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1 1 Slide © 2006 Thomson South-Western. All Rights Reserved. Slides prepared by JOHN LOUCKS St. Edward’s University

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2 2 Slide © 2006 Thomson South-Western. All Rights Reserved. Chapter 4 Decision Analysis n Problem Formulation n Decision Making without Probabilities n Decision Making with Probabilities n Risk Analysis and Sensitivity Analysis n Decision Analysis with Sample Information n Computing Branch Probabilities

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3 3 Slide © 2006 Thomson South-Western. All Rights Reserved. Problem Formulation n The first step in the decision analysis process is problem formulation. n We begin with a verbal statement of the problem. n Then we identify: the decision alternatives the decision alternatives the states of nature (uncertain future events) the states of nature (uncertain future events) the payoff (consequences) associated with each specific combination of: the payoff (consequences) associated with each specific combination of: decision alternativedecision alternative state of naturestate of nature

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4 4 Slide © 2006 Thomson South-Western. All Rights Reserved. Burger Prince Restaurant is considering opening a new restaurant on Main Street. The company has three different building designs (A, B, and C), each with a different seating capacity. Burger Prince estimates that the average number of customers arriving per hour will be 40, 60, or 80. n Example: Burger Prince Restaurant Problem Formulation

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5 5 Slide © 2006 Thomson South-Western. All Rights Reserved. Problem Formulation d 1 = use building design A d 2 = use building design B d 3 = use building design C n Decision Alternatives n States of Nature s 1 = an average of 40 customers arriving per hour s 2 = an average of 60 customers arriving per hour s 3 = an average of 80 customers arriving per hour

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6 6 Slide © 2006 Thomson South-Western. All Rights Reserved. n The consequence resulting from a specific combination of a decision alternative and a state of nature is a payoff. n A table showing payoffs for all combinations of decision alternatives and states of nature is a payoff table. n Payoffs can be expressed in terms of profit, cost, time, distance or any other appropriate measure. Problem Formulation n Payoff Table

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7 7 Slide © 2006 Thomson South-Western. All Rights Reserved. n Payoff Table (Payoffs are Profit Per Week) $ 6,000 $16,000 $21,000 $ 8,000 $18,000 $12,000 $10,000 $15,000 $14,000 Average Number of Customers Per Hour Customers Per Hour s 1 = 40 s 2 = 60 s 3 = 80 Design A Design B Design C Problem Formulation

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8 8 Slide © 2006 Thomson South-Western. All Rights Reserved. n An influence diagram is a graphical device showing the relationships among the decisions, the chance events, and the consequences. n Squares or rectangles depict decision nodes. n Circles or ovals depict chance nodes. n Diamonds depict consequence nodes. n Lines or arcs connecting the nodes show the direction of influence. n Influence Diagram Problem Formulation

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9 9 Slide © 2006 Thomson South-Western. All Rights Reserved. RestaurantDesignRestaurantDesignProfitProfit Average Number of Customers Per Hour Average Number of Customers Per Hour n Influence Diagram Problem Formulation Design A ( d 1 ) Design B ( d 2 ) Design C ( d 3 ) DecisionAlternatives 40 customers per hour ( s 1 ) 60 customers per hour ( s 2 ) 80 customers per hour ( s 3 ) States of Nature Consequence Profit

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10 Slide © 2006 Thomson South-Western. All Rights Reserved. n A decision tree is a chronological representation of the decision problem. n A decision tree has two types of nodes: round nodes correspond to chance events round nodes correspond to chance events square nodes correspond to decisions square nodes correspond to decisions n Branches leaving a round node represent the different states of nature; branches leaving a square node represent the different decision alternatives. n At the end of a limb of the tree is the payoff attained from the series of branches making up the limb. Problem Formulation n Decision Tree

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11 Slide © 2006 Thomson South-Western. All Rights Reserved ,000 10,000 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, Problem Formulation 40 customers per hour ( s 1 ) Design A ( d 1 ) Design B ( d 2 ) Design C ( d 3 ) n Decision Tree 60 customers per hour ( s 2 ) 80 customers per hour ( s 3 ) 40 customers per hour ( s 1 ) 60 customers per hour ( s 2 ) 80 customers per hour ( s 3 ) 40 customers per hour ( s 1 ) 60 customers per hour ( s 2 ) 80 customers per hour ( s 3 )

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12 Slide © 2006 Thomson South-Western. All Rights Reserved. Decision Making without Probabilities Three commonly used criteria for decision Three commonly used criteria for decision making when probability information regarding the likelihood of the states of nature is unavailable are: the optimistic (maximax) approach the optimistic (maximax) approach the conservative (maximin) approach the conservative (maximin) approach the minimax regret approach. the minimax regret approach. n Criteria for Decision Making

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13 Slide © 2006 Thomson South-Western. All Rights Reserved. n The optimistic approach would be used by an optimistic decision maker. n The decision with the overall largest payoff is chosen. n If the payoff table is in terms of costs, the decision with the overall lowest cost will be chosen (hence, a minimin approach). n Optimistic (Maximax) Approach Decision Making without Probabilities

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14 Slide © 2006 Thomson South-Western. All Rights Reserved. The decision that has the largest single value in the payoff table is chosen. Maximaxpayoff Maximax decision n Optimistic (Maximax) Approach Decision Making without Probabilities States of Nature States of Nature Decision (Customers Per Hour) Decision (Customers Per Hour) Alternative 40 s 1 60 s 2 80 s 3 Design A d 1 10,000 15,000 14,000 Design B d 2 8,000 18,000 12,000 Design C d 3 6,000 16,000 21,000

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15 Slide © 2006 Thomson South-Western. All Rights Reserved. n The conservative approach would be used by a conservative decision maker. n For each decision the minimum payoff is listed. n The decision corresponding to the maximum of these minimum payoffs is selected. n If payoffs are in terms of costs, the maximum costs will be determined for each decision and then the decision corresponding to the minimum of these maximum costs will be selected. (Hence, a minimax approach) Decision Making without Probabilities n Conservative (Maximin) Approach

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16 Slide © 2006 Thomson South-Western. All Rights Reserved. List the minimum payoff for each decision. Choose the decision with the maximum of these minimum payoffs. n Conservative (Maximin) Approach Decision Making without Probabilities Maximinpayoff Maximin decision Decision Minimum Decision Minimum Alternative Payoff Design A d 1 10,000 Design B d 2 8,000 Design C d 3 6,000

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17 Slide © 2006 Thomson South-Western. All Rights Reserved. n The minimax regret approach requires the construction of a regret table or an opportunity loss table. n This is done by calculating for each state of nature the difference between each payoff and the largest payoff for that state of nature. n Then, using this regret table, the maximum regret for each possible decision is listed. n The decision corresponding to the minimum of the maximum regrets is chosen. Decision Making without Probabilities n Minimax Regret Approach

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18 Slide © 2006 Thomson South-Western. All Rights Reserved. First compute a regret table by subtracting each payoff in a column from the largest payoff in that column. The resulting regret table is: Decision Making without Probabilities n Minimax Regret Approach States of Nature States of Nature Decision (Customers Per Hour) Decision (Customers Per Hour) Alternative 40 s 1 60 s 2 80 s 3 Design A d 1 0 3,000 7,000 Design B d 2 2, ,000 Design C d 3 4,000 2,000 0

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19 Slide © 2006 Thomson South-Western. All Rights Reserved. For each decision list the maximum regret. Choose the decision with the minimum of these values. Decision Making without Probabilities n Minimax Regret Approach Minimax regret Minimax decision Decision Maximum Decision Maximum Alternative Regret Design A d 1 7,000 Design B d 2 9,000 Design C d 3 4,000

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20 Slide © 2006 Thomson South-Western. All Rights Reserved. Decision Making with Probabilities n Once we have defined the decision alternatives and states of nature for the chance events, we focus on determining probabilities for the states of nature. n The classical, relative frequency, or subjective method of assigning probabilities may be used. n Because only one of the N states of nature can occur, the probabilities must satisfy two conditions: P ( s j ) > 0 for all states of nature n Assigning Probabilities

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21 Slide © 2006 Thomson South-Western. All Rights Reserved. Decision Making with Probabilities n We use the expected value approach to identify the best or recommended decision alternative. n The expected value of each decision alternative is calculated (explained on the next slide). n The decision alternative yielding the best expected value is chosen. n Expected Value Approach

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22 Slide © 2006 Thomson South-Western. All Rights Reserved. Expected Value Approach where: N = the number of states of nature P ( s j ) = the probability of state of nature s j P ( s j ) = the probability of state of nature s j V ij = the payoff corresponding to decision alternative d i and state of nature s j V ij = the payoff corresponding to decision alternative d i and state of nature s j n The expected value (EV) of decision alternative d i is defined as n The expected value of a decision alternative is the sum of the weighted payoffs for the decision alternative.

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23 Slide © 2006 Thomson South-Western. All Rights Reserved. Expected Value Approach n Calculate the expected value (EV) for each decision. n The decision tree on the next slide can assist in this calculation. n Here d 1, d 2, d 3 represent the decision alternatives of Designs A, B, and C. n And s 1, s 2, s 3 represent the states of nature of 40, 60, and 80 customers per hour. n The decision alternative with the greatest EV is the optimal decision.

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24 Slide © 2006 Thomson South-Western. All Rights Reserved. Expected Value Approach 11 10,000 10,000 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, Design A ( d 1 ) Design B ( d 2 ) Design C ( d 3 ) n Decision Tree 40 customers ( s 1 ) P ( s 1 ) =.4 60 customers ( s 2 ) P ( s 2 ) =.2 80 customers ( s 3 ) P ( s 3 ) =.4 40 customers ( s 1 ) P ( s 1 ) =.4 60 customers ( s 2 ) P ( s 2 ) =.2 80 customers ( s 3 ) P ( s 3 ) =.4 40 customers ( s 1 ) P ( s 1 ) =.4 60 customers ( s 2 ) P ( s 2 ) =.2 80 customers ( s 3 ) P ( s 3 ) =.4

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25 Slide © 2006 Thomson South-Western. All Rights Reserved. Expected Value Approach EV( d 1 ) =.4(10,000) +.2(15,000) EV( d 1 ) =.4(10,000) +.2(15,000) +.4(14,000) = $12, (14,000) = $12,600 EV( d 2 ) =.4(8,000) +.2(18,000) EV( d 2 ) =.4(8,000) +.2(18,000) +.4(12,000) = $11, (12,000) = $11,600 EV( d 3 ) =.4(6,000) +.2(16,000) EV( d 3 ) =.4(6,000) +.2(16,000) +.4(21,000) = $14, (21,000) = $14,000 Design A d 1 Design B d 2 Design C d Choose the decision alternative with the largest EV: Design C 33

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26 Slide © 2006 Thomson South-Western. All Rights Reserved. Expected Value of Perfect Information n Frequently information is available which can improve the probability estimates for the states of nature. n The expected value of perfect information (EVPI) is the increase in the expected profit that would result if one knew with certainty which state of nature would occur. n The EVPI provides an upper bound on the expected value of any sample or survey information.

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27 Slide © 2006 Thomson South-Western. All Rights Reserved. n Expected value of perfect information is defined as Expected Value of Perfect Information EVPI = |EVwPI – EVwoPI| EVPI = expected value of perfect information EVPI = expected value of perfect information EVwPI = expected value with perfect information EVwPI = expected value with perfect information about the states of nature about the states of nature EVwoPI = expected value without perfect information about the states of nature where:

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28 Slide © 2006 Thomson South-Western. All Rights Reserved. Expected Value of Perfect Information n EVPI Calculation Step 1: Step 1: Determine the optimal return corresponding to each state of nature. Step 2: Step 2: Compute the expected value of these optimal returns. Step 3: Step 3: Subtract the EV of the optimal decision from the amount determined in step (2).

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29 Slide © 2006 Thomson South-Western. All Rights Reserved. Calculate the expected value for the optimum payoff for each state of nature and subtract the EV of the optimal decision. Calculate the expected value for the optimum payoff for each state of nature and subtract the EV of the optimal decision. Expected Value of Perfect Information n EVPI Calculation EVPI=.4(10,000) +.2(18,000) +.4(21,000) 14,000 = $2,000

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30 Slide © 2006 Thomson South-Western. All Rights Reserved. Risk Analysis n Risk analysis helps the decision maker recognize the difference between: the expected value of a decision alternative, and the expected value of a decision alternative, and the payoff that might actually occur the payoff that might actually occur n The risk profile for a decision alternative shows the possible payoffs for the decision alternative along with their associated probabilities.

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31 Slide © 2006 Thomson South-Western. All Rights Reserved. Risk Analysis n Risk Profile for Decision Alternative d Probability Profit ($ thousands)

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32 Slide © 2006 Thomson South-Western. All Rights Reserved. Sensitivity Analysis n Sensitivity analysis can be used to determine how changes to the following inputs affect the recommended decision alternative: probabilities for the states of nature probabilities for the states of nature values of the payoffs values of the payoffs n If a small change in the value of one of the inputs causes a change in the recommended decision alternative, extra effort and care should be taken in estimating the input value.

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33 Slide © 2006 Thomson South-Western. All Rights Reserved. Sensitivity Analysis EV( d 1 ) ,40012,65412,20011,50010,80014,00014,80015,600 P(s1)P(s1)P(s1)P(s1) n Resolving Using Different Values for the Probabilities of the States of Nature Probabilities of the States of Nature 12,90012,98712,70012,25011,80013,50013,80014,10014,25014,31913,50012,25011,00014,75015,00015,250 P(s2)P(s2)P(s2)P(s2) P(s3)P(s3)P(s3)P(s3) EV( d 2 ) EV( d 3 ) d 3 d 1 and d 3 d 1 d 3 d 2 Optimal d

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34 Slide © 2006 Thomson South-Western. All Rights Reserved. Decision Analysis With Sample Information n Knowledge of sample (survey) information can be used to revise the probability estimates for the states of nature. n Prior to obtaining this information, the probability estimates for the states of nature are called prior probabilities. n With knowledge of conditional probabilities for the outcomes or indicators of the sample or survey information, these prior probabilities can be revised by employing Bayes' Theorem. n The outcomes of this analysis are called posterior probabilities or branch probabilities for decision trees.

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35 Slide © 2006 Thomson South-Western. All Rights Reserved. Burger Prince must decide whether to purchase a Burger Prince must decide whether to purchase a marketing survey from Stanton Marketing for $1,000. The results of the survey are "favorable“ or "unfavorable". The branch probabilities corresponding to all the chance nodes are listed on the next slide. Decision Analysis With Sample Information n Example: Burger Prince

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36 Slide © 2006 Thomson South-Western. All Rights Reserved. Decision Analysis With Sample Information n Branch Probabilities P(80 customers per hour | unfavorable) =.087 P(60 customers per hour | unfavorable) =.217 P(40 customers per hour | unfavorable) =.696 P(80 customers per hour | favorable) =.667 P(60 customers per hour | favorable) =.185 P(40 customers per hour | favorable) =.148 P(Unfavorable market survey) =.46 P(Favorable market survey) =.54

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37 Slide © 2006 Thomson South-Western. All Rights Reserved. Restaurant Design (seating capacity) Restaurant Design (seating capacity) ProfitProfit Avg. Number of Customers Per Hour Avg. Number of Customers Per Hour MarketSurveyResultsMarketSurveyResults MarketSurveyMarketSurvey DecisionChanceConsequence Decision Analysis With Sample Information n Influence Diagram

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38 Slide © 2006 Thomson South-Western. All Rights Reserved. n Decision Tree (top half) Decision Analysis With Sample Information 22 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, d1d1d1d1 d2d2d2d2 d3d3d3d3 s 1 P ( s 1 | I 1 ) =.148 s 2 P ( s 2 | I 1 ) =.185 s 3 P ( s 3 | I 1 ) =.667 s 1 P ( s 1 | I 1 ) =.148 s 2 P ( s 2 | I 1 ) =.185 s 3 P ( s 3 | I 1 ) =.667 s 1 P ( s 1 | I 1 ) =.148 s 2 P ( s 2 | I 1 ) =.185 s 3 P ( s 3 | I 1 ) = P ( I 1 ) =.54 I1I1I1I1

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39 Slide © 2006 Thomson South-Western. All Rights Reserved. n Decision Tree (bottom half) Decision Analysis With Sample Information 33 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, d1d1d1d1 d2d2d2d2 d3d3d3d3 s 1 P ( s 1 | I 2 ) =.696 s 2 P ( s 2 | I 2 ) =.217 s 3 P ( s 3 | I 2 ) =.087 s 1 P ( s 1 | I 2 ) =.696 s 2 P ( s 2 | I 2 ) =.217 s 3 P ( s 3 | I 2 ) =.087 s 1 P ( s 1 | I 2 ) =.696 s 2 P ( s 2 | I 2 ) =.217 s 3 P ( s 3 | I 2 ) = P ( I 2 ) =.46 I2I2I2I2

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40 Slide © 2006 Thomson South-Western. All Rights Reserved. A decision strategy is a sequence of decisions and chance outcomes. A decision strategy is a sequence of decisions and chance outcomes. The sequence of decisions chosen depends on the yet to be determined outcomes of chance events. The sequence of decisions chosen depends on the yet to be determined outcomes of chance events. The EVwSI is calculated by making a backward pass through the decision tree. The EVwSI is calculated by making a backward pass through the decision tree. Decision Analysis With Sample Information n Decision Strategy The optimal decision strategy is based on the Expected Value with Sample Information (EVwSI). The optimal decision strategy is based on the Expected Value with Sample Information (EVwSI).

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41 Slide © 2006 Thomson South-Western. All Rights Reserved. Step 1: Step 1: Determine the optimal decision strategy and its expected returns for the possible outcomes of the sample using the posterior probabilities for the states of nature. Step 2: Step 2: Compute the expected value of these optimal returns. n Expected Value with Sample Information (EVwSI) Decision Analysis With Sample Information

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42 Slide © 2006 Thomson South-Western. All Rights Reserved. I 2 I 2(.46) d1d1d1d1 d2d2d2d2 d3d3d3d3 EV =.696(10,000) +.217(15,000) +.087(14,000) = $11, (14,000) = $11,433 EV =.696(8,000) +.217(18,000) +.087(12,000) = $10, (12,000) = $10,554 EV =.696(6,000) +.217(16,000) +.087(21,000) = $9, (21,000) = $9,475 I 1 I 1(.54) d1d1d1d1 d2d2d2d2 d3d3d3d3 EV =.148(10,000) +.185(15,000) +.667(14,000) = $13, (14,000) = $13,593 EV =.148 (8,000) +.185(18,000) +.667(12,000) = $12, (12,000) = $12,518 EV =.148(6,000) +.185(16,000) +.667(21,000) = $17, (21,000) = $17, $17,855 $11,433 Decision Analysis With Sample Information

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43 Slide © 2006 Thomson South-Western. All Rights Reserved. I 2 I 2(.46) d1d1d1d1 d2d2d2d2 d3d3d3d3 $11,433 $10,554 $ 9,475 I 1 I 1(.54) d1d1d1d1 d2d2d2d2 d3d3d3d3 $13,593 $12,518 $17, $17,855 $11,433 Decision Analysis With Sample Information EVwSI =.54(17,855) +.46(11,433) +.46(11,433) = $14,900.88

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44 Slide © 2006 Thomson South-Western. All Rights Reserved. n If the outcome of the survey is "favorable”, choose Design C. If the outcome of the survey is “unfavorable”, choose Design A. If the outcome of the survey is “unfavorable”, choose Design A. EVwSI =.54($17,855) +.46($11,433) = $14, Decision Analysis With Sample Information n Expected Value with Sample Information (EVwSI) n Optimal Decision Strategy

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45 Slide © 2006 Thomson South-Western. All Rights Reserved. Decision Analysis With Sample Information Payoff I 1 =.54 I 2 =.46 $ 6,000 $16,000$21,000 $10,000$15,000$14, n Risk Profile for Optimal Decision Strategy P(I)P(I)P(I)P(I) P ( s 1 | I 1 ) =.148 P ( s 2 | I 1 ) =.185 P ( s 3 | I 1 ) =.667 P ( s 1 | I 2 ) =.696 P ( s 2 | I 2 ) =.217 P ( s 3 | I 2 ) =.087 P(s|I)P(s|I)P(s|I)P(s|I) Payoffs for d 3 Payoffs for d 1

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46 Slide © 2006 Thomson South-Western. All Rights Reserved. Decision Analysis With Sample Information n Risk Profile for Optimal Decision Strategy Probability Profit ($ thousands)

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47 Slide © 2006 Thomson South-Western. All Rights Reserved. Expected Value of Sample Information n The expected value of sample information (EVSI) is the additional expected profit possible through knowledge of the sample or survey information. EVSI = |EVwSI – EVwoSI| EVSI = expected value of sample information EVSI = expected value of sample information EVwSI = expected value with sample information EVwSI = expected value with sample information about the states of nature about the states of nature EVwoSI = expected value without sample information about the states of nature where:

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48 Slide © 2006 Thomson South-Western. All Rights Reserved. Expected Value of Sample Information Subtract the EVwoSI (the value of the optimal decision obtained without using the sample information) from the EVwSI. n EVSI Calculation EVSI = $14, $14,000 = $ n Conclusion Because the EVSI ($900.88) is less than the cost of the survey ($ ), the survey should not be purchased.

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49 Slide © 2006 Thomson South-Western. All Rights Reserved. Efficiency of Sample Information n The efficiency rating (E) of sample information is the ratio of EVSI to EVPI expressed as a percent. n The efficiency rating (E) of the market survey for Burger Prince Restaurant is:

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50 Slide © 2006 Thomson South-Western. All Rights Reserved. n Bayes’ Theorem can be used to compute branch probabilities for decision trees. n For the computations we need to know: the initial (prior) probabilities for the states of nature, the initial (prior) probabilities for the states of nature, the conditional probabilities for the outcomes or indicators of the sample information, given each state of nature. the conditional probabilities for the outcomes or indicators of the sample information, given each state of nature. n A tabular approach is a convenient method for carrying out the computations. Computing Branch Probabilities

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51 Slide © 2006 Thomson South-Western. All Rights Reserved. n Step 1 For each state of nature, multiply the prior probability by its conditional probability for the indicator. This gives the joint probabilities for the states and indicator. n Step 2 Sum these joint probabilities over all states. This gives the marginal probability for the indicator. n Step 3 For each state, divide its joint probability by the marginal probability for the indicator. This gives the posterior probability distribution. Computing Branch Probabilities

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52 Slide © 2006 Thomson South-Western. All Rights Reserved. Recall that Burger Prince is considering purchasing a marketing survey from Stanton Marketing. The results of the survey are "favorable“ or "unfavorable". The conditional probabilities are: n Example: Burger Prince Restaurant P(favorable |80 customers per hour) =.9 P(favorable |60 customers per hour) =.5 P(favorable |40 customers per hour) =.2 Compute the branch (posterior) probability Compute the branch (posterior) probabilitydistribution. Computing Branch Probabilities

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53 Slide © 2006 Thomson South-Western. All Rights Reserved. State Prior Conditional Joint Posterior Favorable n Posterior Probability Distribution Computing Branch Probabilities P(favorable) =.54 Total.54.08/.54

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54 Slide © 2006 Thomson South-Western. All Rights Reserved. Total.46 State Prior Conditional Joint Posterior Unfavorable n Posterior Probability Distribution Computing Branch Probabilities P(unfavorable) =.46.32/.46

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55 Slide © 2006 Thomson South-Western. All Rights Reserved. End of Chapter 4

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