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1 BMAppendixA.indd Page /03/14 9:46 PM user APPENDIXA Operational Decision-Making Tools: Decision Analysis LEARNING OBJECTIVES < Decision Analysis (With and Without Probabilities) At the operational level, hundreds of decisions are made in order to achieve local outcomes that contribute to the achievement of a company s overall strategic goal. These local outcomes are usually not measured directly in terms of profit, but instead are measured in terms of quality, cost-effectiveness, efficiency, productivity, and so forth. Achieving good results for local outcomes is an important objective for individual operational units and individual operations managers. However, all these decisions are interrelated and must be coordinated for the purpose of attaining the overall company goals. Decision making is analogous to a great stage play or opera, in which all the actors, the costumes, the props, the music, the orchestra, and the script must be choreographed and staged by the director, the stage managers, the author, and the conductor so that everything comes together for the performance. For many topics in operations management, there are quantitative models and techniques available that help managers make decisions. Some techniques simply provide information that the operations manager might use to help make a decision; other techniques recommend a decision to the manager. Some techniques are specific to a particular aspect of operations management; others are more generic and can be applied to a variety of decision-making categories. These different models and techniques are the tools of the operations manager. Simply having these tools does not make someone an effective operations manager, just as owning a saw and a hammer does not make someone a carpenter. An operations manager must know how to use decision-making tools. How these tools are used in the decision-making process is an important and necessary part of the study of operations management. In this supplement and others throughout this book, we examine several different aspects of operational decision making using these tools. Decision Analysis With and Without Probabilities In this supplement we demonstrate a quantitative technique called decision analysis for decisionmaking situations in which uncertainty exists. Decision analysis is a generic technique that can be applied to a number of different types of operational decision-making areas.

2 < BMAppendixA.indd Page /03/14 9:46 PM user Appendix A: Operational Decision-Making Tools Decision Analysis 593 Many decision-making situations occur under conditions of uncertainty. For example, the demand for a product may not be 100 units next week but may vary between 0 and 200 units, depending on the state of the market, which is uncertain. Decision analysis is a set of quantitative decision-making techniques to aid the decision maker in dealing with a decision situation in which there is uncertainty. However, the usefulness of decision analysis for decision making is also a beneficial topic to study because it reflects a structured, systematic approach to decision making that many decision makers follow intuitively without ever consciously thinking about it. Decision analysis represents not only a collection of decision-making techniques but also an analysis of logic underlying decision making. DECISION MAKING WITHOUT PROBABILITIES A decision-making situation includes several components the decisions themselves and the events that may occur in the future, known as states of nature. Future states of nature may be high or low demand for a product or good or bad economic conditions. At the time a decision is made, the decision maker is uncertain which state of nature will occur in the future and has no control over these states of nature. When probabilities can be assigned to the occurrence of states of nature in the future, the situation is referred to as decision making under risk. When probabilities cannot be assigned to the occurrence of future events, the situation is called decision making under uncertainty. We discuss the latter case next. To facilitate the analysis of decision situations, they are organized into payoff tables. A payoff table is a means of organizing and illustrating the payoffs from the different decisions, given the various states of nature, and has the general form shown in Table A.1. Each decision, 1 or 2, in Table A.1 will result in an outcome, or payoff, for each state of nature that will occur in the future. Payoffs are typically expressed in terms of profit, revenues, or cost (although they may be expressed in terms of a variety of quantities). For example, if decision 1 is to expand a production facility and state of nature a is good economic conditions, payoff 1a could be $100,000 in profit. Once the decision situation has been organized into a payoff table, several criteria are available to reflect how the decision maker arrives at a decision, including maximax, maximin, minimax regret, Hurwicz, and equal likelihood. These criteria reflect different degrees of decision-maker conservatism or liberalism. On occasion they result in the same decision; however, they often yield different results. These decision-making criteria are demonstrated by the following example. TABLE A.1 Decision Payoff Table States of Nature a b 1 Payoff 1a Payoff 1b 2 Payoff 2a Payoff 2b EXAMPLE A.1 The Southern Textile Company is contemplating the future of one of its plants located in South Carolina. Three alternative decisions are being considered: (1) Expand the plant and produce lightweight, durable materials for possible sale to the military, a market with little foreign competition; (2) maintain the status quo at the plant, continuing production of textile goods that are DECISION-MAKING CRITERIA UNDER UNCERTAINTY (Continued)

3 < BMAppendixA.indd Page /03/14 9:46 PM user 594 Appendix subject to heavy foreign competition; or (3) sell the plant now. If one of the first two alternatives is chosen, the plant will still be sold at the end of the year. The amount of profit that could be earned by selling the plant in a year depends on foreign market conditions, including the status of a trade embargo bill in Congress. The following payoff table describes this decision situation. States of Nature Good Foreign Poor Foreign Decision Competitive Competitive Expand $800,000 $500,000 Maintain status quo 1,300, ,000 Sell now 320, ,000 Determine the best decision using each of the decision criteria. 1. Maximax 2. Maximin 3. Minimax regret 4. Hurwicz 5. Equal likelihood SOLUTION 1. Maximax The decision is selected that will result in the maximum of the maximum payoffs. This is how this criterion derives its name the maximum of the maxima. The maximax criterion is very optimistic. The decision maker assumes that the most favorable state of nature for each decision alternative will occur. Thus, for this example, the company would optimistically assume that good competitive conditions will prevail in the future, resulting in the following maximum payoffs and decisions: Expand: $800,000 Status quo: 1,300,000 d Maximum Sell: 320,000 Decision: Maintain status quo 2. Maximin The maximin criterion is pessimistic. With the maximin criterion, the decision maker selects the decision that will reflect the maximum of the minimum payoffs. For each decision alternative, the decision maker assumes that the minimum payoff will occur; of these, the maximum is selected as follows: Expand: $500,000 d Maximum Status quo: 2150,000 Sell: 320,000 Decision: Expand 3. Minimax Regret Criterion The decision maker attempts to avoid regret by selecting the decision alternative that minimizes the maximum regret. A decision maker first selects the maximum payoff under each state of nature; then all other payoffs under the respective states of nature are subtracted from these amounts, as follows:

4 < BMAppendixA.indd Page /03/14 9:46 PM user Appendix A: Operational Decision-Making Tools Decision Analysis 595 Good Competitive Poor Competitive $1,300, , ,000 $500, , ,300, ,300, ,000 2 (2150,000) 5 650,000 1,300, , , , , ,000 These values represent the regret for each decision that would be experienced by the decision maker if a decision were made that resulted in less than the maximum payoff. The maximum regret for each decision must be determined, and the decision corresponding to the minimum of these regret values is selected as follows: Expand: $500,000 d Minimum Status quo: 650,000 Sell: 980,000 Decision: Expand 4. Hurwicz A compromise is made between the maximax and maximin criteria. The decision maker is neither totally optimistic (as the maximax criterion assumes) nor totally pessimistic (as the maximin criterion assumes). With the Hurwicz criterion, the decision payoffs are weighted by a coefficient of optimism, a measure of the decision maker s optimism. The coefficient of optimism, defined as a, is between 0 and 1 (i.e., 0, a, 1.0). If a 5 1.0, the decision maker is completely optimistic; if a 5 0, the decision maker is completely pessimistic. (Given this definition, 1 2 a is the coefficient of pessimism.) For each decision alternative, the maximum payoff is multiplied by a and the minimum payoff is multiplied by 1 2 a. For our investment example, if a equals 0.3 (i.e., the company is slightly optimistic) and 1 2 a 5 0.7, the following decision will result: Expand: $800,000(0.3) 1 500,000(0.7) 5 $590,000 d Maximum Status quo: 1,300,000(0.3) 2 150,000(0.7) 5 285,000 Sell: 320,000(0.3) 1 320,000(0.7) 5 320,000 Decision: Expand 5. Equal Likelihood The equal likelihood (or Laplace ) criterion weights each state of nature equally, thus assuming that the states of nature are equally likely to occur. Since there are two states of nature in our example, we assign a weight of 0.50 to each one. Next, we multiply these weights by each payoff for each decision and select the alternative with the maximum of these weighted values. Expand: $800,000(0.50) 1 500,000(0.50) 5 $650,000 d Maximum Status quo: 1,300,000(0.50) 2 150,000(0.50) 5 575,000 Sell: 320,000(0.50) 1 320,000(0.50) 5 320,000 Decision: Expand The decision to expand the plant was designated most often by four of the five decision criteria. The decision to sell was never indicated by any criterion. This is because the payoffs for expansion, under either set of future economic conditions, are always better than the payoffs for selling. Given any situation with these two alternatives, the decision to expand will always be made over the decision to sell. The sell decision alternative could have been eliminated from consideration under each of our criteria. The alternative of selling is said to be dominated by the alternative of expanding. In general, dominated decision alternatives can be removed from the payoff table and not considered when the various decision-making criteria are applied, which reduces the complexity of the decision analysis.

5 BMAppendixA.indd Page /03/14 9:46 PM user 596 Appendix Different decision criteria often result in a mix of decisions. The criteria used and the resulting decisions depend on the decision maker. For example, the extremely optimistic decision maker might disregard the preceding results and make the decision to maintain the status quo, because the maximax criterion reflects his or her personal decision-making philosophy. DECISION ANALYSIS WITH EXCEL Exhibit A.1 shows the Excel spreadsheet solutions for the different decision-making criteria in Example A.1. The call-out boxes displayed on and around the spreadsheet define the cell formulas used to compute the criteria values. For example, the spreadsheet formula used to compute the maximum payoff value for the decision to Expand, 5 MAX( C6:D6), is embedded in cell E6 and is also shown on the toolbar at the top of the spreadsheet. The formula for the Maximax decision, 5 MAX( E6:E8), is embedded in cell C10. The Excel file for Exhibit A.1 and the Excel files for all of the exhibits in this text are contained on the text website. Students and instructors can download this file to see how the spreadsheet was constructed as well as the individual cell formulas. This spreadsheet can also be used as a guideline or template to solve the homework problems at the end of the chapter using Excel. DECISION ANALYSIS WITH OM TOOLS OM Tools is an Excel-based software package published by Wiley that was specifically designed for use with this text. It includes solution modules for most of the quantitative techniques in this text. After downloading OM Tools, modules can be selected by clicking on the OM Tools button on the tool bar at the top of the page, which provides a drop-down list of modules, and then EXHIBIT A.1 EXCEL FILE ExhibitA.1.Decision Analysis [Compatibility Mode] - Excel Example A.1: Decision-Making Criteria Under Uncertainty =MAX(C6:D6) =MAX(D6:D8) F7 =MIN(C8:D8) =MAX(F6:F8) =MAX(G7:H7) =C19*E7+C20*F7 =.5*E8+.5*F8

6 BMAppendixA.indd Page /03/14 9:46 PM user Appendix A: Operational Decision-Making Tools Decision Analysis 597 clicking on the specific module you want to use. In this case we want to use the Decision Analysis module with Decision Making Under Uncertainty. A window for providing the initial problem data, including the problem name and the number of decision alternatives and states of nature, will then be displayed. Exhibit A.2 shows the OM Tools Excel spreadsheet for Example A.1 with all of the problem data input into the cells. Notice that the difference between this spreadsheet and the one in Exhibit A.1 is that the spreadsheet has already been set up with all of the Excel formulas for the various decision criteria in the cells. Thus, all you have to do to solve the problem is type in the problem data. DECISION MAKING WITH PROBABILITIES For the decision-making criteria we just used, we assumed no available information regarding the probability of the states of nature. However, it is often possible for the decision maker to know enough about the future states of nature to assign probabilities that each will occur, which is decision making under conditions of risk. The most widely used decisionmaking criterion under risk is expected value, computed by multiplying each outcome by the probability of its occurrence and then summing these products according to the following formula: where EV(x) a n i 1 x i outcome i p(x i ) probability of outcome i p(x i )x i EXHIBIT A.2 OM TOOLS ExhibitA.2.Decision Analysis [Compatibility Mode] - Excel OM Student - Example A.2

7 < BMAppendixA.indd Page /03/14 9:46 PM user 598 Appendix EXAMPLE A.2 EXPECTED VALUE EXPECTED VALUE OF PERFECT INFORMATION Occasionally, additional information is available, or can be purchased, regarding future events, enabling the decision maker to make a better decision. For example, a company could hire an economic forecaster to determine more accurately the economic conditions that will occur in the future. However, it would be foolish to pay more for this information than it stands to gain in extra profit from having the information. The information has some maximum value that is the limit of what the decision maker would be willing to spend. This value of information can be computed as an expected value hence its name, the expected value of perfect information (EVPI). To compute the expected value of perfect information, first look at the decisions under each state of nature. If information that assured us which state of nature was going to occur (i.e., perfect information) could be obtained, the best decision for that state of nature could be selected. For example, in the textile company example, if the company executives knew for sure that good competitive conditions would prevail, they would maintain the status quo. If they knew for sure that poor competitive conditions will occur, then they would expand. Assume that it is now possible for the Southern Textile Company to estimate a probability of 0.70 that good foreign competitive conditions will exist and a probability of 0.30 that poor conditions will exist in the future. Determine the best decision using expected value. SOLUTION The expected values for each decision alternative are computed as follows. EV(expand) 5 $800,000(0.70) 1 500,000(0.30) 5 $710,000 EV(status quo) 5 1,300,000(0.70) 2 150,000(0.30) 5 865,000 EV(sell) 5 320,000(0.70) 1 320,000(0.30) 5 320,000 d Maximum The decision according to this criterion is to maintain the status quo, since it has the highest expected value. The Excel spreadsheet solution for Example A.2 is shown in Exhibit A.3. Note that the values contained in cells D6, D7, and D8 were computed using the expected value formulas embedded in these cells. For example, the formula for cell D6 is shown on the formula bar on the Excel screen. EXHIBIT A.3 EXCEL FILE ExhibitA.3.Decision Analysis [Compatibility Mode] - Excel Example A.2: Expected Value Formula for expected value computed in cell D6

8 < BMAppendixA.indd Page /03/14 9:46 PM user Appendix A: Operational Decision-Making Tools Decision Analysis 599 The probabilities of each state of nature (i.e., 0.70 and 0.30) indicate that good competitive conditions will prevail 70% of the time and poor competitive conditions will prevail 30% of the time (if this decision situation is repeated many times). In other words, even though perfect information enables the investor to make the right decision, each state of nature will occur only a certain portion of the time. Thus, each of the decision outcomes obtained using perfect information must be weighted by its respective probability: $1,300,000(0.70) 1 (500,000)(0.30) $1,060,000 The amount of $1,060,000 is the expected value of the decision given perfect information, not the expected value of perfect information. The expected value of perfect information is the maximum amount that would be paid to gain information that would result in a decision better than the one made without perfect information. Recall from Example A.2 that the expectedvalue decision without perfect information was to maintain the status quo and the expected value was $865,000. The expected value of perfect information is computed by subtracting the expected value without perfect information from the expected value given perfect information: EVPI expected value given perfect information 2 expected value without perfect information For our example, the EVPI is computed as EVPI $1,060, ,000 $195,000 The expected value of perfect information, $195,000, is the maximum amount that the investor would pay to purchase perfect information from some other source, such as an economic forecaster. Of course, perfect information is rare and is usually unobtainable. Typically, the decision maker would be willing to pay some smaller amount, depending on how accurate (i.e., close to perfection) the information is believed to be. SEQUENTIAL DECISION TREES A payoff table is limited to a single decision situation. If a decision requires a series of decisions, a payoff table cannot be created, and a sequential decision tree must be used. We demonstrate the use of a decision tree in the following example. EXAMPLE A.3 The Southern Textile Company is considering two alternatives: to expand its existing production operation to manufacture a new line of lightweight material, or to purchase land on which to construct a new facility in the future. Each of these decisions has outcomes based on product market in the future that will result in another set of decisions (during a 10-year planning horizon), as shown in the following figure of a sequential decision tree. In this figure the square nodes represent decisions, and the circle nodes reflect different states of nature and their probabilities. The first decision facing the company is whether to expand or buy land. If the company expands, two states of nature are possible. Either the market will grow (with a probability of 0.60) or it will not grow (with a probability of 0.40). Either state of nature will result in a payoff. On the other hand, if the company chooses to purchase land, three years in the future, another decision will have to be made regarding the development of the land. A SEQUENTIAL DECISION TREE

9 < BMAppendixA.indd Page /03/14 9:46 PM user 600 Appendix 1 Expand ( $800,000) Purchase land ( $200,000) Market 0.40 No market Market (3 years, $0 payoff) No market (3 years, $0 payoff) 4 5 $225,000 Expand ( $800,000) Sell land Warehouse ( $600,000) Sell land Market No market $450,000 Market No market $210,000 $2,000,000 $3,000,000 $700,000 $2,300,000 $1,000,000 At decision node 1, the decision choices are to expand or to purchase land. Notice that the costs of the ventures ($800,000 and $200,000, respectively) are shown in parentheses. If the plant is expanded, two states of nature are possible at probability node 2: The market will grow, with a probability of 0.60, or it will not grow or will decline, with a probability of If the market grows, the company will achieve a payoff of $2,000,000 over a 10-year period. However, if no occurs, a payoff of only $225,000 will result. If the decision is to purchase land, two states of nature are possible at probability node 3. These two states of nature and their probabilities are identical to those at node 2; however, the payoffs are different. If market occurs for a three-year period, no payoff will occur, but the company will make another decision at node 4 regarding development of the land. At that point, either the plant will be expanded at a cost of $800,000 or the land will be sold, with a payoff of $450,000. The decision situation at node 4 can occur only if market occurs first. If no market occurs at node 3, there is no payoff, and another decision situation becomes necessary at node 5: A warehouse can be constructed at a cost of $600,000 or the land can be sold for $210,000. (Notice that the sale of the land results in less profit if there is no market than if there is.) If the decision at decision node 4 is to expand, two states of nature are possible: The market may grow, with a probability of 0.80, or it may not grow, with a probability of The probability of market is higher (and the probability of no is lower) than before because there has already been for the first three years, as shown by the branch from node 3 to node 4. The payoffs for these two states of nature at the end of the 10-year period are $3,000,000 and $700,000, respectively. If the company decides to build a warehouse at node 5, two states of nature can occur: Market can occur, with a probability of 0.30 and an eventual payoff of $2,300,000, or no can occur, with a probability of 0.70 and a payoff of $1,000,000. The probability of market is low (i.e., 0.30) because there has already been no market, as shown by the branch from node 3 to node 5. SOLUTION We start the decision analysis process at the end of the decision tree and work backward toward a decision at node 1.

10 < BMAppendixA.indd Page /03/14 9:46 PM user Appendix A: Operational Decision-Making Tools Decision Analysis 601 First, we must compute the expected values at nodes 6 and 7: EV(node 6) 0.80($3,000,000) ($700,000) $2,540,000 EV(node 7) 0.30(2,300,000) ($1,000,000) $1,390,000 Expand ( $800,000) 1 Purchase land ( $200,000) $1,160,000 $1,290, No market Market (3 years, $0 payoff) $1,360, No market (3 years, $0 payoff) Market 4 5 $225,000 $1,740,000 $790,000 Expand ( $800,000) Sell land Market $2,540, No market $450,000 Market $1,390,000 Warehouse 0.30 ( $600,000) No market Sell land $210,000 $2,000,000 $3,000,000 $700,000 $2,300,000 $1,000,000 These expected values (as well as all other nodal values) are shown in boxes in the figure. At decision nodes 4 and 5, a decision must be made. As with a normal payoff table, the decision is made that results in the greatest expected value. At node 4 the choice is between two values: $1,740,000, the value derived by subtracting the cost of expanding ($800,000) from the expected payoff of $2,540,000, and $450,000, the expected value of selling the land computed with a probability of 1.0. The decision is to expand, and the value at node 4 is $1,740,000. The same process is repeated at node 5. The decisions at node 5 result in payoffs of $790,000 (i.e., $1,390, ,000 5 $790,000) and $210,000. Since the value $790,000 is higher, the decision is to build a warehouse. Next, the expected values at nodes 2 and 3 are computed: EV(node 2) 0.60($2,000,000) ($225,000) $1,290,000 EV(node 3) 0.60($1,740,000) ($790,000) $1,360,000 (Note that the expected value for node 3 is computed from the decision values previously determined at nodes 4 and 5.) Now the final decision at node 1 must be made. As before, we select the decision with the greatest expected value after the cost of each decision is subtracted. Expand: $1,290, ,000 $490,000 Land: $1,360, ,000 $1,160,000 Since the highest net expected value is $1,160,000, the decision is to purchase land, and the payoff of the decision is $1,160,000. Decision trees allow the decision maker to see the logic of decision making by providing a picture of the decision process. Decision trees can be used for problems more complex than this example without too much difficulty.

11 BMAppendixA.indd Page /03/14 9:46 PM user 602 Appendix SUMMARY In this appendix, we have provided a general overview of decision analysis. To a limited extent, we have also shown that the logic of such operational decisions throughout the organization is interrelated to achieve strategic goals. KEY FORMULAS Expected Value n EV(x) a p(x i )x i i 1 Expected Value of Perfect Information EVPI expected value given perfect information 2 expected value without perfect information SOLVED PROBLEMS Consider the following payoff table for three product decisions (A, B, and C) and three future market conditions (payoffs 5 $ millions). MARKET CONDITIONS DECISION A $1.0 $2.0 $0.5 B C SOLUTION Step 1. Maximax criterion Determine the best decision using the following decision criteria. 1. Maximax 2. Maximin Step 2. Maximin criterion MINIMUM PAYOFFS A 0.5 B 0.8 d Maximum C 0.7 Decision: Product B MAXIMUM PAYOFFS A $2.0 d Maximum B 1.2 C 1.7 Decision: Product A PROBLEMS A-1. Telecomp is a U.S.-based manufacturer of cellular telephones. It is planning to build a new manufacturing and distribution facility in either South Korea, China, Taiwan, Poland, or Mexico. The cost of the facility will differ between countries and will even vary within countries depending on the economic and political climate, including monetary exchange rates. The company has estimated the facility cost (in $ millions) in each country under three different future economic/ political climates as follows. Economic/Political Climate Country Decline Same Improve South Korea China Economic/Political Climate Country Decline Same Improve Taiwan Poland Mexico Determine the best decision using the following decision criteria. (Note that since the payoff is cost, the maximax criteria becomes minimin and maximin becomes minimax.) a. Minimin b. Minimax c. Hurwicz ( a ) d. Equal likelihood

12 BMAppendixA.indd Page /03/14 9:46 PM user Appendix A: Operational Decision-Making Tools Decision Analysis 603 A-2. A -3. A -4. A -5. A global economist hired by Telecomp, the U.S.-based computer manufacturer in Problem A-1, estimates that the probability that the economic and political climate overseas and in Mexico will decline during the next five years is 0.30, the probability that it will remain approximately the same is 0.40, and the probability that it will improve is Determine the best country to construct the new facility in and the expected value of perfect information. Leevi Starch, an apparel company with a global supply chain, is adding a new supplier for several new styles of its denim jeans, and the suppliers it s considering are in China, India, the Philippines, Brazil, and Mexico. A major factor in the company s decision is transportation and shipping costs, which are dependent on future oil prices. The following payoff table summarizes the total monthly costs (in $100,000s), including manufacturing and shipping costs for the suppliers in each of the countries given the future state of oil prices. Oil Prices Supplier Decrease Same Increase China $2.7 $3.9 $6.3 India Philippines Brazil Mexico Determine the best decision using each of the following criteria. a. Minimin b. Minimax c. Equal likelihood d. Minimax regret Leevi Starch in Problem A-3 estimates that the probabilities of future global changes in oil prices are 0.09 that they will decrease, 0.27 that they will remain the same, and 0.64 that they will decrease. a. Determine the best supplier for the company using expected value. b. If the company wants to hire an energy analyst to help it determine more accurately what future oil prices will do, what is the maximum amount it should pay the analyst? Telecomp, a computer manufacturer with a global supply chain, is adding a new supplier for some of its component parts, and the suppliers it s considering are in China, India, Thailand, and the Philippines. As part of its risk management program Telecomp wants to assess the possible impact of a supplier shutdown in the event of a natural disaster, such as a flood, fire, or an earthquake. The following payoff table summarizes Telecomp s losses (in millions of dollars) for supplier shutdowns given different levels of event severity. Event Severity Supplier Country Low Moderate High China $8 $11 $21 India Thailand Philippines Determine the best decision using each of the following criteria. a. Minimin b. Minimax c. Equal likelihood d. Minimax regret A -6. Telecomp in Problem A-5 estimates that the probabilities of the severity of events in each of the countries are as follows: Event Severity Supplier Country Low Moderate High China India Thailand Philippines Determine the best decision for Telecomp using expected value. A- 7. The Dynamax Company is going to introduce one of three new products: a widget, a hummer, or a nimnot. The market conditions (favorable, stable, or unfavorable) will determine the profit or loss the company realizes, as shown in the following payoff table. Market Product Favorable 0.2 Stable 0.5 Unfavorable 0.3 Widget $160,000 $90,000 2$50,000 Hummer 70,000 40,000 20,000 Nimnot 45,000 35,000 30,000 a. Compute the expected value for each decision and select the best one. b. Determine how much the firm would be willing to pay to a market research firm to gain better information about future market conditions. c. Assume that probabilities cannot be assigned to future market conditions, and determine the best decision using the maximax, maximin, minimax regret, and equal likelihood criteria. A- 8. John Wiley & Sons, Inc. publishes an operations management textbook that is scheduled for a revision. The book has been moderately successful, but each year, as more new books enter the market, some existing books are dropped by publishers, and various innovative pedagogical approaches are introduced by authors and publishers, so that the competitive market is always highly uncertain. In addition, the role that the Internet will play in future textbook publishing is an unknown. As a result, Wiley is trying to decide whether to publish the next edition of the OM book as a smaller paperback, publish a new edition very similar in size and content to the current edition, significantly revise the book with an emphasis on services and processes, or make a major revision with significant physical changes including adding color and more graphics. The following payoff table summarizes the possible revision decisions with profits (or losses) for the three-year lifecycle of the new edition, and the future states of nature relative to the competitive market. Competitive Market Publication Decision Unfavorable Same Favorable Paperback $68,000 $170,000 $395,000 Similar revision 24, , ,000 Major content 31, , ,000 revision Major physical revision 2105, , ,000

13 BMAppendixA.indd Page /03/14 9:46 PM user 604 Appendix Determine the best decision for the publisher using the following criteria. a. Maximax b. Minimax c. Equal likelihood d. Hurwicz ( 5.35) A- 9. In Problem A-8, if Wiley is able to assign probabilities of occurrence of 0.23 to unfavorable market conditions, 0.46 for the same market conditions, and 0.31 for favorable market conditions, what is the best decision using expected value? Based on the results in Problem A-8 and the expected value result in this problem, does there appear to be an overall best decision? Compute the expected value of perfect information, and explain its meaning. A-10. Amtrex International is a major U.S.-based electronics firm that manufactures a number of electronic components for domestic and global consumer electronics companies. It imports most of its materials and the components used in its products to the United States from overseas suppliers. Amtrex is in the process of trying to improve its global supply chain operations, and as part of this process the company wants to determine a single supplier located at one of the major ports around the world to contract with for the majority of its business. The company is considering six suppliers, each located at one of the following ports: Hong Kong, Singapore, Shanghai, Busan, and Kaohsiung. The company has estimated the possible profit (or loss) it might achieve with each of the potential suppliers depending on a variety or possible future company and port conditions, including IT capability, port and expansion, ship and container availability, security, regional market and political environment, and transport to the port from the supplier s suppliers. Depending on these various factors, further supplier and port conditions could decline, grow and expand, or remain the same. The following payoff table summarizes the increased outcomes (in $ millions) for the potential suppliers and the possible future states of nature for a specific time frame. States of Nature Decision Declining Same Growth Hong Kong 2$31 $28 $67 Singapore Shanghai Busan Kaohsiung Determine the best decision using each of the following criteria. a. Maximax b. Maximin c. Equal likelihood d. Hurwicz ( ) e. Minimax regret A-1 1. In Problem A-10, suppose Amtrex is able to assign probabilities to each of the states of nature for each of the suppliers/ports as follows: States of Nature Decision Declining Same Growth Hong Kong Singapore States of Nature Decision Declining Same Growth Shanghai Busan Kaohsiung a. Using expected value, determine the port/supplier Amtrex should use. b. Based on the results from Problem A-10, and the result from part a, is there a best overall decision? A- 12. The Midtown Market purchases apples from a local grower. The apples are purchased on Monday at $2.00 per pound, and the market sells them for $3.00 per pound. Any apples left over at the end of the week are sold to a local zoo for $0.50 per pound. The possible demands for apples and the probability for each are as follows: Demand (lb) Probability a. The market must decide how many apples to order in a week. Construct a payoff table for this decision situation and determine the amount of apples that should be ordered using expected value. b. Assuming that probabilities cannot be assigned to the demand values, what would the best decision be using the maximax and maximin criteria? A A machine shop owner is attempting to decide whether to purchase a new drill press, a lathe, or a grinder. The return from each will be determined by whether the company succeeds in getting a government military contract. The profit or loss from each purchase and the probabilities associated with each contract outcome are shown in the following payoff table. Compute the expected value for each purchase and select the best one. Purchase Contract 0.40 No Contract 0.60 Drill press $40,000 2$8,000 Lathe 20,000 4,000 Grinder 12,000 10,000 A The Extron Oil Company is considering making a bid for a shale oil development contract to be awarded by the federal government. The company has decided to bid $110 million. The company estimates that it has a 60% chance of winning the contract with this bid. If the firm wins the contract, it can choose one of three methods for getting the oil from the shale: It can develop a new method for oil extraction, use an existing (inefficient) process, or subcontract the processing out to a number of smaller companies once the shale has been excavated. The results from these alternatives are given as follows:

14 BMAppendixA.indd Page /03/14 9:46 PM user Appendix A: Operational Decision-Making Tools Decision Analysis 605 Develop New Process Outcomes Probability Profit (millions) Great success 0.30 $600 Moderate success Failure Use Present Process Outcomes Probability Profit (millions) Great success 0.50 $300 Moderate success Failure Subcontract Outcomes Probability Profit (millions) Moderate success 1.00 $250 The cost of preparing the contract proposal is $2,000,000. If the company does not make a bid, it will invest in an alternative venture with a guaranteed profit of $30 million. Construct a sequential decision tree for this decision situation and determine whether the company should make a bid. A-15. The director of career advising at Grand Valley Community College wants to use decision analysis to provide information to help students decide which two-year degree program they should pursue. The director has set up the following payoff table for six of the most popular and successful degree programs at GVCC that shows the estimated five-year gross income ($) from each degree for four future economic conditions: Economic Degree Program Recession Average Good Robust Graphic Design 115, , , ,000 Nursing 140, , , ,000 Real Estate 95, , , ,000 Medical Technology 120, , , ,000 Culinary Technology 85, , , ,000 Computer Information Technology 125, , , ,000 Determine the best degree program in terms of projected income, using the following decision criteria: a. Maximax b. Maximin c. Equal likelihood d. Hurwicz ( 5.25) A In Problem A-15 the director of career advising at Grand Valley Community College has paid a local economic forecasting firm to indicate a probability for each future economic condition over the next five years. The firm estimates that there is a.15 probability of a recession, a.50 probability that the economy will be average, a.25 probability that the economy will be good, and a.10 probability that it will be robust. Using expected value determine the best degree program in terms of projected income. If you were the director of career advising which degree program would you recommend? A- 17. Federated Electronics, Ltd., manufactures display screens and monitors for computers and televisions, which it sells to companies around the world. It wants to construct a new warehouse and distribution center in Asia to serve emerging markets there. It has identified potential sites in the port cities of Shanghai, Singapore, Pusan, Kaohsiung, and Hong Kong and has estimated the possible revenues for each (minus construction costs, which are higher in some cities like Hong Kong). At each site the projected revenues are primarily based on these factors: (1) the economic conditions at the port including the projected traffic, infrastructure, labor rates and availability, and expansion and modernization; and (2) the future government situation, which includes the political stability, fees, tariffs, duties, and trade regulations. Following is a payoff table that shows the projected revenues (in $ billions) for six years given the four possible combinations for positive and negative port and government conditions: Negative/ Government Negative Negative/ Government Positive Positive/ Government Negative Positive/ Government Positive Shanghai 2$0.271 $0.437 $0.523 $1.08 Singapore Pusan Kaoshiung Hong Kong Determine the port city Federated should select for its new distribution center using the following decision criteria: a. Maximax b. Maximin c. Equal likelihood d. Hurwicz ( a 5.55) A- 18. In Problem A-17 Federated Electronics, Ltd. has hired a Washington, D.C.-based global trade research firm to assess the probabilities of each combination of port and government conditions for the five ports. The research firm probability estimates for the five ports are as follows: Negative/ Government Negative/ Government Positive/ Government Positive/ Government Negative Positive Negative Positive Shanghai Singapore Pusan Kaoshiung Hong Kong (a) Using expected value, determine the best port to construct the distribution center. (b) Using any decision criterion, determine the port you think would be the best location for the distribution center, and justify your answer. A- 19. The management of State Union Bank was concerned about the potential loss that might occur in the event of a physical catastrophe

15 BMAppendixA.indd Page /03/14 9:46 PM user 606 Appendix such as a power failure or a fire. The bank estimated that the loss from one of these incidents could be as much as $100 million, including losses due to interrupted service and customer relations. One project the bank is considering is the installation of an emergency power generator at its operations headquarters. The cost of the emergency generator is $900,000, and if it is installed no losses from this type of incident will be incurred. However, if the generator is not installed, there is a 10% chance that a power outage will occur during the next year. If there is an outage, there is a 0.04 probability that the resulting losses will be very large, or approximately $90 million in lost earnings. Alternatively, it is estimated that there is a 0.96 probability of only slight losses of around $2 million. Using decision tree analysis, determine whether the bank should install the new power generator. A-20. Allegheny Mountain Power and Light is an electric utility company with a large fleet of vehicles including automobiles, light trucks, and construction equipment. The company is evaluating four alternative strategies for maintaining its vehicles at the lowest cost: (1) take no preventive maintenance at all and repair vehicle components when they fail; (2) take oil samples at regular intervals and perform whatever preventive maintenance is indicated by the oil analysis; (3) change the vehicle oil on a regular basis and perform repairs when needed; and (4) change the oil at regular intervals and take oil samples regularly, performing maintenance repairs as indicated by the sample analysis. For autos and light trucks, strategy 1 (no preventive maintenance) costs nothing to implement and results in two possible outcomes: There is a 0.08 probability that a defective component will occur, requiring emergency maintenance at a cost of $1,600, or there is 0.92 probability that no defects will occur and no maintenance will be necessary. Strategy 2 (take oil samples) costs $40 to implement (i.e., take a sample), and there is a 0.08 probability that there will be a defective part and 0.92 probability that there will not be a defect. If there is actually a defective part, there is a 0.70 probability the sample will correctly identify it, resulting in preventive maintenance at a cost of $500. However, there is a 0.30 probability that the sample will not identify the defect and indicate everything is okay, resulting in emergency maintenance later at a cost of $1,600. On the other hand, if there are actually no defects, there is a 0.20 probability that the sample will erroneously indicate that there is a defect, resulting in unnecessary maintenance at a cost of $250. There is a 0.80 probability that the sample will correctly indicate there are no defects, resulting in no maintenance and no costs. Strategy 3 (changing the oil regularly) costs $34.80 to implement and has two outcomes: a 0.04 probability of a defective component, which will require emergency maintenance at a cost of $1,600, and a 0.96 probability that no defects will occur, resulting in no maintenance and no cost. Strategy 4 (changing the oil and sampling) costs $54.80 to implement and results in the same probabilities of defects and no defects as strategy 3. If there is a defective component, there is a 0.70 probability that the sample will detect it and $500 in preventive maintenance costs will be incurred. Alternatively, there is a 0.30 probability that the sample will not detect the defect, resulting in emergency maintenance at a cost of $1,600. If there is no defect, there is a 0.20 probability the sample will indicate there is a defect, resulting in an unnecessary maintenance cost of $250, and a 0.80 probability that the sample will correctly indicate no defects, resulting in no cost. Develop a decision strategy for Allegheny Mountain Power and Light and indicate the expected value of this strategy. 1 A-21. In Problem A-20, the decision analysis is for automobiles and light trucks. Allegheny Mountain Power and Light would like to reformulate 1 This problem is based on J. Mellichamp, D. Miller, and O-J. Kwon, The Southern Company Uses a Probability Model for Cost Justification of Oil Sample Analysis, Interfaces 23, issue 3 (May June 1993): pp the problem for its heavy construction equipment. Emergency maintenance is much more expensive for heavy equipment, costing $15,000. Required preventive maintenance costs $2,000 and unnecessary maintenance costs $1,200. The cost of an oil change is $200 and the cost of taking an oil sample and analyzing it is $50. All the probabilities remain the same. Determine the strategy the company should use for its heavy equipment. A-22. Tech is playing State in the last conference game of the season. Tech is trailing State 21 to 14 with 7 seconds left in the game, when they score a touchdown. Still trailing 21 to 20, Tech can either go for two points and win or go for one point to send the game into overtime. The conference championship will be determined by the outcome of this game. If Tech wins they will go to the Sugar Bowl, with a payoff of $9.2 million; if they lose they will go to the Gator Bowl, with a payoff of $1.5 million. If Tech goes for two points there is a 30% chance they will be successful and win (and a 70% chance they will fail and lose). If they go for one point there is a 0.98 probability of success and a tie and a 0.02 probability of failure. If they tie they will play overtime, in which Tech believes they have only a 20% chance of winning because of fatigue. a. Use decision tree analysis to determine if Tech should go for one point or two points. b. What would Tech s probability of winning the game in overtime have to be to make Tech indifferent between going for one point or two points? A -23. Mary Decker is suing the manufacturer of her car because of a defect that she believes caused her to have an accident, and kept her out of work for a year. She is suing the company for $3.5 million. The company has offered her a settlement of $700,000, of which Mary would receive $600,000 after attorneys fees. Her attorney has advised her that she has a 50% chance of winning her case. If she loses she will incur attorneys fees and court costs of $75,000. If she wins she is not guaranteed her full requested settlement. Her attorney believes that if she wins, there is a 50% chance she could receive the full settlement, in which case Mary would get $2 million after her attorney takes his cut, and a 50% chance that the jury will award her a lesser amount of $1,000,000, of which Mary would get $500,000. Using decision tree analysis, decide if Mary should sue the manufacturer. A-24. State University has three healthcare plans for its faculty and staff to choose from, as follows: Plan 1 monthly cost of $32 with a $500 deductible; the participants pay the first $500 of medical payments for the year, the insurer pays 90% of all remaining expenses. Plan 2 monthly cost of $5 but a deductible of $1,200, with the insurer paying 90% of medical expenses after the insured pays the first $1,200 in a year. Plan 3 monthly cost of $24 with no deductible, the participants pay 30% of all expenses with the remainder paid by the insurer. Tracy McCoy, an administrative assistant in the management department, estimates that her annual medical expenses are defined by the following probability distribution. Annual Medical Expenses Probability $ , , , , Determine which medical plan Tracy should select.

16 BMAppendixA.indd Page /03/14 9:46 PM user Appendix A: Operational Decision-Making Tools Decision Analysis 607 A-25. The Orchard Wine Company purchases grapes from one of two nearby growers each season to produce a particular red wine. It purchases enough grapes to produce 3,000 bottles of the wine. Each grower supplies a certain portion of poor-quality grapes that will result in a percentage of bottles being used as fillers for cheaper table wines according to the following probability distribution. Probability of % Defective Percentage Defective Grower A Grower B The two growers charge a different price for their grapes and because of differences in taste, the company charges different prices for their wine depending on which grapes they use. The annual profit from the wine produced from each grower s grapes for each percentage defective is as follows: Profit Defective Grower A Grower B 2% $44,200 $42, ,200 40,300 Profit Defective Grower A Grower B 6 36,200 38, ,200 35, ,200 33,400 Use decision tree analysis to determine from which grower the company should purchase grapes. A- 26. Huntz Food Products is attempting to decide if it should introduce a new line of salad dressings called Special Choices. The company can test market the salad dressings in selected geographic areas or bypass the test market and introduce the product nationally. The cost of the test market is $150,000. If the company conducts the test market, it must wait to see the results before deciding whether or not to introduce the salad dressings nationally. The probability of a positive test market result is estimated to be 0.6. Alternatively, the company cannot conduct the test market and make the decision to introduce the dressings or not. If the salad dressings are introduced nationally and are a success, the company estimates it will realize an annual profit of $1,600,000 while if the dressings fail it will incur a loss of $700,000. The company believes the probability of success for the salad dressings is 0.50 if it is introduced without the test market. If the company does conduct the test market and it is positive, the probability of successfully introducing the salad dressings increases to 0.8. If the test market is negative and the company introduces the salad dressings anyway, the probability of success drops to Using decision tree analysis, determine if the company should conduct the test market. CASE STUDY Evaluating Projects at Nexcom Systems Nexcom Systems develops information technology systems for commercial sale. Each year it considers and evaluates a number of different projects to undertake. It develops a road map for each project in the form of a decision tree that identifies the different decision points in the development process from the initial decision to invest in a project s development through the actual commercialization of the final product. The first decision point in the development process is whether or not to fund a proposed project for one year. If the decision is no, then there is no resulting cost; if the decision is yes, then the project proceeds at an incremental cost to the company. The company establishes specific short-term, early technical milestones for its projects after one year. If the early milestones are achieved, the project proceeds to the next phase of project development; if the milestones are not achieved, the project is abandoned. In its planning process, the company develops probability estimates of achieving and not achieving the early milestones. If the early milestones are achieved, then the project is funded for further development during an extended time frame specific to a project. At the end of this time frame, a project is evaluated according to a second set of (later) technical milestones. Again the company attaches probability estimates for achieving and not achieving these later milestones. If the late milestones are not achieved, the project is abandoned. If the late milestones are achieved, this means that technical uncertainties and problems have been overcome and the company next assesses the project s ability to meet its strategic business objectives. At this stage the company wants to know if the eventual product coincides with the company s competencies, and if there appears to be an eventual clear market for the product. It invests in a product prelaunch to ascertain the answers to these questions. The outcomes of the prelaunch are that either there is a strategic fit or there is not, and the company assigns probability estimates to each of these two possible outcomes. If there is not a strategic fit at this point, the project is abandoned and the company loses its investment in the prelaunch process. If it is determined that there is a strategic fit, th e n three possible decisions result. (1) The company can invest in the product s launch and a successful or unsuccessful outcome will result, each with an estimated probability of occurrence. (2) The company can delay the product s launch and

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