Maximizing Expected Utility and Decision Trees

As explained in this week’s resources, maximization of expected utility involves the following five steps of decision making:

1-Identifying future conditions, along with the likelihood of the condition being realized

2-Listing possible alternatives

3-Estimating the payoff or utility for each alternative under each future condition

4-Calculating the expected utility

5-Selecting the best alternative

Decision trees can be used to illustrate all of this information in a graphic manner.

In this Assignment, apply the information from this week’s resources to solve decision problems and complete a Decision Analysis for those problems. Solve problems 5, 10, and 11 on pp. 231–234 of the Stevenson text by creating decision trees, determining expected utilities of the decision alternatives, and offering recommended decisions based on the decision tree analysis.

Approximate length: 4–5 pages

Submit your Decision Analysis for the assigned problems

Hastie, R., & Dawes, R. M. (2010). Rational choice in an uncertain world (2nd ed.). Thousand Oaks, CA: Sage. Chapter 2, “What Is Decision Making?” (pp. 23–43)

Hastie, R., & Dawes, R. M. (2010). Rational choice in an uncertain world (2nd ed.). Thousand Oaks, CA: Sage. Chapter 11.1, “Formally Defining Rationality” (pp. 237–241)

Decision Theory LEARNING OBJECTIVES SUPPLEMENT OUTLINE 5S.5 Decision Making under Uncertainty, 219 After completing this supplement, you 5S.6 Decision Making under Risk, 220 5S.1 Introduction, 216 should be able to: 5S.7 Decision Trees, 221 5S.2 The Decision Process and Causes of L05S.1 Outline the steps in the decision Poor Decisions, 217 5S.8 Expected Value of Perfect process. Information, 223 5S.3 Decision Environments, 218 L05S.2 Name some causes of poor 5S.9 Sensitivity Analysis, 224 decisions. 5S.4 Decision Making under Certainty, 218 L05S.3 Describe and use techniques that apply to decision making under uncertainty. L05S.4 Describe and use the expectedvalue approach. L05S.5 Construct a decision tree and use it to analyze a problem. L05S.6 Compute the expected value of perfect information. L05S.7 Conduct sensitivity analysis on a simple decision problem. 55.1 INTRODUCTION – Decision theory represents a general approach to decision making. It is suitable for a wide range of operations management decisions. Among them are capacity planning, product and service design, equipment selection, and location planning. Decisions that lend themselves to a decision theory approach tend to be characterized by the following elements: 1. A set of possible future conditions that will have a bearing on the results of the decision. 2. A list of alternatives for the manager to choose from. 3. A known payoff for each alternative under each possible future condition. To use this approach, a decision maker would employ this process: 1. Identify the possible future conditions (e.g., demand will be low, medium, or high; the competitor will or will not introduce a new product). These are called states of nature. 2. Develop a list of possible alternatives, one of which may be to do nothing. t 3. Determine or estimate the payoff associated with each alternative for every possible future condition. —- —————————————-~– 216 Supplement to Chapter Five Decision Theory 217 If possible, estimate the likelihood of each possible future condition. 5. Evaluate alternatives according to some decision criterion (e.g., maximize expected profit), and select the best alternative. The information for a decision is often summarized in a payoff table, which shows the expected payoffs for each alternative under the various possible states of nature. These tables are helpful in choosing among alternatives because they facilitate comparison of alternatives. Consider the following payoff table, which illustrates a capacity planning problem. POSSIBLE FUTURE DEMAND Alternatives Low Moderate High Small facility $10* $10 $10 Medium facility 7 12 12 Large facil ity (4) 2 16 ‘Present value in $ millions. The payoffs are shown in the body of the table. In this instance, the payoffs are in terms of present values, which represent equivalent current dollar values of expected future income less costs. This is a convenient measure because it places all alternatives on a comparable basis. If a small facility is built, the payoff will be the same for all three possible states of nature. For a medium facility, low demand will have a present value of $7 million, whereas both moderate and high demand will have present values of $12 million. A large facility will have a loss of $4 million if demand is low, a present value of $2 million if demand is moderate, and a present value of $16 million if demand is high. The problem for the decision maker is to select one of the alternatives, taking the present value into account. Evaluation of the alternatives differs according to the degree of certainty associated with the possible future conditions. 5S.2 THE DECISION PROCESS AND CAUSES OF POOR DECISIONS Despite the best efforts of a manager, a decision occasionally turns out poorly due to unforeseeable circumstances. Luckily, such occurrences are not common. Often, failures can be traced to a combination of mistakes in the decision process, to bounded rationality, or to suboptimization. The decision process consists of these steps: 1. Identify the problem. 2. Specify objectives and criteria for a solution. 3. Develop suitable alternatives. 4. Analyze and compare alternatives. 5. Select the best alternative. 6. Implement the solution. 7. Monitor to see that desired result is achieved. In many cases, managers fail to appreciate the importance of each step in the decisionmaking process. They may skip a step or not devote enough effort to completing it before jumping to the next step. Sometimes this happens owing to a manager’s style of making quick decisions or a failure to recognize the consequences of a poor decision. The manager’s ego can be a factor. This sometimes happens when the manager has experienced a series of successesimportant decisions that turned out right. Some managers then get the impression that they can do no wrong. But they soon run into trouble, which is usually enough to bring them back down Payoff table Table showing the expected payoffs for each alternative in every possible state of nature. L05S.1 Outline the steps in the decision process. L05S.2 Name some causes of poor decisions. •• 218 Bounded rationality The limitations on decision making caused by costs, human abilities, time, technology, and availability of information. Suboptimization The result of different departments each attempting to reach a solution that is optimum for that department. Certainty Environment in which relevant parameters have known values. Risk Environment in which certain future events have probable outcomes. Uncertainty Environment in which it is impossible to assess the likelihood of various future events. EXAMPLE 55-1 Supplement to Chapter Five Decision Theory to earth. Other managers seem oblivious to negative results and continue the process they associate with their previous successes, not recognizing that some of that success may have been due more to luck than to any special abilities of their own. A part of the problem may be the manager’s unwillingness. to admit a mistake. Yet other managers demonstrate an inability to make a decision; they stall long past the time when the decision should have been rendered . Of course, not all managers fall into these traps-it seems safe to say that the majority do not. Even so, this does not necessarily mean that every decision works out as expected. Another factor with which managers must contend is bounded rationality, or the limits imposed on decision making by costs, human abilities, time, technology, and the availability of information. Because of these limitations, managers cannot always expect to reach decisions that are optimal in the sense of providing the best possible outcome (e.g., highest profit, least cost). Instead, they must often resort to achieving a satisfactory solution. Still another cause of poor decisions is that organizations typically departmentalize decisions. Naturally, there is a great deal of justification for the use of departments in terms of overcoming span-of-control problems and human limitations. However, suboptimization can occur. This is a result of different departments’ attempts to reach a solution that is optimum for each. Unfortunately, what is optimal for one department may not be optimal for the organization as a whole. If you are familiar with the theory of constraints (see Chapter 16), suboptimization and local optima are conceptually the same, with the same negative consequences. 5S.3 DECISION ENVIRONMENTS Operations management decision environments are classified according to the degree of certainty present. There are three basic categories: certainty, risk, and uncertainty. . Certainty means that relevant parameters such as costs, capacity, and demand have known values. Risk means that certain parameters have probabilistic outcomes. Uncertainty means that it is impossible to assess the likelihood of various possible future events. Consider these situations: 1. Profit per unit is $5. You have an order for 200 units. How much profit will you make? (This is an example of certainty since unit profits and total demand are known.) 2. Profit is $5 per unit. Based on previous experience, there is a 50 percent chance of an order for 100 units and a 50 percent chance of an order for 200 units. What is expected profit? (This is an example of risk since demand outcomes are probabilistic.) 3. Profit is $5 per unit. The probabilities of potential demands are unknown. (This is an example of uncertainty.) The importance of these different decision environments is that they require different analysis techniques. Some techniques are better suited for one category than for others. 5S.4 DECISION MAKING UNDER CERTAINTY When it is known for certain which of the possible future conditions will actually happen, the decision is usually relatively straightforward: Simply choose the alternative that has the best payoff under that state of nature. Example 5S-1 illustrates this. Determine the best alternative in the payoff table on the previous page for each of-the cases: It is known with certainty that demand will be (a) low, (b) moderate, (c) high. 219 ‘——~ Supplement to Chapter Five Decision Theory Choose the alternative with the highest payoff. Thus, if we know demand will be low, we would elect to build the small facility and realize a payoff of : 10 million. If we know demand will be moderate, a medium factory would yield the highest payoff ($12 million versus either $10 million or $2 million). For high demand, a large facility would provide the highest payoff. Although complete certainty is rare in such situations, this kind of exercise provides some perspective on the analysis. Moreover, in some instances, there may be an opportunity to consider allocation of funds to research efforts, which may reduce or remove some of the uncertainty surrounding the states of nature, converting uncertainty to risk or to certainty. 5S.5 DECISION MAKING UNDER UNCERTAINTY At the opposite extreme is complete uncertainty: 0 information is available on how likely the various states of nature are. Under those conditions, four possible decision criteria are maximin, maximax, Laplace, and minimax regret. These approaches can be defined as follows: Maximin-Determine the worst possible payoff for each alternative, and choose the alternative that has the “best worst.” The maximin approach is essentially a pessimistic one because it takes into account only the worst possible outcome for each alternative. The actual outcome may not be as bad as that, but this approach establishes a “guaranteed minimum.” Maximax-Determine the best possible payoff, and choose the alternative with that payoff. The maximax approach is an optimistic, “go for it” strategy; it does not take into account any payoff other than the best. Laplace-Determine the average payoff for each alternative, and choose the alternative with the best average. The Laplace approach treats the states of nature as equally likely. Minimax regret-Determine the worst regret for each alternative, and choose the alternative with the “best worst.” This approach seeks to minimize the difference between the payoff that is realized and the best payoff for each state of nature. The next two examples illustrate these decision criteria. Referring to the payoff table on page 217, determine which alternative would be chosen under each of these strategies: a. Maximin b. Maximax c. Laplace a. Using maximin, the worst payoffs for the alternatives are as follows: Small facility: $10 million Medium facility: 7 million Large facility: -4 million Hence, since $10 million is the best, choose to build the small facility using the maximin strategy. b. Using maximax, the best payoffs are as follows: Small facility: $10 million Medium facility: 12 million Large facility: 16 million The best overall payoff is the $16 million in the third row. Hence, the maximax criterion leads to building a large facility. SOLUTION L05S.3 Describe and use techniques that apply to decision making under uncertainty. Maximin Choose the alternative with the best of the worst possible payoffs. Maximax Choose the alternative with the best possible payoff. Laplace Choose the alternative with the best average payoff of any of the alternatives. Minimax regret Choose the alternative that has the least of the worst regrets. EXAMPLE 5S-2 eXcel mhhe.com/stevenson12e SOLUTION 220 I # EXAMPLE 5S-3 SOLUTION Regret (opportunity loss) The differencebetweena given payoff and the best payofffor a state of nature. L05S.4 Describe and use the expected-value approach. Expected monetary value (EMV) criterion The best expected value among the alternatives. Supplement to Chapter Five Decision Theory c. For the Laplace criterion, first find the row totals, and then divide each of those amounts by the number of states of nature (three in this case). Thus, we have Row Total (in $ millions) Row Average (in $ millions) Small facility Medium facility Large facility $30 31 14 $10.00 10.33 4.67 Because the medium facility has the highest average, it would be chosen under the Laplace criterion. Determine which alternative would be chosen using a minimax regret approach to the capacity planning program. The first step in this approach is to prepare a table of regrets (or opportunity losses). To do this, subtract every payoff in each column from the best payoff in that column. For instance, in the first column, the best payoff is 10, so each of the three numbers in that column must be subtracted from 10. Going down the column, the regrets will be 10 – 10 = 0, 10 – 7 = 3, and 10 – (-4) = 14. In the second column, the best payoff is 12. Subtracting each payoff from 12 yields 2, 0, and 10. In the third column, 16 is the best payoff. The regrets are 6, 4, and 0. These results are summarized in a regret table: REGRETS (IN $ MILLIONS) Alternatives Low Moderate High Worst ,Small facility $0 $2 $6 $6 Medium facility 3 0 4 4 Large facility 14 10 0 14 The second step is to identify the worst regret for each alternative. For the first alternative, the worst is 6; for the second, the worst is 4; and for the third, the worst is 14. The best of these worst regrets would be chosen using minimax regret. The lowest regret is 4, which is for a medium facility. Hence, that alternative would be chosen. Solved Problem 6 at the end of this supplement illustrates decision making under uncertainty when the payoffs represent costs. The main weakness of these approaches (except for Laplace) is that they do not take into account all of the payoffs. Instead, they focus on the worst or best, and so they lose some information. Still, for a given set of circumstances, each has certain merits that can be helpful to a decision maker. 5S.6 DECISION MAKING UNDER RISK Between the two extremes of certainty and uncertainty lies the case of risk: The probability of occurrence for each state of nature is known. (Note that because the states are mutually exclusive and collectively exhaustive, these probabilities must add to 1.00.) A widely used approach under such circumstances is the expected monetary value criterion. The expected value is computed for each alternative, and the one with the best expected value is selected. The expected value is the sum of the payoffs for an alternative where each payoff is weighted by the probability for the relevant state of nature. Thus, the approach is Expected monetary value (EMV) criterion-Determine the expected ‘payoff or each alternative, and choose the alternative that has the best expected payoff. Supplement to Chapter Five Decision Theory Using the expected monetary value criterion, identify the best alternative for the previous payoff table for these probabilities: low = .30, moderate = .50, and high = .20. Find the expected value of each alternative by multiplying the probability of occurrence for each state of nature by the payoff for that state of nature and summing them: = .30($10) = .30($7) = .30($-4) EVsmall EVmedium EVlarge .50($10) .50($12) .50($2) + .20($10) = $10 .20($12) = $10.5 .20($16) = $3 + + + + + Hence, choose the medium facility because it has the highest expected value. The expected monetary value approach is most appropriate when a decision maker is neither risk averse nor risk seeking, but is risk neutral. Typically, well-established organizations with numerous decisions of this nature tend to use expected value because it provides an indication of the long-run, average payoff. That is, the expected-value amount (e.g., $10.5 million in the last example) is not an actual payoff but an expected or average amount that would be approximated if a large number of identical decisions were to be made. Hence, if a decision maker applies this criterion to a large number of similar decisions, the expected payoff for the total will approximate the sum of the individual expected payoffs. 55.7 DECISION TREES In health care the array of treatment options and medical costs makes tools such as decision trees particularly valuable in diagnosing and prescribing treatment plans. For example, if a 20-year-old and a 50-year-old both are brought into an emergency room complaining of chest pains, the attending physician, after asking each some questions on family history, patient history, general health, and recent events and activities, will use a decision tree to sort through the options to arrive at the appropriate decision for each patient. Decision trees are tools that have many practical applications, not only in health care but also in legal cases and a wide array of management decision making, including credit card fraud; loan, credit, and insurance risk analysis; decisions on new product or service development; and location analysis. A decision tree is a schematic representation of the alternatives available to a decision maker and their possible consequences. The term gets its name from the treelike appearance of the diagram (see Figure 5S.1). Although tree diagrams can be used in place of a payoff Payoff 1 Payoff 2 >-, ~e CJ”,o ° Payoff 3 Initial decision Payoff 4 C-? 006′(9 ~ Payoff 5 Payoff 6 • Decision point o Chance event 221 EXAMPLE 55-4 mhhe.com/stevenson12e l05S.5 Construct a decision tree and use it to analyze a problem. Decision tree A schematic representation of the available alternatives and their possible consequences. FIGURE 5S.1 Format of a decision tree eXcel mhhe.com/stevenson12e 222 EXAMPLE 55-5 SOLUTION Supplement to Chapter Five Decision Theory table, they are particularly useful for analyzing situations that involve sequential decisions. For instance, a manager may initially decide to build a small facility only to discover that demand is much higher than anticipated. In this case, the manager may then be called upon to make a subsequent decision on whether to expand or build an additional facility. A decision tree is composed of a number of nodes that have branches emanating from them (see Figure 5S.1). Square nodes denote decision points, and circular nodes denote chance events. Read the tree from left to right. Branches leaving square nodes represent alternatives; branches leaving circular nodes represent chance events (i.e., the possible states of nature). After the tree has been drawn, it is analyzed from right to left; that is, starting with the last decision that might be made. For each decision, choose the alternative that will yield the greatest return (or the lowest cost). If chance events follow a decision, choose the alternative that has the highest expected monetary value (or lowest expected cost). A manager must decide on the size of a video arcade to construct. The manager has narrowed the choices to two: large or small. Information has been collected on payoffs, and a decision tree has been constructed. Analyze the decision tree and determine which initial alternative (build small or build large) should be chosen in order to maximize expected monetary value. o ./;’ $40 {’O~ N’l0e ..0 ,,~~ $40 ~0’V. )0 Overtime $50 E:’JrIJ Cillo $55 ,,~~ ($10) ~0’V. )0 o ./;’ {’O~ Reo. N’l0e ..0 liCe IJric $50 es High oerr, Cillo (.6) $70 S,.. “110 -: ge The dollar amounts at the branch ends indicate the estimated payoffs if the sequence of chance events and decisions that is traced back to the initial decision occurs. For example, if the initial decision is to build a small facility and it turns out that demand is low, the payoff will be $40 (thousand). Similarly, if a small facility is built, demand turns out high, and a later decision is made to expand, the payoff will be $55 (thousand). The figures in parentheses on branches leaving the chance nodes indicate the probabilities of those states of nature. Hence, the probability of low demand is .4, and the probability of high demand is .6. Payoffs in parentheses indicate losses. Analyze the decisions from right to left: 1. Determine which alternative would be selected for each possible second decision. For a small facility with high demand, there are three choices: do nothing, work overtime, and expand. Because expand has the highest payoff, you would choose it. Indicate this by placing a double slash through each of the other alternatives. Similarly, for a large facility with low demand, there are two choices: do nothing and reduce prices’Xcsa would choose reduce prices because it has the higher expected value, so a double slash is placed on the other branch. Supplement to Chapter Five Decision Theory 223 Determine the product of the chance probabilities and their respective payoffs for the remaining branches: Build small Low demand .4($40) = $16 High demand .6($55) = $33 Build large Low demand .4($50) = 20 High demand .6($70) = $42 Determine the expected value of each initial alternative: Build small $16 + $33 = $49 Build large $20 + $42 = .62 Hence, the choice should be to build the large facility because it has a larger expected value than the small facility . .8 EXPECTED VALUE OF PERFECT INFORMATION certain situations, it is possible to ascertain which state of nature will actually occur in the e. For instance, the choice of location for a restaurant may weigh heavily on whether a – . highway will be constructed or whether a zoning permit will be issued. A decision maker have probabilities for these states of nature; however, it may be possible to delay a deciuntil it is clear which state of nature will occur. This might involve taking an option to _.; the land. If the state of nature is favorable, the option can be exercised; if it is unfavorable, option can be allowed to expire. The question to consider is whether the cost of the option ill be less than the expected gain due to delaying the decision (i.e., the expected payoff ve the expected value). The expected gain is the expected value of perfect information, EVPI. Other possible ways of obtaining perfect information depend somewhat on the nature of – decision being made. Information about consumer preferences might come from market ch, additional information about a product could come from product testing, or legal rts might be called on. There are two ways to determine the EVPL One is to compute the expected payoff under enainty and subtract the expected payoff under risk. That is, Expected value of Expected payoff Expected payoff (5S-1) perfect information under certainty under risk “‘:-singthe information from Example 5S-4, determine the expected value of perfect informan using Formula 5S-1. -=- t, compute the expected payoff under certainty. To do this, identify the best payoff under h state of nature. Then combine these by weighting each payoff by the probability of that e of nature and adding the amounts. Thus, the best payoff under low demand is $10, the t under moderate demand is 12, and the best under high demand is $16. The expected yoff under certainty is, then, .30($10) + .50($12) + .20($16) = $12.2 L05S.6 Compute the expected value of perfect information. Expected value of perfect information (EVPI) The difference between the expected payoff with perfect information and the expected payoff under risk. EXAMPLE 5S-6 SOLUTION # 224 I~ EXAMPLE 5S-7 eXcel mhhe.com/stevenson12e SOLUTION Ul5S.7 Conduct sensitivity analysis on a simple decision problem. Sensitivity analysis Determining the range of probability for which an alternative has the best expected payoff. EXAMPLE 5S-8 eXcel mhhe.com/stevenson12e SOLUTION Supplement to Chapter Five Decision Theory The expected payoff under risk, as computed in Example 5S–4, is $10.5. The EVPI is the difference between these: EVPI = $12.2 – $10.5 = $1.7 This figure indicates the upper limit on the amount the decision maker should be willing to spend to obtain perfect information in this case. Thus, if the cost equals or exceeds this amount, the decision maker would be better off not spending additional money and simply going with the alternative that has the highest expected payoff. A second approach is to use the regret table to compute the EVPI. To do this, find the expected regret for each alternative. The minimum expected regret is equal to the EVPI. Determine the expected value of perfect information for the capacity-planning problem using the expected regret approach. Using information from Examples 5S-2, 5S-3, and 5S–4, we can compute the expected regret for each alternative. Thus: ‘ Small facility Medium facility Large facility + .20(6) = 2.2 + .20(4) = 1.7 [minimum] + .20(0) = 9.2 .30(0) .30(3) .30(14) + .50(2) + .50(0) + .50(10) The lowest expected regret is l.7, which is associated with the second alternative. Hence, the EVPI is $l.7 million, which agrees with the previous example using the other approach. 5S.9 SENSITIVITY ANALYSIS Generally speaking, both the payoffs and the probabilities in this kind of a decision problem are estimated values. Consequently, it can be useful for the decision maker to have some indication of how sensitive the choice of an alternative is to changes in one or more of these values. Unfortunately, it is impossible to consider all possible combinations of every variable in a typical problem. Nevertheless, there are certain things a decision maker can do to judge the sensitivity of probability estimates. Sensitivity analysis provides a range of probability over which the choice of alternatives would remain the same. The approach illustrated here is useful when there are two states of nature. It involves constructing a graph and then using algebra to determine a range of probabilities for which a given solution is best. In effect, the graph provides a visual indication of the range of probability over which the various alternatives are optimal, and the algebra provides exact values of the endpoints of the ranges. Example 5S-8 illustrates the procedure . •.•. Given the following table, determine the range of probability for state of nature #2, that is, P(2), for which each alternative is optimal under the expected-value approach. STATE OF NATURE #1 #2 A 4 12 Alternative B 16 2 C 12 8 First, plot each alternative relative to P(2). To do this, plot the #1 value on the left side of the graph and the #2 value on the right side. For instance, for alternative A, plot 4 on the left side of the graph and 12 on the right side. Then connect these two points with a straight line. The three alternatives are plotted on the graph as shown on the next page. Supplement to Chapter Five Decision Theory 225 16 16 14 12 10 #2 8 Payoff 6 4 2 .8 1.0 14 12 10 #1 Payoff 8 6 I I 4 I I 2 I –B best–~C best-i~– o .2 .4 .6 P(2) The graph shows the range of values of P(2) over which each alternative is optimal. Thus, low values of P(2) [and thus high values of P(l), since P(l) + P(2) = 1.0], alternative B have the highest expected value; for intermediate values of P(2), alternative C is best; and higher values of P(2), alternative A is best. To find exact values of the ranges, determine where the upper parts of the lines intersect. e that at the intersections, the two alternatives represented by the lines would be equivalent terms of expected value. Hence, the decision maker would be indifferent between the two :nat point. To determine the intersections, you must obtain the equation of each line. This relatively simple to do. Because these are straight lines, they have the form y = a + bx, – e a is the y-intercept value at the left axis, b is the slope of the line, and x is P(2). Slope fined as the change in y for a one-unit change in x. In this type of problem, the distance ‘een the two vertical axes is 1.0. Consequently, the slope of each line is equal to the rightvalue minus the left-hand value. The slopes and equations are as follows: #1 #2 Slope Equation 4 12 12 – 4 = + 8 4 + 81’1:2) 16 2 2 -16 = -14 16 – 141’1:2) 12 8 8-12=- 4 12-41’1:2) From the graph, we can see that alternative B is best from P(2) 5 0 to the point where that ight line intersects the straight line of alternative C, and that begins the region where C is r. To find that point, solve for the value of P(2) at their intersection. This requires setting two equations equal to each other and solving for P(2). Thus, 16 – 14P(2) = 12 – 4P(2) Rearranging terms yields 4 = lOP(2) olving yields P(2) = AO. Thus, alternative B is best from P(2) = 0 up to P(2) = AO. B C are equivalent at P(2) = AO. Alternative C is best from that point until its line intersects alternative A’s line. To find that ection, set those two equations equal and solve for P(2). Thus, + 8P(2) = 12 – 4P(2) Rearranging terms results in 12P(2) = 8 olving yields P(2) = .67. Thus, alternative C is best from P(2) > AO up to P(2) = .67, re A and C are equivalent. For values of P(2) greater than .67 up to P(2) = 1.0, A is best. . ote: If a problem calls for ranges with respect to P(l), find the P(2) ranges as above, and subtract each P(2) from 1.00 (e.g., AO becomes .60, and .67 becomes .33). 226 SUMMARY .. ” KEY TERMS SOLVED PROBLEMS Problem 1 eXcel mhhe.com/stevenson12e Solution Problem 2 eXcel mhhe.com/stevenson12e Solution Supplement to Chapter Five Decision Theory Decision making is an integral part of operations management. Decision theory is a general approach to decision making that is useful in many different aspects of operations management. Decision theory provides a framework for the analysis of decisions. It includes a number of techniques that can be classified according to the degree of uncertainty associated with a particular decision problem. Two visual tools useful for analyzing some decision problems are decision trees and graphical sensitivity analysis. bounded rationality, 218 Laplace, 219 risk, 218 certainty, 218 maximax, 219 sensitivity analysis, 224 decision tree, 221 maximin, 219 suboptimization, 218 expected monetary value minimax regret, 219 uncertainty, 218 (EMV) criterion, 220 payoff table, 217 expected value of perfect regret {opportunity information (EVPI), 223 loss), 220 The following solved problems refer to this payoff table: New No Bridge New Built Bridge Alternative capacity A 1 14 for new store B 2 10 C 4 6 where A = small, B = medium, and C = large. Assume the payoffs represent profits. Determine the alternative that would be chosen under each of these decision criteria: a. Maximin. b. Maximax. c. Laplace. New No New Maximin Maximax Laplace Bridge Bridge (worst) (best) (average) A 1 14 1 14 [best] 15 7 2 = 7.5 [best] B 2 10 2 10 1272=6 C 4 6 4 [best] 6 1072=5 Thus, the alternatives chosen would be C under maximin, A under maximax, and A under Laplace. Using graphical sensitivity analysis, determine the probability for no new bridge for which each alternative would be optimal. Plot a straight line for each alternative. Do this by plotting the payoff for new bridge on the left axis and the payoff for no new bridge on the right axis and then connecting the two points. Each line represents the expected profit for an alternative for the entire range of probability of no new bridge. Because the lines represent expected profit, the line that is highest for a given value of P (no new bridge) is optimal. Thus, from the graph, you can see that for low values of this probability, alternative C is best, and for higher values, alternative A is best (B is never the highest line, so it is never optimal). Supplement to Chapter Five Decision Theory 227 Payoff for new bridge 14 Payoff for no new bridge 10 6 4 2 1 o .27 P(no new bridge) The dividing line between the ranges where C and A are optimal occurs where the two lines inter- .e::t. To find that probability, first formulate the equation for each line. To do this, let the intersection ith the left axis be the y intercept; the slope equals the right-side payoff minus the left-side payoff. Thus, for C you have 4 + (6 – 4)P, which is 4 + 2P For A, I + (14 – I)P, which is I + 13P Setting these two equal to each other, you can solve for P: 4 + 2P = 1 + 13P Solving, P = .27. Therefore, the ranges for P(no new bridge) are A: .27 < P :s 1.00 B: never optimal C: o s P < .27 Using the information in the payoff table, develop a table of regrets, and then Problem 3 a. Determine the alternative that would be chosen under minimax regret. b. Determine the expected value of perfect information using the regret table, assuming that the eXcel probability of a new bridge being built is .60. mhhe.com/stevenson12e To obtain the regrets, subtract all payoffs in each column from the best payoff in the column. The Solution regrets are New No New Bridge Bridge A B C 3 2 o o 4 8 a. Minimax regret involves finding the worst regret for each alternative and then choosing native that has the “best” worst. Thus, you would choose A: the alterA B C Worst 3 [best] 4 8 iI’ …•. 228 Supplement to Chapter Five Decision Theory b. Once the regret table has been developed, you can compute the EVPI as the smallest expected regret. Since the probability of a new bridge is given as .60, we can deduce that the probability of no new bridge is l.00 – .60 = .40. The expected regrets are A: .60(3) + .40(0) = l.80 B: .60(2) + .40(4) = 2.80 C: .60(0) + .40(8) = 3.20 Hence, the EVPI is l.80. mhhe.com/stevenson12e Using the probabilities of .60 for a new bridge and .40 for no new bridge, a. Compute the expected value of each alternative in the payoff table, and identify the alternative that would be selected under the expected-value approach. b. Construct a decision tree for the problem showing expected values. Problem 4 eXcel Solution a. A: .60(1) + .40(14) = 6.20 [best] B: .60(2) + .40(10) = 5.20 C: .60(4) + .40(‘6) = 4.80 b. Expected values A ‘O’i::J $1 e” ‘Q~llj, ~e-l’l No /)ew 6 . rid ge (.40) $14 e .’O)1 $2 ‘Q~og ~e-l’l !to /)ew 6riq, {Jer. .40) $10 ~o) $4 ‘0ge . ‘>J’I ‘o~ ~e !to /)ew 6riq, »: ,40) $6 .60(1) = .60 $6.20 (best) .40( 14) = 5.60 .60(2) = 1.20 $5.20 B c .40(10) = 4.00 .60(4) = 2.40 $4.80 .40(6) = 2.40 Problem 5 Compute the EVPI using the information from the previous problem. Solution Using Formula 5S-1, the EVPI is the expected payoff under certainty minus the maximum expected value. The expected payoff under certainty involves multiplying the best payoff i~ each column by Supplement to Chapter Five Decision Theory the c?lumn probability and then summing those amounts. The best payoff in the first column is 4, and the b’est in the second is 14. Thus, Expected payoff under certainty .60(4) + .40(14) 8.00 Then EVPI = 8.00 – 6.20 = 1.80 (This agrees with the result obtained in Solved Problem 3b.) Excel solution: Home Inust ~a9f ll)’OUl. Formt.Wis Dau RfriN’ …w. _ .._——- ·10 . 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