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1 1 Slide © 2001 South-Western College Publishing/Thomson Learning Anderson Sweeney Williams Anderson Sweeney Williams Slides Prepared by JOHN LOUCKS QUANTITATIVE METHODS FOR BUSINESS 8e QUANTITATIVE METHODS FOR BUSINESS 8e

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2 2 Slide Chapter 4 Decision Analysis, Part B n Expected Value of Perfect Information n Decision Analysis with Sample Information n Developing a Decision Strategy n Expected Value of Sample Information

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3 3 Slide Example: Burger Prince Burger Prince Restaurant is contemplating opening a new restaurant on Main Street. It has three different models, each with a different seating capacity. Burger Prince estimates that the average number of customers per hour will be 80, 100, or 120. The payoff table for the three models is as follows: Average Number of Customers Per Hour Average Number of Customers Per Hour s 1 = 80 s 2 = 100 s 3 = 120 s 1 = 80 s 2 = 100 s 3 = 120 Model A $10,000 $15,000 $14,000 Model A $10,000 $15,000 $14,000 Model B $ 8,000 $18,000 $12,000 Model B $ 8,000 $18,000 $12,000 Model C $ 6,000 $16,000 $21,000 Model C $ 6,000 $16,000 $21,000

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4 4 Slide Example: Burger Prince n Expected Value Approach Calculate the expected value for each decision. The decision tree on the next slide can assist in this calculation. Here d 1, d 2, d 3 represent the decision alternatives of models A, B, C, and s 1, s 2, s 3 represent the states of nature of 80, 100, and 120.

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5 5 Slide Example: Burger Prince n Decision Tree 11.2.4.4.4.2.4.4.2.4 d1d1d1d1 d2d2d2d2 d3d3d3d3 s1s1s1s1 s1s1s1s1 s1s1s1s1 s2s2s2s2 s3s3s3s3 s2s2s2s2 s2s2s2s2 s3s3s3s3 s3s3s3s3 Payoffs 10,000 15,000 14,000 8,000 18,000 12,000 6,000 16,000 21,000 22 33 44

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6 6 Slide Example: Burger Prince n Expected Value For Each Decision Choose the model with largest EV, Model C. 33 d1d1d1d1 d2d2d2d2 d3d3d3d3 EMV =.4(10,000) +.2(15,000) +.4(14,000) = $12,600 = $12,600 EMV =.4(8,000) +.2(18,000) +.4(12,000) = $11,600 = $11,600 EMV =.4(6,000) +.2(16,000) +.4(21,000) = $14,000 = $14,000 Model A Model B Model C 22 11 44

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7 7 Slide 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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8 8 Slide Expected Value of Perfect Information n EVPI Calculation Step 1:Step 1: Determine the optimal return corresponding to each state of nature. Determine the optimal return corresponding to each state of nature. Step 2:Step 2: Compute the expected value of these optimal returns. 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). Subtract the EV of the optimal decision from the amount determined in step (2).

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9 9 Slide Example: Burger Prince n Expected Value of Perfect Information 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. EVPI=.4(10,000) +.2(18,000) +.4(21,000) - 14,000 = $2,000 EVPI=.4(10,000) +.2(18,000) +.4(21,000) - 14,000 = $2,000

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10 Slide Example: Burger Prince n Spreadsheet for Expected Value of Perfect Information

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11 Slide Risk Analysis n Risk analysis helps the decision maker recognize the difference between: the expected value of a decision alternative the expected value of a decision alternativeand the payoff that might actually occurthe 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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12 Slide Example: Burger Prince n Risk Profile for the Model C Decision Alternative.10.20.30.40.50 5 10 15 20 25 Probability Profit ($thousands)

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13 Slide 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 natureprobabilities for the states of nature values of the payoffsvalues 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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14 Slide Bayes Theorem and Posterior Probabilities n Knowledge of sample or 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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15 Slide Computing Branch Probabilities n Branch (Posterior) Probabilities Calculation Step 1: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. 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. Step 2:Step 2: Sum these joint probabilities over all states -- this gives the marginal probability for the indicator. Sum these joint probabilities over all states -- this gives the marginal probability for the indicator. Step 3:Step 3: For each state, divide its joint probability by the marginal probability for the indicator -- this gives the posterior probability distribution. For each state, divide its joint probability by the marginal probability for the indicator -- this gives the posterior probability distribution.

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16 Slide 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.

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17 Slide Expected Value of Sample Information n EVSI Calculation Step 1:Step 1: Determine the optimal decision and its expected return for the possible outcomes of the sample or survey using the posterior probabilities for the states of nature. Determine the optimal decision and its expected return for the possible outcomes of the sample or survey using the posterior probabilities for the states of nature. Step 2: Compute the expected value of these optimal returns. Compute the expected value of these optimal returns. Step 3:Step 3: Subtract the EV of the optimal decision obtained without using the sample information from the amount determined in step (2). Subtract the EV of the optimal decision obtained without using the sample information from the amount determined in step (2).

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18 Slide Efficiency of Sample Information n Efficiency of sample information is the ratio of EVSI to EVPI. n As the EVPI provides an upper bound for the EVSI, efficiency is always a number between 0 and 1.

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19 Slide Example: Burger Prince n Sample Information Burger Prince must decide whether or not to purchase a marketing survey from Stanton Marketing for $1,000. The results of the survey are "favorable" or "unfavorable". The conditional probabilities are: P(favorable | 80 customers per hour) =.2 P(favorable | 100 customers per hour) =.5 P(favorable | 120 customers per hour) =.9 Should Burger Prince have the survey performed by Stanton Marketing?

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20 Slide Example: Burger Prince n Influence Diagram RestaurantSize Profit Avg. Number of Customers Per HourMarketSurveyResults Legend:DecisionChanceConsequence MarketSurvey

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21 Slide Example: Burger Prince n Posterior Probabilities Favorable State Prior Conditional Joint Posterior State Prior Conditional Joint Posterior 80.4.2.08.148 80.4.2.08.148 100.2.5.10.185 100.2.5.10.185 120.4.9.36.667 120.4.9.36.667 Total.54 1.000 Total.54 1.000 P(favorable) =.54 P(favorable) =.54

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22 Slide Example: Burger Prince n Posterior Probabilities Unfavorable State Prior Conditional Joint Posterior State Prior Conditional Joint Posterior 80.4.8.32.696 80.4.8.32.696 100.2.5.10.217 100.2.5.10.217 120.4.1.04.087 120.4.1.04.087 Total.46 1.000 Total.46 1.000 P(unfavorable) =.46 P(unfavorable) =.46

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23 Slide Example: Burger Prince n Formula Spreadsheet for Posterior Probabilities

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24 Slide Example: Burger Prince n Solution Spreadsheet for Posterior Probabilities

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25 Slide Example: Burger Prince n Decision Tree (top half) s 1 (.148) s 2 (.185) s 3 (.667) $10,000 $15,000 $14,000 $8,000 $18,000 $12,000 $6,000 $16,000 $21,000 I 1 I 1(.54) d1d1d1d1 d2d2d2d2 d3d3d3d3 22 44 55 66 11

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26 Slide Example: Burger Prince n Decision Tree (bottom half) s 1 (.696) s 2 (.217) s 3 (.087) $10,000 $15,000 $18,000 $14,000 $8,000 $12,000 $6,000 $16,000 $21,000 I 2 I 2(.46) d1d1d1d1 d2d2d2d2 d3d3d3d3 77 99 88 33 11

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27 Slide Example: Burger Prince I 2 I 2(.46) d1d1d1d1 d2d2d2d2 d3d3d3d3 EMV =.696(10,000) +.217(15,000) +.087(14,000)= $11,433 +.087(14,000)= $11,433 EMV =.696(8,000) +.217(18,000) +.087(12,000) = $10,554 +.087(12,000) = $10,554 EMV =.696(6,000) +.217(16,000) +.087(21,000) = $9,475 +.087(21,000) = $9,475 I 1 I 1(.54) d1d1d1d1 d2d2d2d2 d3d3d3d3 EMV =.148(10,000) +.185(15,000) +.667(14,000) = $13,593 +.667(14,000) = $13,593 EMV =.148 (8,000) +.185(18,000) +.667(12,000) = $12,518 +.667(12,000) = $12,518 EMV =.148(6,000) +.185(16,000) +.667(21,000) = $17,855 +.667(21,000) = $17,855 44 55 66 77 88 99 22 33 11 $17,855 $11,433

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28 Slide Example: Burger Prince n Expected Value of Sample Information If the outcome of the survey is "favorable" choose Model C. If it is unfavorable, choose model A. EVSI =.54($17,855) +.46($11,433) - $14,000 = $900.88 Since this is less than the cost of the survey, the survey should not be purchased.

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29 Slide Example: Burger Prince n Efficiency of Sample Information The efficiency of the survey: EVSI/EVPI = ($900.88)/($2000) =.4504

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30 Slide The End of Chapter 4, Part B

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