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Decision-Making Analysis1 THE COMPONENTS OF A DECISION PROBLEM .................................................. 1 Decision Making under Uncertainty ............................................................................... 3 Maximax Procedure ........................................................................................................ 3 Maximin procedure ......................................................................................................... 4 Equal Likelihood Procedure ........................................................................................... 5 Minimax Regret Procedure ............................................................................................. 5 Decision Making under Risk ............................................................................................ 6 Maximizing Expected Gain ............................................................................................ 6 Minimizing Expected Regret .......................................................................................... 7 The Expected Value of Perfect Information ................................................................... 8 Decision Trees: An Alternative to Payoff Tables ......................................................... 10 Nodes and Branches ...................................................................................................... 10 Paths and Payoffs .......................................................................................................... 11 Analyzing a Decision Tree .............................................................................................. 11 Multistage Decision Trees .............................................................................................. 13 Other Approaches to Analyzing a Decision Tree ......................................................... 14 Risk analysis ................................................................................................................. 16 Extreme Case Analysis ................................................................................................. 16 UTILITY: AN ALTERNATIVE TO DOLLAR VALUE ........................................... 17 SUMMARY ..................................................................................................................... 17 DISCUSSION QUESTIONS .......................................................................................... 18 SOLVED PROBLEMS ................................................................................................... 18 PROBLEMS .................................................................................................................... 24 CASE: TRANSFORMER REPLACEMENT AT MOUNTAIN STATES ELECTRIC SERVICE ................................................................................................... 31 KEY TERMS ................................................................................................................... 32 The Components of a Decision Problem The three basic components of a decision problem are the decision alternatives; the possible environments, conditions, or states of nature within which a decision is to be implemented; and the 1 Excerpted from Integrated Operations Management, Mark Hanna and W. Rocky Newman, Prentice Hall, 2003. Supplementary lecture notes for educational purposes only. outcomes or payoffs that will result from each possible combination of alternatives and states of nature. The decision alternatives are those actions from among which the decision maker must choose. That is, they are the aspects of the decision situation over which the decision maker has direct control. The states of nature are those aspects of the decision situation over which the decision maker does not have direct control. They certainly include things over which there is no control, such as the weather, the general state of the economy, or actions taken by the national government. They may also include things which can be influenced to some extent but not completely. The market response to a new product (which is partly determined by the price charged and advertising) or a competitor's actions (which may be at least partly determined by choices made by the decision maker) are conditions the decision maker cannot determine completely. Making a decision and implementing it within a state of nature will result in some outcome, which may be either a gain or a loss. The payoff function specifies what that outcome is for every possible alternative/state-of-nature combination. The payoff function may be given either as an equation or, when the possible decision alternatives and states of nature are relatively limited, as a payoff table or matrix. For example, Table B.1 shows a payoff table for a proposed cardiac catheterization service at Cheryl's hospital. Example B.1 Cheryl has been approached by the head of the hospital's cardiac care unit about the possibility of starting a cardiac catheterization service at the hospital. Doing so would require either the conversion of existing space in the hospital (which would allow the development of a small unit) or the construction of an addition (which would make a larger unit possible). Additional equipment would also have to be purchased. Before taking this idea to the capital expenditure review committee for the chain that owns the hospital, Cheryl gets information on the costs of the two possibilities from the hospital architect and asks the hospital's marketing department for a preliminary assessment of the possible levels of demand for this type of service. Putting this information together, Cheryl creates the payoff table in Table B.1. The hospital chain uses a five- year planning horizon for major capital expenditures. The five-year returns shown in the table are stated in millions of dollars of net present value over that time horizon. Table B.1 Payoff Table for Cardiac Catheterization Service Demand Level Alternative Low Medium High Build addition –3 1 7 Convert space –1 2 5 Do nothing 0 0 0 There are three basic types of decision problems, each with its own mode of analysis: 1. Decision making under certainty: In this type of problem, the state of nature is known, so the payoff table has only one column. In principle, the analysis is simple: Choose the alternative with the best payoff. In practice, it is not always that easy, since an "alternative" may actually represent the values for many decision variables. However, solution procedures have been developed for many problems of this type. For example, Supplement C presents an introduction to linear programming, a popular technique for decision making under certainty. 2. Decision making under uncertainty: In this type of problem, decision makers have the payoff table, but no information on the relative likelihood of the states of nature. A number of approaches to this type of problem are possible, several of which will be discussed in this supplement. 3. Decision making under risk: In this type of problem, in addition to the payoff table, decision makers have the probabilities for the different states of nature. The standard approach is expected value analysis, which will be discussed in this supplement. DECISION MAKING UNDER UNCERTAINTY As noted above, in decision making under uncertainty we have a payoff table but no information about the relative likelihood of the different states of nature. A number of procedures have been developed for choosing an alternative under these conditions, each based on a different philosophy about what constitutes a good choice. We shall consider four: maximax, maximin, equal likelihood, and minimax regret. Maximax Procedure The maximax procedure (called the minimin procedure if the payoffs are costs or losses rather than profits or gains) is an optimistic approach. It assumes that no matter what alternative we choose, "nature" will smile favorably on us and choose the state of nature that will benefit us the most (or hurt us the least). Thus, we choose the alternative with the largest maximum gain (or the smallest minimum loss). Example B.2 Find the maximax solution to Cheryl's cardiac catheterization service problem described in Example B.1. Solution: From Exhibit B.1, the maximum payoffs for the decision alternatives are: Build addition: 7 Convert space: 5 Do nothing: 0 The largest maximum payoff is 7, so the "maximax" decision is to build an addition. Exhibit B.1 Maximin procedure The maximin procedure (called the minimax procedure if the payoffs are costs or losses) is a pessimistic approach. It assumes that, no matter what alternative we choose, "nature" will choose the state of nature that benefits us the least (or hurts us the most). Thus, we choose the alternative with the largest minimum gain (or the smallest maximum loss). In fact, if "nature" is a competitor who gets to choose his alternative after we have chosen ours, this could be a very good strategy. (This gets into an area called game theory, which we will not discuss.) Example B.3 Find the maximin solution to Cheryl's cardiac catheterization service problem described in Example B.1. Solution: From Exhibit B.1, the minimum payoffs for the alternatives are: Build addition: 3 Convert space: 1 Do nothing: 0 The largest minimum payoff is 0, so the "maximin" decision is to do nothing. Equal Likelihood Procedure One criticism of the maximax and maximin strategies is that they focus on only one payoff for each alternative, ignoring all other states of nature and their payoffs. The equal likelihood procedure is based on the assumption that if we cannot determine the relative likelihood of the states of nature, then it is rational to presume that they are equally likely. If this is the case, then we should choose the alternative with the largest average payoff (or the smallest average payoff for losses). Example B.4 Find the equal likelihood solution to Cheryl's cardiac catheterization service problem described in Example B.1. Solution: From Exhibit B.1, the averages of the rows of the payoff table are: Build addition: 1.667 Convert space: 2.000 Do nothing: 0 The largest average payoff is 2.0, so the "equal likelihood" decision is to convert space. Minimax Regret Procedure We have all had the experience of making a decision and then, after observing the result, wishing that we had made another choice instead, so that our payoff would have been better (or less bad). That is the concept behind the minimax regret procedure: We choose the alternative that will yield the smallest possible maximum regret. The procedure has two steps: 1. Construct a regret table by subtracting each payoff from the maximum payoff in its column. (For a loss table, subtract the smallest loss in each column from each entry in its column.) 2. Apply the minimax procedure to the regret table: Find the maximum regret in each row; then choose the alternative with the smallest maximum regret. Example B.5 Find the minimax regret solution to Cheryl's cardiac catheterization service problem described in Example B.1. Solution: Step 1: Construct the regret table from the payoff table in Table B.1. As shown in Exhibit B.2, this is done in two steps. First, as shown in Exhibit B.2(a), find the maximum entry in each column of the payoff table. Second, as shown in Exhibit B.2(b), each payoff table entry is subtracted from the maximum in its column to get the regret table. Step 2: Find the maximum regret for each alternative, as shown to the right of the regret table in Figure B.2(b). The alternative with the smallest maximum regret is to convert space, with a maximum regret of 2. Exhibit B.2 DECISION MAKING UNDER RISK There are two basic approaches to decision making under risk: (1) maximizing expected gain or minimizing expected loss, and (2) minimizing expected regret. Maximizing Expected Gain Under some reasonable assumptions about the characteristics of a rational decision maker, it can be shown that when probabilities for the states of nature are known, the alternative that maximizes the expected gain (or minimizes the expected loss) is the appropriate decision. To determine the expected value of a decision alternative, multiply that alternative's payoff under each different state of nature by that state of nature's probability and sum the resulting products. Minimizing Expected Regret As an alternative to using the payoff table to maximize the expected gain (or minimize the expected loss), we can choose the alternative that minimizes the expected regret. As shown in Examples B.6 and B.7, the two approaches lead to the same decision. Example B.6 (Refer to Example B.1.) Cheryl has gone back to the hospital's marketing department to get some additional information about the relative likelihood of the different levels of demand for the proposed cardiac catheterization service. After review of their marketing research, the department gives Cheryl the following probability estimates: P(low) = .2, P(medium) = .7, P(high) = .1. Using these probabilities, determine which alternative maximizes the expected value return to the hospital. Solution: The projected returns are given by the rows of Table B.1. The expected returns for the three alternatives are: Build addition: .2(–3) + .7(1) + .1(7) = –.6 + .7 + .7 = .8 Convert space: .2(–1) + .7(2) + .1(5) = –.2 + 1.4 + .5 = 1.7 Do nothing: .2(0) + .7(0) + .1(0) = 0 The alternative with the highest expected payoff is to convert space. Exhibit B.3 shows the Excel computation of the expected values. Exhibit B.3 Example B.7 (Refer to Example B.1.) The regret matrix for the cardiac catheterization lab decision problem was developed in Example B.5. Determine which decision alternative minimizes the hospital management's expected regret. Solution: Using the regret matrix from Exhibit B.2(b) and the probabilities given in Example B.6, the expected regrets for the three decision alternatives are: Build addition: .2(–3) + .7(1) + .1(0) = .6 + .7 + 0 = 1.3 Convert space: .2(–1) + .7(0) + .1(2) = –.2 + 0 + .2 = .4 Do nothing: .2(0) + .7(2) + .1(7) = 0 + 1.4 + .7 = 2.1 The alternative with the lowest expected regret is to convert space. Comparing the expected regrets found in Example B.7 with the expected payoffs in Example B.6, we see that, for each pair of alternatives, the difference between the values of the expected gains and the difference between the values of the expected regrets are identical. For example, the difference between the expected gains for "build addition" and "convert space" is 1.7 – .8 = .9. The difference between the expected regrets for the same pair is 1.3 – .4 = .9. This finding is not specific to this particular example, but is a general result. The Expected Value of Perfect Information If, before having to make a final decision, we could find out exactly what the state of nature was going to be, we could improve the quality of the decision. The expected value of perfect information (EVPI) is a measure of the expected current worth to the decision maker of being able to find out, just before having to make the decision, what the state of nature will be. It combines the current assessments of the probabilities of the states of nature with the improved value of knowing the state of nature. Thus, the EVPI is computed as the difference between the expected value of making the decision with perfect information (EWPI), given by the weighted combination of the best payoffs for each state of nature, and the expected value of the best decision with current information. Example B.8 (Refer to Example B.1.) Assuming it would be possible to find out the demand level for the cardiac catheterization service before making a final decision about whether to introduce the service and, if so, how large a unit to build, determine the current expected value of having this perfect information. Solution: The best decisions for each state of nature (demand level) and their values, as given in Exhibit B.1, are: Demand Best Decision Value Low Do nothing 0 Medium Convert space 2 High Build addition 7 Using the probabilities of the demand levels given in Example B.6, the expected value with perfect information (EWPI) is: EWPI = .2(0) +.7(2) + .1(7) = 0 + 1.4 + .7 = 2.1 The best decision without having perfect information, as found in Example B.6, is to convert space, which has an expected value of 1.7. Thus, the expected value of perfect information is: EVPI = EWPI – E(best decision) = 2.1 – 1.7 = .4 That is, the hospital could increase its expected net present value by $.4 million if it were able to find out what the demand for this new service would be before committing to whether and how to provide it. Note that the expected value of perfect information in Example B.8 is exactly the same as the expected regret from Example B.7. Given that EVPI and expected regret are both computed by comparing, for each state of nature, the value of a specific decision with the value of the best decision for that state of nature, this result is not an accident. It will always be the case. Since it is never possible (at least legally) to get perfect information before having to make a decision, the expected value of perfect information is useful mainly as an upper bound on the expected value of sample or imperfect information (such as market surveys or economic forecasts), which is often available before a final decision must be made. DECISION TREES: AN ALTERNATIVE TO PAYOFF TABLES An alternative to using a payoff to compute the expected values of decision alternatives is to use a decision tree. As shown in Figure B.1, a decision tree consists of two or more stages or levels, shown in time order from left to right. Figure B.1 Decision Tree for Cardiac Catheterization Service Proposal Nodes and Branches Each level of a decision tree consists of nodes (the squares and circles) and branches, which represent alternatives. The square nodes represent decision points; each branch from a decision node represents one of the decision alternatives available at that point. The circles, or chance nodes, represent problem features that are determined by chance or probability, such as the states of nature. Figure B.1 is actually the decision tree for Cheryl's problem of whether to propose starting a cardiac catheterization service and, if so, how large a unit to build. The nodes have been numbered to facilitate the description of the tree. Node 1 is the decision node. There are three alternatives, represented by the three branches: (1) build an addition, (2) convert existing space, and (3) do nothing. Node 2 is a chance node representing the possible states of nature that might follow a decision to build an addition. There are three possibilities: (1) low demand, which has a probability of .2, (2) medium demand, which has a probability of .7, and (3) high demand, which has a probability of .1. Notice that each state of nature's probability has been written next to its branch. Node 3 is another chance node, representing the possible states of nature that might follow a decision to convert existing space. The branching is identical to the one for node 2. Node 4 is the chance node that represents what might happen after a decision to do nothing. While it could have branches identical to those for nodes 2 and 3, it is simpler to represent this way. That is, with probability 1, nothing is going to happen since no action is being taken. Paths and Payoffs A connected series of branches that starts at the extreme left side of the tree (node 1) and goes through all levels of the tree is called a path. Each path represents one of the possible sequences of decisions and chance results for the problem represented by the tree. For example, "build addition" from node 1 and "medium demand" from node 2 is one possible path or decision alternative and demand level sequence. At the extreme right edge of the tree is the net payoff for each of these possible paths, as given originally in the payoff table in Table B.1. For example, building an addition and experiencing medium demand for the service will result in a net present value of $1 million. ANALYZING A DECISION TREE The standard approach to analyzing a decision tree is to choose on the basis of expected value. The procedure for doing this is called averaging out and folding back. Averaging out means replacing each branching by a single number. For a chance branching, the number used is the expected value, which is found by multiplying the probability of each branch by the value at its right end and summing. For a decision branching, the number used is the value of the best alternative in the branching. Folding back means that this process starts at the right-hand edge of the tree and proceeds back to the start of the tree, working from right to left. Example B.9 Use the process of averaging out and folding back to analyze the decision tree shown in Figure B.1. Determine which decision alternative maximizes the expected return for the cardiac catheterization decision problem described in Example B.1. Solution: The first round of averaging out consists of replacing each chance branching in the right-most level of the tree by its expected value: The chance branching from node 2 is replaced by its expected value: .2(–3) + .7(1) + .1(7) = –.6 + .7 + .7 = .8 The chance branching from node 3 is replaced by its expected value: .2(–1) + .7(2) + .1(5) = –.2 + 1.4 + .5 = 1.7 The chance branching from node 4 is replaced by its expected value, which is 0. The result of this first round of averaging out is the reduced tree shown in Figure B.2(a), in which each second-level chance branching has been replaced by its expected value. Figure B.2 Analysis of Cardiac Catheterization Service Proposal Decision Tree We now back up one level in the tree (folding back). In the second round of "averaging out," node 1, which is in the new right-most level of the tree of the reduced tree shown in Figure B.2(a), is replaced by the value of the best alternative, which is 1.7 for "convert space." To show this, the value of the alternative chosen has been written above the decision node and the branches not chosen have been marked out with slashes as shown in Figure B.2(b). Since the tree now has only one decision node, we can readily see that the decision strategy that maximizes the expected return to the hospital over the planning horizon is: Convert space for an expected net present value of $1.7 million. In practice, we would consolidate the entire analysis process into a single decision tree rather than redrawing successively smaller trees after each round of averaging out and folding back. To do this, the result of each averaging out is written above the node to which it applies, with, as suggested in Example B.9, the decision alternatives not chosen being marked out with slashes. The result of applying this summarization process to the analysis in Example B.9 is shown in Figure B.3, with the averaging out results being shown in red. Figure B.3 Analyzed Decision Tree for Cardiac Catheterization Service Proposal MULTISTAGE DECISION TREES While we can certainly use a decision tree to analyze a one-decision problem, as just illustrated for Cheryl's cardiac catheterization service proposal, we really will not gain anything we could not get from a payoff table analysis. The real benefit of using a decision tree comes in more complicated problems with multiple levels of decisions or states of nature, particularly if the decision alternatives, the states of nature, or their probabilities depend on what precedes them in the tree. While we could still use payoff tables to analyze these types of problems, structuring the tables would be difficult. A tree shows the structure and relationships in such a problem much better than a table does. Example B.10 Fred has just returned from a weeklong trip to China, where he was part of a team sent by his company to explore a joint venture with a Chinese company to manufacture and market consumer electronics in China and Southeast Asia. Part of the decision about whether and how to enter into the arrangement is the decision about the size of the facility to build. The two alternatives discussed were (1) to build a large plant initially or (2) to build a small plant and then, if warranted, expand later or, if business is not good enough, sell out to the Chinese partner. The possible decisions—along with preliminary estimates of the sales levels, their probabilities, and the resulting net present values of the different combinations—are shown in the decision tree in Figure B.4. Notice two particular features of this tree: It is possible to have two or more successive chance nodes, as in the "high" result of node 2 (sales level during the first two years) being followed by node 4 rather than a decision. (Technically, there is a decision to "stay" or "sell out," but the decision is obvious and is not included in the analysis.) The probabilities of the states of nature for nodes 4, 8, 9, 10, and 11, all of which deal with the sales level during years 3–5, depend on the sales level during the first two years. That is, the later years' sales probabilities are conditional on the earlier years' results. Determine what Fred's strategy should be. That is, determine whether they should initially build a large or small plant and, subsequently, what to do after the end of the first two years. Solution: The averaging out and folding back analysis of this decision is shown in red on the tree in Figure B.4. The best strategy is to build a large plant (node 1 choice) and, whether the first two years' sales are high or low (node 4), to stay with the project. The overall expected net present value from following this strategy is $20.0 million. Figure B.4 Decision Tree for Chinese Joint Venture Proposal OTHER APPROACHES TO ANALYZING A DECISION TREE While expected value analysis is the standard approach to analyzing a decision tree, it is not the only method possible. Two alternative approaches are risk analysis, which recognizes the various possible outcomes and their probabilities, and extreme case analysis, which, like the maximax and maximin strategies, takes either an optimistic or pessimistic view rather than working with probabilities. Explanations of both approaches follow. Example B.11 Refer back to the decision tree in Figure B.1. Construct cumulative probability distributions for the payoffs for the three decision alternatives. Solution: "Build addition" has possible payoffs of –3, with probability .2, 1 with probability .7, and 7, with probability 1. The cumulative probability distribution for "build addition" is, therefore: Cumulative Value Probability –3 .2 1 .9 7 .1 Similarly, the cumulative probability distributions for the alternatives "convert space" and "do nothing" are: Convert space Do nothing Cumulative Cumulative Value Value Probability Probability –1 .2 0 1.0 2 .9 5 1.0 Figure B.5 shows graphs of these three cumulative probability distributions. The decision maker can choose among the three alternatives by comparing the cumulative distributions on whatever basis he or she prefers. Figure B.5 Risk Analysis Distributions Risk analysis This approach to analyzing a decision tree recognizes each possible decision combination and the set of possible payoffs and their probabilities. These are then converted into a cumulative probability distribution and graphed for comparison purposes. Extreme Case Analysis This approach combines the averaging out and folding back approach described for expected value analysis with the procedures used earlier for decision making under certainty. The folding back part is the same, but in averaging out, rather than replacing a chance branching by its expected value, use the highest branch value (for the "best case" analysis) or use the lowest branch value (for the "worst case" analysis). Example B.12 Refer back to Figure B.1. Perform a "best case" analysis on the cardiac catheterization service problem. (The "worst case" analysis will be left for the problems.) Solution: The results of averaging out and folding back for a best case analysis are shown in red in the tree diagram in Figure B.6. The probability values from Figure B.1 have been eliminated since they are not relevant for a best case analysis. In the first round of averaging out: The branching from node 2 is replaced by 7, the value for high demand. The branching from node 3 is replaced by 5, the value for high demand. The branching from node 4 is replaced by 0, which is the only possible payoff. The second round of averaging out uses the results from the first round. Since node 1 is a decision node, its branching is replaced with the value of the alternative chosen. Given that we are following a maximax or "best case" approach, that alternative is "build addition," with value 7. Decision Tree for Cardiac Catheterization Proposal, Using "Best Case" Figure B.6 Analysis Utility: An Alternative to Dollar Value In some decisions, money is not an appropriate measure of the quality of the various outcomes. For example, how could you put a dollar value on losing your health or winning the top international award in your field? In other cases, the dollar value of an outcome may have implications beyond its monetary significance. A big loss, for example, may result in the bankruptcy of the organization, which is more significant than the dollar value of the loss. When dollar value alone is not an adequate measure of the value of an outcome, an alternative measure, utility, may be used. How to determine values for a utility function for a given situation is outside the scope of this supplement. However, a fairly complete discussion may be found in any text on decision theory, such as Raiffa, Decision Analysis: Introductory Lectures on Choices Under Uncertainty (Addison-Wesley, 1970). Making decisions based on utility, however, is no different from making decisions based on money, as described in this supplement. Summary All decision problems have three basic components: decision alternatives, implementation environments or states of nature, and a payoff function that gives the gain or loss for each alternative/state-of-nature combination. In decision making under uncertainty, we assume that these three components are all that are known. A number of procedures based on different philosophies of what constitutes a good approach— maximax, maximin, equal likelihood, and minimax regret—were presented and illustrated. Maximization of expected value or minimization of expected cost is the standard procedure for decision making under risk, in which probabilities for the states of nature are known in addition to the three basic components described. An alternative to using payoff tables for decision making under risk is the decision tree, in which branchings represent either the decision alternatives or the states of nature relevant at a given point in the process. Decision trees are more flexible than payoff tables and can be more easily used to represent multistage decision problems and situations in which the probabilities of the states of nature depend on what has happened up to that point in the tree. The standard analysis approach for a decision tree with probabilities is averaging out and folding back, in which, starting at the right-most side of the tree, each branching is replaced by a single number—the expected value of a chance branching or the value of the alternative chosen in a decision branching. Two other possible approaches to analyzing a decision tree were also presented: risk analysis and best case or worst case analysis. Discussion Questions 1. Identify the basic components of a decision-making problem. 2. Describe the differences among decision making under certainty, uncertainty, and risk. 3. Identify four different procedures for decision making under uncertainty and what the basic concept or philosophy is for each. 4. Describe the basic idea of the expected value of perfect information. 5. Why might using a decision tree be preferable to using a decision table? 6. Describe the process of averaging out and folding back. Solved Problems 1. A high school band's parents organization operates a Christmas tree lot every year to raise funds. Trees are bought in batches of 100 for $1,000 and sold for $20 each. Leftover trees are taken away by a landscaping company (at no cost or revenue to the band) to be shredded for mulch. Based on past experience, the organization estimates that they can sell either 3, 4, 5, or 6 batches of trees. a. Find the maximax, maximin, equally likely, and minimax regret decisions based on profits. b. Based on prior years' experience, the organization estimates that there is a 10% chance of selling 3 batches, a 30% chance of selling 4 batches, a 40% chance of selling 5 batches, and a 20% chance of selling 6 batches. Based on these probability estimates, determine the number of batches to buy to maximize expected profits. Answers: a. First, develop a profit payoff table as shown in Table B.2. The table entries are determined as follows: Profit = Revenue – Cost = 2,000(sold) – 1,000(bought), so: If demand < bought, profit = 2,000(demand) – 1,000(bought). If demand > bought, profit = (2,000 – 1,000)(bought). The maximum, minimum, and average profits for the alternatives are shown in Exhibit B.4a. Table B.2 Payoff Table for Solved Problem 1 Demand (batches) Bought (batches) 3 4 5 6 3 3,000 3,000 3,000 3,000 4 2,000 4,000 4,000 4,000 5 1,000 3,000 5,000 5,000 6 0 2,000 4,000 6,000 Maximax: The largest maximum profit is $6,000 from stocking 6 batches. Maximin: The largest minimum profit is $3,000 from stocking 3 batches. Equally likely: The largest average profit is $3,500 from stocking either 4 or 5 batches. Minimax regret: The regret matrix and the maximum regret for each possible stocking level are shown in Exhibit B.4b. The smallest maximum regret is $2,000 from stocking either 4 or 5 batches. Exhibit B.4 c. The expected values of the different possible stocking levels are: Stock 3: .1(3,000) + .3(3,000) + .4(3,000) + .2(3,000) = 3,000 Stock 4: .1(2,000) + .3(4,000) + .4(4,000) + .2(4,000) = 3,800 Stock 5: .1(1,000) + .3(3,000) + .4(5,000) + .2(5,000) = 4,000 Stock 6: .1(0) + .3(2,000) + .4(4,000) + .2(6,000) = 3,400 The stocking level that maximizes expected profits is 5 batches for an expected profit of $4,000. 2. A resort development company has the opportunity to buy all or a portion of the acreage surrounding a lake for development. They must decide the size of the development for which they should buy land and put in roads and utilities. The company's owners are considering either a small or large development. Similarly, they have identified two general levels of market acceptance of their project: low or high. Estimates of the cost of land, road development, and utility installation for the different development sizes and of how many lots would be sold for each market acceptance level result in the profit estimates (in millions of dollars) for each combination of size and acceptance level shown in Table B.3. Table B.3 Payoff Table for Lakeside Development Problem Acceptance Level Size of Development Low High Small 5 7.5 Large –5 25 a. Assuming that the company cannot make probability estimates for the different market acceptance levels, determine the appropriate decisions with the maximax, maximin, equally likely, and minimax regret procedures. b. Using some basic market research, the owners estimate that the probabilities of the different levels of acceptance are P(low) = .6 and P(high) = .4. Using these probabilities, find the alternative that maximizes expected profits and the expected value of perfect information. c. Construct and analyze a decision tree using the expected profit criterion. d. The developers' chief financial officer (CFO) has proposed that the company consider adopting a two-stage development approach. Under this approach, the company will initially buy enough property for a small development and also purchase a three-year option on the rest of the property. At the end of the three years, the company can, if it seems warranted, then buy the rest of the property and increase the size of the development. Due to the cost of the option, all final payoffs in Table B.3 will be reduced by 1 ($1 million), but the opportunity to take advantage of the initial acceptance results to change the size decision may make up for this reduction. The CFO believes that the probabilities of the two market acceptance levels during the first three years will remain as currently estimated: P(low) = .6 and P(high) = .4. However, he also believes that the probabilities of the final levels of market acceptance can be re-estimated on the basis of the experience during the first three years. Specifically, he estimates these probabilities as follows: P(low final acceptance|low initial acceptance) = .8 P(high final acceptance|low initial acceptance) = .2 P(low final acceptance|high initial acceptance) = .3 P(high final acceptance|high initial acceptance) = .7 Develop a decision tree to model the CFO's alternative proposal. Analyze this tree to determine whether this two-stage development strategy is preferable to the decision reached in part c. Solution: a. The maximum, minimum, and average rows are shown in Exhibit B.5(a). From them we can determine that the decisions for the different procedures are: Maximax: The larger row maximum is 25, so the decision is a large development. Maximin: The larger row minimum is 5, so the decision is a small development. Equally likely: The larger row average is 10, which comes from a small development. Exhibit B.5 The regret matrix and its maximum rows are shown in Exhibit B.5(b). The alternative with the smallest maximum regret is a large development, with a maximum regret of 10. b. The expected profits of the alternatives are: Alternative Expected Profit Small .6(5) + .4(7.5) = 6 Large .6(–5)+ .4(25) = 7 c. The alternative with the higher expected profit is a large development. Using the highest payoffs for the different states of nature, the expected value with perfect information (EWPI) is: EWPI = .6(5) + .4(25) = 13 The expected value of perfect information is: EVPI = EWPI – E(Large development) = 13 – 7 = 6 or $6 million. d. The decision tree and the analysis (in red) are shown in Figure B.7. The averaging out and folding back goes as follows: Node 2: The expected profits are .6(5) + .4(7.5) = 6. Node 3: The expected profits are .6(–5) + .4(25) = 7. Moving back to the first level: Node 1: The alternative with the highest expected value is a large development, so mark out (with slashes) the other alternative. The alternative with the highest expected profit is a large development with expected profit = 7 or $7,000,000. Figure B.7 Decision Tree for Basic Resort Development Problem d. The decision tree (with analysis in red) is shown in Figure B.8. The expected payoffs for the two original decision possibilities (small development without an option to expand and large development) are taken from the decision tree in Figure B.7. Based on the analysis shown in Figure B.8, the company should initially build a small development with an option to expand after three years. If the initial level of acceptance is low, the company should not expand. If, however, the initial level of acceptance is high, the company should expand. Following this two-stage development strategy increases the expected profit from $7 million for initially adopting a large development, to $8.7 million. Figure B.8 Decision Tree for Expanded Resort Development Problem Problems B.1. A toy company has developed a new toy for the upcoming Christmas season. Since this toy is considerably different from the ones it has manufactured previously, the company will need to develop a new production facility for it. Three facility sizes—small, medium, and large—are under consideration. Given the nature of the toy market, the company is unsure as to what demand level it will encounter. The preliminary analysis is to be based on the demand being low, average, or high. A small amount of subcontracting will be available, so that if the production facility is undersized it will be possible to meet some of the excess demand. The accompanying table shows the estimated profits, in $1,000s, of the various facility-size–demand-level combinations. Demand Level Facility Size Low Average High Small 750 900 900 Medium 350 1,100 1,300 Large –250 600 2,000 a. Determine the best production facility size using maximax, maximin, equally likely, and minimax regret. b. The company's initial assessment of the probabilities of the different market sizes is: P(low) = .5, P(average) = .3, P(high) = .2. Determine the production facility size that maximizes expected profits and find the expected value of perfect information. B.2. A publisher has received an unsolicited manuscript of a first novel. The decision is whether to offer the author a contract. Based on an initial reading of the manuscript, the publisher estimates the following profits if a contract is offered: If the sales level is high, profits will be $100,000; if moderate, profits will be $20,000; if low, they will lose $30,000. The publisher estimates the probabilities for the sales level to be: P(high) = .1, P(moderate) = .4, P(low) = .5. Determine, based on expected profits, whether the author should be offered a contract or not. Determine the expected value of perfect information. B.3. A plumbing contractor has the opportunity to bid on a contract to do the plumbing work for a new office building. After reviewing the blueprints and specifications, the contractor estimates that the job will cost $300,000. The possible bids the contractor might make and his estimates of the probability of winning the contract at each bid level are: Bid Probability Win $330,000 .90 $350,000 .75 $375,000 .50 $400,000 .25 $425,000 0 What should the contractor bid if he wishes to maximize his expected profits? B.4. A developer is planning a new office complex, which may include some retail space. The possible percentages of retail space that the developer is considering are: none, 20%, or 40%. The desirability of the various percentages of retail space depends on the demand for office space. The estimated yearly profits (in $1,000s) for the different retail percentages and office space demand levels are given in the accompanying table. Office Space Demand Retail Percentage Low Medium High None –100 100 250 20 percent 150 200 200 40 percent 300 150 100 a. Determine what the percentage allocation of retail space should be using the maximax, maximin, equally likely, and minimax regret procedures. b. The developer's assessments of the probabilities of the different office space demand levels are: P(low) = .3, P(moderate) = .4, P(high) = .3. Determine the percentage allocation of retail space that maximizes expected yearly profits. Find the expected value of perfect information. B.5. A recent business school finance graduate has just received an inheritance of $10,000. Trying to decide how she should invest the money, she has identified three possible alternatives: stocks, commodities, and T-bills. The success of any alternative will depend on the performance of the economy over the next year. The accompanying table shows the gain (in $100s) from each investment alternative for each performance level of the economy. Performance of Economy Investment Alternatives Recession Stagnant Growth Stocks –10 0 20 Commodities –50 5 50 T-Bills 7 7 7 a. Determine the best way to invest the money using the maximax, maximin, equally likely, and minimax regret approaches. b. Her estimates of the probabilities of the different possible performance levels of the economy are: P(recession) = .1, P(stagnant) = .6, P(growth) = .3. Determine how to invest the money to maximize her expected gain. Find the expected value of perfect information. B.6. Conduct a "worst case" analysis of the decision tree for the cardiac catheterization service proposal in Figure B.1 and Example B.1. B.7. A hardware store orders snow blowers during the summer for delivery in the fall. Each snow blower costs the store $400 and, if sold prior to or during the winter, sells for $550. The store's manager doesn't want to carry any unsold snow blowers in inventory from one year to the next, so he reduces the price to $350 in the spring in order to get rid of any leftovers. Based on past experience, the store's manager expects that the demand for snow blowers at full price will be between 6 and 10. a. Identify the decision alternatives and states of nature and construct a payoff table for the store manager's snow blower stocking problem. b. Determine the number of snow blowers to stock if the maximax criterion is used. Repeat for maximin, equally likely, and minimax regret. c. Based on the long-range forecasts for the upcoming winter's weather, the store manager estimates the probabilities of selling different numbers of snow blowers at full price to be: P(6) = .35, P(7) = .30, P(8) = .20, P(9) = .10, and P(10) = .05. Determine the number of snow blowers to stock to maximize expected profits. d. Using the same probabilities, find the number of snow blowers to stock to minimize expected regret. Verify that the decision is the same as in part c. e. Using the probabilities in part c, find the expected value of perfect information. B.8. Dot'z Bakery bakes fresh apple pies each morning for sale that day. A pie costs $2 to make and sells for $4. Any pies left at the end of the day are sold the following day at a discounted price of $1.50. Based on her past experience, the bakery's manager expects to sell between 8 and 12 pies per day. a. Identify the decision alternatives and states of nature and construct a payoff table for the bakery manager's apple pie stocking problem. b. Determine the number of apple pies to bake if the maximax criterion is used. Repeat for maximin, equally likely, and minimax regret. c. Based on historical sales records, the bakery manager estimates the probabilities of the different apple pie demand levels as: P(8) = .1, P(9) = .2, P(10) = .4, P(11) = .2, P(12) = .1. Determine the number of apple pies to bake to maximize expected profits. d. Using the same probabilities, find the number of apple pies to bake to minimize expected regret. Verify that the decision is the same as in part (c). e. Using the probabilities in part c, find the expected value of perfect information. B.9. Considerable research and a great deal of practical experience show that the production of goods or services is generally subject to learning or experience curve effects. That is, records kept on the operation of many manufacturing and service delivery systems show that the time and cost required to produce a unit of output decrease at a fairly predictable rate as experience with that production increases. Given this predictable improvement, some companies follow a strategy of initially pricing a new product below its production cost as a way of building demand for the product, recognizing that, due to the learning curve effects, the cost will eventually be lower than the sale price, generating profits that will more than compensate for the initial losses. A company is considering using this type of pricing strategy for a new product it is about to introduce but is uncertain as to what learning rate to expect and, therefore, what pricing strategy to use. If a low initial price is set based on the assumption of a high rate of improvement and the improvement is slower, then the long-term profits will be lower than anticipated or nonexistent. If, however, a higher initial price is set, then demand would probably be lower and, even if the rate of learning is high, profits will not meet the hoped-for levels. Initial assessments of the profit levels under various combinations of pricing and learning rates are summarized in the accompanying table. Rate of Learning Pricing Strategy Low Moderate High Price Low –10 –3 12 Price Medium –4 3 10 Price High 1 6 7 a. Determine the pricing strategy to follow if the maximax criterion is used. Repeat for maximin, equally likely, and minimax regret. b. Based on the company's experience with other new products, the operations manager estimates the probabilities of the different rates of learning to be: P(low) = .2, P(moderate) = .6, P(high) = .2. Determine the pricing strategy to follow to maximize expected profits. c. Using the same probabilities, find the pricing strategy to follow to minimize expected regret. Verify that the decision is the same as in part c. d. Using the probabilities in part c, find the expected value of perfect information. B.10. Tom's airline is considering a coupon promotion to increase business. Under this program, the airline will give anyone purchasing a full-fare ticket a coupon worth half off on any future flight. The airline is considering doing this for two weeks, a month, or not all. The possible results of undertaking this program are that it will be very successful, moderately successful, or not successful at all in developing new business over the next year after the program is over. Using other airlines' experiences with this type of program, the airline marketing manager estimates the net profits (in millions of dollars) for the various combinations of program times and market responses shown in the accompanying table. Level of Success Program Length Not Moderate Very Month –5 –1 10 Two Weeks –2.5 2 7 Do nothing 0 0 0 a. Determine the amount of time for which to run the promotion if the maximax criterion is used. Repeat for maximin, equally likely, and minimax regret. b. Again based on other airlines' experiences with this type of coupon program, the airline's marketing manager estimates the probabilities of the different levels of program success to be: P(very successful) = .20, P(moderately successful) = .25, P(not successful) = .55. Which alternative should the airline adopt if management wishes to maximize expected net profits from the program? Determine the expected value of perfect information. B.11. A distributor of greeting cards and related products has the opportunity to participate in the merchandising activities associated with a forthcoming children's movie. Due to the production lead- time and the relatively short expected product life, the distributor must make a decision now about how much of this special party package to order. The distributor's marketing manager estimates that demand for this product will be between 400 and 800 units, in increments of 100 units. A unit, which consists of 1,000 party packages, will cost $3,000 and sell for $5,000 at full price. If demand is too low, the selling price will be cut to $2,000 to clear out the excess inventory. a. Determine the number of units of product to order if the maximax criterion is used. Repeat for maximin, equally likely, and minimax regret. b. Based on her past experience with similar types of movie-based special products, the marketing manager estimates that the probabilities of the different possible demand levels are: P(400) = .2, P(500) = .4, P(600) = .2, P(700) = .1, and P(800) = .1. Determine the number of units to order to maximize expected profit. Also determine the expected value of perfect information. c. The manufacturer of the party packages will offer the distributor a discounted cost of $2,500 per unit if a minimum of 600 units are ordered. Does the availability of this quantity discount make any difference in the decision about how many units to order if the expected profit criterion is used? B.12. A company is being sued for damages as a result of injuries incurred due to product failure under normal use. The management is consulting with lawyers to determine how much to offer the plaintiff as a settlement to avoid having to go to trial. The lawyers have suggested taking a low/medium/high offer approach. In an attempt to get off relatively cheaply, the company could initially offer the plaintiff $100,000, an amount that the lawyers believe would have only a 20% chance of being accepted. Alternatively, they could offer $150,000, which the lawyers believe would have a 50-50 chance of acceptance, or offer $200,000, which the lawyers believe would definitely be accepted. If the plaintiff does not accept the initial offer, the company would then make a second, higher offer, which the lawyers believe would have to be higher than a comparable offer made initially to get the same probability of acceptance. Specifically, the lawyers believe that the company would have to offer $175,000 to have a 50-50 chance of acceptance in the second stage and offer $225,000 to be guaranteed of acceptance. Finally, if the plaintiff accepts neither the first nor the second offer, the lawyers believe that a final offer of $250,000 would definitely be accepted. a. Determine what strategy the company should follow to minimize the expected cost of the settlement. That is, determine what the initial offer should be and, if it is not accepted, what the second offer should be. b. Suppose that there is a 30% chance of the plaintiff accepting an initial offer of $100,000 (nothing else in the description given changes). Would this affect the strategy that minimizes the expected cost? c. Determine the minimum probability of acceptance of an initial offer of $100,000 that would make the strategy starting with that amount optimal. B.13. The city manager of Bridgeport is developing a recommendation to the city council on when to undertake repairs needed on one of the main bridges crossing the river that runs through the middle of town. The city engineer has informed her that the bridge is deteriorating at a rate that will make major repairs necessary sometime within the next three years. If the repairs are done this year, the cost is estimated to be $1.2 million. If the repairs are delayed until next year and the condition of the bridge does not get significantly worse, then inflation and increased minor deterioration are estimated to raise the cost by 10%. If, however, there is significant additional deterioration, which is a function of the severity of the winter, the repairs will have to be done and the cost will increase to $1.8 million. At this point, the weather service to which the city subscribes estimates only a 20% chance that the winter will be severe. If the upcoming winter is not severe, then the city will again have the option of delaying the repairs for another year. Again, if the following winter is not severe, inflation and normal increased deterioration will raise the cost by an additional 10% over the cost of repair in the second year. If the winter between the second and third years is severe, the added deterioration to the bridge will raise the cost to $2.4 million. Normal weather patterns suggest that there is a 50-50 chance of a severe winter two years from now. If this were all there were to the decision, the city manager would not have a problem. Since the cost keeps going up, regardless of how severe the winter is, the repairs should be done this year. However, the city manager has been informed by the local state representative that there is a possibility that the city could get a $600,000 grant from the state that would help pay for the repairs, thus reducing the cost to the city. While the grant will not be available for the current year, the representative estimates a 60% chance that the grant will be available in the second year. Furthermore, if the city does not get a grant in year 2, there is a 75% chance they will get one in year 3. a. Using this information, develop a recommendation for the city manager that will minimize the expected cost to the city of repairing the bridge. b. Changing nothing else, what is the minimum probability of getting a grant in year 3 (given that one had not been received in year 2) that would make delaying the repair to year 3 (if possible) the optimal strategy? B.14. The Pittsburgh Steelers have just won the AFC title, and the owner of a sports specialty store in Pittsburgh has to decide how many of the special "AFC Champions" T-shirts he should order. The T- shirt manufacturer will only sell the shirts in cases of 100 at $1,000 per case. The shirts will be priced for sale at $15 each, with any left over by the time of the Super Bowl (which will be in two weeks) being sold on the discount table at $6 each. To simplify the analysis, assume that demand for the T- shirts is in whole cases. The sports store owner believes that his store will sell between one and three cases of shirts. a. Determine the number of cases of shirts the store owner should order if the maximax criterion is used. Repeat for maximin, equally likely, and minimax regret. b. Based on T-shirt sales during previous Steeler appearances in the Super Bowl, the store owner estimates the probabilities of the different demand levels as: P(1) = .12, P(2) = .48, P(3) = .40. Determine the number of cases of T-shirts to order to maximize expected profits. c. Using the same probabilities, find the number of cases of T-shirts to order to minimize expected regret. Verify that the decision is the same as in part b. d. Using the probabilities in part b, find the expected value of perfect information. B.15. (Continuation of Problem B.14.) To provide some protection against getting stuck with too many unsold T-shirts, the store owner has asked the T-shirt producer about the possibility of buying some shirts now and, if desired, more a week later. The producer says he is willing to do this, but, because he will have to set his equipment up to produce a second batch later, the reorder will cost $1,200 per case rather than $1,000. The store owner believes that during the first week there is a 60% chance that there will be demand for one case of shirts and a 40% chance of demand for two cases. He also believes that the first week's demand will be a good indication of what the demand will be in the second week. Specifically, he estimates the second week's demand (in cases) to be: If the first week's demand is for one case P(2nd week's demand = 0) = .2 P(2nd week's demand = 1) = .6 P(2nd week's demand = 2) = .2 If the first week's demand is for two cases P(2nd week's demand = 0) = .3 P(2nd week's demand = 1) = .7 a. Using a decision tree, determine the ordering strategy that will maximize the store owner's expected profit. b. Compare the expected profit from following the optimal two-stage ordering strategy developed in part a with the expected profit if all shirts have to be ordered initially, as determined in part b of Problem B.14. CASE: Transformer Replacement at Mountain States Electric Service Mountain States Electric Service is an electrical utility company serving several states in the Rocky Mountain region. It is considering replacing some of its equipment at a generating substation and is attempting to decide whether it should replace an older, existing PCB transformer. (PCB is a toxic chemical known formally as polychlorinated biphenyl.) Even though the PCB generator meets all current regulations, if an incident occurred, such as a fire, and PCB contamination caused harm either to neighboring businesses or farms or to the environment, the company would be liable for damages. Recent court cases have shown that simply meeting utility regulations does not relieve a utility of liability if an incident causes harm to others. Also, courts have been awarding large damages to individuals and businesses harmed by hazardous incidents. If the utility replaces the PCB transformer, no PCB incidents will occur, and the only cost will be that of the transformer, $85,000. Alternatively, if the company decides to keep the existing PCB transformer, then management estimates there is a 50-50 chance of there being a high likelihood of an incident or a low likelihood of an incident. For the case in which there is a high likelihood that an incident will occur, there is a .004 probability that a fire will occur sometime during the remaining life of the transformer and a .996 probability that no fire will occur. If a fire occurs, there is a .20 probability that it will be bad and the utility will incur a very high cost of approximately $90 million for the cleanup, whereas there is a .80 probability that the fire will be minor and a cleanup can be accomplished at a low cost of approximately $8 million. If no fire occurs, then no cleanup costs will occur. For the case in which there is a low likelihood of an incident occurring, there is a .001 probability that a fire will occur during the life of the existing transformer and a .999 probability that a fire will not occur. If a fire does occur, then the same probabilities exist for the incidence of high and low cleanup costs, as well as the same cleanup costs, as indicated for the previous case. Similarly, if no fire occurs, there is no cleanup cost. Perform a decision tree analysis of this problem for Mountain States Electric Service and indicate the recommended solution. Is this the decision you believe the company should make? Explain your reasons.1 1 This case was adapted from W. Balson, J. Welsh, and D. Wilson, "Using Decision Analysis and Risk Analysis to Manage Utility Environmental Risk," Interfaces 22, no. 6 (November-December 1992): 126- 39. Key Terms alternative The possible environments, conditions, or states of nature within which a decision is to be implemented. averaging out The process of replacing each branching by a single number. For a chance branching, the number used is the expected value, which is found by multiplying the probability of each branch by the value at its right end and summing. For a decision branching, the number used is the value of the best alternative in the branching. chance node Problem features that are determined by chance or probability, such as the states of nature. decision making under certainty A basic type of decision problem where decision makers have the payoff table, but no information on the relative likelihood of the states of nature. In principle, the analysis is simple: Choose the alternative with the best payoff. In practice, it is not always that easy, since an "alternative" may actually represent the values for many decision variables. decision making under risk A basic type of decision problem where the state of nature is known, so the payoff table has only one column. In principle, the analysis is simple: Choose the alternative with the best payoff. decision making under uncertainty A basic type of decision problem where decision makers have the payoff table, but no information on the relative likelihood of the states of nature. decision node A tool that facilitates the description of the decision tree. There are three alternatives, represented by the three branches: (1) build an addition, (2) convert existing space, and (3) do nothing. decision tree An alternative to using a payoff to compute the expected values of decision alternatives, which consists of two or more stages or levels, shown in time order from left to right. environment The conditions, or states of nature, within which a decision is to be implemented. equal likelihood A procedure based on the assumption that if we cannot determine the relative likelihood of the states of nature, then it is rational to presume that they are equally likely. expected value of perfect information (EVPI) A measure of the expected current worth to the decision maker of being able to find out, just before having to make the decision, what the state of nature will be. folding back The process which starts at the right-hand edge of the tree and proceeds back to the start of the tree, working from right to left. maximax An optimistic approach that assumes that no matter what alternative we choose, "nature" will smile favorably on us and choose the state that will benefit us the most. Thus, we choose the alternative with the largest maximum gain. maximin A pessimistic approach that assumes that no matter what alternative we choose, "nature" will choose the state that benefits us the least. Thus, we choose the alternative with the largest minimum gain. minimax A procedure taken if the payoffs are costs or losses. A pessimistic approach that assumes that no matter what alternative we choose, "nature" will choose the state that benefits us the least. Thus, we choose the alternative with the largest minimum gain (or the smallest maximum loss). minimax regret The approach in which we choose the alternative that will yield the smallest possible maximum regret. minimin An approach that assumes that no matter what alternative we choose, "nature" will smile favorably on us and choose the state of nature that will hurt us the least. Thus, we choose the alternative with the smallest minimum loss. outcome Payoffs that will result from each possible combination of alternatives and states of nature. payoff Outcomes that will result from each possible combination of alternatives and states of nature. payoff function A tool which specifies what that outcome is for every possible alternative/state-of-nature combination. It may be given either as an equation or, when the possible decision alternatives and states of nature are relatively limited, as a payoff table or matrix. payoff table A matrix that specifies the outcome for every possible alternative/state-of-nature when the states-of-nature are relatively limited. risk analysis An evaluation of the various possible outcomes and their probabilities, and extreme case analysis, which, like the maximax and maximin strategies, takes either an optimistic or pessimistic view rather than working with probabilities. states of nature Aspects of the decision situation over which the decision maker does not have direct control. utility An alternative measure when dollar value alone is not an adequate measure of the value of an outcome.

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