PART IV QUANTITATIVE MODULES L EARNING O BJECTIVES When you complete this module you should be able to IDENTIFY OR DEFINE: Decision trees and decision tables Highest monetary value Expected value of perfect information Sequential decisions DESCRIBE OR EXPLAIN: Decision making under risk Decision making under uncertainty Decision making under certainty A Decision-Making Tools Module Outline THE DECISION PROCESS IN OPERATIONS FUNDAMENTALS OF DECISION MAKING DECISION TABLES TYPES OF DECISION-MAKING ENVIRONMENTS Decision Making Under Uncertainty Decision Making Under Risk Decision Making Under Certainty Expected Value of Perfect Information (EVPI) DECISION TREES A More Complex Decision Tree Using Decision Trees in Ethical Decision Making SUMMARY KEY TERMS USING SOFTWARE FOR DECISION MODELS SOLVED PROBLEMS INTERNET AND STUDENT CD-ROM EXERCISES DISCUSSION QUESTIONS PROBLEMS INTERNET HOMEWORK PROBLEMS CASE STUDIES: TOM TUCKER’S LIVER TRANSPLANT ; SKI RIGHT CORP . ADDITIONAL CASE STUDIES BIBLIOGRAPHY Quantitative Module
Transcript
P A R T I VQUANTITATIVE MODULES
LEARNING OBJECTIVESWhen you complete this module youshould be able to
IDENTIFY OR DEFINE:
Decision trees and decision tables
Highest monetary value
Expected value of perfect information
Sequential decisions
DESCRIBE OR EXPLAIN:
Decision making under risk
Decision making under uncertainty
Decision making under certainty
ADecision-Making Tools
Module OutlineTHE DECISION PROCESS IN OPERATIONS
FUNDAMENTALS OF DECISION MAKING
DECISION TABLES
TYPES OF DECISION-MAKINGENVIRONMENTS
Decision Making Under Uncertainty
Decision Making Under Risk
Decision Making Under Certainty
Expected Value of Perfect Information (EVPI)
DECISION TREES
A More Complex Decision Tree
Using Decision Trees in Ethical DecisionMaking
SUMMARY
KEY TERMS
USING SOFTWARE FOR DECISION MODELS
SOLVED PROBLEMS
INTERNET AND STUDENT CD-ROM EXERCISES
DISCUSSION QUESTIONS
PROBLEMS
INTERNET HOMEWORK PROBLEMS
CASE STUDIES: TOM TUCKER’S LIVER TRANSPLANT; SKI RIGHT CORP.
ADDITIONAL CASE STUDIES
BIBLIOGRAPHY
Quantitative Module
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674 MO D U L E A DE C I S I O N-MA K I N G TO O L S
“The business executive isby profession a decisionmaker. Uncertainty is hisopponent. Overcoming itis his mission.”
John McDonald
Operations managers are decision makers. To achieve the goals of their organizations, managersmust understand how decisions are made and know which decision-making tools to use. To a greatextent, the success or failure of both people and companies depends on the quality of their deci-sions. Bill Gates, who developed the DOS and Windows operating systems, became chairman of themost powerful software firm in the world (Microsoft) and a billionaire. In contrast, the Firestonemanager who headed the team that designed the flawed tires that caused so many accidents withFord Explorers in the late 1990s is not working there anymore.
THE DECISION PROCESS IN OPERATIONSWhat makes the difference between a good decision and a bad decision? A “good” decision—onethat uses analytic decision making—is based on logic and considers all available data and possiblealternatives. It also follows these six steps:
1. Clearly define the problem and the factors that influence it.2. Develop specific and measurable objectives.3. Develop a model—that is, a relationship between objectives and variables (which are mea-
surable quantities).4. Evaluate each alternative solution based on its merits and drawbacks.5. Select the best alternative.6. Implement the decision and set a timetable for completion.
Throughout this book, we have introduced a broad range of mathematical models and tools to helpoperations managers make better decisions. Effective operations depend on careful decision mak-ing. Fortunately, there are a whole variety of analytic tools to help make these decisions. This mod-
The wildcatter’s decision was a tough one. Which of his new Kentucky lease areas—Blair East or
Blair West—should he drill for oil? A wrong decision in this type of wildcat oil drilling could mean
the difference between success and bankruptcy for the company. Talk about decision making under
uncertainty and pressure! But using a decision tree, Tomco Oil President Thomas E. Blair identified 74
different options, each with its own potential net profit. What had begun as an overwhelming number of
geological, engineering, economic, and political factors now became much clearer. Says Blair, “Decision
tree analysis provided us with a systematic way of planning these decisions and clearer insight into the
numerous and varied financial outcomes that are possible.”1
1J. Hosseini, “Decision Analysis and Its Application in the Choice between Two Wildcat Ventures,” Interfaces, Vol. 16, no. 2.Reprinted by permission, INFORMS, 901 Elkridge Landing Road, Suite 400, Linthicum, Maryland 21090 USA.
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DE C I S I O N TA B L E S 675
Unfavorable market
Favorable market
A decision node A state of nature node
1
2Unfavorable market
Favorable marketConstruct
large plant
Do nothing
Constructsmall plant
FIGURE A.1 � Getz Products Decision Tree
ule introduces two of them—decision tables and decision trees. They are used in a wide number ofOM situations, ranging from new-product analysis (Chapter 5), to capacity planning (Supplement7), to location planning (Chapter 8), to scheduling (Chapter 15), and to maintenance planning(Chapter 17).
FUNDAMENTALS OF DECISION MAKINGRegardless of the complexity of a decision or the sophistication of the technique used to analyze it,all decision makers are faced with alternatives and “states of nature.” The following notation will beused in this module:
1. Terms:a. Alternative—a course of action or strategy that may be chosen by a decision maker
(for example, not carrying an umbrella tomorrow).b. State of nature—an occurrence or a situation over which the decision maker has little
or no control (for example, tomorrow’s weather).2. Symbols used in a decision tree:
a. □—decision node from which one of several alternatives may be selected.b. �—a state-of-nature node out of which one state of nature will occur.
To present a manager’s decision alternatives, we can develop decision trees using the above sym-bols. When constructing a decision tree, we must be sure that all alternatives and states of natureare in their correct and logical places and that we include all possible alternatives and states ofnature.
DECISION TABLESWe may also develop a decision or payoff table to help Getz Products define its alternatives. Forany alternative and a particular state of nature, there is a consequence or outcome, which isusually expressed as a monetary value. This is called a conditional value. Note that all of thealternatives in Example A2 are listed down the left side of the table, that states of nature (out-comes) are listed across the top, and that conditional values (payoffs) are in the body of thedecision table.
Decision tableA tabular means ofanalyzing decisionalternatives and states of nature.
“Management means, in the last analysis, thesubstitution of thought for brawn and muscle, of knowledge for folkloreand tradition, and ofcooperation for force.”
Peter Drucker
Getz Products Company is investigating the possibility of producing and marketing backyard storage sheds.Undertaking this project would require the construction of either a large or a small manufacturing plant.The market for the product produced—storage sheds—could be either favorable or unfavorable. Getz, ofcourse, has the option of not developing the new product line at all. A decision tree for this situation ispresented in Figure A.1.
Example A1A simple decision tree
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TABLE A.1 � Decision Table with Conditional Values for Getz Products
STATES OF NATURE
ALTERNATIVES FAVORABLE MARKET UNFAVORABLE MARKET
Construct large plant $200,000 �$180,000Construct small plant $100,000 �$ 20,000Do nothing $ 0 $ 0
There are optimisticdecision makers(“maximax”) andpessimistic ones(“maximin”). Maximax andmaximin present bestcase–worst case planningscenarios.
TABLE A.2 � Decision Table for Decision Making under Uncertainty
We construct a decision table for Getz Products (Table A.1), including conditional values based on thefollowing information. With a favorable market, a large facility will give Getz Products a net profit of$200,000. If the market is unfavorable, a $180,000 net loss will occur. A small plant will result in a net profitof $100,000 in a favorable market, but a net loss of $20,000 will be encountered if the market is unfavorable.
The toughest part ofdecision tables is gettingthe data to analyze.
In Examples A3 and A4, we see how to use decision tables.
TYPES OF DECISION-MAKING ENVIRONMENTSThe types of decisions people make depend on how much knowledge or information they have
about the situation. There are three decision-making environments:
• Decision making under uncertainty• Decision making under risk• Decision making under certainty
Decision Making Under UncertaintyWhen there is complete uncertainty as to which state of nature in a decision environment may occur (that is,when we cannot even assess probabilities for each possible outcome), we rely on three decision methods:
1. Maximax—this method finds an alternative that maximizes the maximum outcome for everyalternative. First, we find the maximum outcome within every alternative, and then we pickthe alternative with the maximum number. Because this decision criterion locates the alterna-tive with the highest possible gain, it has been called an “optimistic” decision criterion.
2. Maximin—this method finds the alternative that maximizes the minimum outcome for everyalternative. First, we find the minimum outcome within every alternative, and then we pickthe alternative with the maximum number. Because this decision criterion locates the alterna-tive that has the least possible loss, it has been called a “pessimistic” decision criterion.
3. Equally likely—this method finds the alternative with the highest average outcome. First,we calculate the average outcome for every alternative, which is the sum of all outcomesdivided by the number of outcomes. We then pick the alternative with the maximum num-ber. The equally likely approach assumes that each state of nature is equally likely to occur.
Example A3 applies each of these approaches to the Getz Products Company.
MaximaxA criterion that finds analternative that maximizesthe maximum outcome.
MaximinA criterion that finds analternative that maximizesthe minimum outcome.
Equally likelyA criterion that assignsequal probability to eachstate of nature.
Example A2A decision table
Given Getz’s decision table of Example A2, determine the maximax, maximin, and equally likely decisioncriteria (see Table A.2).
Example A3A decision table analysisunder uncertainty
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TABLE A.3 � Decision Table for Getz Products
STATES OF NATURE
ALTERNATIVES FAVORABLE MARKET UNFAVORABLE MARKET
Construct large plant (A1) $200,000 �$180,000Construct small plant (A2) $100,000 �$ 20,000Do nothing (A3) $ 0 $ 0
Probabilities .50 .50
2To review these and other statistical terms, refer to the CD-ROM Tutorial 1, “Statistical Review for Managers.”
1. The maximax choice is to construct a large plant. This is the maximum of the maximum numberwithin each row, or alternative.
2. The maximin choice is to do nothing. This is the maximum of the minimum number within eachrow, or alternative.
3. The equally likely choice is to construct a small plant. This is the maximum of the average outcomeof each alternative. This approach assumes that all outcomes for any alternative are equally likely.
Decision Making Under RiskDecision making under risk, a more common occurrence, relies on probabilities. Several possiblestates of nature may occur, each with an assumed probability. The states of nature must be mutuallyexclusive and collectively exhaustive and their probabilities must sum to 1.2 Given a decision tablewith conditional values and probability assessments for all states of nature, we can determine theexpected monetary value (EMV) for each alternative. This figure represents the expected value ormean return for each alternative if we could repeat the decision a large number of times.
The EMV for an alternative is the sum of all possible payoffs from the alternative, each weightedby the probability of that payoff occurring.
Example A4 illustrates how to compute the maximum EMV.
EMV (Alternative ) Payoff of 1st state of nature) (Probability of 1st state of nature)
(Payoff of 2nd state of nature) (Probability of 2nd state of nature)
(Payoff of last state of nature) (Probability of last state of nature)
i =×
+×
+ +×
(
L
Decision Making Under CertaintyNow suppose that the Getz operations manager has been approached by a marketing research firm thatproposes to help him make the decision about whether to build the plant to produce storage sheds. Themarketing researchers claim that their technical analysis will tell Getz with certainty whether the mar-ket is favorable for the proposed product. In other words, it will change Getz’s environment from oneof decision making under risk to one of decision making under certainty. This information could pre-vent Getz from making a very expensive mistake. The marketing research firm would charge Getz$65,000 for the information. What would you recommend? Should the operations manager hire the firm to make the study? Even if the information from the study is perfectly accurate, is it worth $65,000? What might it be worth? Although some of these questions are difficult to answer,
Expected monetaryvalue (EMV)The expected payout orvalue of a variable thathas different possiblestates of nature, eachwith an associatedprobability.
EVPI places an upper limiton what you should payfor information.
Excel OM Data FileModAEx4.xla
Example A4Expected monetary value
Getz Products operations manager believes that the probability of a favorable market is exactly the same asthat of an unfavorable market; that is, each state of nature has a .50 chance of occurring. We can nowdetermine the EMV for each alternative (see Table A.3):
The maximum EMV is seen in alternative A2. Thus, according to the EMV decision criterion, Getz wouldbuild the small facility.
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678 MO D U L E A DE C I S I O N-MA K I N G TO O L S
determining the value of such perfect information can be very useful. It places an upper bound on whatyou would be willing to spend on information, such as that being sold by a marketing consultant. Thisis the concept of the expected value of perfect information (EVPI), which we now introduce.
Expected Value of Perfect Information (EVPI)If a manager were able to determine which state of nature would occur, then he or she would knowwhich decision to make. Once a manager knows which decision to make, the payoff increasesbecause the payoff is now a certainty, not a probability. Because the payoff will increase withknowledge of which state of nature will occur, this knowledge has value. Therefore, we now look athow to determine the value of this information. We call this difference between the payoff undercertainty and the payoff under risk the expected value of perfect information (EVPI).
EVPI = Expected value under certainty � Maximum EMV
To find the EVPI, we must first compute the expected value under certainty, which is the expected(average) return if we have perfect information before a decision has to be made. To calculate thisvalue, we choose the best alternative for each state of nature and multiply its payoff times the prob-ability of occurrence of that state of nature.
In Example A5 we use the data and decision table from Example A4 to examine the expectedvalue of perfect information.
Expected value under certainty Best outcome or consequence for 1st state of nature) (Probability of 1st state of nature)
(Best outcome for 2nd state of nature) (Probability of 2nd state of nature)
(Best outcome for last state of nature) (Probability of last state of nature)
=×
+×
+ +×
(
L
Expected value under certaintyThe expected (average)return if perfectinformation is available.
Expected value ofperfect information(EVPI)The difference betweenthe payoff under certaintyand the payoff under risk.
Decision treeA graphical means ofanalyzing decisionalternatives and states of nature.
By referring back to Table A.3, the Getz operations manager can calculate the maximum that he would payfor information—that is, the expected value of perfect information, or EVPI. He follows a two-stageprocess. First, the expected value under certainty is computed. Then, using this information, EVPI iscalculated. The procedure is outlined as follows:
1. The best outcome for the state of nature “favorable market” is “build a large facility” with a pay-off of $200,000. The best outcome for the state of nature “unfavorable market” is “do nothing”with a payoff of $0. Expected value under certainty = ($200,000)(0.50) + ($0)(0.50) = $100,000.Thus, if we had perfect information, we would expect (on the average) $100,000 if the decisioncould be repeated many times.
2. The maximum EMV is $40,000, which is the expected outcome without perfect information. Thus:
In other words, the most Getz should be willing to pay for perfect information is $60,000. This conclusion,of course, is again based on the assumption that the probability of each state of nature is 0.50.
DECISION TREESDecisions that lend themselves to display in a decision table also lend themselves to display in adecision tree. We will therefore analyze some decisions using decision trees. Although the use of adecision table is convenient in problems having one set of decisions and one set of states of nature,many problems include sequential decisions and states of nature. When there are two or moresequential decisions, and later decisions are based on the outcome of prior ones, the decision treeapproach becomes appropriate. A decision tree is a graphic display of the decision process thatindicates decision alternatives, states of nature and their respective probabilities, and payoffs foreach combination of decision alternative and state of nature.
Expected monetary value (EMV) is the most commonly used criterion for decision tree analysis.One of the first steps in such analysis is to graph the decision tree and to specify the monetary con-sequences of all outcomes for a particular problem.
EVPI = Expected value under certainty Maximum EMV−= − =$ , $ , $ ,100 000 40 000 60 000
Example A5Expected value of perfectinformation
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Example A6Solving a tree for EMV
DE C I S I O N TR E E S 679
Decision tree software is a relatively new
advance that permits users to solve decision-
analysis problems with flexibility, power, and
ease. Programs such as DPL, Tree Plan, and
Supertree allow decision problems to be
analyzed with less effort and in greater depth
than ever before. Full-color presentations of the
options open to managers always have impact.
In this photo, wildcat drilling options are
explored with DPL, a product of Syncopation
Software.
Analyzing problems with decision trees involves five steps:
1. Define the problem.2. Structure or draw the decision tree.3. Assign probabilities to the states of nature.4. Estimate payoffs for each possible combination of decision alternatives and states of nature.5. Solve the problem by computing expected monetary values (EMV) for each state-of-nature
node. This is done by working backward—that is, by starting at the right of the tree andworking back to decision nodes on the left.
Unfavorable market (.5)
Favorable market (.5)
1
2Unfavorable market (.5)
Favorable market (.5)Construct la
rge plant
Do nothing
Construct small plant
EMV for node 1= $10,000
EMV for node 2= $40,000
= (.5) ($200,000) + (.5) (–$180,000)
= (.5) ($100,000) + (.5) (–$20,000)
Payoffs
$200,000
–$180,000
$100,000
20,000–$
$0
FIGURE A.2 � Completed and Solved Decision Tree for Getz Products
A completed and solved decision tree for Getz Products is presented in Figure A.2. Note that the payoffs areplaced at the right-hand side of each of the tree’s branches. The probabilities (first used by Getz in ExampleA4) are placed in parentheses next to each state of nature. The expected monetary values for each state-of-nature node are then calculated and placed by their respective nodes. The EMV of the first node is $10,000.This represents the branch from the decision node to “construct a large plant.” The EMV for node 2, to“construct a small plant,” is $40,000. The option of “doing nothing” has, of course, a payoff of $0. Thebranch leaving the decision node leading to the state-of-nature node with the highest EMV will be chosen.In Getz’s case, a small plant should be built.
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Example A7A decision tree withsequential decisions
680 MO D U L E A DE C I S I O N-MA K I N G TO O L S
Surv
eyre
sults
favo
rabl
e
First DecisionPoint
Second DecisionPoint
1
2
3 Unfavorable market
Favorable marketLarge plant
No plant
Smallplant
Unfavorable market
Favorable market$190,000
–$190,000
$ 90,000
–$ 30,000
–$ 10,000
Payoffs
4
5 Unfavorable market
Favorable marketLarge plant
Smallplant
Unfavorable market
Favorable market$190,000
–$190,000
$ 90,000
–$ 30,000
–$ 10,000
6
7 Unfavorable market
Favorable marketSmallplant
Unfavorable market
Favorable market$200,000
–$180,000
$100,000
20,000–$
$0
Survey
results
negative
Con
duct
mar
ket s
urve
y
Do not conduct survey
$106
,400
$49,
200
$2,4
00$4
0,00
0
$49,200
No plant
No plant
Large plant
$106,400
$63,600
–$87,400
$2,400
$10,000
$40,000
(.22)
(.78)
(.22)
(.78)
(.73)
(.27)
(.73)
(.27)
(.5)
(.5)
(.5)
(.5)
(.45)
(.55)
FIGURE A.3 � Getz Products Decision Tree with Probabilities and EMVs Shown
The short parallel lines mean “prune” that branch, as it is less favorable thananother available option and may be dropped.
A More Complex Decision TreeWhen a sequence of decisions must be made, decision trees are much more powerful tools than aredecision tables. Let’s say that Getz Products has two decisions to make, with the second decision depen-dent on the outcome of the first. Before deciding about building a new plant, Getz has the option of con-ducting its own marketing research survey, at a cost of $10,000. The information from this survey couldhelp it decide whether to build a large plant, to build a small plant, or not to build at all. Getz recognizesthat although such a survey will not provide it with perfect information, it may be extremely helpful.
Getz’s new decision tree is represented in Figure A.3 of Example A7. Take a careful look at thismore complex tree. Note that all possible outcomes and alternatives are included in their logicalsequence. This procedure is one of the strengths of using decision trees. The manager is forced toexamine all possible outcomes, including unfavorable ones. He or she is also forced to make deci-sions in a logical, sequential manner.
There is a widespread useof decision trees beyondOM. Managers oftenappreciate a graphicaldisplay of a toughproblem.
Examining the tree in Figure A.3, we see that Getz’s first decision point is whether to conduct the $10,000market survey. If it chooses not to do the study (the lower part of the tree), it can either build a large plant,a small plant, or no plant. This is Getz’s second decision point. If the decision is to build, the market will beeither favorable (.50 probability) or unfavorable (also .50 probability). The payoffs for each of the possibleconsequences are listed along the right-hand side. As a matter of fact, this lower portion of Getz’s tree isidentical to the simpler decision tree shown in Figure A.2.
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DE C I S I O N TR E E S 681
The upper part of Figure A.3 reflects the decision to conduct the market survey. State-of-nature nodenumber 1 has 2 branches coming out of it. Let us say there is a 45% chance that the survey results will indi-cate a favorable market for the storage sheds. We also note that the probability is .55 that the survey resultswill be negative.
The rest of the probabilities shown in parentheses in Figure A.3 are all conditional probabilities. For exam-ple, .78 is the probability of a favorable market for the sheds given a favorable result from the market survey.Of course, you would expect to find a high probability of a favorable market given that the research indicatedthat the market was good. Don’t forget, though: There is a chance that Getz’s $10,000 market survey did notresult in perfect or even reliable information. Any market research study is subject to error. In this case, thereremains a 22% chance that the market for sheds will be unfavorable given positive survey results.
Likewise, we note that there is a 27% chance that the market for sheds will be favorable given negativesurvey results. The probability is much higher, .73, that the market will actually be unfavorable given a neg-ative survey.
Finally, when we look to the payoff column in Figure A.3, we see that $10,000—the cost of the market-ing study—has been subtracted from each of the top 10 tree branches. Thus, a large plant constructed in afavorable market would normally net a $200,000 profit. Yet because the market study was conducted, thisfigure is reduced by $10,000. In the unfavorable case, the loss of $180,000 would increase to $190,000.Similarly, conducting the survey and building no plant now results in a �$10,000 payoff.
With all probabilities and payoffs specified, we can start calculating the expected monetary value ofeach branch. We begin at the end or right-hand side of the decision tree and work back toward the origin.When we finish, the best decision will be known.
1. Given favorable survey results,
The EMV of no plant in this case is �$10,000. Thus, if the survey results are favorable, a largeplant should be built.
2. Given negative survey results,
The EMV of no plant is again �$10,000 for this branch. Thus, given a negative survey result,Getz should build a small plant with an expected value of $2,400.
3. Continuing on the upper part of the tree and moving backward, we compute the expected value ofconducting the market survey.
The EMV of no plant is $0. Thus, building a small plant is the best choice, given the marketingresearch is not performed.
5. Because the expected monetary value of conducting the survey is $49,200—versus an EMV of$40,000 for not conducting the study—the best choice is to seek marketing information. If the sur-vey results are favorable, Getz should build the large plant; if they are unfavorable, it should buildthe small plant.
Using Decision Trees in Ethical Decision MakingDecision trees can also be a useful tool to aid ethical corporate decision making. The decision treeillustrated in Example A8, developed by Harvard Professor Constance Bagley, provides guidance asto how managers can both maximize shareholder value and behave ethically. The tree can be appliedto any action a company contemplates, whether it is expanding operations in a developing countryor reducing a workforce at home.
EMV (node 6)
EMV (node 7)
= + − == + − =
(. )($ , ) (. )( $ , ) $ ,
(. )($ , ) (. )( $ , ) $ ,
50 200 000 50 180 000 10 000
50 100 000 50 20 000 40 000
EMV (node 4)
EMV (node 5)
= + − = −= + − =
(. )($ , ) (. )( $ , ) $ ,
(. )($ , ) (. )( $ , ) $ ,
27 190 000 73 190 000 87 400
27 90 000 73 30 000 2 400
EMV (node 2)
EMV (node 3)
= + − == + − =
(. )($ , ) (. )( $ , ) $ ,
(. )($ , ) (. )( $ , ) $ ,
78 190 000 22 190 000 106 400
78 90 000 22 30 000 63 600
You can reduce complexityby viewing and solving anumber of smaller trees—start at the end branchesof a large one. Take onedecision at a time.
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682 MO D U L E A DE C I S I O N-MA K I N G TO O L S
S U M M A R Y
Yes
Yes
Yes
No
No
No
Action outcome
Do it
Don't do it
Don't do it
Is actionlegal?
Does actionmaximizecompanyreturns?
Yes
No
Don't do it
Do it,but notifyappropriateparties
Is it ethical?(Weigh the effecton employees,
customers, suppliers,community versus
shareholder benefit.)
Is it ethical not to takeaction? (Weigh the
harm to shareholdersversus benefits to
other stakeholders.)
FIGURE A.4 �
Smithson’s Decision Tree for Ethical Dilemma
Source: Modified from Constance E. Bagley, “The Ethical Leader’s Decision Tree,” Harvard BusinessReview (January–February 2003): 18–19.
K E Y T E R M S Decision table (p. 675)Maximax (p. 676)Maximin (p. 676)Equally likely (p. 676)Expected monetary value (EMV) (p. 677)
Expected value of perfect information (EVPI)(p. 678)
Expected value under certainty (p. 678)Decision tree (p. 678)
Smithson Corp. is opening a plant in Malaysia, a country with much less stringent environmental laws thanthe U.S., its home nation. Smithson can save $18 million in building the manufacturing facility—and boostits profits—if it does not install pollution-control equipment that is mandated in the U.S. but not inMalaysia. But Smithson also calculates that pollutants emitted from the plant, if unscrubbed, could damagethe local fishing industry. This could cause a loss of millions of dollars in income as well as create healthproblems for local inhabitants.
Figure A.4 outlines the choices management can consider. For example, if in management’s best judg-ment the harm to the Malaysian community by building the plant will be greater than the loss in companyreturns, the response to the question “Is it ethical?” will be no.
Now, say Smithson proposes building a somewhat different plant, one with pollution controls, despite anegative impact on company returns. That decision takes us to the branch “Is it ethical not to take action?”If the answer (for whatever reason) is no, the decision tree suggests proceeding with the plant but notifyingthe Smithson Board, shareholders, and others about its impact.
Ethical decisions can be quite complex: What happens, for example, if a company builds a pol-luting plant overseas, but this allows the company to sell a life-saving drug at a lower cost aroundthe world? Does a decision tree deal with all possible ethical dilemmas? No—but it does providemanagers with a framework for examining those choices.
Example A8Ethical decision making
This module examines two of the most widely used decision techniques—decision tables and deci-sion trees. These techniques are especially useful for making decisions under risk. Many decisionsin research and development, plant and equipment, and even new buildings and structures can beanalyzed with these decision models. Problems in inventory control, aggregate planning, mainte-nance, scheduling, and production control are just a few other decision table and decision treeapplications.
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SO LV E D PRO B L E M S 683
Find the best outcomefor each measure using= MAX(G8:G10).
= MIN(B8:C8)
= E14 – E11
= MAX(B8:C8)
To calculate the EVPI,find the best outcomefor each scenario.= MAX(B8:B10)
= SUMPRODUCT(B$7:C$7,B14:C14)
Compute the EMV for each alternativeusing = SUMPRODUCT(B$7:C$7, B8:C8).
PROGRAM A.1 � Using Excel OM to Compute EMV and Other Measures for Getz
USING SOFTWARE FOR DECISION MODELS
Analyzing decision tables is straightforward with Excel, Excel OM, and POM for Windows. When decisiontrees are involved, commercial packages such as DPL, Tree Plan, and Supertree provide flexibility, power, andease. POM for Windows will also analyze trees but does not have graphic capabilities.
U s i n g E x c e l O M
Excel OM allows decision makers to evaluate decisions quickly and to perform sensitivity analysis on theresults. Program A.1 uses the Getz data to illustrate input, output, and selected formulas needed to compute theEMV and EVPI values.
SOLVED PROBLEMSSolved Problem A.1Stella Yan Hua is considering the possibility of opening a small dressshop on Fairbanks Avenue, a few blocks from the university. She haslocated a good mall that attracts students. Her options are to open asmall shop, a medium-sized shop, or no shop at all. The market for adress shop can be good, average, or bad. The probabilities for thesethree possibilities are .2 for a good market, .5 for an average market,and .3 for a bad market. The net profit or loss for the medium-sizedor small shops for the various market conditions are given in the fol-lowing table. Building no shop at all yields no loss and no gain. Whatdo you recommend?
GOOD AVERAGE BAD
MARKET MARKET MARKET
ALTERNATIVES ($) ($) ($)
Small shop 75,000 25,000 �40,000Medium-sized
shop 100,000 35,000 �60,000No shop 0 0 0
Probabilities .20 .50 .30
SOLVED PROBLEMS
U s i n g P O M f o r W i n d o w s
POM for Windows can be used to calculate all of the information described in the decision tables and decisiontrees in this module. For details on how to use this software, please refer to Appendix IV.
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Demand is 5 cases
Demand is 6 cases
Demand is 7 cases
Demand is 5 cases
Demand is 6 casesStock 6 cases
Demand is 7 cases
Demand is 5 cases
Demand is 6 cases
Demand is 7 cases
Stock 5 ca
ses
Stock 7 cases
INTERNET AND STUDENT CD-ROM EXERCISESINTERNET AND STUDENT CD-ROM EXERCISESVisit our Companion Web site or use your student CD-ROM to help with this material in this module.
On Your Student CD-ROMOn Our Companion Web site, www.prenhall.com/heizer
Solved Problem A.2Daily demand for cases of Tidy Bowl cleaner at Ravinder Nath’sSupermarket has always been 5, 6, or 7 cases. Develop a decision treethat illustrates her decision alternatives as to whether to stock 5, 6, or 7cases.
Solution
The decision tree is shown in Figure A.5.
• Self-Study Quizzes
• Practice Problems
• Internet Homework Problems
• Internet Cases
• PowerPoint Lecture
• Practice Problems
• Excel OM
• Excel OM Example Data File
• POM for Windows
DISCUSSION QUESTIONS
1. Identify the six steps in the decision process.2. Give an example of a good decision you made that resulted in a
bad outcome. Also give an example of a bad decision you madethat had a good outcome. Why was each decision good or bad?
3. What is the equally likely decision model?4. Discuss the differences between decision making under certainty,
under risk, and under uncertainty.5. What is a decision tree?
FIGURE A.5 � Demand at Ravinder Nath’s Supermarket
Solution
The problem can be solved by computing the expected monetary value (EMV) for each alternative.
A.2 Even though independent gasoline stations have been having a difficult time, Susan Helms has been thinkingabout starting her own independent gasoline station. Susan’s problem is to decide how large her station shouldbe. The annual returns will depend on both the size of her station and a number of marketing factors related tothe oil industry and demand for gasoline. After a careful analysis, Susan developed the following table:
SIZE OF
FIRST STATION GOOD MARKET ($) FAIR MARKET ($) POOR MARKET ($)
Small 50,000 20,000 �10,000Medium 80,000 30,000 �20,000Large 100,000 30,000 �40,000Very large 300,000 25,000 �160,000
For example, if Susan constructs a small station and the market is good, she will realize a profit of $50,000.a) Develop a decision table for this decision.b) What is the maximax decision?c) What is the maximin decision?d) What is the equally likely decision?e) Develop a decision tree. Assume each outcome is equally likely, then find the highest EMV.
A.3 Clay Whybark, a soft-drink vendor at Hard Rock Cafe’s annual Rockfest, created a table of conditional valuesfor the various alternatives (stocking decision) and states of nature (size of crowd):
If the probabilities associated with the states of nature are 0.3 for a big demand, 0.5 for an average demand, and0.2 for a small demand, determine the alternative that provides Clay Whybark the greatest expected monetaryvalue (EMV).
A.4 For Problem A.3, compute the expected value of perfect information (EVPI).
6. Explain how decision trees might be used in several of the 10 OMdecisions.
7. What is the expected value of perfect information?8. What is the expected value under certainty?9. Identify the five steps in analyzing a problem using a decision tree.
10. Why are the maximax and maximin strategies considered to beoptimistic and pessimistic, respectively?
11. The expected value criterion is considered to be the rational crite-rion on which to base a decision. Is this true? Is it rational to con-sider risk?
12. When are decision trees most useful?
*Note: means the problem may be solved with POM for Windows; means the problem may be solved with Excel
OM; and means the problem may be solved with POM for Windows and/or Excel OM.P
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A.5 H. Weiss, Inc., is considering building a sensitive new airport scanning device. His managers believe that thereis a probability of 0.4 that the ATR Co. will come out with a competitive product. If Weiss adds an assemblyline for the product and ATR Co. does not follow with a competitive product, Weiss’s expected profit is$40,000; if Weiss adds an assembly line and ATR follows suit, Weiss still expects $10,000 profit. If Weiss addsa new plant addition and ATR does not produce a competitive product, Weiss expects a profit of $600,000; ifATR does compete for this market, Weiss expects a loss of $100,000.
Determine the EMV of each decision.
A.6 For Problem A.5, compute the expected value of perfect information.
A.7 The following payoff table provides profits based on various possible decision alternatives and various levelsof demand at Amber Gardner’s software firm:
The probability of low demand is 0.4, whereas the probability of high demand is 0.6.a) What is the highest possible expected monetary value?b) What is the expected value under certainty?c) Calculate the expected value of perfect information for this situation.
A.8 Leah Johnson, director of Legal Services of Brookline, wants to increase capacity to provide free legal advicebut must decide whether to do so by hiring another full-time lawyer or by using part-time lawyers. The tablebelow shows the expected costs of the two options for three possible demand levels:
A.9 Chung Manufacturing is considering the introduction of a family of new products. Long-term demand for theproduct group is somewhat predictable, so the manufacturer must be concerned with the risk of choosing aprocess that is inappropriate. Chen Chung is VP of operations. He can choose among batch manufacturing orcustom manufacturing, or he can invest in group technology. Chen won’t be able to forecast demand accuratelyuntil after he makes the process choice. Demand will be classified into four compartments: poor, fair, good, andexcellent. The table below indicates the payoffs (profits) associated with each process/demand combination, aswell as the probabilities of each long-term demand level.
a) Based on expected value, what choice offers the greatest gain?b) What would Chen Chung be willing to pay for a forecast that would accurately determine the level of demand
in the future?
A.10 Julie Resler’s company is considering expansion of its current facility to meet increasing demand. If demandis high in the future, a major expansion will result in an additional profit of $800,000, but if demand is lowthere will be a loss of $500,000. If demand is high, a minor expansion will result in an increase in profits of $200,000, but if demand is low, there will be a loss of $100,000. The company has the option of notexpanding. If there is a 50% chance demand will be high, what should the company do to maximize long-runaverage profits?
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A.11 The University of Dallas bookstore stocks textbooks in preparation for sales each semester. It normally relieson departmental forecasts and preregistration records to determine how many copies of a text are needed.Preregistration shows 90 operations management students enrolled, but bookstore manager Curtis Kettermanhas second thoughts, based on his intuition and some historical evidence. Curtis believes that the distribution ofsales may range from 70 to 90 units, according to the following probability model:
This textbook costs the bookstore $82 and sells for $112. Any unsold copies can be returned to the publisher,less a restocking fee and shipping, for a net refund of $36.
a) Construct the table of conditional profits.b) How many copies should the bookstore stock to achieve highest expected value?
A.12 Palmer Cheese Company is a small manufacturer of several different cheese products. One product is a cheesespread sold to retail outlets. Susan Palmer must decide how many cases of cheese spread to manufacture eachmonth. The probability that demand will be 6 cases is .1, for 7 cases it is .3, for 8 cases it is .5, and for 9 casesit is .1. The cost of every case is $45, and the price Susan gets for each case is $95. Unfortunately, any cases notsold by the end of the month are of no value as a result of spoilage. How many cases should Susan manufactureeach month?
A.13 Ronald Lau, chief engineer at South Dakota Electronics, has to decide whether to build a new state-of-the-artprocessing facility. If the new facility works, the company could realize a profit of $200,000. If it fails, SouthDakota Electronics could lose $180,000. At this time, Lau estimates a 60% chance that the new process will fail.
The other option is to build a pilot plant and then decide whether to build a complete facility. The pilotplant would cost $10,000 to build. Lau estimates a 50-50 chance that the pilot plant will work. If the pilotplant works, there is a 90% probability that the complete plant, if it is built, will also work. If the pilot plantdoes not work, there is only a 20% chance that the complete project (if it is constructed) will work. Laufaces a dilemma. Should he build the plant? Should he build the pilot project and then make a decision?Help Lau by analyzing this problem.
A.14 Karen Villagomez, president of Wright Industries, is considering whether to build a manufacturing plant in theOzarks. Her decision is summarized in the following table:
ALTERNATIVES FAVORABLE MARKET UNFAVORABLE MARKET
Build large plant $400,000 �$300,000Build small plant $ 80,000 �$ 10,000Don’t build $ 0 $ 0
Market probabilities 0.4 0.6
a) Construct a decision tree.b) Determine the best strategy using expected monetary value (EMV).c) What is the expected value of perfect information (EVPI)?
A.15 Deborah Kellogg buys Breathalyzer test sets for the Denver Police Department. The quality of the test setsfrom her two suppliers is indicated in the following table:
PERCENT PROBABILITY PROBABILITY
DEFECTIVE FOR LOOMBA TECHNOLOGY FOR STEWART-DOUGLAS ENTERPRISES
1 .70 .303 .20 .305 .10 .40
For example, the probability of getting a batch of tests that are 1% defective from Loomba Technology is .70.Because Kellogg orders 10,000 tests per order, this would mean that there is a .7 probability of getting 100defective tests out of the 10,000 tests if Loomba Technology is used to fill the order. A defective Breathalyzertest set can be repaired for $0.50. Although the quality of the test sets of the second supplier, Stewart-DouglasEnterprises, is lower, it will sell an order of 10,000 test sets for $37 less than Loomba.
a) Develop a decision tree.b) Which supplier should Kellogg use?
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A.16 Deborah Hollwager, a concessionaire for the Des Moines ballpark, has developed a table of conditional valuesfor the various alternatives (stocking decision) and states of nature (size of crowd).
If the probabilities associated with the states of nature are 0.3 for a large crowd, 0.5 for an average crowd, and0.2 for a small crowd, determine:
a) The alternative that provides the greatest expected monetary value (EMV).b) The expected value of perfect information (EVPI).
A.17 Joseph Biggs owns his own sno-cone business and lives 30 miles from a California beach resort. The sale ofsno-cones is highly dependent on his location and on the weather. At the resort, his profit will be $120 per dayin fair weather, $10 per day in bad weather. At home, his profit will be $70 in fair weather and $55 in badweather. Assume that on any particular day, the weather service suggests a 40% chance of foul weather.
a) Construct Joseph’s decision tree.b) What decision is recommended by the expected value criterion?
A.18 Kenneth Boyer is considering opening a bicycle shop in North Chicago. Boyer enjoys biking, but this is to bea business endeavor from which he expects to make a living. He can open a small shop, a large shop, or no shopat all. Because there will be a 5-year lease on the building that Boyer is thinking about using, he wants to makesure he makes the correct decision. Boyer is also thinking about hiring his old marketing professor to conducta marketing research study to see if there is a market for his services. The results of such a study could be eitherfavorable or unfavorable. Develop a decision tree for Boyer.
A.19 Kenneth Boyer (of Problem A.18) has done some analysis of his bicycle shop decision. If he builds a large shop, hewill earn $60,000 if the market is favorable; he will lose $40,000 if the market is unfavorable. A small shop willreturn a $30,000 profit with a favorable market and a $10,000 loss if the market is unfavorable. At the present time,he believes that there is a 50-50 chance of a favorable market. His former marketing professor, Y. L. Yang, willcharge him $5,000 for the market research. He has estimated that there is a .6 probability that the market survey willbe favorable. Furthermore, there is a .9 probability that the market will be favorable given a favorable outcome ofthe study. However, Yang has warned Boyer that there is a probability of only .12 of a favorable market if the mar-keting research results are not favorable. Expand the decision tree of Problem A.18 to help Boyer decide what to do.
A.20 Dick Holliday is not sure what he should do. He can build either a large video rental section or a small one in hisdrugstore. He can also gather additional information or simply do nothing. If he gathers additional information, theresults could suggest either a favorable or an unfavorable market, but it would cost him $3,000 to gather the infor-mation. Holliday believes that there is a 50-50 chance that the information will be favorable. If the rental market isfavorable, Holliday will earn $15,000 with a large section or $5,000 with a small. With an unfavorable video-rentalmarket, however, Holliday could lose $20,000 with a large section or $10,000 with a small section. Without gath-ering additional information, Holliday estimates that the probability of a favorable rental market is .7. A favorablereport from the study would increase the probability of a favorable rental market to .9. Furthermore, an unfavorablereport from the additional information would decrease the probability of a favorable rental market to .4. Of course,Holliday could ignore these numbers and do nothing. What is your advice to Holliday?
A.21 Problem A.8 dealt with a decision facing Legal Services of Brookline. Using the data in that problem, provide:a) The appropriate decision tree showing payoffs and probabilities.b) The best alternative using expected monetary value (EMV).
A.22 The city of Segovia is contemplating building a second airport to relieve congestion at the main airport and is con-sidering two potential sites, X and Y. Hard Rock Hotels would like to purchase land to build a hotel at the new air-port. The value of land has been rising in anticipation and is expected to skyrocket once the city decides betweensites X and Y. Consequently, Hard Rock would like to purchase land now. Hard Rock will sell the land if the citychooses not to locate the airport nearby. Hard Rock has four choices: (1) buy land at X, (2) buy land at Y, (3) buyland at both X and Y, or (4) do nothing. Hard Rock has collected the following data (which are in millions of euros):
SITE X SITE Y
Current purchase price 27 15Profits if airport and hotel built at this site 45 30Sales price if airport not built at this site 9 6
Hard Rock determines there is a 45% chance the airport will be built at X (hence, a 55% chance it will be built at Y).a) Set up the decision table.b) What should Hard Rock decide to do to maximize total net profit?
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CA S E ST U DY 689
INTERNET HOMEWORK PROBLEMS
See our Companion Web site at www.prenhall.com/heizer for these additional homework problems: A.24through A.31.
CASE STUDYTom Tucker’s Liver Transplant
Tom Tucker, a robust 50-year-old executive living in the northernsuburbs of St. Paul, has been diagnosed by a University of Minnesotainternist as having a decaying liver. Although he is otherwise healthy,Tucker’s liver problem could prove fatal if left untreated.
Firm research data are not yet available to predict the likelihoodof survival for a man of Tucker’s age and condition without surgery.However, based on her own experience and recent medical journalarticles, the internist tells him that if he elects to avoid surgical treat-ment of the liver problem, chances of survival will be approximatelyas follows: only a 60% chance of living 1 year, a 20% chance of sur-viving for 2 years, a 10% chance for 5 years, and a 10% chance of
living to age 58. She places his probability of survival beyond age 58without a liver transplant to be extremely low.
The transplant operation, however, is a serious surgical proce-dure. Five percent of patients die during the operation or its recoverystage, with an additional 45% dying during the first year. Twentypercent survive for 5 years, 13% survive for 10 years, and 8%, 5%,and 4% survive, respectively, for 15, 20, and 25 years.
D i s c u s s i o n Q u e s t i o n s
1. Do you think that Tucker should select the transplant operation?2. What other factors might be considered?
CASE STUDY
CASE STUDY
CASE STUDY
A.23 Louisiana is busy designing new lottery “scratch-off” games. In the latest game, Bayou Boondoggle, the player isinstructed to scratch off one spot: A, B, or C. A can reveal “Loser, ” “Win $1,” or “Win $50.” B can reveal “Loser”or “Take a Second Chance.” C can reveal “Loser” or “Win $500.” On the second chance, the player is instructedto scratch off D or E. D can reveal “Loser” or “Win $1.” E can reveal “Loser” or “Win $10.” The probabilities atA are .9, .09, and .01. The probabilities at B are .8 and .2. The probabilities at C are .999 and .001. The probabili-ties at D are .5 and .5. Finally, the probabilities at E are .95 and .05. Draw the decision tree that represents this sce-nario. Use proper symbols and label all branches clearly. Calculate the expected value of this game.
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Ski Right Corp.
After retiring as a physician, Bob Guthrie became an avid downhillskier on the steep slopes of the Utah Rocky Mountains. As an ama-teur inventor, Bob was always looking for something new. With therecent deaths of several celebrity skiers, Bob knew he could use hiscreative mind to make skiing safer and his bank account larger. Heknew that many deaths on the slopes were caused by head injuries.Although ski helmets have been on the market for some time, mostskiers consider them boring and basically ugly. As a physician, Bobknew that some type of new ski helmet was the answer.
Bob’s biggest challenge was to invent a helmet that was attrac-tive, safe, and fun to wear. Multiple colors and using the latest fash-ion designs would be musts. After years of skiing, Bob knew thatmany skiers believe that how you look on the slopes is more impor-tant than how you ski. His helmets would have to look good and fit inwith current fashion trends. But attractive helmets were not enough.Bob had to make the helmets fun and useful. The name of the new skihelmet, Ski Right, was sure to be a winner. If Bob could come upwith a good idea, he believed that there was a 20% chance that themarket for the Ski Right helmet would be excellent. The chance of agood market should be 40%. Bob also knew that the market for hishelmet could be only average (30% chance) or even poor (10%chance).
The idea of how to make ski helmets fun and useful came toBob on a gondola ride to the top of a mountain. A busy executive onthe gondola ride was on his cell phone trying to complete a compli-cated merger. When the executive got off the gondola, he droppedthe phone and it was crushed by the gondola mechanism. Bobdecided that his new ski helmet would have a built-in cell phone andan AM/FM stereo radio. All the electronics could be operated by acontrol pad worn on a skier’s arm or leg.
Bob decided to try a small pilot project for Ski Right. Heenjoyed being retired and didn’t want a failure to cause him to goback to work. After some research, Bob found Progressive Products(PP). The company was willing to be a partner in developing the SkiRight and sharing any profits. If the market was excellent, Bobwould net $5,000 per month. With a good market, Bob would net$2,000. An average market would result in a loss of $2,000, and apoor market would mean Bob would be out $5,000 per month.
Another option for Bob was to have Leadville Barts (LB) makethe helmet. The company had extensive experience in making bicyclehelmets. Progressive would then take the helmets made by LeadvilleBarts and do the rest. Bob had a greater risk. He estimated that he couldlose $10,000 per month in a poor market or $4,000 in an average mar-ket. A good market for Ski Right would result in $6,000 profit for Bob,and an excellent market would mean a $12,000 profit per month.
(continued)
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ADDITIONAL CASE STUDIES
BIBLIOGRAPHY
Brown, R. V. “The State of the Art of Decision Analysis.” Interfaces22, 6 (November–December 1992): 5–14.
Collin, Ian. “Scale Management and Risk Assessment for DeepwaterDevelopments.” World Oil 224, no. 5 (May 2003): 62.
Hammond, J. S., R. L. Kenney, and H. Raiffa. “The Hidden Traps in Decision Making.” 76, no. 5 Harvard Business Review(September–October 1998): 47–60.
Jbuedj, C. “Decision Making under Conditions of Uncertainty.”Journal of Financial Planning (October 1997): 84.
Keefer, Donald L. “Balancing Drug Safety and Efficacy for a Go/No-Go Decision.” Interfaces 34, no. 2 (March–April 2004):113–116.
Kirkwood, C. W. “An Overview of Methods for Applied Decision
Analysis.” Interfaces 22, 6 (November–December 1992): 28–39.Perdue, Robert K., William J. McAllister, Peter V. King, and Bruce G.
Berkey. “Valuation of R and D Projects Using Options Pricingand Decision Analysis Models.” Interfaces 29, 6 (November1999): 57–74.
Raiffa, H. Decision Analysis: Introductory Lectures on Choices UnderCertainty. Reading, MA: Addison-Wesley (1968).
Render, B., R. M. Stair, Jr., and R. Balakrishnan. Managerial DecisionModeling with Spreadsheets. 2nd ed. Upper Saddle River, NJ:Prentice Hall (2006).
Render, B., R. M. Stair Jr., and M. Hanna. Quantitative Analysis forManagement, 9th ed. Upper Saddle River, NJ: Prentice Hall (2006).
Schlaifer, R. Analysis of Decisions Under Certainty. New York:McGraw-Hill (1969).
A third option for Bob was to use TalRad (TR), a radio companyin Tallahassee, Florida. TalRad had extensive experience in makingmilitary radios. Leadville Barts could make the helmets, andProgressive Products could do the rest of production and distribution.Again, Bob would be taking on greater risk. A poor market wouldmean a $15,000 loss per month, and an average market would mean a$10,000 loss. A good market would result in a net profit of $7,000 forBob. An excellent market would return $13,000 per month.
Bob could also have Celestial Cellular (CC) develop the cellphones. Thus, another option was to have Celestial make the phonesand have Progressive do the rest of the production and distribution.Because the cell phone was the most expensive component of the hel-met, Bob could lose $30,000 per month in a poor market. He could lose$20,000 in an average market. If the market was good or excellent, Bobwould see a net profit of $10,000 or $30,000 per month, respectively.
Bob’s final option was to forget about Progressive Productsentirely. He could use Leadville Barts to make the helmets, Celestial
Cellular to make the phones, and TalRad to make the AM/FM stereoradios. Bob could then hire some friends to assemble everything andmarket the finished Ski Right helmets. With this final alternative,Bob could realize a net profit of $55,000 a month in an excellentmarket. Even if the market was just good, Bob would net $20,000.An average market, however, would mean a loss of $35,000. If themarket was poor Bob would lose $60,000 per month.
D i s c u s s i o n Q u e s t i o n s
1. What do you recommend?2. Compute the expected value of perfect information.3. Was Bob completely logical in how he approached this decision
problem?
Source: B. Render, R. M. Stair, and M. Hanna, Quantitative Analysis forManagement, 9th ed. Upper Saddle River, N.J.: Prentice Hall (2006).Reprinted by permission of Prentice Hall, Inc.
See our Companion Web site at www.prenhall.com/heizer for these additional free case studies:• Arctic, Inc.: A refrigeration company has several major options with regard to capacity and expansion.
• Toledo Leather Company: This firm is trying to select new equipment based on potential costs.
