SUPPLEMENTARY CHAPTER E Decision Analysis · courses of action—choice of cleaning agent, whether...

Chapter Outline

Applying Decision Analysis Tools

Structuring Decision ProblemsSelecting Decision Alternatives

One-Time Decisions Without EventProbabilities

Repeated Decisions With EventProbabilities

Expected Value of PerfectInformation

Decision Trees

OM Spotlight: How ComputersPlay Chess

OM Spotlight: Collegiate AthleticDrug Testing

Solved ProblemsKey Terms and ConceptsQuestions for Review and

DiscussionProblems and ActivitiesCases

Trendy’s PiesService Guarantee Decisions

for McCord HotelsEndnotes

Learning Objectives

• To identify characteristics of management decisions where decisionanalysis techniques are used and to define the elements of a decision problem.

• To evaluate risk in making decisions and apply decision criteria toselect an appropriate decision alternative.

• To construct simple decision trees and use them to select optimalexpected value decisions.

Decision AnalysisSUPPLEMENTARYCHAPTER E

Decision analysis is the formal study of how people make decisions, particularly whenfaced with uncertain information, as well as a collection of techniques to supportthe analysis of decision problems. For example, the manufacturer of a new style orline of seasonal clothing would like to manufacture large quantities of the productif consumer acceptance and, consequently, demand for the product are going to behigh. Unfortunately, the seasonal clothing items require the manufacturer to makea production-quantity decision before the actual demand is known. Most decisionsthat we face in business and in our personal lives require a choice in the face of anuncertain future.

Decision analysis has many applications in product selection, facility capacityexpansion and location, inventory analysis, technology and process selection, andother areas of operations management. The two opening episodes are some exam-ples. In fact, Virgil Carter, a former NFL quarterback, and Robert Machol applieddecision analysis to evaluate football strategies. They found, for example, that theexpected value of having the ball with first down and 10 yards to go varies by fieldposition. If the ball is close to one’s own goal line, then the team’s expected scoringvalue is �1.64, indicating that their opponent is more likely to score as a result ofgetting the ball back in good field position. As field position moves closer to theopponent’s goal line, the expected value becomes positive and increases. A furtheranalysis of field goal attempts showed that inside the 30-yard line, the run is pre-ferred to the field goal attempt if there are 1 or 2 yards to go, and possibly with3. Inside the 10-yard line, the run is preferred to the field goal attempt with up to5 yards to go. These results were contrary to practice, but many coaches contin-ued to employ the field goal far more than the analysis indicated.2

E2

• “What do you think we should do? We’re down by 10 with 5 minutes left—plenty of time to get the ball back,” pondered Ken Kendall, head coach ofWest High in talking to offensive coach Craig Russell. West was facing fourthdown and short yardage for another first down from their opponent’s 9-yardline. “Should we try for the first down or go for the field goal?” Craig notedthat statistically a run is better than a field goal attempt inside the 10-yard line.Ken wasn’t so sure, trying to weigh the risk of not getting the first down or atouchdown instead of an almost sure field goal.

• Electric utilities face decisions that can have important impacts on the environ-ment. The impacts stem from the by-products of combustion and otherchemicals, equipment, and processes that utilities use to produce electricity.For example, utilities use large boilers to boil water and make steam to gener-ate electricity. The cleaning process results in a waste solution that may behazardous. Whether or not the waste stream will be hazardous is uncertain,as are the costs and effects of the various management strategies. Severalcourses of action—choice of cleaning agent, whether or not to include a pre-rinse stage, treatment and disposal method, and cleaning frequency—are avail-able. Using techniques of decision analysis, the consulting firm Decision FocusIncorporated developed a strategy that would save a utility $119,000 for oneboiler over a 20-year horizon.1

Decision analysis is the formalstudy of how people makedecisions, particularly whenfaced with uncertaininformation, as well as acollection of techniques tosupport the analysis ofdecision problems.

Supplementary Chapter E: Decision Analysis E3

Learning ObjectiveTo identify characteristics ofmanagement decisions wheredecision analysis techniques areused and to define the elementsof a decision problem.

APPLYING DECISION ANALYSIS TOOLSDecision analysis tools should not be used in every decision situation. Characteristicsof management decisions where decision analysis techniques apply are summarizedas follows:3

1. They must be important. Decision analysis techniques would not be appropri-ate for minor decisions where the consequences of a mistake are so small thatit is not worth our time to study the situation carefully. The consequences ofmany decisions, such as building a major facility, are not felt immediately butmay cover a long time period.

2. They are probably unique. Decisions that recur can be programmed and thendelegated. But the ones that are unusual and perhaps occur only one time can-not be handled this way.

3. They allow some time for study. For example, decision analysis techniqueswould not be useful in making a decision in the emergency room or when a jetfighter flames out during takeoff.

4. They are complex. Practical decision problems involve multiple objectives, requir-ing the evaluation of trade-offs among the objectives. For example, in evaluat-ing routes for proposed pipelines, a decision maker would want to minimizeenvironmental impact, minimize health and safety hazards, maximize economicbenefit, and maximize social impact. Decisions involve many intangibles, suchas the goodwill of a client, employee morale, and governmental regulations, andmay involve several stakeholders. For instance, to build a plant in a new area,corporate management may require approval from stockholders, regulatoryagencies, community zoning boards, and perhaps even the courts. Finally, mostdecisions are closely allied to other decisions. Choices today affect both thealternatives available in the future and the desirability of those alternatives.Thus, a sequence of decisions must often be made.

5. They involve uncertainty and risk. Uncertainty refers to not knowing what willhappen in the future. An advertising campaign may fail, a reservoir may break,or a new product may be a complete failure. Uncertainty is further complicatedwhen little or no data are available, or some data are very expensive or time-consuming to obtain. Faced with such uncertainties, different people view thesame set of information in different ways. Risk is the uncertainty associated withan undesirable outcome, such as financial loss. To appreciate the importance ofrisk, consider the fact that it takes hundreds of millions of dollars and about10 years for a pharmaceutical company to bring a drug to market. Once there,seven of ten products fail to return the company’s cost of capital. Decisions in-volving capital investment and continuation of research over the long develop-ment cycle do not lend themselves to traditional financial analysis.4

Structuring Decision ProblemsTo illustrate the process of defining a decision problem, we present an example of amedium-size producer of industrial chemical products, Commonwealth ChemicalsCompany, that is facing a decision about capacity expansion. The company hasrecently developed a new synthetic industrial lubricant that will increase tool lifefor machining operations in metal-fabrication industries. A new factory would benecessary to produce the lubricant on a large scale, but expanding the existingfacilities would allow production on a smaller scale.

Managers are uncertain which decision to choose. Clearly, the best decisiondepends on future demand. If the demand for the product is high, the expansion

Uncertainty refers to notknowing what will happen inthe future.

Risk is the uncertaintyassociated with anundesirable outcome, such asfinancial loss.

alternative will not provide enough capacity to meet all the demand and profits willbe lost. If demand is low, and a new factory is built, the excess capacity will sub-stantially reduce the return on investment. With an unstable economy, it is diffi-cult to predict actual demand for the product.

The first step in structuring a decision problem is to define the decision alter-natives. Decision alternatives represent the choices that a decision maker can make.In this case, the alternatives are whether to expand the existing plant or to build anew factory. Let

d1 � decision to expand the existing plantd2 � decision to build a new plant

The second step is to define the events that might occur after a decision is made.Events represent the future outcomes that can occur after a decision is made andthat are not under the control of the decision maker. For each combination ofproduction-volume decision and subsequent event, a payoff can be computed. Forinstance, if the manufacturer decides to produce 10,000 units, but demand is low,the manufacturer will incur the cost of producing the 10,000 units but will receiverevenue for sales of only 5,000; the remaining units will have to be disposed of ata loss. On the other hand, if sales are medium or high, all 10,000 units will be sold,and the net profit can be computed. The payoff would be the net profit.

For instance, in deciding to expand an existing plant or build a new one, Com-monwealth Chemicals needs to consider the future demand for the product. Dif-ferent possible levels of demand represent the events. Demand might be expressedquantitatively in sales units or dollars. In this example, events might be designatedas “high demand,” “medium demand,” and “low demand.” Alternatively, theymight be quantified as “demand estimated as 15,000 units,” “demand estimated as10,000 units,” and “demand estimated as 5,000 units.” If you are planning a springbreak vacation to Florida in January, you might define events as the weather thatyou might encounter. Uncertain weather-related outcomes might be defined quali-tatively, for example, sunny and warm, sunny and cold, rainy and warm, or rainyand cold. For the Commonwealth Chemicals decision problem, we will define theevents as

s1 � low product demands2 � high product demand

Next, we need well-defined decision criteria on which to evaluate potential op-tions. Decision criteria might be net profit, customer service, cost, social benefits,or any other measure of output that may be appropriate for the particular situa-tion being analyzed. A numerical value associated with a decision coupled withsome event is called a payoff. Using the best information available, the managers ofCommonwealth Chemicals have estimated the payoffs, expressed as profits, shownin Exhibit E.1. A table of this form is referred to as a payoff table. The notationwe use for the entries in the payoff table is V(di, sj), which denotes the payoff, V,associated with decision alternative di and event sj. Using this notation, we see thatV(d2, s1) � $100,000.

E4 Supplementary Chapter E: Decision Analysis

Decision alternatives representthe choices that a decisionmaker can make.

Events represent the futureoutcomes that can occur aftera decision is made and thatare not under the control ofthe decision maker.

Possible Future Events

Decision Alternative Low Product Demand (s1) High Product Demand (s2)

Expand existing plant (d1) $200,000 $300,000Build new plant (d2) $100,000 $450,000

Exhibit E.1Payoff Table for CommonwealthChemicals

A numerical value associatedwith a decision coupled withsome event is called a payoff.


Learning ObjectiveTo evaluate risk in makingdecisions and apply decisioncriteria to select an appropriatedecision alternative.

In many decision problems, the probabilities of events can be estimated, eitherfrom historical data or managerial judgment. Knowing the likelihood of the occur-rence of events helps to assess risk when making a decision. In some cases, how-ever, event probabilities may not be available or appropriate to try to assess. Wewill provide examples of both situations in the following sections.

In summary, the elements of a decision problem are (1) decision alternatives,(2) events, (3) estimated payoffs for each combination of decision alternatives andevents, and possibly (4) probabilities of the events.

SELECTING DECISION ALTERNATIVESMaking decisions with uncertain future consequences is often quite frustrating anda source of anxiety for individuals and managers alike. We run the risk that anydecision we choose may result in undesirable consequences once we see what thefuture holds in store. There are two principal ways of viewing a decision strategy,and these depend on the frequency with which the decision will be made. For one-time decisions, managers must take into account the risk associated with makingthe wrong decision. However, for decisions that are repeated over and over, man-agers can choose decisions based on the expected payoffs that might occur.

One-Time Decisions Without Event ProbabilitiesThe Commonwealth Chemicals decision is clearly a one-time decision. So howshould the choice be made? Different criteria can be used to reflect different atti-tudes toward risk, and they may result in different decision recommendations. Fora problem in which the payoff is profit, as it is in the Commonwealth Chemicalsproblem, three common criteria are

1. Maximax—choose the decision that will maximize the maximum possible profitamong all events. This is an aggressive, or risk-taking, approach.

2. Maximin—choose the decision that will maximize the minimum possible profitamong all events. This is a conservative, or risk-averse, approach.

3. Minimax regret—choose the decision that will minimize the maximum oppor-tunity loss associated with the events. Opportunity loss represents the regret, orill-feeling, that people often have after making a nonoptimal decision (“I shouldhave bought that stock years ago . . .”). This approach is neither aggressive norconservative, but focuses on not erring too much in either direction.

We will apply these criteria for the Commonwealth Chemicals problem. For themaximax criterion, we see that if d1 is selected, the maximum payoff is $300,000,and it occurs for s2. If d2 is selected, the maximum payoff is $450,000, also for s2.The decision maker should choose d2, build a new plant, since it results in thelargest possible payoff.

For the maximin criterion, we see that if d1 is chosen, the minimum payoff is$200,000, whereas if d2 is selected, the minimum payoff is $100,000. Thus, to max-imize the minimum payoff, the decision maker should choose d1, expand the exist-ing plant.

To apply the minimax-regret criterion, we must first construct a regret or opportunity-loss matrix. The opportunity loss associated with a particular decision,di, and state of nature, sj, is the difference between the best payoff that the deci-sion maker can receive by making the optimal decision d* corresponding to sj,V(d*, sj), and the payoff for choosing any arbitrary decision di and having sj occur,V(di, sj). For example, if we know that s1 will occur, the best decision is to choose

d* � d1 and receive a payoff of $200,000; the opportunity loss will be zero. If wechoose d2, we will receive only $100,000 and will lose the opportunity to receive$200,000 � $100,000 � $100,000. Similarly, if we know that s2 will occur, thebest decision is d* � d2; an opportunity loss of $450,000 � $300,000 � $150,000will occur if we choose d1. Exhibit E.2 shows the complete opportunity-loss matrixfor this situation. We see that the smallest maximum opportunity loss occurs ford2, so using this criterion, Commonwealth should build the new plant.

We see that different criteria can result in different decisions; which to use ispurely a judgment call on the part of the decision maker and reflects the person’svalues and attitudes toward risk.

For problems in which the payoff is cost, the criteria change somewhat. Theaggressive decision criterion is minimin—minimize the minimum possible payoffover all events. The conservative decision is minimax—minimize the maximum pos-sible payoff over all events. Finally, the minimax-regret criterion does not change,since opportunity loss is always a cost. However, care is needed in computing theopportunity loss correctly. It is still the difference between the best possible payoff(received by making the optimal decision) and the payoff of any other decision. Theonly difference when the output measure is cost is that the “best” payoff is the low-est cost, not the highest profit. The difference must be viewed as an absolute value,that is, the savings in cost, since it does not make sense for opportunity losses tobe negative.

Repeated Decisions With Event ProbabilitiesIf an individual or business faces the same decision problem repeatedly, then overthe long run, the decision can be made based on expected value. The expected value

approach is to select the decision alternative with the best expected payoff. Theexpected value criterion requires probability estimates for the events. In many sit-uations, good probability estimates can be developed from historical data or judg-mentally. Let

P(sj) � probability of occurrence for event sj

N � number of events

Since one and only one of the N states of nature can occur, the associated proba-bilities must satisfy these two conditions:

P(sj) � 0 for all jP(sj) � P(s1) � P(s2) � . . . � P(sN) � 1

The expected value for decision alternative di is given by

EV(di) � �jP(sj)V(di, sj) (E.1)

The EV criterion is used in revenue management applications (see Chapter 10).Most airlines, for example, offer discount fares for advanced purchase. Assume thatonly two fares are available: full and discount. The airline must make the decisionof whether or not to accept the next request for a discount seat. If it accepts thediscount request, the revenue it earns is the discount fare. If it rejects the discountrequest, two outcomes are possible. First, the seat may remain empty and the air-


Low Product High Product MaximumDecision Demand (s1) Demand (s2) Opportunity Loss

Expand existing plant (d1) 0 $150,000 $150,000Build new plant (d2) $100,000 0 $100,000

Exhibit E.2Opportunity-Loss Matrix forCommonwealth Chemicals

The expected value approach

is to select the decisionalternative with the bestexpected payoff.

line will not realize additional revenue. Alternatively, the remaining seat may befilled by a full-fare passenger, either because full-fare passenger demand is suffi-cient to fill the seats or because discount-fare passengers choose to pay full farewhen told the discount fare is not available.

This decision situation is illustrated by an example in Exhibit E.3. Suppose thata full-fare ticket is $560 and the discount fare is $400. The decision depends onthe probability, p, of getting a full-fare request when a discount request is rejected.The expected value of rejecting the discount seat request is p times the full-farevalue. Thus, if p � .75, the expected value of rejecting the discount request is.25(0) � .75 ($560) � $420. Since this is higher than the discount fare, the dis-count request should be rejected. Since an airline makes hundreds or thousandsof such decisions each day, the expected value criterion is appropriate.

Expected Value of Perfect InformationBy perfect information, we mean knowing in advance what state of nature will oc-cur. Although we never have perfect information in practice, it is worth knowinghow much we could improve the value of our decision if we had such information.This is called the expected value of perfect information, or EVPI, which is the differencebetween the expected payoff under perfect information and the expected payoff ofthe optimal decision without perfect information. We compute EVPI by asking thefollowing question: If each event occurs, what would be the best decision and pay-off? Then we weight these payoffs by the probabilities associated with the eventsto obtain the expected payoff under perfect information.

Suppose the airline somehow knew in advance that it could not sell the full-fareticket to a particular customer (perhaps based on demographic profiles and analy-sis of past behavior). Then clearly the best decision would be to accept the discountrequest and receive revenue of $400. On the other hand, if it knows that it can sellthe full-fare ticket, then obviously it should reject the request and receive $560.However, on average, we know that only 75 percent of customers will buy the full-fare ticket if the request is rejected and 25 percent will not. So the expected valueof having perfect information would be

(.75)(560) � (.25)(400) � $520

Recall that without the perfect information, the best decision is to always choosed1, which has an expected value of $420. By having perfect information about whata particular customer might do, we see that the value of the expected payoff canbe increased by

$520 � 420 � $100

This difference is the expected value of perfect information (EVPI), and it repre-sents the maximum amount the company should be willing to pay for any infor-mation about the events, no matter how good it is. In this case, we might interpretit as the maximum incentive that the airline might give to a customer who is un-willing to purchase the full-fare ticket.


Events

Sell Do Not SellDecision Full-Fare Ticket Full-Fare Ticket Expected Value

Reject request $560 $0 $560 � .75 � $420Accept request $400 $400 $400Probability of event .75 .25

Exhibit E.3Airline Discount-Fare RequestDecision

Expected Value of Perfect

Information, or EVPI, which is the difference between theexpected payoff under perfectinformation and the expectedpayoff of the optimal decisionwithout perfect information.

DECISION TREESDecision problems can be depicted graphically using a decision tree. A decision tree

is a graphical schematic of the logical order with which decisions are made andevents occur. In the terminology associated with decision trees, nodes refer to theintersections, or junction points, of the tree. Arcs are the connectors between thenodes. Arcs are sometimes called branches. When the branches leaving a given nodeare decision branches, we refer to the node as a decision node. Decision nodes areusually denoted by squares. Similarly, when the branches leaving a given node areevent branches, we refer to the node as an event node. Event nodes are denoted bycircles. The number at each endpoint of the tree represents the payoff associatedwith a particular chain of events.

Exhibit E.4 is a decision tree of the airline fare request decision. Note that thetree shows the natural, or logical, progression of the decision-making process. First,the firm must make its decision (d1 or d2); then, once the decision is implemented,an event (s1 or s2) occurs. Note that in this case, if the request is accepted it doesnot matter if the airline could have sold the full fare or not, so we do not have toinclude event branches for this decision. The OM Spotlight: How Computers PlayChess provides another interesting application of decision trees.


Learning ObjectiveTo construct simple decisiontrees and use them to selectoptimal expected valuedecisions.

A decision tree is a graphicalschematic of the logical orderwith which decisions aremade and events occur.

Nodes refer to theintersections, or junctionpoints, of the tree.

Arcs are the connectorsbetween the nodes.

Reject request

Accept request

Sell full fare

Do not sell

$560

$0

$400

P � 0.75

1 � p � .25

Decision Event PayoffExhibit E.4Airline Discount-Fare RequestDecision

Exhibit E.5How Computers Play Chess

Decision trees are useful for more complex business decisions. For example, anationwide restaurant franchise that frequently introduces new products mightdevelop the decision tree shown in Exhibit E.6 to help make a decision on howto best market the products. Even if the tree is not used analytically to evaluateexpected payoffs, it can be of substantial benefit in helping decision makers to log-ically determine what decisions need to be made and how to react to external forcessuch as competitor strategies or economic changes beyond their control.

Expected value calculations can be made directly on the tree to arrive at the bestdecision strategy. Working backward through the decision tree, we first compute


Exhibit E.6New Product IntroductionDecision Tree

O M S P O T L I G H T

How Computers Play Chess5

Humans play chess by learning thenuances of the game, reading booksand learning patterns of play, and ul-timately develop strategies and tac-

tics about how they will play. When computers play chessthey do none of this. In fact, the computer is calculatingstrategies through sets of formulas based in full or in parton decision trees. Exhibit E.5 shows a decision tree for thestart of a chess game.

Assume there are 20 possible moves to begin the gamefor the white chess pieces. The player chooses from those 20moves and makes the move. Then the player with the black

pieces must choose among 20 possible moves. When blackmoves, white must respond, and so on. For just a sequenceof three moves as shown in the figure, there are 20 � 20� 20 � 8,000 possible combinations. If you were to carrythese computations out for a typical chess game, you end upwith at least 10120 moves, give or take a few. No computeror computers are ever going to be able to calculate the en-tire tree, so complex algorithms are used to prune the treebranches and only evaluate the most promising branchesof the decision tree. So what some people think is intelli-gence is completely mechanical and involves no thoughtwhatsoever—hence the name artificial intelligence!

the expected monetary value of each event node by weighting the possible payoffsby their chances of occurrence. In the airline fare example, the expected value forthe event node corresponding to the reject request decision is

EV � .75 (560) � .25 (0) � $420

shown in the box in Exhibit E.7. We continue backward through the tree to thedecision node. At this point we compare the expected value for rejecting the re-quest with the value of the branch associated with accepting the request. Becausethe value for rejecting the request is higher, it corresponds to the best decision.

Many unique uses of decision trees exist, such as the one described in the OMSpotlight: Collegiate Athletic Drug Testing.


Reject request

Accept request

Sell full fare

Do not sell

$560

$0

$400

p � 0.75

1 � p � .25

Decision Event Payoff

$420

$420

Exhibit E.7Calculation of Optimal DecisionStrategy

O M S P O T L I G H T

Collegiate Athletic Drug Testing6

The athletic board of Santa Clara Uni-versity had to decide whether torecommend implementing a drug-testing program for intercollegiate

athletes. One of the board members, who was a manage-ment science professor, developed a simple decision modelto address the question of whether or not to test a singleindividual for the presence of drugs. The model focused onthe key issue of the reliability of the testing procedures, con-sequences of testing errors, and the benefits of identifyinga drug user compared with the costs of false accusationsand nonidentification of users.

Exhibit E.8 shows the decision tree developed for testingan individual for drug use. The two main alternatives are “test”or “don’t test.” The model evaluates the expected cost oftesting for drug use compared with that of not testing. If test-ing is chosen, the test is given and the result, positive or neg-ative, is observed. If the result is positive, action is taken.Since not all those who test positively are actually users, there

is some chance of a false accusation, which costs an amountC1. If the result is negative, then some drug users are notidentified, which costs C2. Nonusers who test negativelymight be expected to experience some cost, C3, perhapsbased on invasion of privacy. Following the lower path of thetree, if the alternative “don’t test” were selected, the ex-pected cost is just the cost of an unidentified user, C2, mul-tiplied by the prior probability that an individual is a drug user.

The model’s results surprised many board members. Forinstance, the model showed that if a test that is 95 percentreliable is applied to a population of 5 percent drug users,only 50 percent of all those who tested positively will actu-ally be drug users. Most board members had read about thereliability of drug tests in various publications and agreed that95 percent reliability was a representative value. As a result,the board concluded that a false accusation was more seri-ous than not identifying drug users and rejected the pro-posal. The university administration later accepted thisrecommendation.


Exhibit E.8Decision Tree for Drug-UseTesting

SOLVED PROBLEMS

SOLVED PROBLEM #1

Maling Manufacturing needs to purchase a new pieceof machining equipment. The two choices are a con-ventional (labor-intensive) machine and an automated(computer-controlled) machine. Profitability will de-pend on demand volume. The following data providean estimate of profits over the next three years.

Demand VolumeDecision Low (s1) High (s2)

Conventional machine (d1) $15,000 $21,000Automated machine (d2) $ 9,000 $35,000

What decisions would be indicated by maximax, max-imin, and minimax-regret criteria?

Solution:

Decision Maximum Profit Minimum Profit

Conventional (d1) $21,000 $15,000Automated (d2) $35,000 $ 9,000

Maximax decision � d2 Maximin decision � d1

Opportunity-Loss MatrixDecision Low High Maximum

Conventional (d1) 0 $14,000 $14,000Automated (d2) $6,000 0 $ 6,000

Minimax-regret decision � d2

SOLVED PROBLEM #2

Martin’s Service Station is considering investing in aheavy-duty snowplow this fall. Martin has analyzed thesituation carefully and feels that this would be a veryprofitable investment if the snowfall is heavy, somewhatprofitable if the snowfall is moderate, and would resultin a loss if the snowfall is light. Specifically, Martin fore-casts a profit of $7,000 if snowfall is heavy and $2,000if it is moderate, and a $9,000 loss if it is light. Fromthe Weather Bureau’s long-range forecast, Martin esti-mates that P(heavy snowfall) � .4, P(moderate snow-fall) � .3, and P(light snowfall) � .3.

a. Prepare a decision tree for Martin’s problem.

b. Using the EV criterion, would you recommend thatMartin invest in the snowplow?

c. Discuss the value of using EV for this situation.

Solution:

a. Exhibit E.9 is the decision tree and the variables aredefined as follows: d1 � invest, d2 � do not invest,s1 � heavy, s2 � moderate, s3 � light, P(s1) � .4,P(s2) � .3, and P(s3) � .3.

b. Recommended decision: d1 (invest) since EV (d1) �.4(7,000) � .3(2,000) � .3(29,000) � $700 and EV(d2) � 0.

c. Although the decision tree helps structure the prob-lem, the fact remains that this is a one-time decision.The expected value criterion does not incorporaterisk into the decision.


KEY TERMS AND CONCEPTS

Arcs/branchesDecision alternativesDecision analysisDecision treesEventsExpected valueExpected value approachExpected value of perfect information (EVPI)Five decision analysis characteristicsMaximax criterion

Maximin criterionMinimax regret criterionNodesOpportunity-loss matrixPayoff matrixRegretRiskThree decision analysis elementsUncertainty

Exhibit E.9Decision Tree for SolvedProblem 2

QUESTIONS FOR REVIEW AND DISCUSSION

1. Describe two situations where decision analysiscould be applied in operations to support manage-ment decision-making.

2. What are five characteristics of management deci-sions for which decision analysis techniques shouldbe used? Do all five characteristics have to exist toapply decision analysis techniques?

3. Define the four major elements of a decision problem.

4. Explain the difference between uncertainty and risk.Provide some examples from OM of each of theseconcepts.

5. What information is provided in a payoff table?

6. How do managers determine the probabilities forexpected value decision-making?

7. Describe how the following criteria are applied to adecision problem in which the objective is maxi-mization:

a. maximaxb. maximinc. minimax regret

8. How do the criteria in Question 7 change if the ob-jective is minimization?

9. Explain the concept of regret, or opportunity loss,in decision analysis.

10. In what situations would the expected monetaryvalue criterion be useful? In what situations wouldit not be useful?

11. What does the expected value of perfect informa-tion provide to a decision maker?

12. Explain the structure and purpose of a decision tree.


PROBLEMS AND ACTIVITIES

1. Suppose a decision maker faced with four decisionalternatives and four states of nature develops theprofit-payoff table shown as follows:

State of NatureDecision s1 s2 s3 s4

d1 14 9 10 5d2 11 10 8 7d3 9 10 10 11d4 8 10 11 13

a. If the decision maker knows nothing about thechances or probability of occurrence of the fourstates of nature, what decision would be indi-cated by the maximax, maximin, and minimax-regret criteria?

b. Which decision criterion do you prefer? Explain.Should the decision maker establish the most ap-propriate decision criterion before analyzing theproblem? Explain.

c. Assume the payoff table provides cost, ratherthan profit, payoffs. What is the recommendeddecision using the optimistic, conservative, andminimax-regret decision criteria?

2. Suppose the decision maker in Problem 1 obtainsinformation that enables these probability estimatesto be made: P(s1) � .5, P(s2) � .2, P(s3) � .2, P(s4)� .1.

a. Use the expected value (EV) criterion to deter-mine the optimal decision.

b. Now assuming the entries in the payoff tableare costs, use the EV criterion to determine theminimum-cost solution.

3. Southland Corporation’s decision to produce a newline of recreational products has resulted in the needto construct either a small plant or a large plant.

The decision as to which size to select depends onthe marketplace reaction to the new product line.To conduct an analysis, marketing managers havedecided to view the possible long-run demand aslow, medium, or high. The payoff table gives theprojected profits in millions of dollars as follows:

Long-Run DemandDecision Low Medium High

Small plant $150 $200 $200Large plant 50 200 500

a. Construct a decision tree for this problem anddetermine the best decisions using the maximax,maximin, and minimax-regret decision criteria.

b. Assume that the best estimate of the probabilityof low long-run demand is .20, of medium long-run demand is .15, and of high long-run demandis .65. What is the best decision using the EVcriterion?

4. Milford Trucking, located in Chicago, has requeststo haul two shipments, one to St. Louis and one toDetroit. Because of a scheduling problem, Milfordwill be able to accept only one of these assignments.The St. Louis customer has guaranteed a return ship-ment, and the Detroit customer has not. Thus, ifMilford accepts the Detroit shipment and cannotfind a Detroit-to-Chicago return shipment, the truckwill return to Chicago empty. The payoff table forprofit is shown as follows:

Return Shipment No Return ShipmentDestination from Detroit (s1) from Detroit (s1)

St. Louis (d1) $2,000 $2,000Detroit (d2) 2,500 1,000

If the probability of a Detroit return shipment is .4,what should Milford do?

5. McHuffter Condominiums, Inc., of Pensacola,Florida, recently purchased land near the Gulf ofMexico and is attempting to determine the size ofthe condominium development it should build there.Three sizes of developments are being considered:small, d1, medium, d2, and large, d3. At the sametime an uncertain economy makes it difficult to as-certain the demand for the new condominiums.McHuffter’s managers realize that a large develop-ment followed by a low demand could be very costlyto the company. However, if McHuffter makes aconservative, small-development decision and thenfinds high demand, the firm’s profits will be lowerthan they might have been. With the three levels ofdemand—low, medium, and high—McHuffter’smanagers prepared the payoff table as follows:

Demand (in Thousands of Dollars)Decision Low Medium High

Small $400 $400 $400Medium 100 600 600Large �300 300 900

If P(low) � .20, P(medium) � .35, and P(high) �.45, what decision is recommended by the EV cri-terion?

6. Construct a decision tree for the McHuffter Condo-miniums problem (Problem 5). What is the expectedvalue at each state-of-nature node? What is the op-timal decision?

7. Construct a decision tree for Solved Problem 1. Sup-pose the probabilities of low- and high-demand vol-ume are estimated to be .7 and .3, respectively. Whatdecision would you recommend?

8. Refer again to the investment problem faced by Mar-tin’s Service Station (Solved Problem 2). Martin canpurchase a blade to attach to his service truck thatcan be used to plow driveways and parking lots.Since this truck would also need to be available tostart cars and do other tasks, Martin would not beable to generate as much revenue plowing snow ifhe elects this alternative, but he would keep his losssmaller if there is light snowfall. Under this alter-native Martin forecasts a profit of $3,500 if snow-fall is heavy, $1,000 if it is moderate, and a loss of$1,500 if snowfall is light.

a. Prepare a new decision tree showing all three al-ternatives.

b. Using the EV approach, what is the optimaldecision?

9. The Gorman Manufacturing Company must decidewhether to purchase a component part from a sup-plier or to manufacture the component at its ownplant. If demand is high, it would be to Gorman’sadvantage to manufacture the component. If de-mand is low, however, Gorman’s unit manufactur-ing cost will be high because of underutilization ofequipment. The projected profit in thousands ofdollars for Gorman’s make-or-buy decision is asfollows:

DemandDecision Low Medium High

Manufacture component $220 $40 $100Purchase component 210 45 70

The states of nature have these probabilities: P(lowdemand) � .35, P(medium demand) � .35, andP(high demand) � .30. Use a decision tree to rec-ommend a decision.

10. A firm produces a perishable food product at a costof $10 per case. The product sells for $15 per case.For planning purposes, the company is consideringpossible demands of 100, 200, and 300 cases. If thedemand is less than production, the excess produc-tion is discarded. If demand is more than produc-tion, the firm, in an attempt to maintain a goodservice image, will satisfy the excess demand with aspecial production run at a cost of $18 per case. Theproduct, however, always sell at $15 per case.

a. Set up the payoff table for this problem.b. If P(100) � .2, P(200) � .2, and P(300) � .6,

should the company produce 100, 200, or 300cases?

11. Sealcoat, Inc. has a contract with one of its cus-tomers to supply a unique liquid chemical productused in the manufacture of a lubricant for airplaneengines. Because of the chemical process Sealcoatuses, batch sizes for the product must be 1,000pounds. The customer has agreed to adjust manu-facturing to the full-batch quantities and will ordereither one, two, or three batches every three months.Since production includes a one-month agingprocess, Sealcoat must make its production (howmuch to make) decision before the customer placesan order. Thus the product demand alternativesare 1,000, 2,000, and 3,000 pounds, but the exactdemand is unknown.

Sealcoat’s manufacturing costs are $150 perpound, and the product sells at the fixed contractprice of $200 per pound. If the customer ordersmore than Sealcoat has produced, Sealcoat hasagreed to absorb the added cost of filling the order


by purchasing a higher-quality substitute productfrom another chemical firm. The substitute product,including transportation expenses, will cost Sealcoat$240 per pound. Since the product cannot be storedmore than two months without spoilage, Sealcoatcannot inventory excess production until the cus-tomer’s next three-month order. Therefore, if thecustomer’s current order is less than Sealcoat hasproduced, the excess production will be reprocessedand will then be valued at $50 per pound.

The decision in this problem is: How muchshould Sealcoat produce given the costs and the pos-sible demands of 1,000, 2,000, and 3,000 pounds?From historical data and analysis of the customer’sfuture demands, Sealcoat has developed the proba-bility distribution for demand as follows.

Demand Probability

1,000 .32,000 .53,000 .2

a. Develop a payoff table for the problem.b. How many batches should Sealcoat produce

every three months?

12. A quality control procedure involves 100-percentinspection of parts received from a supplier. His-torical records show the observed defect rates asfollows.

Percent Defective Probability

0 .151 .252 .403 .20

The cost to inspect 100 percent of the parts receivedis $250 for each shipment of 500 parts. If the ship-ment is not 100-percent inspected, defective partswill cause rework problems later in the productionprocess. The rework cost is $25 per each defectivepart.

a. Complete the payoff table shown here, in whichentries represent the total cost of inspection andreworking.

Percent DefectiveDecision 0 1 2 3

100% inspection $250 $250 $250 $250No inspection ? ? ? ?

b. The plant manager is considering eliminating theinspection process to save the $250 inspectioncost per shipment. Do you support this action?Use EV to justify your answer.

c. Show the decision tree for this problem.

13. The R&D manager of the Beck Company is tryingto decide whether or not to fund a project to de-velop a new lubricant. It is assumed that the projectwill be a major technical success, a minor technicalsuccess, or a failure. The company estimates thevalue of a major technical success as $150,000, sincethe lubricant could be used in a number of productsthe company is making. If the project is a minortechnical success, its value is estimated as $10,000,since Beck feels the knowledge gained will benefitsome other ongoing projects. If the project is a fail-ure, it will cost the company $100,000. Based onthe opinion of the scientists involved and the man-ager’s own subjective assessment, the followingprobabilities are assigned:

P(major success) � .15P(minor success) � .45P(failure) � .40

a. According to the EV criterion, should the pro-ject be funded?

b. Suppose a group of expert scientists from a re-search institute could be hired as consultants tostudy the project and make a recommendation.If this study would cost $30,000, should theBeck Company hire the consultants?

14. Consider again the problem faced by the Beck Com-pany R&D manager (Problem 13). Suppose an ex-periment can be conducted to shed some light onthe technical feasibility of the project. There arethree possible outcomes of the experiment:

I1 � prototype lubricant works well at all temperatures

I2 � prototype lubricant works well only at temperatures above 10°F

I3 � prototype lubricant does not work well atany temperature

How would the decision tree be modified to includethis information?

15. Explain how decision analysis techniques can be im-plemented on spreadsheets. Design spreadsheets forthe examples in this chapter.



CASES

TRENDY’S PIES

Trendy’s is a national chain specializing in selling pies,either whole or by the slice, from small facilities withdrive-through capabilities. Trendy’s corporate kitchenstaff has developed a new type of pie and needs to makea decision on whether to introduce it nationally acrossthe chain or to try a regional test market first. TinaTrendy, the franchise founder, and her staff sketchedout the decision tree described earlier in the chapter inExhibit E.6.

Based on various research reports and industryknowledge and judgment, Trendy and her staff came upwith the following financial estimates and risk proba-bilities. If they decide to roll the product out nationally,they would incur costs of $200,000. A high consumerresponse would result in expected revenues of $700,000,with a .6 probability; whereas a low consumer responsewould result in only $150,000 of revenue, with a .4probability. If they first introduce the product in a re-gional test market, they would incur $30,000 in costsand expect a 70 percent chance of a high regional re-sponse and a 30 percent chance of a low regional re-sponse. Regardless of the outcome, they still have tomake a decision whether to remain regional with theproduct (thereby avoiding potential risks of nationalfailure), market the product nationally, or drop the idea.If the regional response is high, they anticipate that re-

maining regional would result in revenues of $200,000;remaining regional with a low regional response wouldresult in revenues of only $100,000. If they decide tomarket nationally after a high regional test market re-sponse, Trendy estimates that there is a .9 probability ofa high national response that would result in revenuesof $700,000 and a .1 probability of a low national re-sponse with revenues of $150,000. If they market na-tionally after a low regional test market response, theprobability of a high national response is only .05; theprobability of a low national response would be .95(revenue estimates would remain the same).

a. Use these cost, revenue, and probability estimatesalong with the decision tree to identify the bestdecision strategy for Trendy’s Pies.

b. Suppose that Trendy is concerned about herprobability estimates of the consumer responseto the regional test market. Although her esti-mates are .7 for a high response and .3 for a lowresponse, she is not very confident of these val-ues. Determine how the decision strategy wouldchange if the probability of a high responsevaries from .1 to .9 in increments of .1. Howsensitive is the best strategy in part a to this prob-ability assumption?

SERVICE GUARANTEE DECISIONS FOR MCCORD HOTELS

McCord Hotels is a small chain of 25 hotels located infour states—Indiana, Kentucky, Ohio, and West Vir-ginia. The hotel prides itself on superior customer ser-vice and is well known in the region. Gregory Hamlet,the chief executive officer of McCord Hotels, is con-sidering whether to offer a service guarantee at all Mc-Cord hotels. If he decides to offer a service guaranteedue to service upsets and mistakes, he will also have todecide whether to incur the cost of service-recoverytraining. This type of training develops a list of the top20 types of service upsets and trains all employees whattheir response should be. The service-recovery trainingprogram includes studying training manuals, in-classexercises and videos, and out-of-class reading for allemployees. The probabilities for states of nature areshown in Exhibit E.10.

Revenue-producing room-nights for all 25 hotels foreach payoff scenario are shown in Exhibit E.11. The pay-off matrix is stated in total annual room-nights becausethe hotel reservation system and pricing policies are very

Variable/State of Nature Probabilities

Adopt service guarantees (ASG) —No service guarantees (NSG) —Do service recovery training (DST) .67No service recovery training (NST) .33High Demand (HD) .60Low Demand (LD) .40

Exhibit E.10Probabilities for McCord Hotels Service Guarantee Decision

consistent in all 25 hotel properties. A room-night is de-fined as one customer staying one night in a hotel room.Therefore, a room-night generates revenue; if the roomis empty, it generates zero revenue. Hotel rooms repre-sent perishable service capacity that is time-dependent.Service guarantees were described in Chapter 6. If Mc-Cord Hotels does not adopt a service guarantee, it is

a. Construct a decision tree to help make this decision.Using expected value, what decision does it support?Explain.

b. If the average room-night was valued at $80 of rev-enue, total annual service-guarantee training cost forall 25 hotels at a cost of $312,500, new marketingand advertising materials at a cost of $200,000 an-nually, and the economic loss from service upset pay-outs were estimated to average less than $150,000per year, evaluate the economics of the situation.What other costs might be considered?

c. What are your final recommendations to Hamlet?


Incremental AnnualDecision Tree Branches Room-Night Payoffs

ASG—DST—HD 75,000ASG—DST—LD 20,000ASG—NST—HD �50,000ASG—NST—LD �15,000NSG �2,500

Exhibit E.11 Room-Night Payoff Matrix for McCord Hotels

ENDNOTES

1 Balson, William E., Welsh, Justin L., and Wilson, Donald S., “Using Decision Analysis and Risk Analysis to Manage Utility EnvironmentalRisk,” Interfaces 22, no. 6, November–December 1992, pp. 126–139.2 Carter, Virgil, and Machol, Robert E., “Optimal Strategies on Fourth Down,” Management Science 24, no. 16, December 1978, pp.1758–1762.3 Baird, Bruce F., Managerial Decisions Under Uncertainty, New York: John Wiley & Sons, 1989, p. 6; and Keeney, Ralph L., “DecisionAnalysis: An Overview,” Operations Research 30, no. 5, September–October 1982, pp. 803–838.4 Nichols, Nancy A., “Scientific Management at Merck: An Interview with CFO Judy Lewent,” Harvard Business Review January–February1994, pp. 89–99.5 Brain, M., “How Chess Computers Work,” http://www.ibs.howstuffworks.com/ibs//chess1.htm, October 20, 2004.6 Adapted from: Feinstein, Charles D., “Deciding Whether to Test Student Athletes for Drug Use,” Interfaces 20, no. 3, May–June 1990, pp.80–87.

expected to lose 12,500 room-nights annually for its 25hotels because some competing hotels are offering suchguarantees.

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