1 1 Slide © 2009 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University

2 2 Slide © 2009 Thomson South-Western. All Rights Reserved Chapter 19 Decision Analysis n Problem Formulation n Decision Making with Probabilities n Decision Analysis with Sample Information with Sample Information n Computing Branch Probabilities Using Bayes’ Theorem Using Bayes’ Theorem

3 3 Slide © 2009 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 payoffs (consequences) associated with each the payoffs (consequences) associated with each specific combination of: specific combination of: decision alternativedecision alternative state of naturestate of nature

4 4 Slide © 2009 Thomson South-Western. All Rights Reserved Problem Formulation n A decision problem is characterized by decision alternatives, states of nature, and resulting payoffs. n The decision alternatives are the different possible strategies the decision maker can employ. n The states of nature refer to future events, not under the control of the decision maker, which may occur. n States of nature should be defined so that they are mutually exclusive and collectively exhaustive.

5 5 Slide © 2009 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. Payoff Tables 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.

6 6 Slide © 2009 Thomson South-Western. All Rights Reserved Decision Trees n A decision tree provides a graphical representation showing the sequential nature of the decision- making process. n Each decision tree has two types of nodes: round nodes correspond to the states of nature round nodes correspond to the states of nature square nodes correspond to the decision alternatives square nodes correspond to the decision alternatives

7 7 Slide © 2009 Thomson South-Western. All Rights Reserved Decision Trees n The branches leaving each round node represent the different states of nature while the branches leaving each square node represent the different decision alternatives. n At the end of each limb of a tree are the payoffs attained from the series of branches making up that limb.

8 8 Slide © 2009 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 method, relative frequency method, or subjective method of assigning probabilities may be used. n Because one and 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

9 9 Slide © 2009 Thomson South-Western. All Rights Reserved Decision Making with Probabilities n Then 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.

10 Slide © 2009 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 weighted payoffs for the decision alternative.

11 Slide © 2009 Thomson South-Western. All Rights Reserved Expected Value Approach Burger Prince Restaurant is considering opening a new restaurant on Main Street. It has three different restaurant layout models (A, B, and C), each with a different seating capacity. n Example: Burger Prince Burger Prince estimates that the average number of customers served per hour will be 80, 100, or 120. The payoff table for the three models is on the next slide.

12 Slide © 2009 Thomson South-Western. All Rights Reserved Expected Value Approach n Payoff Table $ 6,000 $16,000 $21,000 $ 8,000 $18,000 $12,000 $10,000 $15,000 $14,000 Average Number of Customers Per Hour s 1 = 80 s 2 = 100 s 3 = 120 Model A Model B Model C

13 Slide © 2009 Thomson South-Western. All Rights Reserved Expected Value Approach n Calculate the expected value 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 models A, B, and C. n And s 1, s 2, s 3 represent the states of nature of 80, 100, and 120 customers per hour.

14 Slide © 2009 Thomson South-Western. All Rights Reserved 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, Expected Value Approach Decision Tree

15 Slide © 2009 Thomson South-Western. All Rights Reserved Expected Value Approach 33 d1d1d1d1 d2d2d2d2 d3d3d3d3 EMV =.4(10,000) +.2(15,000) +.4(14,000) = $12, (14,000) = $12,600 EMV =.4(8,000) +.2(18,000) +.4(12,000) = $11, (12,000) = $11,600 EMV =.4(6,000) +.2(16,000) +.4(21,000) = $14, (21,000) = $14,000 Model A Model B Model C Choose the model with largest EV, Model C

16 Slide © 2009 Thomson South-Western. All Rights Reserved Expected Value Approach n Excel Formula Worksheet

17 Slide © 2009 Thomson South-Western. All Rights Reserved Expected Value Approach n Excel Value Worksheet

18 Slide © 2009 Thomson South-Western. All Rights Reserved Expected Value of Perfect Information n Frequently, information is available that 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.

19 Slide © 2009 Thomson South-Western. All Rights Reserved n The 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 about the states of nature where:

20 Slide © 2009 Thomson South-Western. All Rights Reserved 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 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). Step 2: Step 2: Compute the expected value of these optimal returns. Compute the expected value of these optimal returns. Expected Value of Perfect Information n EVPI Calculation

21 Slide © 2009 Thomson South-Western. All Rights Reserved EVPI =.4(10,000) +.2(18,000) +.4(21,000) - 14,000 = $2,000 Expected Value of Perfect Information n Calculate the expected value for the optimum payoff for each state of nature and subtract the EV of the optimal decision.

22 Slide © 2009 Thomson South-Western. All Rights Reserved Expected Value of Perfect Information n Excel Formula Worksheet

23 Slide © 2009 Thomson South-Western. All Rights Reserved Expected Value of Perfect Information n Excel Value Worksheet

24 Slide © 2009 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.

25 Slide © 2009 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 decisions chosen depend on the yet to be determined outcomes of chance events. The decisions chosen depend on the yet to be determined outcomes of chance events. The approach used to determine the optimal decision strategy is based on a backward pass through the decision tree. The approach used to determine the optimal decision strategy is based on a backward pass through the decision tree. Decision Analysis With Sample Information n Decision Strategy

26 Slide © 2009 Thomson South-Western. All Rights Reserved At Chance Nodes: At Chance Nodes: Compute the expected value by multiplying the payoff at the end of each branch by the corresponding branch probability. At Decision Nodes: At Decision Nodes: Select the decision branch that leads to the best expected value. This expected value becomes the expected value at the decision node. Decision Analysis With Sample Information n Backward Pass Through the Decision Tree

27 Slide © 2009 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 conditional probabilities are: Decision Analysis With Sample Information n Example: Burger Prince P(favorable | 120 customers per hour) =.9 P(favorable | 100 customers per hour) =.5 P(favorable | 80 customers per hour) =.2

28 Slide © 2009 Thomson South-Western. All Rights Reserved 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 d3d3d3d Decision Analysis With Sample Information

29 Slide © 2009 Thomson South-Western. All Rights Reserved n Decision Tree (bottom half) s 1 (.696) s 2 (.217) s 3 (.087) $10,000 $15,000 $14,000 $8,000 $18,000 $12,000 $6,000 $16,000 $21,000 d1d1d1d1 d2d2d2d2 d3d3d3d I 2 I 2(.46) Decision Analysis With Sample Information

30 Slide © 2009 Thomson South-Western. All Rights Reserved I 2 I 2(.46) d1d1d1d1 d2d2d2d2 d3d3d3d3 EMV =.696(10,000) +.217(15,000) +.087(14,000)= $11, (14,000)= $11,433 EMV =.696(8,000) +.217(18,000) +.087(12,000) = $10, (12,000) = $10,554 EMV =.696(6,000) +.217(16,000) +.087(21,000) = $9, (21,000) = $9,475 I 1 I 1(.54) d1d1d1d1 d2d2d2d2 d3d3d3d3 EMV =.148(10,000) +.185(15,000) +.667(14,000) = $13, (14,000) = $13,593 EMV =.148 (8,000) +.185(18,000) +.667(12,000) = $12, (12,000) = $12,518 EMV =.148(6,000) +.185(16,000) +.667(21,000) = $17, (21,000) = $17, $17,855 $11,433 Decision Analysis With Sample Information

31 Slide © 2009 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 about the states of nature where:

32 Slide © 2009 Thomson South-Western. All Rights Reserved Step 1: Step 1: Determine the optimal decision and its expected return for the possible outcomes of the sample using the posterior probabilities for the states of nature. Determine the optimal decision and its expected return 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. Compute the expected value of these optimal returns. Expected Value of Sample Information n EVwSI Calculation

33 Slide © 2009 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

34 Slide © 2009 Thomson South-Western. All Rights Reserved Expected Value of Sample Information n If the outcome of the survey is "favorable”, choose Model C. If the outcome of the survey is “unfavorable”, choose Model A. If the outcome of the survey is “unfavorable”, choose Model A. EVwSI =.54($17,855) +.46($11,433) = $14,900.88

35 Slide © 2009 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 =.54($17,855) +.46($11,433) - $14,000 = $ n Conclusion Because the EVSI is less than the cost of the survey, the survey should not be purchased.

36 Slide © 2009 Thomson South-Western. All Rights Reserved Computing Branch Probabilities Using Bayes’ Theorem 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.

37 Slide © 2009 Thomson South-Western. All Rights Reserved n Step 1 Computing Branch Probabilities Using Bayes’ Theorem 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.

38 Slide © 2009 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: Decision Analysis With Sample Information n Example: Burger Prince P(favorable | 120 customers per hour) =.9 P(favorable | 100 customers per hour) =.5 P(favorable | 80 customers per hour) =.2 Compute the branch (posterior) probability Compute the branch (posterior) probabilitydistribution.

39 Slide © 2009 Thomson South-Western. All Rights Reserved Posterior Probabilities P(favorable) =.54 Total.54 State Prior Conditional Joint Posterior Favorable /

40 Slide © 2009 Thomson South-Western. All Rights Reserved Posterior Probabilities P(unfavorable) =.46 Total.46 State Prior Conditional Joint Posterior Unfavorable /

41 Slide © 2009 Thomson South-Western. All Rights Reserved Computing Branch Probabilities Using Bayes’ Theorem n Excel Formula Worksheet

42 Slide © 2009 Thomson South-Western. All Rights Reserved Computing Branch Probabilities Using Bayes’ Theorem n Excel Value Worksheet

43 Slide © 2009 Thomson South-Western. All Rights Reserved End of Chapter 19