1. 1 1 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Slides by John Loucks St. Edward’s University

2. 2 2 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 13 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

3. 3 3 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Decision Analysis n Decision analysis can be used to develop an optimal strategy when a decision maker is faced with several decision alternatives and an uncertain or risk-filled pattern of future events. n Even when a careful decision analysis has been conducted, the uncertain future events make the final consequence uncertain. n The risk associated with any decision alternative is a direct result of the uncertainty associated with the final consequence. Good decision analysis includes risk analysis that provides probability information about the favorable as well as the unfavorable consequences that may occur. Good decision analysis includes risk analysis that provides probability information about the favorable as well as the unfavorable consequences that may occur.

4. 4 4 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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. States of nature should be defined so that they are mutually exclusive and collectively exhaustive.

5. 5 5 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Pittsburgh Development Corporation (PDC) purchased land that will be the site of a new luxury condominium complex. PDC commissioned preliminary architectural drawings for three different projects: one with 30, one with 60, and one with 90 condominiums. Pittsburgh Development Corporation (PDC) purchased land that will be the site of a new luxury condominium complex. PDC commissioned preliminary architectural drawings for three different projects: one with 30, one with 60, and one with 90 condominiums. The financial success of the project depends upon the size of the condominium complex and the chance event concerning the demand for the condominiums. The statement of the PDC decision problem is to select the size of the new complex that will lead to the largest profit given the uncertainty concerning the demand for the condominiums. The financial success of the project depends upon the size of the condominium complex and the chance event concerning the demand for the condominiums. The statement of the PDC decision problem is to select the size of the new complex that will lead to the largest profit given the uncertainty concerning the demand for the condominiums. Example: Pittsburgh Development Corp.

6. 6 6 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Influence Diagrams 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.

7. 7 7 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. ComplexSize Profit Demand for the Condominiums DecisionChanceConsequence Influence Diagrams

8. 8 8 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Payoff Tables 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.

9. 9 9 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Consider the following problem with three decision alternatives and two states of nature with the following payoff table representing profits: States of Nature States of Nature Strong Demand Weak Demand Strong Demand Weak Demand Decision Alternative s 1 s 2 Decision Alternative s 1 s 2 Small complex, d 1 8 7 Small complex, d 1 8 7 Medium complex, d 2 14 5 Medium complex, d 2 14 5 Large complex, d 3 20 -9 Large complex, d 3 20 -9 PAYOFF TABLE Example: Pittsburgh Development Corp.

10. 10 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Decision Making without Probabilities n Three commonly used criteria for decision making when probability information regarding the likelihood of the states of nature is unavailable are: the optimistic approach the optimistic approach the conservative approach the conservative approach the minimax regret approach. the minimax regret approach.

11. 11 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Optimistic Approach n The optimistic approach would be used by an optimistic decision maker. n The decision with the largest possible payoff is chosen. (“Maximum of the maximum”) n If the payoff table was in terms of costs, the decision with the lowest cost would be chosen.

12. 12 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Optimistic Approach An optimistic decision maker would use the optimistic ( maximax ) approach. We choose the decision that has the largest single value in the payoff table. Maximum Maximum Decision Payoff Decision Payoff d 1 8 d 1 8 d 2 14 d 2 14 d 3 20 d 3 20 Maximaxpayoff Maximax decision Strong Demand

13. 13 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Conservative Approach n The conservative approach would be used by a conservative decision maker. n For each decision the minimum payoff is listed and then the decision corresponding to the maximum of these minimum payoffs is selected. (Hence, the minimum possible payoff is maximized.) n If the payoff was in terms of costs, the maximum costs would be determined for each decision and then the decision corresponding to the minimum of these maximum costs is selected. (Hence, the maximum possible cost is minimized.)

14. 14 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Conservative Approach A conservative decision maker would use the conservative (maximin) approach. List the minimum payoff for each decision. Choose the decision with the maximum of these minimum payoffs. Minimum Minimum Decision Payoff Decision Payoff d 1 7 d 1 7 d 2 5 d 2 5 d 3 -9 d 3 -9 Maximindecision Maximinpayoff Weak Demand

15. 15 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Minimax Regret Approach 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 chosen is the one corresponding to the minimum of the maximum regrets.

16. 16 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. For the minimax regret approach, first compute a regret table by subtracting each payoff in a column from the largest payoff in that column. In this example, in the first column subtract 8, 14, and 20 from 20; etc. The resulting regret table is: States of Nature States of Nature Strong Demand Weak Demand Strong Demand Weak Demand Decision Alternative s 1 s 2 Decision Alternative s 1 s 2 Small complex, d 1 12 0 Small complex, d 1 12 0 Medium complex, d 2 6 2 Medium complex, d 2 6 2 Large complex, d 3 0 16 Large complex, d 3 0 16 Example: Minimax Regret Approach REGRET TABLE

17. 17 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. For each decision list the maximum regret. Choose the decision with the minimum of these values. Maximum Maximum Decision Regret Decision Regret d 1 12 d 1 12 d 2 6 d 2 6 d 3 16 d 3 16 Example: Minimax Regret Approach Minimaxdecision Minimaxregret From strong or weak demand

18. 18 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Decision Making with Probabilities n Expected Value Approach If probabilistic information regarding the states of nature is available, one may use the expected value (EV) approach. If probabilistic information regarding the states of nature is available, one may use the expected value (EV) approach. Here the expected return for each decision is calculated by summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring. Here the expected return for each decision is calculated by summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring. The decision yielding the best expected return is chosen. The decision yielding the best expected return is chosen.

19. 19 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n The expected value of a decision alternative is the sum of weighted payoffs for the decision alternative. n The expected value (EV) of decision alternative d i is defined as: 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 Expected Value of a Decision Alternative

20. 20 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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, and d 3 represent the decision alternatives of building a small, medium, and large complex, while s 1 and s 2 represent the states of nature of strong demand and weak demand.

21. 21 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Decision Trees n A decision tree is a chronological representation of the decision problem. n Each decision tree has two types of nodes; round nodes correspond to the states of nature while square nodes correspond to the decision alternatives. 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.

22. 22 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Decision Tree 11.8.2.8.2.8.2 d1d1d1d1 d2d2d2d2 d3d3d3d3 s1s1s1s1 s1s1s1s1 s1s1s1s1 s2s2s2s2 s2s2s2s2 s2s2s2s2 Payoffs $8 mil $7 mil $14 mil $5 mil $20 mil -$9 mil 22 33 44

23. 23 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Expected Value for Each Decision Choose the decision alternative with the largest EV. Build the large complex. 33 d1d1d1d1 d2d2d2d2 d3d3d3d3 EMV =.8(8 mil) +.2(7 mil) = $7.8 mil EMV =.8(14 mil) +.2(5 mil) = $12.2 mil EMV =.8(20 mil) +.2(-9 mil) = $14.2 mil Small Medium Large 22 11 44

24. 24 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

25. 25 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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).

26. 26 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Expected Value with Perfect Information (EVwPI) n Expected Value without Perfect Information (EVwoPI) n Expected Value of Perfect Information (EVPI) Expected Value of Perfect Information EVwPI =.8(20 mil) +.2(7 mil) = $17.4 mil EVwoPI =.8(20 mil) +.2(-9 mil) = $14.2 mil EVPI = |EVwPI – EVwoPI| = |17.4 – 14.2| = $3.2 mil Maximax minimax Highest EV

27. 27 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

28. 28 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Risk Profile n Large Complex Decision Alternative.20.40.60.80 1.00 -10 -5 0 5 10 15 20 Probability Profit ($ millions)

29. 29 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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, we could calculate the new solution, or create some tools for us to use n If the EV for each decision is linear, can be represented in graph form.

30. 30 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis Graph n Convert each EV(d) into a formula, by making p the probability of one of the states of nature: P(s 2 ) = 1 – P(s 1 ) = 1 – p n Sub in payoff values (table 13.1) for each decision to get the EV formula: EV(d1) = P(s 1 )*8 + P(s 2 )*7 EV(d1) = P(s 1 )*8 + P(s 2 )*7 »p8 + (1 – p)*7 »8p + 7 – 7p »p + 7

31. 31 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis Graph n EV(d2) = P(s 1 )*14 + P(s 2 )*5 n 14p + (1 - p)*5 n 14p + 5 – 5p n 9p + 5 n EV(d3) = P(s 1 )*20 + P(s 2 )*-9 n 20p + (1 – p)*-9 n 20p - 9 + 9p n 29p - 9

32. 32 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis Graph As “p” increases, different decisions become optimal, according to the EV of each decision

33. 33 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis Graph n Solve for intercepts to see where one solution becomes optimal: EV(d1) = EV(d2) EV(d1) = EV(d2) p + 7 = 9p + 5p + 7 = 9p + 5 8p = 28p = 2 p = 2/8 = 0.25p = 2/8 = 0.25 EV(d2) = EV(d3) EV(d2) = EV(d3) 9p + 5 = 29p – 99p + 5 = 29p – 9 -20p = -14-20p = -14 p = 14/20 = 0.7p = 14/20 = 0.7

34. 34 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis Graph p = 0.25 p = 0.7

35. 35 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis Graph EV = 7.8 EV = 12.2 * From figure 13.4

36. Sensitivity Analysis with Payoffs n Remember EV(d3) >= 12.2 EV(d3) >= 12.2 n Substitute in p (0.8) and (1 – p) (0.2) to the EV(d3) equation, leaving the profit as variables: EV(d3) = 0.8S + 0.2W >= 12.2 EV(d3) = 0.8S + 0.2W >= 12.2 n Find the “strong” payoff, by subbing in the weak 0.8S + 0.2 (-9) >=12.2 0.8S + 0.2 (-9) >=12.2 0.8S >= 14 0.8S >= 14 S >= 17.5 S >= 17.5 n d3 will remain optimal when demand is strong, as long as the payoff is at least 17.5 million n Compare this to the original 20 million projected

37. 37 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sensitivity Analysis with Payoffs n Remember EV(d3) = 0.8(20) + 0.2W >= 12.2 EV(d3) = 0.8(20) + 0.2W >= 12.2 n Solve for W 16 + 0.2W >= 12.2 16 + 0.2W >= 12.2 0.2W >= -3.8 0.2W >= -3.8 W >= -19 W >= -19 n d3 will remain optimal if the payoff in weak demand is at least -19 million n Compare to -9 million

38. 38 Slide © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter 13