2Chapter 4 Decision Analysis, Part B Expected Value of Perfect InformationDecision Analysis with Sample InformationDeveloping a Decision StrategyExpected Value of Sample Information
3Example: 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 The payoff table for the three models is as follows:Average Number of Customers Per Hours1 = s2 = s3 = 120Model A $10, $15, $14,000Model B $ 8, $18, $12,000Model C $ 6, $16, $21,000
4Example: Burger Prince Expected Value ApproachCalculate the expected value for each decision. The decision tree on the next slide can assist in this calculation. Here d1, d2, d3 represent the decision alternatives of models A, B, C, and s1, s2, s3 represent the states of nature of 80, 100, and 120.
5Example: Burger Prince Decision TreePayoffs.4s110,000s2.2215,000s3.4d114,000.4s18,000d213s2.218,000d3s3.412,000.4s16,0004s2.216,000s3.421,000
6Example: Burger Prince Expected Value For Each DecisionChoose the model with largest EV, Model C.EMV = .4(10,000) + .2(15,000) + .4(14,000)= $12,6002d1Model Ad2EMV = .4(8,000) + .2(18,000) + .4(12,000)= $11,600Model B13d3Model CEMV = .4(6,000) + .2(16,000) + .4(21,000)= $14,0004
7Expected Value of Perfect Information Frequently information is available which can improve the probability estimates for the states of nature.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.The EVPI provides an upper bound on the expected value of any sample or survey information.
8Expected Value of Perfect Information EVPI CalculationStep 1:Determine the optimal return corresponding to each state of nature.Step 2:Compute the expected value of these optimal returns.Step 3:Subtract the EV of the optimal decision from the amount determined in step (2).
9Example: Burger Prince Expected Value of Perfect InformationCalculate 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 = $2,000
10Example: Burger Prince Spreadsheet for Expected Value of Perfect Information
11Risk AnalysisRisk analysis helps the decision maker recognize the difference between:the expected value of a decision alternativeandthe payoff that might actually occurThe risk profile for a decision alternative shows the possible payoffs for the decision alternative along with their associated probabilities.
12Example: Burger Prince Risk Profile for the Model C Decision Alternative.50.40Probability.30.20.10Profit ($thousands)
13Sensitivity AnalysisSensitivity analysis can be used to determine how changes to the following inputs affect the recommended decision alternative:probabilities for the states of naturevalues of the payoffsIf 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.
14Bayes’ Theorem and Posterior Probabilities Knowledge of sample or survey information can be used to revise the probability estimates for the states of nature.Prior to obtaining this information, the probability estimates for the states of nature are called prior probabilities.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.The outcomes of this analysis are called posterior probabilities or branch probabilities for decision trees.
15Computing Branch Probabilities Branch (Posterior) Probabilities CalculationStep 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.Step 2:Sum these joint probabilities over all states -- this gives the marginal probability for the indicator.Step 3:For each state, divide its joint probability by the marginal probability for the indicator -- this gives the posterior probability distribution.
16Expected Value of Sample Information The expected value of sample information (EVSI) is the additional expected profit possible through knowledge of the sample or survey information.
17Expected Value of Sample Information EVSI CalculationStep 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.Step 2:Compute the expected value of these optimal returns.Step 3:Subtract the EV of the optimal decision obtained without using the sample information from the amount determined in step (2).
18Efficiency of Sample Information Efficiency of sample information is the ratio of EVSI to EVPI.As the EVPI provides an upper bound for the EVSI, efficiency is always a number between 0 and 1.
19Example: Burger Prince Sample InformationBurger 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) = .2P(favorable | 100 customers per hour) = .5P(favorable | 120 customers per hour) = .9Should Burger Prince have the survey performed by Stanton Marketing?
20Example: Burger Prince Influence DiagramLegend:DecisionChanceConsequenceMarketSurveyResultsAvg. Numberof CustomersPer HourMarketSurveyRestaurantSizeProfit
28Example: Burger Prince Expected Value of Sample InformationIf 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 = $900.88Since this is less than the cost of the survey, the survey should not be purchased.
29Example: Burger Prince Efficiency of Sample InformationThe efficiency of the survey:EVSI/EVPI = ($900.88)/($2000) =