exclusive and collectivelyexhaustive.exhaustive.

Influence DiagramsInfluence Diagrams

AnAn influence diagraminfluence diagram is a graphical device showingis a graphical device showingthe relationships among the decisions, the chancethe relationships among the decisions, the chanceevents, and the consequences.events, and the consequences.

Squares or rectanglesSquares or rectangles depict decision nodes.depict decision nodes. Circles or ovalsCircles or ovals depict chance nodes.depict chance nodes. DiamondsDiamonds depict consequence nodes.depict consequence nodes. Lines or arcsLines or arcs connecting the nodes show the directionconnecting the nodes show the direction

of influence.of influence.

22

Payoff TablesPayoff Tables

The consequence resulting from a specificThe consequence resulting from a specificcombination of a decision alternative and a state ofcombination of a decision alternative and a state ofnature is anature is a payoffpayoff..

A table showing payoffs for all combinations ofA table showing payoffs for all combinations ofdecision alternatives and states of nature is adecision alternatives and states of nature is a payoffpayofftabletable..

Payoffs can be expressed in terms ofPayoffs can be expressed in terms of profitprofit,, costcost,, timetime,,distancedistance or any other appropriate measure.or any other appropriate measure.

Decision TreesDecision Trees

AA decision treedecision tree is a chronological representation ofis a chronological representation ofthe decision problem.the decision problem.

Each decision tree has two types of nodes;Each decision tree has two types of nodes; roundroundnodesnodes correspond to the states of nature whilecorrespond to the states of nature while squaresquarenodesnodes correspond to the decision alternatives.correspond to the decision alternatives.

TheThe branchesbranches leaving each round node represent theleaving each round node represent thedifferent states of nature while the branches leavingdifferent states of nature while the branches leavingeach square node represent the different decisioneach square node represent the different decisionalternatives.alternatives.

At the end of each limb of a tree are the payoffsAt the end of each limb of a tree are the payoffsattained from the series of branches making up thatattained from the series of branches making up thatlimb.limb.

Decision Making without ProbabilitiesDecision Making without Probabilities

Three commonly used criteria for decision makingThree commonly used criteria for decision makingwhen probability information regarding thewhen probability information regarding thelikelihood of the states of nature is unavailable are:likelihood of the states of nature is unavailable are:•• thethe optimisticoptimistic approachapproach•• thethe conservativeconservative approachapproach•• thethe minimax regretminimax regret approach.approach.

33

Optimistic ApproachOptimistic Approach

TheThe optimistic approachoptimistic approach would be used by anwould be used by anoptimistic decision maker.optimistic decision maker.

TheThe decision with the largest possible payoffdecision with the largest possible payoff isischosen.chosen.

If the payoff table was in terms of costs, theIf the payoff table was in terms of costs, the decisiondecisionwith the lowest costwith the lowest cost would be chosen.would be chosen.

Conservative ApproachConservative Approach

TheThe conservative approachconservative approach would be used by awould be used by aconservative decision maker.conservative decision maker.

For each decision the minimum payoff is listed andFor each decision the minimum payoff is listed andthen the decision corresponding to the maximumthen the decision corresponding to the maximumof these minimum payoffs is selected. (Hence, theof these minimum payoffs is selected. (Hence, theminimum possible payoff is maximizedminimum possible payoff is maximized.).)

If the payoff was in terms of costs, the maximumIf the payoff was in terms of costs, the maximumcosts would be determined for each decision andcosts would be determined for each decision andthen the decision corresponding to the minimumthen the decision corresponding to the minimumof these maximum costs is selected. (Hence, theof these maximum costs is selected. (Hence, themaximum possible cost is minimizedmaximum possible cost is minimized.).)

Minimax Regret ApproachMinimax Regret Approach

The minimax regret approach requires theThe minimax regret approach requires theconstruction of aconstruction of a regret tableregret table or anor an opportunityopportunityloss tableloss table..

This is done by calculating for each state of natureThis is done by calculating for each state of naturethe difference between each payoff and the largestthe difference between each payoff and the largestpayoff for that state of nature.payoff for that state of nature.

Then, using this regret table, the maximum regretThen, using this regret table, the maximum regretfor each possible decision is listed.for each possible decision is listed.

The decision chosen is the one corresponding toThe decision chosen is the one corresponding tothethe minimum of the maximum regretsminimum of the maximum regrets..

44

ExampleExample

Consider the following problem with threeConsider the following problem with threedecision alternatives and three states of nature withdecision alternatives and three states of nature withthe following payoff table representing profits:the following payoff table representing profits:

States of NatureStates of Naturess11 ss22 ss33

dd11 4 44 4 --22DecisionsDecisions dd22 0 30 3 --11

dd33 1 51 5 --33

Example: Optimistic ApproachExample: Optimistic Approach

An optimistic decision maker would use theAn optimistic decision maker would use theoptimistic (maximax) approach. We choose theoptimistic (maximax) approach. We choose thedecision that has the largest single value in thedecision that has the largest single value in thepayoff table.payoff table.

MaximumMaximumDecisionDecision PayoffPayoffdd11 44dd22 33dd33 55

MaximaxMaximaxpayoffpayoff

MaximaxMaximaxdecisiondecision

Example: Optimistic ApproachExample: Optimistic Approach

Formula SpreadsheetFormula SpreadsheetA B C D E F

123 Decision Maximum Recommended4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 =MAX(B5:D5) =IF(E5=$E$9,A5,"")6 d2 0 3 -1 =MAX(B6:D6) =IF(E6=$E$9,A6,"")7 d3 1 5 -3 =MAX(B7:D7) =IF(E7=$E$9,A7,"")89 =MAX(E5:E7)

State of Nature

Best Payoff

PAYOFF TABLE

55

Example: Optimistic ApproachExample: Optimistic Approach

Solution SpreadsheetSolution SpreadsheetA B C D E F

123 Decision Maximum Recommended4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 46 d2 0 3 -1 37 d3 1 5 -3 5 d389 5

State of Nature

Best Payoff

PAYOFF TABLE

Example: Conservative ApproachExample: Conservative Approach

A conservative decision maker would use theA conservative decision maker would use theconservative (maximin) approach. List the minimumconservative (maximin) approach. List the minimumpayoff for each decision. Choose the decision withpayoff for each decision. Choose the decision withthe maximum of these minimum payoffs.the maximum of these minimum payoffs.

MinimumMinimumDecisionDecision PayoffPayoffdd11 --22dd22 --11dd33 --33

MaximinMaximindecisiondecision

MaximinMaximinpayoffpayoff

Example: Conservative ApproachExample: Conservative Approach

Formula SpreadsheetFormula SpreadsheetA B C D E F

123 Decision Minimum Recommended4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 =MIN(B5:D5) =IF(E5=$E$9,A5,"")6 d2 0 3 -1 =MIN(B6:D6) =IF(E6=$E$9,A6,"")7 d3 1 5 -3 =MIN(B7:D7) =IF(E7=$E$9,A7,"")89 =MAX(E5:E7)

State of Nature

Best Payoff

PAYOFF TABLE

66

Example: Conservative ApproachExample: Conservative Approach

Solution SpreadsheetSolution SpreadsheetA B C D E F

123 Decision Minimum Recommended4 Alternative s1 s2 s3 Payoff Decision5 d1 4 4 -2 -26 d2 0 3 -1 -1 d27 d3 1 5 -3 -389 -1

State of Nature

Best Payoff

PAYOFF TABLE

For the minimax regret approach, first compute aFor the minimax regret approach, first compute aregret table by subtracting each payoff in a columnregret table by subtracting each payoff in a columnfrom the largest payoff in that column. In thisfrom the largest payoff in that column. In thisexample, in the first column subtract 4, 0, and 1 fromexample, in the first column subtract 4, 0, and 1 from4; etc. The resulting regret table is:4; etc. The resulting regret table is:

ss11 ss22 ss33

dd11 0 1 10 1 1dd22 4 2 04 2 0dd33 3 0 23 0 2

Example: Minimax Regret ApproachExample: Minimax Regret Approach

For each decision list the maximum regret.For each decision list the maximum regret.Choose the decision with the minimum of theseChoose the decision with the minimum of thesevalues.values.

MaximumMaximumDecisionDecision RegretRegretdd11 11dd22 44dd33 33

Example: Minimax Regret ApproachExample: Minimax Regret Approach

MinimaxMinimaxdecisiondecision

MinimaxMinimaxregretregret

77

Example: Minimax Regret ApproachExample: Minimax Regret Approach

Formula SpreadsheetFormula SpreadsheetA B C D E F

12 Decision3 Altern. s1 s2 s34 d1 4 4 -25 d2 0 3 -16 d3 1 5 -3789 Decision Maximum Recommended10 Altern. s1 s2 s3 Regret Decision11 d1 =MAX($B$4:$B$6)-B4 =MAX($C$4:$C$6)-C4 =MAX($D$4:$D$6)-D4 =MAX(B11:D11) =IF(E11=$E$14,A11,"")12 d2 =MAX($B$4:$B$6)-B5 =MAX($C$4:$C$6)-C5 =MAX($D$4:$D$6)-D5 =MAX(B12:D12) =IF(E12=$E$14,A12,"")13 d3 =MAX($B$4:$B$6)-B6 =MAX($C$4:$C$6)-C6 =MAX($D$4:$D$6)-D6 =MAX(B13:D13) =IF(E13=$E$14,A13,"")14 =MIN(E11:E13)Minimax Regret Value

State of NaturePAYOFF TABLE

State of NatureOPPORTUNITY LOSS TABLE

Solution SpreadsheetSolution SpreadsheetA B C D E F

12 Decision3 Alternative s1 s2 s34 d1 4 4 -25 d2 0 3 -16 d3 1 5 -3789 Decision Maximum Recommended10 Alternative s1 s2 s3 Regret Decision11 d1 0 1 1 1 d112 d2 4 2 0 413 d3 3 0 2 314 1Minimax Regret Value

State of NaturePAYOFF TABLE

State of NatureOPPORTUNITY LOSS TABLE

Example: Minimax Regret ApproachExample: Minimax Regret Approach

Decision Making with ProbabilitiesDecision Making with Probabilities

Expected Value ApproachExpected Value Approach•• If probabilistic information regarding the statesIf probabilistic information regarding the states

of nature is available, one may use theof nature is available, one may use the expectedexpectedvalue (EV) approachvalue (EV) approach..

•• Here the expected return for each decision isHere the expected return for each decision iscalculated by summing the products of thecalculated by summing the products of thepayoff under each state of nature and thepayoff under each state of nature and theprobability of the respective state of natureprobability of the respective state of natureoccurring.occurring.

•• The decision yielding theThe decision yielding the best expected returnbest expected return isischosen.chosen.

88

TheThe expected value of a decision alternativeexpected value of a decision alternative is theis thesum of weighted payoffs for the decision alternative.sum of weighted payoffs for the decision alternative.

The expected value (EV) of decision alternativeThe expected value (EV) of decision alternative ddii isisdefined as:defined as:

where:where: NN = the number of states of nature= the number of states of naturePP((ssjj ) = the probability of state of nature) = the probability of state of nature ssjjVVijij = the payoff corresponding to decision= the payoff corresponding to decision

alternativealternative ddii and state of natureand state of nature ssjj

Expected Value of a Decision AlternativeExpected Value of a Decision Alternative

EV( ) ( )d P s Vi j ijj

N

1

EV( ) ( )d P s Vi j ijj

N

1

Example: Burger PrinceExample: Burger Prince

Burger Prince Restaurant is considering openingBurger Prince Restaurant is considering openinga new restaurant on Main Street. It has threea new restaurant on Main Street. It has threedifferent models, each with a differentdifferent models, each with a differentseating capacity. Burger Princeseating capacity. Burger Princeestimates that the average number ofestimates that the average number ofcustomers per hour will be 80, 100, orcustomers per hour will be 80, 100, or120. The payoff table for the three120. The payoff table for the threemodels is on the next slide.models is on the next slide.

Payoff TablePayoff Table

Average Number of Customers Per HourAverage Number of Customers Per Hourss11 = 80= 80 ss22 = 100= 100 ss33 = 120= 120

Model A $10,000 $15,000 $14,000Model A $10,000 $15,000 $14,000Model B $ 8,000 $18,000 $12,000Model B $ 8,000 $18,000 $12,000Model C $ 6,000 $16,000 $21,000Model C $ 6,000 $16,000 $21,000

99

Expected Value ApproachExpected Value Approach

Calculate the expected value for each decision.Calculate the expected value for each decision.The decision tree on the next slide can assist in thisThe decision tree on the next slide can assist in thiscalculation. Herecalculation. Here dd11,, dd22,, dd33 represent the decisionrepresent the decisionalternatives of models A, B, C, andalternatives of models A, B, C, and ss11,, ss22,, ss33 representrepresentthe states of nature of 80, 100, and 120.the states of nature of 80, 100, and 120.

Decision TreeDecision Tree

1111

.2.2

.4.4

.4.4

.4.4

.2.2

.4.4

.4.4

.2.2

.4.4

dd11

dd22

dd33

ss11

ss11

ss11

ss22ss33

ss22

ss22ss33

ss33

PayoffsPayoffs10,00010,00015,00015,00014,00014,0008,0008,000

18,00018,00012,00012,0006,0006,000

16,00016,00021,00021,000

2222

3333

4444

Expected Value for Each DecisionExpected Value for Each Decision

Choose the model with largest EV, Model C.Choose the model with largest EV, Model C.

3333

dd11

dd22

dd33

EMV = .4(10,000) + .2(15,000) + .4(14,000)EMV = .4(10,000) + .2(15,000) + .4(14,000)= $12,600= $12,600

EMV = .4(8,000) + .2(18,000) + .4(12,000)EMV = .4(8,000) + .2(18,000) + .4(12,000)= $11,600= $11,600

EMV = .4(6,000) + .2(16,000) + .4(21,000)EMV = .4(6,000) + .2(16,000) + .4(21,000)= $14,000= $14,000

Model AModel A

Model BModel B

Model CModel C

2222

1111

4444

1010

Expected Value ApproachExpected Value Approach

Formula SpreadsheetFormula SpreadsheetA B C D E F

123 Decision Expected Recommended4 Alternative s1 = 80 s2 = 100 s3 = 120 Value Decision5 d1 = Model A 10,000 15,000 14,000 =$B$8*B5+$C$8*C5+$D$8*D5 =IF(E5=$E$9,A5,"")6 d2 = Model B 8,000 18,000 12,000 =$B$8*B6+$C$8*C6+$D$8*D6 =IF(E6=$E$9,A6,"")7 d3 = Model C 6,000 16,000 21,000 =$B$8*B7+$C$8*C7+$D$8*D7 =IF(E7=$E$9,A7,"")8 Probability 0.4 0.2 0.49 =MAX(E5:E7)

State of Nature

Maximum Expected Value

PAYOFF TABLE

Solution SpreadsheetSolution SpreadsheetA B C D E F

123 Decision Expected Recommended4 Alternative s1 = 80 s2 = 100 s3 = 120 Value Decision5 d1 = Model A 10,000 15,000 14,000 126006 d2 = Model B 8,000 18,000 12,000 116007 d3 = Model C 6,000 16,000 21,000 14000 d3 = Model C8 Probability 0.4 0.2 0.49 14000

State of Nature

Maximum Expected Value

PAYOFF TABLE

Expected Value ApproachExpected Value Approach

Expected Value of Perfect InformationExpected Value of Perfect Information

Frequently information is available which canFrequently information is available which canimprove the probability estimates for the states ofimprove the probability estimates for the states ofnature.nature.

TheThe expected value of perfect informationexpected value of perfect information (EVPI) is(EVPI) isthe increase in the expected profit that wouldthe increase in the expected profit that wouldresult if one knew with certainty which state ofresult if one knew with certainty which state ofnature would occur.nature would occur.

The EVPI provides anThe EVPI provides an upper bound on theupper bound on theexpected value of any sample or surveyexpected value of any sample or surveyinformationinformation..

1111

Expected Value of Perfect InformationExpected Value of Perfect Information

EVPI CalculationEVPI Calculation•• Step 1:Step 1:

Determine the optimal return corresponding toDetermine the optimal return corresponding toeach state of nature.each state of nature.

•• Step 2:Step 2:Compute the expected value of these optimalCompute the expected value of these optimal

returns.returns.•• Step 3:Step 3:

Subtract the EV of the optimal decision from theSubtract the EV of the optimal decision from theamount determined in step (2).amount determined in step (2).

Calculate the expected value for the optimumCalculate the expected value for the optimumpayoff for each state of nature and subtract the EV ofpayoff for each state of nature and subtract the EV ofthe optimal decision.the optimal decision.

EVPI= .4(10,000) + .2(18,000) + .4(21,000)EVPI= .4(10,000) + .2(18,000) + .4(21,000) -- 14,000 = $2,00014,000 = $2,000

Expected Value of Perfect InformationExpected Value of Perfect Information

SpreadsheetSpreadsheetA B C D E F

123 Decision Expected Recommended4 Alternative s1 = 80 s2 = 100 s3 = 120 Value Decision5 d1 = Model A 10,000 15,000 14,000 126006 d2 = Model B 8,000 18,000 12,000 116007 d3 = Model C 6,000 16,000 21,000 14000 d3 = Model C8 Probability 0.4 0.2 0.49 140001011 EVwPI EVPI12 10,000 18,000 21,000 16000 2000

State of Nature

Maximum Expected Value

PAYOFF TABLE

Maximum Payoff

Expected Value of Perfect InformationExpected Value of Perfect Information

1212

Risk AnalysisRisk Analysis

Risk analysisRisk analysis helps the decision maker recognize thehelps the decision maker recognize thedifference between:difference between:•• the expected value of a decision alternative, andthe expected value of a decision alternative, and•• the payoff that might actually occurthe payoff that might actually occur

TheThe risk profilerisk profile for a decision alternative shows thefor a decision alternative shows thepossible payoffs for the decision alternative alongpossible payoffs for the decision alternative alongwith their associated probabilities.with their associated probabilities.

Risk ProfileRisk Profile

Model C Decision AlternativeModel C Decision Alternative

.10.10

.20.20

.30.30

.40.40

.50.50

5 10 15 20 255 10 15 20 25

Prob

abili

tyPr

obab

ility

Sensitivity AnalysisSensitivity Analysis

Sensitivity analysisSensitivity analysis can be used to determine howcan be used to determine howchanges to the following inputs affect thechanges to the following inputs affect therecommended decision alternative:recommended decision alternative:•• probabilities for the states of natureprobabilities for the states of nature•• values of the payoffsvalues of the payoffs

If a small change in the value of one of the inputsIf a small change in the value of one of the inputscauses a change in the recommended decisioncauses a change in the recommended decisionalternative, extra effort and care should be taken inalternative, extra effort and care should be taken inestimating the input value.estimating the input value.

1313

Bayes’ Theorem and Posterior ProbabilitiesBayes’ Theorem and Posterior Probabilities

Knowledge of sample (survey) information can be usedKnowledge of sample (survey) information can be usedto revise the probability estimates for the states of nature.to revise the probability estimates for the states of nature.

Prior to obtaining this information, the probabilityPrior to obtaining this information, the probabilityestimates for the states of nature are calledestimates for the states of nature are called priorpriorprobabilitiesprobabilities..

With knowledge ofWith knowledge of conditional probabilitiesconditional probabilities for thefor theoutcomes or indicators of the sample or surveyoutcomes or indicators of the sample or surveyinformation, these prior probabilities can be revised byinformation, these prior probabilities can be revised byemployingemploying Bayes' TheoremBayes' Theorem..

The outcomes of this analysis are calledThe outcomes of this analysis are called posteriorposteriorprobabilitiesprobabilities oror branch probabilitiesbranch probabilities for decision trees.for decision trees.

Computing Branch ProbabilitiesComputing Branch Probabilities

Branch (Posterior) Probabilities CalculationBranch (Posterior) Probabilities Calculation•• Step 1:Step 1:

For each state of nature, multiply the priorFor each state of nature, multiply the priorprobability by its conditional probability for theprobability by its conditional probability for theindicatorindicator ---- this gives thethis gives the joint probabilitiesjoint probabilities for thefor thestates and indicator.states and indicator.

Computing Branch ProbabilitiesComputing Branch Probabilities

Branch (Posterior) Probabilities CalculationBranch (Posterior) Probabilities Calculation•• Step 2:Step 2:

Sum these joint probabilities over all statesSum these joint probabilities over all states ---- thisthisgives thegives the marginal probabilitymarginal probability for the indicator.for the indicator.

•• Step 3:Step 3:For each state, divide its joint probability by theFor each state, divide its joint probability by the

marginal probability for the indicatormarginal probability for the indicator ---- this givesthis givesthe posterior probability distribution.the posterior probability distribution.

1414

Expected Value of Sample InformationExpected Value of Sample Information

TheThe expected value of sample informationexpected value of sample information (EVSI) is(EVSI) isthe additional expected profit possible throughthe additional expected profit possible throughknowledge of the sample or survey information.knowledge of the sample or survey information.

Expected Value of Sample InformationExpected Value of Sample Information

EVSI CalculationEVSI Calculation•• Step 1:Step 1:

Determine the optimal decision and its expectedDetermine the optimal decision and its expectedreturn for the possible outcomes of the sample usingreturn for the possible outcomes of the sample usingthe posterior probabilities for the states of nature.the posterior probabilities for the states of nature.

•• Step 2:Step 2:Compute the expected value of these optimalCompute the expected value of these optimal

returns.returns.•• Step 3:Step 3:

Subtract the EV of the optimal decision obtainedSubtract the EV of the optimal decision obtainedwithout using the sample information from thewithout using the sample information from theamount determined in step (2).amount determined in step (2).

Efficiency of Sample InformationEfficiency of Sample Information

Efficiency of sample informationEfficiency of sample information is the ratio of EVSIis the ratio of EVSIto EVPI.to EVPI.

As the EVPI provides an upper bound for the EVSI,As the EVPI provides an upper bound for the EVSI,efficiency is always a number between 0 and 1.efficiency is always a number between 0 and 1.

1515

Burger Prince must decide whether or not toBurger Prince must decide whether or not topurchase a marketing survey from Stanton Marketingpurchase a marketing survey from Stanton Marketingfor $1,000. The results of the survey are "favorable" orfor $1,000. The results of the survey are "favorable" or"unfavorable". The conditional probabilities are:"unfavorable". The conditional probabilities are:

P(favorable | 80 customers per hour) = .2P(favorable | 80 customers per hour) = .2P(favorable | 100 customers per hour) = .5P(favorable | 100 customers per hour) = .5P(favorable | 120 customers per hour) = .9P(favorable | 120 customers per hour) = .9

Should Burger Prince have the survey performedShould Burger Prince have the survey performedby Stanton Marketing?by Stanton Marketing?

Sample InformationSample Information

Influence DiagramInfluence Diagram

RestaurantRestaurantSizeSize ProfitProfit

Avg. NumberAvg. Numberof Customersof Customers

Per HourPer Hour

MarketMarketSurveySurveyResultsResults

MarketMarketSurveySurvey

DecisionDecisionChanceChanceConsequenceConsequence

FavorableFavorable

StateState PriorPrior ConditionalConditional JointJoint PosteriorPosterior80 .4 .2 .08 .14880 .4 .2 .08 .148

100 .2 .5 .10 .185100 .2 .5 .10 .185120 .4 .9120 .4 .9 .36.36 .667.667

Total .54 1.000Total .54 1.000

P(favorable) = .54P(favorable) = .54

Posterior ProbabilitiesPosterior Probabilities

1616

UnfavorableUnfavorable

StateState PriorPrior ConditionalConditional JointJoint PosteriorPosterior80 .4 .8 .32 .69680 .4 .8 .32 .696

100 .2 .5 .10 .217100 .2 .5 .10 .217120 .4 .1120 .4 .1 .04.04 .087.087

Total .46 1.000Total .46 1.000

P(unfavorable) = .46P(unfavorable) = .46

Posterior ProbabilitiesPosterior Probabilities

Formula SpreadsheetFormula SpreadsheetA B C D E

12 Prior Conditional Joint Posterior3 State of Nature Probabilities Probabilities Probabilities Probabilities4 s1 = 80 0.4 0.2 =B4*C4 =D4/$D$75 s2 = 100 0.2 0.5 =B5*C5 =D5/$D$76 s3 = 120 0.4 0.9 =B6*C6 =D6/$D$77 =SUM(D4:D6)8910 Prior Conditional Joint Posterior11 State of Nature Probabilities Probabilities Probabilities Probabilities12 s1 = 80 0.4 0.8 =B12*C12 =D12/$D$1513 s2 = 100 0.2 0.5 =B13*C13 =D13/$D$1514 s3 = 120 0.4 0.1 =B14*C14 =D14/$D$1515 =SUM(D12:D14)

Market Research Favorable

P(Favorable) =

Market Research Unfavorable

P(Unfavorable) =

Posterior ProbabilitiesPosterior Probabilities

Solution SpreadsheetSolution SpreadsheetA B C D E

12 Prior Conditional Joint Posterior3 State of Nature Probabilities Probabilities Probabilities Probabilities4 s1 = 80 0.4 0.2 0.08 0.1485 s2 = 100 0.2 0.5 0.10 0.1856 s3 = 120 0.4 0.9 0.36 0.6677 0.548910 Prior Conditional Joint Posterior11 State of Nature Probabilities Probabilities Probabilities Probabilities12 s1 = 80 0.4 0.8 0.32 0.69613 s2 = 100 0.2 0.5 0.10 0.21714 s3 = 120 0.4 0.1 0.04 0.08715 0.46

Market Research Favorable

P(Favorable) =

Market Research Unfavorable

P(Favorable) =

Posterior ProbabilitiesPosterior Probabilities

1717

Decision TreeDecision Tree

Top HalfTop Half

ss11 (.148)(.148)

ss11 (.148)(.148)

ss11 (.148)(.148)ss22 (.185)(.185)

ss22 (.185)(.185)

ss22 (.185)(.185)

ss33 (.667)(.667)

ss33 (.667)(.667)

ss33 (.667)(.667)

$10,000$10,000$15,000$15,000$14,000$14,000$8,000$8,000

$18,000$18,000$12,000$12,000$6,000$6,000

$16,000$16,000

$21,000$21,000

II11

(.54)(.54)

dd11

dd22

dd33

2222

4444

5555

6666

1111

Bottom HalfBottom Half

ss11 (.696)(.696)

ss11 (.696)(.696)

ss11 (.696)(.696)

ss22 (.217)(.217)

ss22 (.217)(.217)

ss22 (.217)(.217)

ss33 (.087)(.087)

ss33 (.087)(.087)

ss33 (.087)(.087)

$10,000$10,000$15,000$15,000

$18,000$18,000

$14,000$14,000$8,000$8,000

$12,000$12,000$6,000$6,000

$16,000$16,000$21,000$21,000

II22(.46)(.46) dd11

dd22

dd33

7777

9999

88883333

1111

Decision TreeDecision Tree

II22(.46)(.46)

dd11

dd22

dd33

EMV = .696(10,000) + .217(15,000)EMV = .696(10,000) + .217(15,000)+.087(14,000)=+.087(14,000)= $11,433$11,433

EMV = .696(8,000) + .217(18,000)EMV = .696(8,000) + .217(18,000)+ .087(12,000) = $10,554+ .087(12,000) = $10,554

EMV = .696(6,000) + .217(16,000)EMV = .696(6,000) + .217(16,000)+.087(21,000) = $9,475+.087(21,000) = $9,475

II11(.54)(.54)

dd11

dd22

dd33

EMV = .148(10,000) + .185(15,000)EMV = .148(10,000) + .185(15,000)+ .667(14,000) = $13,593+ .667(14,000) = $13,593

EMV = .148 (8,000) + .185(18,000)EMV = .148 (8,000) + .185(18,000)+ .667(12,000) = $12,518+ .667(12,000) = $12,518

EMV = .148(6,000) + .185(16,000)EMV = .148(6,000) + .185(16,000)+.667(21,000) =+.667(21,000) = $17,855$17,855

4444

5555

6666

7777

8888

9999

2222

3333

1111

$17,855$17,855

$11,433$11,433

Decision TreeDecision Tree

1818

If the outcome of the survey is "favorable”,If the outcome of the survey is "favorable”,choose Model C. If it is “unfavorable”, choose Model A.choose Model C. If it is “unfavorable”, choose Model A.

EVSI = .54($17,855) + .46($11,433)EVSI = .54($17,855) + .46($11,433) -- $14,000 = $900.88$14,000 = $900.88

Since this is less than the cost of the survey, theSince this is less than the cost of the survey, thesurvey shouldsurvey should notnot be purchased.be purchased.

Expected Value of Sample InformationExpected Value of Sample Information

Efficiency of Sample InformationEfficiency of Sample Information

The efficiency of the survey:The efficiency of the survey:

EVSI/EVPI = ($900.88)/($2000) = .4504EVSI/EVPI = ($900.88)/($2000) = .4504