Decision Analysis REVISED TEACHING SUGGESTIONS ALTERNATIVE EXAMPLES

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1 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page 7 3 C H A P T E R Decision Analysis TEACHING SUGGESTIONS Teaching Suggestion 3.: Using the Steps of the Decision-Making Process. The six steps used in decision theory are discussed in this chapter. Students can be asked to describe a decision they made in the last semester, such as buying a car or selecting an apartment, and describe the steps that they took. This will help in getting students involved in decision theory. It will also help them realize how this material can be useful to them in making important personal decisions. Teaching Suggestion 3.2: Importance of Defining the Problem and Listing All Possible Alternatives. Clearly defining the problem and listing the possible alternatives can be difficult. Students can be asked to do this for a typical decisionmaking problem, such as constructing a new manufacturing plant. Role-playing can be used to make this exercise more interesting. Many students get too involved in the mathematical approaches and do not pay enough attention to the importance of carefully defining the problem and considering all possible alternatives. These initial steps are important. Students need to realize that if they do not carefully define the problem and list all alternatives, most likely their analyses will be wrong. Teaching Suggestion 3.3: Categorizing Decision-Making Types. Decision-making types are discussed in this chapter; decision making under certainty, risk, and uncertainty are included. Students can be asked to describe an important decision they had to make in the past year and categorize the decision type. A good example can be a financial investment of $,000. In-class discussion can help students realize the importance of decision theory and its potential use. Teaching Suggestion 3.4: Starting the EVPI Concept. The material on the expected value of perfect information (EVPI) can be started with a discussion of how to place a value on information and whether or not new information should be acquired. The use of EVPI to place an upper limit on what you should pay for information is a good way to start the section on this topic. Teaching Suggestion 3.5: Starting the Decision-Making Under Uncertainty Material. The section on decision-making under uncertainty can be started with a discussion of optimistic versus pessimistic decision makers. Students can be shown how maximax is an optimistic approach, while maximin is a pessimistic decision technique. While few people use these techniques to solve real problems, the concepts and general approaches are useful. Teaching Suggestion 3.6: Decision Theory and Life-Time Decisions. This chapter investigates large and complex decisions. During one s life, there are a few very important decisions that have a major impact. Some call these life-time decisions. Students can be asked to carefully consider these life-time decisions and how decision theory can be used to assist them. Life-time decisions include decisions about what school to attend, marriage, and the first job. Teaching Suggestion 3.7: Popularity of Decision Trees Among Business Executives. Stress that decision trees are not just an academic subject; they are a technique widely used by top-level managers. Everyone appreciates a graphical display of a tough problem. It clarifies issues and makes a great discussion base. Harvard business students regularly use decision trees in case analysis. Teaching Suggestion 3.8: Importance of Accurate Tree Diagrams. Developing accurate decision trees is an important part of this chapter. Students can be asked to diagram several decision situations. The decisions can come from the end-of-chapter problems, the instructor, or from student experiences. Teaching Suggestion 3.9: Diagramming a Large Decision Problem Using Branches. Some students are intimidated by large and complex decision trees. To avoid this situation, students can be shown that a large decision tree is like having a number of smaller trees or decisions that can be solved separately, starting at the end branches of the tree. This can help students use decision-making techniques on larger and more complex problems. Teaching Suggestion 3.0: Using Tables to Perform Bayesian Analysis. Bayesian analysis can be difficult; the formulas can be hard to remember and use. For many, using tables is the most effective way to learn how to revise probability values. Once students understand how the tables are used, they can be shown that the formulas are making exactly the same calculations. ALTERNATIVE EXAMPLES Alternative Example 3.: Goleb Transport George Goleb is considering the purchase of two types of industrial robots. The Rob (alternative ) is a large robot capable of performing a variety of tasks, including welding and painting. The Rob2 (alternative 2) is a smaller and slower robot, but it has all the capabilities 7

2 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page 8 8 CHAPTER 3 D ECISION A NALYSIS of Rob. The robots will be used to perform a variety of repair operations on large industrial equipment. Of course, George can always do nothing and not buy any robots (alternative 3). The market for the repair could be either favorable (event ) or unfavorable (event 2). George has constructed a payoff matrix showing the expected returns of each alternative and the probability of a favorable or unfavorable market. The data are presented below: EVENT EVENT 2 Probability Alternative 50,000 40,000 Alternative 2 30,000 20,000 Alternative This problem can be solved using expected monetary value. The equations are presented below: EMV (alternative ) ($50,000)(0.6) ( $40,000)(0.4) $4,000 EMV (alternative 2) ($30,000)(0.6) ( $20,000)(0.4) $0,000 EMV (alternative 3) 0 The best solution is to purchase Rob, the large robot. Alternative Example 3.2: George Goleb is not confident about the probability of a favorable or unfavorable market. (See Alternative Example 3..) He would like to determine the equally likely (Laplace), maximax, maximin, coefficient of realism (Hurwicz), and minimax regret decisions. The Hurwicz coefficient should be 0.7. The problem data are summarized below: EVENT EVENT 2 Probability Alternative 50,000 40,000 Alternative 2 30,000 20,000 Alternative The Laplace (equally likely) solution is computed averaging the payoffs for each alternative and choosing the best. The results are shown below. Alternatives and 2 both give the highest average return of $5,000. Average (alternative ) [$50,000 ( $40,000)]/2 $5,000 Average (alternative 2) [$30,000 ( $20,000)]/2 $5,000 Average (alternative 3) 0 The maximin decision (pessimistic) maximizes the minimum payoff outcome for every alternative: these are 40,000; 20,000; and 0. Therefore, the decision is to do nothing. The maximax decision (optimistic) maximizes the maximum payoff for any alternative: these maximums are 50,000; 30,000; and 0. Therefore, the decision is to purchase the large robot (alternative ). The Hurwicz approach uses a coefficient of realism value of 0.7, and a weighted average of the best and the worst payoffs for each alternative is computed. The results are as follows: Weighted average (alternative ) ($50,000)(0.7) ( $40,000)(0.3) $23,000 Weighted average (alternative 2) ($30,000)(0.7) ( $20,000)(0.3) $5,000 Weighted average (alternative 3) 0 The decision would be alternative. The minimax regret decision minimizes the maximum opportunity loss. The opportunity loss table for Goleb is as follows: Favorable Unfavorable Maximum Alternatives Market Market in Row Rob 0 40,000 40,000 Rob2 20,000 20,000 20,000 Nothing 50, ,000 The alternative that minimizes the maximum opportunity loss is the Rob2. This is due to the $20,000 in the last column in the table above. Rob has a maximum opportunity loss of $40,000, and doing nothing has a maximum opportunity loss of $50,000. Alternative Example 3.3: George Goleb is considering the possibility of conducting a survey on the market potential for industrial equipment repair using robots. The cost of the survey is $5,000. George has developed a decision tree that shows the overall decision, as in the figure on the next page. This problem can be solved using EMV calculations. We start with the end of the tree and work toward the beginning computing EMV values. The results of the calculations are shown in the tree. The conditional payoff of the solution is $8,802. Alternative Example 3.4: George (in Alternative Example 3.3) would like to determine the expected value of sample information (EVSI). EVSI is equal to the expected value of the best decision with sample information, assuming no cost to gather it, minus the expected value of the best decision without sample information. Because the cost of the survey is $5,000, the expected value of the best decision with sample information, assuming no cost to gather it, is $23,802. The expected value of the best decision without sample information is found on the lower branch of the decision tree to be $4,000. Thus, EVSI is $9,802. Alternative Example 3.5: This example reveals how the conditional probability values for the George Goleb examples (above) have been determined. The probability values about the survey are summarized in the following table: Results of Favorable Market Unfavorable Market Survey (FM) (UM) Positive (P) P(P FM) 0.9 P(P UM) 0.2 Negative (N) P(N FM) 0. P(N UM) 0.8 Using the values above and the fact that P(FM) 0.6 and P(UM) 0.4, we can compute the conditional probability values of a favorable or unfavorable market given a positive or negative

3 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page 9 CHAPTER 3 D ECISION A NALYSIS 9 First Decision Point Second Decision Point $8,802 Conduct Market Survey Results Negative Survey (0.62) Results Favorable Survey (0.38) $33,390 Rob Rob2 Rob Rob Favorable Market (0.87) Unfavorable Market (0.29) Favorable Market (0.87) Unfavorable Market (0.29) Favorable Market (0.58) Unfavorable Market (0.842) Favorable Market (0.58) Unfavorable Market (0.842) $45,000 $45,000 $25,000 $25,000 $5,000 $45,000 $45,000 $25,000 $25,000 Do Not Conduct Survey $ 5,000 $5,000 $4,000 Rob Rob2 6 7 Favorable Market (0.60) Unfavorable Market (0.40) Favorable Market (0.60) Unfavorable Market (0.40) $50,000 $40,000 $30,000 $20,000 $0 Figure for Alternative Example 3.3 survey result. The calculations are presented in the following two tables. Probability revision given a positive survey result State of Conditional Prior Joint Posterior Nature Probability Prob. Prob. Probability FM / UM / Total Probability given a negative survey result State of Conditional Prior Joint Posterior Nature Probability Prob. Prob. Probability FM / UM / Total Alternative Example 3.6: In the section on utility theory, Mark Simkin used utility theory to determine his best decision. What decision would Mark make if he had the following utility values? Is Mark still a risk seeker? U( $0,000) 0.8 U($0) 0.9 U($0,000) Using the data above, we can determine the expected utility of each alternative as follows: U(Mark plays the game) 0.45() 0.55(0.8) 0.89 U(Mark doesn t play the game) 0.9 Thus, the best decision for Mark is not to play the game with an expected utility of 0.9. Given these data, Mark is a risk avoider.

4 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page CHAPTER 3 D ECISION A NALYSIS SOLUTIONS TO DISCUSSION QUESTIONS AND PROBLEMS 3-. The purpose of this question is to make students use a personal experience to distinguish between good and bad decisions. A good decision is based on logic and all of the available information. A bad decision is one that is not based on logic and the available information. It is possible for an unfortunate or undesirable outcome to occur after a good decision has been made. It is also possible to have a favorable or desirable outcome occur after a bad decision The decision-making process includes the following steps: () define the problem, (2) list the alternatives, (3) identify the possible outcomes, (4) evaluate the consequences, (5) select an evaluation criterion, and (6) make the appropriate decision. The first four steps or procedures are common for all decision-making problems. Steps 5 and 6, however, depend on the decision-making model An alternative is a course of action over which we have complete control. A state of nature is an event or occurrence in which we have no control. An example of an alternative is deciding whether or not to take an umbrella to school or work on a particular day. An example of a state of nature is whether or not it will rain on a particular day The basic differences between decision-making models under certainty, risk, and uncertainty depend on the amount of chance or risk that is involved in the decision. A decision-making model under certainty assumes that we know with complete confidence the future outcomes. Decision-making-under-risk models assume that we do not know the outcomes for a particular decision but that we do know the probability of occurrence of those outcomes. With decision making under uncertainty, it is assumed that we do not know the outcomes that will occur, and furthermore, we do not know the probabilities that these outcomes will occur The techniques discussed in this chapter used to solve decision problems under uncertainty include maximax, maximin, equally likely, coefficient of realism, and minimax regret. The maximax decision-making criterion is an optimistic decision-making criterion, while the maximin is a pessimistic decision-making criterion For a given state of nature, opportunity loss is the difference between the payoff for a decision and the best possible payoff for that state of nature. It indicates how much better the payoff could have been for that state of nature. The minimax regret and the minimum expected opportunity loss are the criteria used with this Alternatives, states of nature, probabilities for all states of nature and all monetary outcomes (payoffs) are placed on the decision tree. In addition, intermediate results, such as EMVs for middle branches, can be placed on the decision tree Using the EMV criterion with a decision tree involves starting at the terminal branches of the tree and working toward the origin, computing expected monetary values and selecting the best alternatives. The EMVs are found by multiplying the probabilities of the states of nature times the economic consequences and summing the results for each alternative. At each decision point, the best alternative is selected A prior probability is one that exists before additional information is gathered. A posterior probability is one that can be computed using Bayes Theorem based on prior probabilities and additional information The purpose of Bayesian analysis is to determine posterior probabilities based on prior probabilities and new information. Bayesian analysis can be used in the decision-making process whenever additional information is gathered. This information can then be combined with prior probabilities in arriving at posterior probabilities. Once these posterior probabilities are computed, they can be used in the decision-making process as any other probability value. 3-. The expected value of sample information (EVSI) is the increase in expected value that results from having sample information. It is computed as follows: EVSI (expected value with sample information) (cost of information) (expected value without sample information) 3-2. The overall purpose of utility theory is to incorporate a decision maker s preference for risk in the decision-making process A utility function can be assessed in a number of different ways. A common way is to use a standard gamble. With a standard gamble, the best outcome is assigned a utility of, and the worst outcome is assigned a utility of 0. Then, intermediate outcomes are selected and the decision maker is given a choice between having the intermediate outcome for sure and a gamble involving the best and worst outcomes. The probability that makes the decision maker indifferent between having the intermediate outcome for sure and a gamble involving the best and worst outcomes is determined. This probability then becomes the utility of the intermediate value. This process is continued until utility values for all economic consequences are determined. These utility values are then placed on a utility curve When a utility curve is to be used in the decision-making process, utility values from the utility curve replace all monetary values at the terminal branches in a decision tree or in the body of a decision table. Then, expected utilities are determined in the same way as expected monetary values. The alternative with the highest expected utility is selected as the best decision A risk seeker is a decision maker who enjoys and seeks out risk. A risk avoider is a decision maker who avoids risk even if the potential economic payoff is higher. The utility curve for a risk seeker increases at an increasing rate. The utility curve for a risk avoider increases at a decreasing rate a. Decision making under uncertainty. b. Maximax criterion. c. Sub 00 because the maximum payoff for this is $300,000. Row Row Equipment Favorable Unfavorable Maximum Minimum Sub , , , ,000 Oiler J 250,000 00, ,000 00,000 Texan 75,000 8,000 75,000 8, Using the maximin criterion, the best alternative is the Texan (see table above) because the worst payoff for this ($ 8,000) is better than the worst payoffs for the other decisions a. Decision making under risk maximize expected monetary value.

5 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page 2 CHAPTER 3 D ECISION A NALYSIS 2 b. EMV (Sub 00) 0.7(300,000) 0.3( 200,000) 50,000 EMV (Oiler J) 0.7(250,000) 0.3( 00,000) 45,000 EMV (Texan) 0.7(75,000) 0.3( 8,000) 47,00 Optimal decision: Sub 00. c. Ken would change decision if EMV(Sub 00) is less than the next best EMV, which is $45,000. Let X payoff for Sub 00 in favorable market. (0.7)(X) (0.3)( 200,000) 45, X 45,000 60, ,000 X (205,000)/ ,857.4 The decision would change if this payoff were less than 292,857.4, so it would have to decrease by about $7, a. The expected value (EV) is computed for each alternative. EV(stock market) 0.5(80,000) 0.5( 20,000) 30,000 EV(Bonds) 0.5(30,000) 0.5(20,000) 25,000 EV(CDs) 0.5(23,000) 0.5(23,000) 23,000 Therefore, he should invest in the stock market. b. EVPI EV(with perfect information) (Maximum EV without P, I) [0.5(80,000) 0.5(23,000)] 30,000 5,500 30,000 2,500 Thus, the most that should be paid is $2, The opportunity loss table is Alternative Good Economy Poor Economy Stock Market 0 43,000 Bonds 50,000 3,000 CDs 57,000 0 EOL(Stock Market) 0.5(0) 0.5(43,000) 2,500* This minimizes EOL. EOL(Bonds) 0.5(50,000) 0.5(3,000) 26,500 EOL(CDs) 0.5(57,000) 0.5(0) 28, a. Market Alternative Condition Good Fair Poor EMV Stock market, Bank deposit Probabilities of market conditions b. Best decision: deposit $0,000 in bank a. Expected value with perfect information is,400(0.4) 900(0.4) 900(0.2),00; the maximum EMV without the information is 900. Therefore, Allen should pay at most EVPI, $200. b. Yes, Allen should pay [,00(0.4) 900(0.4) 900(0.2)] 900 $ a. Opportunity loss table Strong Fair Poor Max. Market Market Market Regret Large 0 9,000 30,000 30,000 Medium 250, , ,000 Small 350,000 29,000 32, ,000 None 550,000 29, ,000 b. Minimax regret decision is to build medium a. Stock Demand (Cases) (Cases) 2 3 EMV Probabilities b. Stock cases. c. If no loss is involved in excess stock, the recommended course of action is to stock 3 cases and to replenish stock to this level each week. This follows from the following decision table. Stock Demand (Cases) (Cases) 2 3 EMV Manu- Demand facture (Cases) (Cases) EMV Probabilities John should manufacture 8 cases of cheese spread Cost of produced case $5. Cost of purchased case $6. Selling price $5.

6 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page CHAPTER 3 D ECISION A NALYSIS Money recovered from each unsold case $3. Supply Demand (Cases) (Cases) EMV 00 00(5) 00(5) (5) 00(5) 300(5) 00(5) (6) (6) (5) 00(3) 200(5) 200(5) (5) 200(5) (5) (6) (5) 200(3) 200(5) 00(3) 300(5) 300(5) (5) (5) 800 Probabilities b. Produce 300 cases each day a. The table presented is a decision table. The basis for the decisions in the following questions is shown in the table below. EQUALLY CRIT. OF MARKET MAXIMAX MAXIMIN LIKELY REALISM Decision Row Row Row Weighted Alternatives Good Fair Poor Maximum Minimum Average Average Small 50,000 20,000 0,000 50,000 0,000 20,000 38,000 Medium 80,000 30,000 20,000 80,000 20,000 30,000 60,000 Large 00,000 30,000 40,000 00,000 40,000 30,000 72,000 Very Large 300,000 25,000 60, ,000 60,000 55, ,000 b. Maximax decision: Very large station. c. Maximin decision: Small station. d. Equally likely decision: Very large station. e. Criterion of realism decision: Very large station. f. Opportunity loss table: MARKET MINIMAX Decision Good Fair Poor Row Alternatives Market Market Market Maximum Small 250,000 0, ,000 Medium 220, , ,000 Large 200, , ,000 Very Large 0 5,000 50,000 50,000 Minimax regret decision: Very large station EMV for node 0.5(00,000) 0.5( 40,000) $30,000. Choose the highest EMV, therefore construct the clinic. Construct Clinic Favorable Market (0.5) Unfavorable Market (0.5) Payoff $00,000 $40,000 $30,000 Do Nothing EMV for no clinic is $0 $0

7 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page 23 CHAPTER 3 D ECISION A NALYSIS a. Payoff CONSTRUCT 2 Favorable Market (0.82) Unfavorable Market (0.8) $95,000 $45,000 Survey Favorable (0.55) $69,800 DO NOT CONSTRUCT $5,000 Conduct Market Survey $36,40 Survey Negative (0.45) CONSTRUCT 3 Favorable Market (0.) Unfavorable Market (0.89) $95,000 $45,000 $36,40 $5,000 DO NOT CONSTRUCT $5,000 Do Not Conduct Survey CONSTRUCT CLINIC 4 Favorable Market (0.5) Unfavorable Market (0.5) $00,000 $40,000 $30,000 DO NOT CONSTRUCT $0 b. EMV(node 2) (0.82)($95,000) (0.8)( $45,000) 77,900 8,00 $69,800 EMV(node 3) (0.)($95,000) (0.89)( $45,000) 0,450 $40,050 $29,600 EMV(node 4) $30,000 EMV(node ) (0.55)($69,800) (0.45)( $5,000) 38,390 2,250 $36,40 The EMV for using the survey $36,40. EMV(no survey) (0.5)($00,000) (0.5)( $40,000) $30,000 The survey should be used. c. EVSI ($36,40 $5,000) $30,000 $,40. Thus, the physicians would pay up to $,40 for the survey.

8 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page CHAPTER 3 D ECISION A NALYSIS Large Shop 2 Favorable Market Unfavorable Market Favorable Survey No Shop Small Shop 3 Favorable Market Unfavorable Market Market Survey Unfavorable Survey Large Shop 4 Favorable Market Unfavorable Market No Shop No Survey Small Shop 5 Favorable Market Unfavorable Market Large Shop 6 Favorable Market Unfavorable Market No Shop Small Shop 7 Favorable Market Unfavorable Market 3-3. a. EMV(node 2) (0.9)(55,000) (0.)( $45,000) 49,500 4,500 $45,000 EMV(node 3) (0.9)(25,000) (0.)( 5,000) 22,500,500 $2,000 EMV(node 4) (0.2)(55,000) (0.88)( 45,000) 6,600 39,600 $33,000 EMV(node 5) (0.2)(25,000) (0.88)( 5,000) 3,000 3,200 $0,200 EMV(node 6) (0.5)(60,000) (0.5)( 40,000) 30,000 20,000 $0,000 EMV(node 7) (0.5)(30,000) (0.5)( 0,000) 5,000 5,000 $0,000 EMV(node ) (0.6)(45,000) (0.4)( 5,000) 27,000 2,000 $25,000 Since EMV(market survey) > EMV(no survey), Jerry should conduct the survey. Since EMV(large shop favorable survey) is larger than both EMV(small shop favorable survey) and EMV(no shop favorable survey), Jerry should build a large shop if the survey is favorable. If the survey is unfavorable, Jerry should build nothing since EMV(no shop unfavorable survey) is larger than both EMV(large shop unfavorable survey) and EMV(small shop unfavorable survey).

9 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page 25 CHAPTER 3 D ECISION A NALYSIS 25 $25,000 Favorable Survey (0.6) $45,000 Large Shop No Shop Small Shop $45,000 2 $2,000 3 Favorable Market (0.9) Unfavorable Market (0.) Favorable Market (0.9) Unfavorable Market (0.) Payoff $55,000 $45,000 $5,000 $25,000 $5,000 Market Survey No Survey Unfavorable Survey (0.4) $5,000 Large Shop No Shop Small Shop $33,000 4 $0,200 5 Favorable Market (0.2) Unfavorable Market (0.88) Favorable Market (0.2) Unfavorable Market (0.88) $55,000 $45,000 $5,000 $25,000 $5,000 $0,000 Large Shop No Shop Small Shop $0,000 6 $0,000 7 Favorable Market (0.5) Unfavorable Market (0.5) Favorable Market (0.5) Unfavorable Market (0.5) $60,000 $40,000 $0 $30,000 $0,000 b. If no survey, EMV 0.5(30,000) 0.5( 0,000) $0,000. To keep Jerry from changing decisions, the following must be true: EMV(survey) EMV(no survey) Let P probability of a favorable survey. Then, P[EMV(favorable survey)] ( P) [EMV(unfavorable survey)] EMV(no survey) This becomes: P(45,000) ( P)( 5,000) $0,000 Solving gives 45,000P 5,000 5,000P 0,000 50,000P 5,000 P 0.3 Thus, the probability of a favorable survey could be as low as 0.3. Since the marketing professor estimated the probability at 0.6, the value can decrease by 0.3 without causing Jerry to change his decision. Jerry s decision is not very sensitive to this probability value.

10 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page CHAPTER 3 D ECISION A NALYSIS $2,750 Information Favorable (0.5) $8,500 A 3 A 4 A 5 $8,500 2 $500 3 (0.9) (0.) (0.9) (0.) Payoff $2,000 $23,000 $2,000 $3,000 $3,000 A Gather More Information Information Unfavorable (0.5) $3,000 A 3 A 4 $9,000 4 $7,000 5 (0.4) (0.6) (0.4) (0.6) $2,000 $23,000 $2,000 $3,000 A2 Do Not Gather More Information A 5 $3,000 $4,500 A 3 A 4 $4,500 6 $500 7 (0.7) (0.3) (0.7) (0.3) $5,000 $20,000 $5,000 $0,000 A 5 $0 A : gather more information A 2 : do not gather more information A 3 : build quadplex A 4 : build duplex A 5 : do nothing EMV(node 2) 0.9(2,000) 0.( 23,000) 8,500 EMV(node 3) 0.9(2,000) 0.( 3,000) 500 EMV(get information and then do nothing) 3,000 EMV(node 4) 0.4(2,000) 0.6( 23,000) 9,000 EMV(node 5) 0.4(2,000) 0.6( 3,000) 7,000 EMV(get information and then do nothing) 3,000 EMV(node ) 0.5(8,500) 0.5(-3,000) 2,750 EMV(build quadplex) 0.7(5,000) 0.3( 20,000) 4,500 EMV(build duplex) 0.7(5,000) 0.3( 0,000) 500 EMV(do nothing) 0 Decisions: do not gather information; build quadplex I : favorable research or information I 2 : unfavorable research S : store successful S 2 : store unsuccessful P(S ) 0.5; P(S 2 ) 0.5 P(I S ) 0.8; P(I 2 S ) 0.2 P(I S 2 ) 0.3; P(I 2 S 2 ) 0.7 a. P(successful store favorable research) P(S I ) PI ( S) PS ( ) PS ( I) PI ( S) PS ( ) + PI ( S) PS ( 2 ) (.) PS ( I) (.) (.) b. P(successful store unfavorable research) P(S I 2 ) PI ( 2 S) PS ( ) PS ( I2) PI ( S) PS ( ) + PI ( S) PS ( 2 ) (. ) PS ( I2) (.) (.) c. Now P(S ) 0.6 and P(S 2 ) (.) PS ( I) (. ) (. ) (. ) PS ( I2) (. ) (. )

11 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page 27 CHAPTER 3 D ECISION A NALYSIS I : favorable survey or information I 2 : unfavorable survey S : facility successful S 2 : facility unsuccessful P(S ) 0.3; P(S 2 ) 0.7 P(I S ) 0.8; P(I 2 S ) 0.2 P(I S 2 ) 0.3; P(I 2 S 2 ) 0.7 P(successful facility favorable survey) P(S I ) PI ( S) PS ( ) PS ( I) PI ( S) PS ( ) + PI ( S) PS ( 2 ) (.) PS ( I) (.) (.) P(successful facility unfavorable survey) P(S I 2 ) PI ( 2 S) PS ( ) PS ( I2) PI ( S) PS ( ) + PI ( S) PS ( 2 ) b. EMV(A) 0,000(0.2) 2,000(0.3) ( 5,000)(0.5) 00 EMV(B) 6,000(0.2) 4,000(0.3) 0(0.5) 2,400 Fund B should be selected. c. Let X payout for Fund A in a good economy. EMV(A) EMV(B) X(0.2) 2,000(0.3) ( 5,000)(0.5) 2, X 4,300 X 4,300/0.2 2,500 Therefore, the return would have to be $2,500 for Fund A in a good economy for the two alternatives to be equally desirable based on the expected values (. ) PS ( I2) (. ) (. ) a. Fund A Good economy 0.2 Fair economy 0.3 Poor economy 0.5 0,000 2,000 5,000 Fund B Good economy 0.2 Fair economy 0.3 Poor economy 0.5 6,000 4,000 0

12 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page CHAPTER 3 D ECISION A NALYSIS a. Payoff Survey Favorable Produce Razor 3 Do Not Produce Razor Favorable Market Unfavorable Market $95,000 $65,000 $5,000 Survey Produce Razor 4 Favorable Market Unfavorable Market $95,000 $65,000 Conduct Survey Unfavorable Study Do Not Produce Razor Produce Razor 5 Favorable Market Unfavorable Market $5,000 $80,000 $80,000 Conduct Pilot Study 2 Favorable Study Do Not Produce Razor Produce Razor 6 Favorable Market Unfavorable Market $20,000 $80,000 $80,000 Neither Test Unfavorable Do Not Produce Razor $20,000 Produce Razor 7 Favorable Market Unfavorable Market $00,000 $60,000 Do Not Produce Razor $0 b. S: survey favorable S2: survey unfavorable S3: study favorable S4: study unfavorable S5: market favorable S6: market unfavorable 0705.(.) PS ( 5 S) (.) (.) P(S 6 S ) (.) PS ( 5 S2) (.) (.) P(S 6 S 2 ) (.) PS ( 5 S3) (.) (.) P(S 6 S 3 ) (. ) PS ( 5 S4) (.) (.) P(S 6 S 4 ) c. EMV(node 3) 95,000(0.78) ( 65,000)(0.22) 59,800 EMV(node 4) 95,000(0.27) ( 65,000)(0.73) 2,800 EMV(node 5) 80,000(0.89) ( 80,000)(0.) 62,400 EMV(node 6) 80,000(0.8) ( 80,000)(0.82) 5,200 EMV(node 7) 00,000(0.5) ( 60,000)(0.5) 20,000 EMV(conduct survey) 59,800(0.45) ( 5,000)(0.55) 24,60 EMV(conduct pilot study) 62,400(0.45) ( 20,000)(0.55) 7,080 EMV(neither) 20,000 Therefore, the best decision is to conduct the survey. If it is favorable, produce the razor. If it is unfavorable, do not produce the razor The following computations are for the decision tree that follows. EU(node 3) 0.95(0.78) 0.5(0.22) 0.85 EU(node 4) 0.95(0.27) 0.5(0.73) 0.62 EU(node 5) 0.9(0.89) 0(0.) 0.80 EU(node 6) 0.9(0.8) 0(0.82) 0.6 EU(node 7) (0.5) 0.55(0.5) 0.78 EU(conduct survey) 0.85(0.45) 0.8(0.55) EU(conduct pilot study) 0.80(0.45) 0.7(0.55) EU(neither test) 0.8 Therefore, the best decision is to conduct the survey. Jim is a risk avoider.

13 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page 29 CHAPTER 3 D ECISION A NALYSIS Conduct Pilot 2 Study Conduct Survey Neither Test Survey Favorable (0.45) Survey Unfavorable (0.55) Study Favorable (0.45) Study Unfavorable (0.55) Produce Razor 3 Do Not Produce Razor Produce Razor Do Not Produce Razor Produce Razor Do Not Produce Razor Produce Razor Do Not Produce Razor Market Favorable (0.78) Market Unfavorable (0.22) Market Favorable (0.27) Market Unfavorable (0.73) Market Favorable (0.89) Market Unfavorable (0.) Market Favorable (0.8) Market Unfavorable (0.82) Utility Produce Razor Market Favorable (0.5) Market Unfavorable (0.5) 0.55 Do Not Produce Razor a. P(good economy prediction of 0806.(.) good economy) (. ) (. ) P(poor economy prediction of 004.(.) good economy) (. ) (. ) P(good economy prediction of (. ) poor economy) (. ) (. ) P(poor economy prediction of 0906.(.) poor economy) (. ) (. ) P(poor economy prediction of 0903.(.) poor economy) (. ) (. ) The expected value of the payout by the insurance company is EV 0(0.999) 00,000(0.00) 00 The expected payout by the insurance company is $00, but the policy costs $200, so the net gain for the individual buying this policy is negative ( $00). Thus, buying the policy does not maximize EMV since not buying this policy would have an EMV of 0, which is better than $00. However, a person who buys this policy would be maximizing the expected utility. The peace of mind that goes along with the insurance policy has a relatively high utility. A person who buys insurance would be a risk avoider. b. P(good economy prediction of 0807.(.) good economy) (.) (.) P(poor economy prediction of 003.(.) good economy) (.) (.) P(good economy prediction of (. ) poor economy) (. ) (. )

14 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page CHAPTER 3 D ECISION A NALYSIS Payoff Utility U 0.76 Conduct Market Do Not Conduct Survey Survey Favorable (0.55) Survey Unfavorable (0.45) U 0.88 Construct 2 Clinic Do Not Construct Clinic U Construct 3 Clinic Do Not Construct Clinic Favorable Market (0.82) Unfavorable Market (0.8) Favorable Market (0.) Unfavorable Market (0.89) $95,000 $45,000 $5,000 $95,000 $45,000 $5, Construct Clinic U Favorable Market (0.5) Unfavorable Market (0.5) $00,000 $40, Do Not Construct Clinic $0 0.9 EU(node 2) (0.82)(0.99) (0.8)(0) 0.88 EU(node 3) (0.)(0.99) (0.89)(0) EU(node 4) 0.5() 0.5(0.) 0.55 EU(node ) (0.55)(0.88) (0.45)(0.7000) EU(no survey) 0.9 The expected utility with no survey (0.9) is higher than the expected utility with a survey (0.765), so the survey should be not used. The medical professionals are risk avoiders EU(large plant survey favorable) 0.78(0.95) 0.22(0) 0.74 EU(small plant survey favorable) 0.78(0.5) 0.22(0.) 0.42 EU(no plant survey favorable) 0.2 EU(large plant survey negative) 0.27(0.95) 0.73(0) EU(small plant survey negative) 0.27(0.5) 0.73(0.0) EU(no plant survey negative) 0.2 EU(large plant no survey) 0.5() 0.5(0.05) EU(small plant no survey) 0.5(0.6) 0.5(0.5) EU(no plant no survey) 0.3 EU(conduct survey) 0.45(0.74) 0.55(0.2565) EU(no survey) John s decision would change. He would not conduct the survey and build the large plant a. Expected travel time on Broad Street 40(0.5) 5(0.5) 27.5 minutes. Broad Street has a lower expected travel time. Expressway Broad Street Congestion (0.5) No Congestion (0.5) 30 Minutes, U Minutes, U Minutes, U 0.9 Utility b. Expected utility on Broad Street 0.2(0.5) 0.9(0.5) Therefore, the expressway maximizes expected utility. c. Lynn is a risk avoider Time (minutes) Selling price $20 per gallon; manufacturing cost $2 per gallon; salvage value $3; handling costs $ per gallon; and advertising costs $3 per gallon. From this information, we get: marginal profit selling price minus the manufacturing, handling, and advertising costs marginal profit $20 $2 $ $3 $4 per gallon If more is produced than is needed, a marginal loss is incurred. marginal loss $3 $2 $ $3 $3 per gallon In addition, there is also a shortage cost. Coren has agreed to fulfill any demand that cannot be met internally. This requires that Coren purchase chemicals from an outside company. Because the cost of obtaining the chemical from the outside company is $25 and the price charged by Coren is $20, this results in shortage cost $5 per gallon In other words, Coren will lose $5 for every gallon that is sold that has to be purchased from an outside company due to a shortage.

15 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page 3 CHAPTER 3 D ECISION A NALYSIS 3 a. A decision tree is shown below: Decision Tree $,500 Stock 500 (0.2) Demand (0.3) (0.4) (0.) 500,000,500 2,000 $2,000 (500)(4) $500 (500)(4) (500)(5) $3,000 (500)(4) (,000)(5) $5,500 (500)(4) (,500)(5) $2,400 Stock,000 $,800 $3,300 Stock,500 Stock 2,000 (0.2) (0.3) (0.4) (0.) (0.2) (0.3) (0.4) (0.) (0.2) (0.3) (0.4) (0.) 500,000,500 2, ,000,500 2, ,000,500 2,000 $500 (500)(4) (500)(3) $4,000 (,000)(4) $,500 (,000)(4) (5)(500) $,000 (,000)(4) (5)(,000) $,000 (500)(4) (3)(,000) $2,500 (,000)(4) (3)(500) $6,000 (,500)(4) $3,500 (,500)(4) (5)(500) $2,500 (500)(4) (3)(,500) $,000 (,000)(4) (3)(,000) $4,500 (,500)(4) (3)(500) $8,000 (2,000)(4) b. The computations are shown in the following table. These numbers are entered into the tree above. The best decision is to stock,500 gallons. Table for Problem 3-43 Demand Stock 500,000,500 2,000 EMV 500 2, ,000 5,500 $,500, ,000,500,000 $,800,500,000 2,500 6,000 3,500 $3,300 2,000 2,500,000 4,500 8,000 $2,400 Maximum 2,000 4,000 6,000 8,000 $4,800 EVwPI Probabilities c. EVwPI (0.2)(2,000) (0.3)(4,000) (0.4)(6,000) (0.)(8,000) $4,800 EVPI EVwPI EMV $4,800 $3,300 $, If no survey is to be conducted, the decision tree is fairly straightforward. There are three main decisions, which are building a small, medium, or large facility. Extending from these decision branches are three possible demands, representing the possible states of nature. The demand for this type of facility could be either low (L), medium (M), or high (H). It was given in the problem that the probability for a low demand is 0.5. The probabilities for a medium and a high demand are 0.40 and 0.45, respectively. The problem also gave monetary consequences for building a small, medium, or large facility when the demand could be low, medium, or high for the facility. These data are reflected in the following decision tree. Decision Tree No Survey Small $500,000 Medium $670,000 Large $580,000 (0.5) (0.40) (0.45) (0.5) (0.40) (0.45) (0.5) (0.40) (0.45) $500,000 $500,000 $500,000 $200,000 $700,000 $800,000 $200,000 $400,000 $,000,000 With no survey, we have: EMV(Small) 500,000; EMV(Medium) 670,000; and EMV(Large) 580,000. The medium facility, with an expected monetary value of $670,000, is selected because it represents the highest expected monetary value. If the survey is used, we must compute the revised probabilities using Bayes theorem. For each alternative facility, three revised probabilities must be computed, representing low, medium, and high demand for a facility. These probabilities can be computed using tables. One table is used to compute the probabilities for low survey results, another table is used for

16 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page CHAPTER 3 D ECISION A NALYSIS medium survey results, and a final table is used for high survey results. These tables are shown below. These probabilities will be used in the decision tree that follows. Decision Tree Survey L 450,000 For low survey results A: State of Nature P(Bi) P(Ai Bj) P(Bj and Ai) P(Bj Ai) B B B P(A) 0.30 Small Medium M H L M 450, ,000 50, ,000 For medium survey results A2: State of Nature P(Bi) P(Ai Bj) P(Bj and Ai) P(Bj Ai) H L 750, ,000 B B B P(A2) For high survey results A3: State of Nature P(Bi) P(Ai Bj) P(Bj and Ai) P(Bj Ai) B B B P(A3) When survey results are low, the probabilities are P(L) 0.339; P(M) 0.56; and P(H) This results in EMV(Small) 450,000; EMV(Medium) 495,000; and EMV(Large) 233,600. When survey results are medium, the probabilities are P(L) 0.082; P(M) 0.548; and P(H) This results in EMV (Small) 450,000; EMV(Medium) 646,000; and EMV(Large) 522,800. When survey results are high, the probabilities are P(L) 0.046; P(M) 0.23; and P(H) This results in EMV(Small) 450,000; EMV(Medium) 70,00; and EMV(Large) 82,000. If the survey results are low, the best decision is to build the medium facility with an expected return of $495,000. If the survey results are medium, the best decision is also to build the medium plant with an expected return of $646,000. On the other hand, if the survey results are high, the best decision is to build the large facility with an expected monetary value of $82,000. The expected value of using the survey is computed as follows: EMV(with Survey) 0.30(495,000) 0.365(646,000) 0.325(82,000) 656,065 Because the expected monetary value for not conducting the survey is greater (670,000), the decision is not to conduct the survey and to build the medium-sized facility. $495,000 Low (0.30) $646,000 Medium (0.365) $82,000 High (0.325) Large Small Medium Large Small Medium Large M H L M H L M H L M H L M H L M H L M H 350, , , , ,000 50, , , , , , , , ,000 50, , , , , ,000

17 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page 33 CHAPTER 3 D ECISION A NALYSIS a. $75,000 Succeed (0.5) Payoff $250,000 Downtown Mall $40,000 2 Don t Succeed (0.5) Succeed (0.6) Don t Succeed (0.4) $00,000 $300,000 $00,000 Traffic Circle $250,000 Succeed (0.75) $400,000 No Grocery Store 3 Don t Succeed (0.25) $200,000 $0 Mary should select the traffic circle location (EMV $250,000). b. Use Bayes Theorem to compute posterior probabilities. P(SD SRP) 0.78; P(SD SRP) 0.22 P(SM SRP) 0.84; P(SM SRP) 0.6 P(SC SRP) 0.9; P(SC SRP) 0.09 P(SD SRN) 0.27; P(SD SRN) 0.73 P(SM SRN) 0.36; P(SM SRN) 0.64 P(SC SRN) 0.53; P(SC SRN) 0.47 Example computations: PSRP ( SM) PSM ( ) PSM ( SRP) PSRP ( SM) PSM ( ) + PSR ( P SM) P( SM) 0706.(.) PSM ( SRP) (. ) (. ) EMV(8) $75,000 EMV(9) $40,000 EMV(0) $250,000 EMV(no grocery C) $0 EMV(A) (best of four alternatives) $36,000 EMV(B) (best of four alternatives) $88,000 EMV(C) (best of four alternatives) $250,000 EMV() (0.6)($36,000) (0.4)($88,000) $224,800 EMV(D) (best of two alternatives) $250,000 c. EVSI [EMV() cost] (best EMV without sample information) $254,800 $250,000 $4, (. ) PSC ( SRN) (. ) (. ) These calculations are for the tree that follows: EMV(2) $7,600 $28,600 $43,000 EMV(3) $226,800 $20,800 $206,000 EMV(4) $336,700 $20,700 $36,000 EMV(no grocery A) $30,000 EMV(5) $59,400 $94,900 $35,500 EMV(6) $97,200 $83,200 $4,000 EMV(7) $96,00 $08,00 $88,000 EMV(no grocery B) $30,000

18 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page CHAPTER 3 D ECISION A NALYSIS First Decision Point D Purchase Market Survey Survey Results Positive (0.6) Survey Results Negative (0.4) Do Not Purchase Market Survey Second Decision Point A B C Downtown 2 Mall 3 Circle 4 No Grocery Store Downtown 5 Mall 6 Circle 7 No Grocery Store Downtown 8 Mall 9 Circle 0 No Grocery Store SD (0.78) SD (0.22) SM (0.84) SM (0.6) SC (0.9) SC (0.09) SD (0.27) SD (0.73) SM (0.36) SM (0.64) SC (0.53) SC (0.47) SD (0.5) SD (0.5) SM (0.6) SM (0.4) SC (0.75) SC (0.25) Payoff $220,000 $30,000 $270,000 $30,000 $370,000 $230,000 $30,000 $220,000 $30,000 $270,000 $30,000 $370,000 $230,000 $30,000 $250,000 $00,000 $300,000 $00,000 $400,000 $200,000 $ a. Sue can use decision tree analysis to find the best solution. The results are presented below. In this case, the best decision is to get information. If the information is favorable, she should build the retail store. If the information is not favorable, she should not build the retail store. The EMV for this decision is $29,200. In the following results (using QM for Windows), Branch ( 2) is to get information, Branch 2 ( 3) is the decision to not get information, Branch 3 (2 4) is favorable information, Branch 4 (2 5) is unfavorable information, Branch 5 (3 8) is the decision to build the retail store and get no information, Branch 6 (3 7) is the decision to not build the retail store and to get no information, Branch 7 (4 6) is the decision to build the retail store given favorable information, Branch 8 (4 ) is the decision to not build given favorable information, Branch 9 (6 9) is a good market given favorable information, Branch 0 (6 0) is a bad market given favorable information, Branch (5 7) is the decision to build the retail store given unfavorable information, Branch 2 (5 4) is the decision not to build the retail store given unfavorable information, Branch 3 (7 2) is a successful retail store given unfavorable information, Branch 4 (7 3) is an unsuccessful retail store given unfavorable information, Branch 5 (8 5) is a successful retail store given that no information is obtained, and Branch 6 (8 6) is an unsuccessful retail store given no information is obtained.

19 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page 35 CHAPTER 3 D ECISION A NALYSIS 35 b. The suggested changes would be reflected in Branches 4 and 5. The decision stays the same, but the EMV increases to $46,000. The results are provided in the tables that follow: Results for a. Start Ending Branch Profit Use Node Node Node Node Probability (End Node) Branch? Type Value Start Decision 29,200 Branch Yes Chance 29,200 Branch Decision 28,000 Branch Decision 62,000 Branch Decision 20,000 Branch Yes Chance 28,000 Branch Final 0 Branch Yes Chance 62,000 Branch ,000 Final 20,000 Branch ,000 Final 80,000 Branch ,000 Final 00,000 Branch Chance 64,000 Branch ,000 Yes Final 20,000 Branch ,000 Final 80,000 Branch ,000 Final 00,000 Branch ,000 Final 00,000 Branch ,000 Final 80,000 Results for b. Start Ending Branch Profit Use Node Node Node Node Probability (End Node) Branch? Type Value Start Decision 37,400 Branch Yes Chance 37,400 Branch Decision 28,000 Branch Decision 62,000 Branch Decision 20,000 Branch Yes Chance 28,000 Branch Final 0 Branch Yes Chance 62,000 Branch ,000 Final 20,000 Branch ,000 Final 80,000 Branch ,000 Final 00,000 Branch Chance 64,000 Branch ,000 Yes Final 20,000 Branch ,000 Final 80,000 Branch ,000 Final 00,000 Branch ,000 Final 00,000 Branch ,000 Final 80,000

20 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page CHAPTER 3 D ECISION A NALYSIS c. Sue can determine the impact of the change by changing the probabilities and recomputing EMVs. This analysis shows the decision changes. Given the new probability values, Sue s best decision is build the retail store without getting additional information. The EMV for this decision is $28,000. The results are presented below: Results for c. Start Ending Branch Profit Use Node Node Node Node Probability (End Node) Branch? Type Value Start Decision 28,000 Branch Chance 8,400 Branch Yes Decision 28,000 Branch Decision 44,000 Branch Decision 20,000 Branch Yes Chance 28,000 Branch Final 0 Branch Yes Chance 44,000 Branch ,000 Final 20,000 Branch ,000 Final 80,000 Branch ,000 Final 00,000 Branch Chance 64,000 Branch ,000 Yes Final 20,000 Branch ,000 Final 80,000 Branch ,000 Final 00,000 Branch ,000 Final 00,000 Branch ,000 Final 80,000 d. Yes, Sue s decision would change from her original decision. With the higher cost of information, Sue s decision is to not get the information and build the retail store. The EMV of this decision is $28,000. The results are given below: Results for d. Start Ending Branch Profit Use Node Node Node Node Probability (End Node) Branch? Type Value Start Decision 28,000 Branch Chance 9,200 Branch Yes Decision 28,000 Branch Decision 52,000 Branch Decision 30,000 Branch Yes Chance 28,000 Branch Final 0 Branch Yes Chance 52,000 Branch ,000 Final 30,000 Branch ,000 Final 70,000 Branch ,000 Final 0,000 Branch Chance 74,000 Branch ,000 Yes Final 30,000 Branch ,000 Final 70,000 Branch ,000 Final 0,000 Branch ,000 Final 00,000 Branch ,000 Final 80,000

21 M03_REND6289_0_IM_C03.QXD 5/7/08 3:48 PM Page 37 CHAPTER 3 D ECISION A NALYSIS 37 e. The expected utility can be computed by replacing the monetary values with utility values. Given the utility values in the problem, the expected utility is The utility table represents a risk seeker. The results are given below: Results for e. Start Ending Branch Profit Use Ending Node Node Node Node Probability (End Node) Branch? Node Type Value Start Decision 0.62 Branch Chance Branch Yes 3 Decision 0.62 Branch Decision 0.36 Branch Decision 0. Branch Yes 8 Chance 0.62 Branch Final 0.20 Branch Yes 6 Chance 0.36 Branch Final 0. Branch Final 0.4 Branch Final 0 Branch Chance 0.08 Branch Yes 4 Final 0. Branch Final 0.4 Branch Final 0 Branch Final Branch Final 0.05 f. This problem can be solved by replacing monetary values with utility values. The expected utility is The utility table given in the problem is representative of a risk avoider. The results are presented below: Results for f. Start Ending Branch Profit Use Node Node Node Node Probability (End Node) Branch? Type Value Start Decision 0.80 Branch Chance Branch Yes Decision 0.80 Branch Decision 0.8 Branch Decision 0.60 Branch Yes Chance 0.76 Branch Final 0.80 Branch Yes Chance 0.8 Branch Final 0.60 Branch Final 0.90 Branch Final 0.00 Branch Chance 0.8 Branch Yes Final 0.60 Branch Final 0.90 Branch Final 0.00 Branch Final.00 Branch Final a. The decision table for Chris Dunphy along with the expected profits or expected monetary values (EMVs) for each alternative are shown on the next page.