supplement ch05

SUPPLEMENT OUTLINEIntroduction, 5S-2

Decision Trees, 5S-3Influence Diagrams, 5S-5

Expected Value of PerfectInformation, 5S-6

Sensitivity Analysis, 5S-7

Summary, 5S-9

Key Terms, 5S-9

Solved Problems, 5S-9

Discussion and Review Questions, 5S-11

Problems, 5S-11

Reading: Oglethorpe Power:Example of Decision Analysis, 5S-16

Selected Bibliography and FurtherReading, 5S-16

Decision Analysis

LEARNING OBJECTIVES

After completing this supplement,you should be able to:

1 Model operations decisionsproblems.

2 Describe and use the expected-value approach to solvedecision problems.

3 Construct a decision tree anduse it to analyze a problem.

4 Compute the expected value ofperfect information.

5 Conduct sensitivity analysis ona simple decision problem.

5S-1

SUPPLEMENT TO

CHAPTER5

ste51675_ch05_suppl.qxd 10/11/2006 12:43 PM

INTRODUCTIONDecision modelling and analysis represents a general approach to decision making. It issuitable for a wide range of operations management decisions. Among them are capacityplanning, product and service design, equipment selection, and location planning. Deci-sions that lend themselves to a decision analysis approach tend to be characterized bythese elements in its simplest forms:

1. There is a random variable with a set of possible future conditions that will have abearing on the results of the decision.

2. There is a decision with a list of alternatives for the manager to choose from.

3. A known payoff for each alternative under each possible future condition.

To use this approach, a decision maker would employ this process:

1. Identify the possible future conditions (e.g., demand will be low, medium, or high; thenumber of contracts awarded will be one, two, or three; the competitor will or will notintroduce a new product). These are called states of nature for the random variable.

2. Develop a list of possible alternatives for the decision one of which may be to donothing.

3. Determine or estimate the payoff associated with each alternative for every possiblefuture condition.

4. Estimate the likelihood of each possible future condition.

5. Evaluate alternatives according to some decision criterion (e.g., maximize expectedprofit), and select the best alternative.

The information for a decision is summarized in a payoff table, which shows thepayoffs for each alternative under the various possible states of nature. These tables arehelpful in choosing among alternatives because they facilitate comparison of alternatives.Consider the following payoff table, which illustrates a capacity planning problem.

The payoffs are shown in the body of the table. In this instance, the payoffs are in termsof net present values, which represent equivalent current dollar values of estimated futureincome less costs. This is a convenient measure because it places all alternatives on acomparable basis. If a small facility is built, the payoff will be the same for all threepossible states of nature. For a medium facility, low demand will have a net present valueof $7 million, whereas both moderate and high demand will have net present values of$12 million. A large facility will have a loss of $4 million if demand is low, a net presentvalue of $2 million if demand is moderate, and a net present value of $16 million ifdemand is high.

The problem for the decision maker is to select one of the alternatives, taking the netpresent values into account. In order to do so, one needs to determine the probability ofoccurrence for each state of nature. Because the states should be mutually exclusive andcollectively exhaustive, these probabilities must add to 1.00. These probabilities areusually subjective opinions of experts. The decision analyst interviewing an expert shouldbe careful not to ask leading questions and should be aware of psychological biases suchas overconfidence or intentional underestimation by marketing/sales managers (to makemeeting these goals easier).

POSSIBLE FUTURE DEMAND

Alternatives Low Moderate High

Small facility $10* $10 $10Medium facility 7 12 12Large facility (4) 2 16

*Net present value in $ millions.

5S-2 SUPPLEMENT TO CHAPTER FIVE DECISION ANALYSIS

payoff table Table showingthe payoffs for each alternativeof a decision in every possiblestate of nature of a randomvariable.


Given the probabilities for each state of nature, a widely used approach is the expectedmonetary value criterion. The expected value is computed for each alternative, and theone with the highest expected value is selected. The expected value is the sum of the pay-offs times probabilities for an alternative.

Using the expected monetary value criterion, identify the best alternative for the follow-ing payoff table for these probabilities: low�.30, moderate�.50, and high�.20.

Find the expected value of each alternative by multiplying the probability of occurrencefor each state of nature by the payoff for that state of nature and summing them:

EVsmall � .30($10) � .50($10) � .20($10) � $10

EVmedium � .30($7) � .50($12) � .20($12) � $10.5

EVlarge � .30(–4) � .50($2) � .20($16) � $3

Hence, choose the medium facility because it has the highest expected value.

The expected monetary value approach is most appropriate when a decision maker isneither risk averse nor risk seeking, but is risk neutral. Typically, well-established orga-nizations with numerous decisions of this nature tend to use expected value because itprovides an indication of the long-run, average payoff. That is, the expected-valueamount (e.g., $10.5 million in the last example) is not an actual payoff but an expected oraverage amount that would be approximated if a large number of identical decisions wereto be made. Hence, if a decision maker applies this criterion to a large number of similardecisions, the expected payoff for the total will approximate the sum of the individualexpected payoffs.

Note that this approach only considers the average payoff, and not the spread or risk ofpayoffs. In this case, the minimum payoff, if medium facility is built, $7 million, may stillbe acceptable. If it is not, the decision maker may look for a new alternative (e.g., seekmore information) or choose to build a small facility which has a higher minimum payoff($10 million).

More complex decision problems may have more than one decision variable and morethan one random variable. In this case, the payoffs cannot be represented by a table.Instead, decision trees are used to graphically model the decision problem.

DECISION TREESA decision tree is a schematic representation of the decision variables, random variables,and their payoffs. The term gets its name from the treelike appearance of the diagram (seeFigure 5S–1). Decision trees are particularly useful for analyzing situations that involvesequential decisions. For instance, a manager may initially decide to build a small facil-ity only to discover that demand is much higher than anticipated. In this case, the managermay then be called upon to make a subsequent decision on whether to expand or build anadditional facility.

A decision tree is composed of a number of nodes that have branches emanating fromthem (see Figure 5S–1). Square nodes denote decision points, and circular nodes denote





SUPPLEMENT TO CHAPTER FIVE DECISION ANALYSIS 5S-3

Example S–1

decision tree A schematicrepresentation of the decisionvariables, random variables,and payoffs.

Solution


chance events. Read the tree from left to right. Branches leaving square nodes representalternatives; branches leaving circular nodes represent chance events (i.e., the states ofnature).

After the tree has been drawn, it is analyzed from right to left; that is, starting with thelast decision that might be made. For each decision, choose the alternative that will yieldthe greatest return (or the lowest cost). If chance events follow a decision, choose the alternative that has the highest expected monetary value (or lowest expected cost).

A manager must decide on the size of a video arcade to construct. The manager hasnarrowed the choices to two: large or small. Information has been collected on payoffs,and the following decision tree has been constructed. Analyze the decision tree anddetermine which initial alternative (build small or build large) should be chosen in orderto maximize expected monetary value.

The dollar amounts at the branch ends indicate the estimated payoffs if the sequence ofdecisions and chance events occurs. For example, if the initial decision is to build a smallfacility and it turns out that demand is low, the payoff will be $40 (thousand). Similarly,

Low demand (.4)

High demand (.6)

$40

Do nothing

Expand

Build small

Build large

Do nothing

Reduce pricesLow demand (.4)

High demand (.6)

$55($10)

$50

$40

$70


State of nature 1

State of nature 2

State of nature 1

State of nature 2

Payoff 1

Payoff 2

Payoff 3

Payoff 4

Payoff 5

Payoff 6

2

2

Choose A'1

Choose A'2

1

Choose A1

Choose A2

Initialdecision

Possibleseconddecisions

Choose A'3

Choose A'4

Decision point Chance event

FIGURE 5S–1

Format of a decision tree

Example S–2

Solution


if a small facility is built, and demand turns out high, and a later decision is made toexpand, the payoff will be $55. The figures in parentheses on branches leaving the chancenodes indicate the probabilities of those states of nature. Hence, the probability of lowdemand is .4, and the probability of high demand is .6. Payoffs in parentheses indicatelosses.

Analyze the decisions from right to left:

1. Determine which alternative would be selected for each possible second decision. Fora small facility with high demand, there are two choices: do nothing, and expand.Because expand has higher payoff, you would choose it. Indicate this by placing adouble slash through do nothing alternative. Similarly, for a large facility with lowdemand, there are two choices: do nothing and reduce prices. You would choosereduce prices because it has the higher expected value, so a double slash is placed onthe other branch.

2. Determine the product of the chance probabilities and their respective payoffs for theremaining branches:

Build small

Low demand .4($40)�$16

High demand .6($55)� 33

Build large

Low demand .4($50)�20

High demand .6($70)�42

3. Determine the expected value of each initial alternative:

Build small $16�$33�$49

Build large $20�$42�$62

Hence, the choice should be to build the large facility because it has a larger expectedvalue than the small facility.

The above problem was entered in the Decision Analysis software DPL, fromSyncopation Software (www.syncopationsoftware.com/downloads.html). The solutionis shown below:

Influence DiagramsInfluence diagrams graphically represent complex situations with many random variables(events) and one or more decision variable. They are more concise than decision treesbecause they do not show the alternatives at the decision nodes and states of nature at the chance events. Constructing and validating the influence diagram improves commu-nication and consensus building at the beginning of the decision modelling process. The

Build[62.000]

Large

SmallDemand[49.000]

Demand[62.000]

High

60%

Low

40%

High

60%

Low40%

[40.000]40.000

Do nothing

Expand

[40.000]

40.000[55.000]

Do nothing

Reduce prices

[�10.000]

�10.000[50.000]

50.000

55.000

[70.000]

70.000

React to High Demand[55.000]

React to Low Demand[50.000]



following is an example of the influence diagram representing the decision of whether ornot to introduce a new product. The circles show the random variables (chance events)and the rounded squares show the payoff or part of it.

EXPECTED VALUE OF PERFECT INFORMATIONIn certain situations, it is possible to know with more certainty which state of nature of thecritical random variable will actually occur in the future. For instance, the choice of loca-tion for a restaurant may weigh heavily on whether a new highway will be constructed orwhether a zoning permit will be issued. A decision maker may have probabilities for thesestates of nature; however, it may be possible to delay a decision until it is more clearwhich state of nature will occur. This might involve taking an option to buy the land. Ifthe state of nature is favourable, the option can be exercised; if it is unfavourable, the op-tion can be allowed to expire. The question to consider is whether the cost of the optionwill be less than the expected gain due to delaying the decision. Other possible ways ofobtaining information about a random variable depend somewhat on the nature of therandom variable. Information about consumer preferences might come from market re-search, additional information could come from product testing, or legal experts might becalled on. The expected gain is the expected value of perfect information, or EVPI.

Expected value of perfect information (EVPI)—the difference between the expectedpayoff with perfect information and the expected payoff under risk.

To determine the EVPI, one can compute the expected payoff under certainty andsubtract the expected payoff under risk. That is,

Expected value of Expected payoff Expected payoff � � (5S–1)perfect information under certainty under risk

Using the information from Example S–1, determine the expected value of perfect infor-mation using Formula 5S–1.

First, compute the expected payoff under certainty. To do this, identify the best payoff undereach state of nature. Then combine these by weighting each payoff by the probability ofthat state of nature and adding the amounts. Thus, the best payoff under low demand is$10, the best under moderate demand is $12, and the best under high demand is $16. Theexpected payoff under certainty is, then:

.30($10)�.50($12)�.20($16)�$12.2


RevenueTotalCost

VariableCost

FixedCost

Profit

Price?

IntroduceProduct?

UnitsSold

This influence diagram for anew product decision alsoinvolves a pricing decision. Itwas produced with DPLsoftware. The key uncertaintiesare units sold, which isaffected by the pricingdecision, fixed cost, andvariable cost. Profit is theultimate value, which isinfluenced by the total cost and revenue values

Source: K. Chelst, “Can’t See theForest Because of the Decision Trees:A Critique of Decision Analysis inSurvey Texts,” Interfaces (28)2,March–April 1998, pp. 80–98.Reprinted by permission, copyright1998, the Institute for OperationsResearch and the ManagementSciences (INFORMS), 7240 ParkwayDrive, Suite 310, Hanover, MD 21076USA.

expected value of perfect

information (EVPI)

The difference between theexpected payoff with perfectinformation and the expectedpayoff under risk.

Example S–3

Solution


The expected payoff under risk, as computed in Example S–1, is $10.5. The EVPI is thedifference between these:

EVPI�$12.2�$10.5�$1.7

This figure indicates the upper limit on the amount the decision maker should be willingto spend to obtain information in this case. Thus, if the cost equals or exceeds thisamount, the decision maker would be better off not spending additional money and sim-ply going with the alternative that has the highest expected payoff.

SENSITIVITY ANALYSISGenerally speaking, both the payoffs and the probabilities in a decision problem areestimated values. Consequently, it can be useful for the decision maker to have someindication of how sensitive the choice of an alternative is to changes in one or more ofthese values. Unfortunately, it is impossible to consider all possible combinations ofevery variable in a typical problem. Nevertheless, there are certain things a decisionmaker can do to judge the sensitivity of probability estimates.

Sensitivity analysis provides a range of probability over which the choice of alterna-tives would remain the same. The approach illustrated here is useful when there are twostates of nature. It involves constructing a graph and then using algebra to determine a range of probabilities for which a given solution is best. In effect, the graph provides avisual indication of the range of probability over which the various alternatives areoptimal, and the algebra provides exact values of the endpoints of the ranges. Example S–4illustrates the procedure.

Given the following payoff table, determine the range of probability for state of nature #2,that is, P(2), for which each alternative is optimal using the expected-value criteria.

STATE OFNATURE

#1 #2

A 4 12Alternative B 16 2

C 12 8

First, plot each alternative relative to P(2). To do this, plot the #1 payoff on the left sideof the graph and the #2 payoff on the right side. For instance, for alternative A, plot 4 onthe left side of the graph and 12 on the right side. Then connect these two points with astraight line. The three alternatives are plotted on the graph as shown below.

The graph shows the range of values of P(2) over which each alternative is optimal.Thus, for low values of P(2) [and thus high values of P(1), since P(1)�P(2)�1.0], alter-native B will have the highest expected value; for intermediate values of P(2), alternativeC is best; and for higher values of P(2), alternative A is best.

To find exact values of the ranges, determine where the upper parts of the lines inter-sect. Note that at the intersections, the two alternatives represented by the lines would be






sensitivity analysis

Determining the range ofprobability for which analternative has the bestexpected payoff.

Example S–4

Solution


equivalent in terms of expected value. Hence, the decision maker would be indifferentbetween the two at that point. To determine the intersections, you must obtain theequation of each line. This is relatively simple to do. Because these are straight lines, theyhave the form y � a � bx, where a is the y-intercept value at the left axis, b is the slopeof the line, and x is P(2). Slope is defined as the change in y for a one-unit change in x. Inthis type of problem, the distance between the two vertical axes is 1.0. Consequently, theslope of each line is equal to the right-hand value minus the left-hand value. The slopesand equations are:

#1 #2 Slope Equation

A 4 12 12 � 4 � � 8 4 � 8P(2)B 16 2 2 � 16 � �14 16 � 14P(2)C 12 8 8 � 12 � � 4 12 � 4P(2)

From the graph, we can see that alternative B is best from P(2)�0 to the point wherethat straight line intersects the straight line of alternative C, and that begins the regionwhere C is better. To find that point, solve for the value of P(2) at their intersection. Thisrequires setting the two equations equal to each other and solving for P(2). Thus,

16 � 14P(2)�12�4P(2)

Rearranging terms yields

4�10P(2)

Solving yields P(2)�.40. Thus, alternative B is best from P(2)�0 up to P(2)�.40. B andC are equivalent at P(2)�.40.

Alternative C is best from that point until its line intersects alternative A’s line. To findthat intersection, set those two equations equal and solve for P(2). Thus,

4 � 8P(2)�12 � 4P(2)

Rearranging terms results in

12P(2)�8

Solving yields P(2)�.67. Thus, alternative C is best from P(2) � .40 up to P(2)�.67,where A and C are equivalent. For values of P(2) greater than .67 up to P(2)�1.0, A isbest.

Note: If a problem calls for ranges with respect to P(1), find the P(2) ranges as above,and then subtract each P(2) from 1.00 (e.g., .40 becomes .60, and .67 becomes .33).


16

14

12

10

8

6

4

2

0 1.0.8.6.4.2

#1Payoff

16

14

12

10

8

6

4

2

#2Payoff

B

CA

P(2)

B best A bestC best


Decision making is an integral part of operations management. Decision analysis is a generalapproach to decision making that is useful in many different aspects of operations management.Decision analysis provides a framework for the analysis of decisions. It involves identifying the decision alternatives, chance events, and payoffs. Two visual tools useful for analyzing some decision problems are decision trees and graphical sensitivity analysis.


SUMMARY

decision tree, 5S-3expected value of perfect

information (EVPI), 5S-6

payoff table, 5S-2sensitivity analysis, 5S-7

KEY TERMS

Solved Problems

Problem 1

The following solved problems refer to this payoff table:

New No Bridge NewBuilt Bridge

Alternative capacity A 1 14for new store B 2 10

C 4 6where A � small, B � medium, and C � large.

Using graphical sensitivity analysis, determine the probability for no new bridge for which eachalternative would be optimal.

Plot a straight line for each alternative. Do this by plotting the payoff for new bridge on the leftaxis and the payoff for no new bridge on the right axis and then connecting the two points. Eachline represents the expected profit for an alternative for the entire range of probability of no newbridge. Because the lines represent expected profit, the line that is highest for a given value of P(no new bridge) is optimal. Thus, from the graph, you can see that for low values of this proba-bility, alternative C is best, and for higher values, alternative A is best (B is never the highestline, so it is never optimal).

The dividing line between the ranges where C and A are optimal occurs where the two linesintersect. To find that probability, first formulate the equation for each line. To do this, let theintersection with the left axis be the y intercept; the slope equals the right-side payoff minus

4

21

0 1.0

Payoff ifnew bridge 14

10

6

0

Payoff ifno newbridge

A

A bestC best

B

C

.27P (no new bridge)

Solution


the left-side payoff. Thus, for C you have 4 � (6 � 4)P, which is 4 � 2P. For A, 1 � (14 � 1)P,which is 1 � 13P. Setting these two equal to each other, you can solve for P:

4 � 2P � 1 � 13P

Solving, P � .27. Therefore, the ranges for P(no new bridge) for each alternative to be bestare:

A: .27 � P � 1.00

B: never optimal

C: 0 � P � .27

Using the probabilities of .60 for a new bridge and .40 for no new bridge, compute the expectedvalue of each alternative in the payoff table, and identify the alternative that would be selectedunder the expected-value approach.

A: .60(1) � .40(14) � 6.20 [best]

B: .60(2) � .40(10) � 5.20

C: .60(4) � .40(6) � 4.80

Compute the EVPI using the information from the previous problem.

Using Formula 5S–1, the EVPI is the expected payoff under certainty minus the maximum ex-pected value. The expected payoff under certainty involves multiplying the best payoff in eachcolumn by the column probability and then summing those amounts. The best payoff in the firstcolumn is 4, and the best in the second is 14. Thus,

Expected payoff under certainty � .60(4) � .40(14) � 8.00

Then

EVPI � 8.00 � 6.20 � 1.80

Suppose that the values in the payoff table represent costs instead of profits.

a. Using sensitivity analysis, determine the range of P(no new bridge) for which each alterna-tive would be optimal.

b. If P(new bridge)�.60 and P(no new bridge) � .40, find the alternative chosen to minimizeexpected cost.

a. The graph is identical to that shown in Solved Problem 1. However, the lines now representexpected costs, so the best alternative for a given value of P(no new bridge) is the lowestline. Hence, for very low values of P(no new bridge), A is best; for intermediate values, B isbest; and for high values, C is best. You can set the equations of A and B, and B and C, equalto each other in order to determine the values of P(no new bridge) at their intersections.Thus,

A � B: 1 � 13P � 2 � 8P; solving, P � .20

B � C: 2 � 8P � 4 � 2P; solving, P � .33

Hence, the ranges are:

A best: 0 � P � .20

B best: .20 � P � .33

C best: .33 � P � 1.00

b. Expected-value computations are the same whether the values represent costs or profits.Hence, the expected payoffs for costs are the same as the expected payoffs for profits thatwere computed in Solved Problem 2. However, now you want the alternative that has thelowest expected payoff rather than the one with the highest payoff. Consequently, alternativeC is the best because its expected payoff is the lowest of the three.


Problem 2

Solution

Problem 3

Solution

Problem 4

Solution


4

21

0 1.0

Cost withnew bridge 14

10

6

0

Cost withno newbridge

A

C best

B

C

.20P (no new bridge)

.33

Bbest

Abest


1. List the steps in the decision-making process.

2. What information is contained in a payoff table?

3. What is sensitivity analysis, and how can it be useful to a decision maker?

4. Under what circumstances is expected monetary value appropriate as a decision criterion?When isn’t it appropriate?

5. Define expected value of perfect information.

6. What information does a decision maker need in order to perform an expected monetary-valueanalysis of a problem? What options are available to the decision maker if the probabilities ofthe states of nature are unknown? Can you think of a way you might use sensitivity analysis insuch a case?

7. Suppose a manager is using maximum EMV as a basis for making a capacity decision and, inthe process, obtains a result in which there is a virtual tie between two of the seven alterna-tives. How is the manager to make a decision?

1. A small building contractor has recently experienced two successive years in which workopportunities exceeded the firm’s capacity. The contractor must now make a decision oncapacity for next year. Estimated profits under each of the two possible states of nature fornext year’s demand are as shown in the table below for next year’s demand.

NEXT YEAR’S DEMAND

Alternative Low High

Do nothing $50* $60Expand 20 80Subcontract 40 70

*Profit in $ thousands.

Suppose after a certain amount of discussion, the contractor is able to subjectively assess theprobabilities of low and high demand: P(low)�.3 and P(high)�.7.

a. Determine the expected profit of each alternative. Which alternative is best?

b. Analyze the problem using a decision tree. Show the expected profit of each alternative onthe tree.

DISCUSSION AND REVIEW QUESTIONS

PROBLEMS


c. Compute the expected value of perfect information. How could the contractor use thisknowledge?

2. Refer to Problem 1. Construct a graph that will enable you to perform sensitivity analysis onthe problem. Over what range of P(high) would the alternative of doing nothing be best?Expand? Subcontract?

3. A firm that plans to expand its product line must decide whether to build a small or a largefacility to produce the new products. If it builds a small facility and demand is low, the netpresent value after deducting for building costs will be $400,000; If demand is high, the firmcan either maintain the small facility or expand it. Expansion would have a net present valueof $450,000, and maintaining the small facility would have a net present value of $50,000.

If a large facility is built and demand is high, the estimated net present value is $800,000;If demand turns out to be low, the net present value will be �$10,000.

The probability that demand will be high is estimated to be .60, and the probability of lowdemand is estimated to be .40.

a. Analyze using a tree diagram.

b. Compute the EVPI. How could this information be used?

c. Determine the range over which each alternative would be best in terms of the value ofP(demand low).

4. Determine the course of action that has the highest expected payoff for the decision tree below.

5. The lease of Theme Park, Inc., is about to expire. Management must decide whether to renewthe lease for another 10 years or to relocate near the site of a proposed motel. The town plan-ning board is currently debating the merits of granting approval to the motel. A consultant hasestimated the net present value of Theme Park’s two alternatives under each state of nature ofthe planning board’s decision as shown below.

Motel MotelOptions Approved Rejected

Renew $500,000 $4,000,000Relocate 5,000,000 100,000

Subc

ontra

ct

ExpandSmall demand (.4)

Large demand (.1)

Small demand (.4)

Medium demand (.5) Large demand (.1)

Medium demand (.5)

Medium demand (.5)

Large demand (.1)

Do nothing

* Net present value in millions

Other use #2

Do nothing

Other use #1

Other use #2

Do nothing

Subcontract

Build

Do nothing

Other use #1

Other use #2

Do nothing

Expand

Build

Do nothing

Expand

$1.0*

$1.3

$1.3

$1.5

$1.6

$1.8

$0.7

$1.5

$1.0

$1.6

$1.6

$1.5

$1.7

($0.9)

$1.4

$1.0

$1.0

$1.1

$0.9

$2.4

Build

Other use #1

Small

deman

d (.4)



Suppose that the management of Theme Park, Inc., has decided that there is a .35 probabilitythat the motel’s application will be approved.

a. If management uses maximum expected monetary value as the decision criterion, whichalternative should it choose?

b. Represent this problem in the form of a decision tree.

c. If management has been offered the option of a temporary lease while the town planningboard considers the motel’s application, would you advise management to sign the lease?The lease will cost $24,000.

6. Construct a graph that can be used for sensitivity analysis for the preceding problem.

a. How sensitive is the solution to the problem in terms of the probability estimate of .35?

b. Suppose that, after consulting with a member of the town planning board, managementdecides that an estimate of approval is approximately .45. How sensitive is the solution tothis revised estimate? Explain.

c. Suppose the management is confident of all the estimated payoffs except for $4 million. Ifthe probability of approval is .35, for what range of payoff for renew/rejected will thealternative selected using maximum expected value remain the same?

7. A firm must decide whether to construct a small, medium, or large plant. A consultant’s reportindicates a .20 probability that demand will be low and an .80 probability that demand will be high.

If the firm builds a small facility and demand turns out to be low, the net present valuewill be $42 million; if demand turns out to be high, the firm can either subcontract andrealize the net present value of $42 million or expand for a net present value of $48 million.

The firm could build a medium-size facility as a hedge: if demand turns out to be low, itsnet present value is estimated at $22 million; if demand turns out to be high, the firm could donothing and realize a net present value of $46 million, or it could expand and realize a netpresent value of $50 million.

If the firm builds a large facility and demand is low, the net present value will be �$20 million,whereas high demand will result in a net present value of $72 million.

a. Analyze this problem using a decision tree. What is the maximum EMV alternative?

b. Compute the EVPI and interpret it.

c. Perform sensitivity analysis on P(high).

8. A manager must decide how many machines of a certain type to buy. The machines will beused to manufacture a new gear for which there is increased demand. The manager has nar-rowed the decision to two alternatives: buy one machine or buy two. If only one machine ispurchased and demand is more than it can handle, a second machine can be purchased at a latertime. However, the cost per machine would be lower if the two machines were purchased atthe same time.

The estimated probability of low demand is .30, and the estimated probability of highdemand is .70.

The net present value associated with the purchase of two machines initially is $75,000 ifdemand is low and $130,000 if demand is high.

The net present value for one machine and low demand is $90,000; if demand is high, thereare three options: one option is to do nothing, which would have a net present value of$90,000; a second option is to subcontract, which would have a net present value of $110,000;the third option is to purchase a second machine, which would have a net present value of$100,000.

How many machines should the manager purchase initially? Use a decision tree to analyzethis problem.



9. Determine the course of action that has the highest EMV for the following tree diagram.

10. The director of social services of a province has learned that a new Act has mandatedadditional information requirements. This will place an additional burden on the agency. Thedirector has identified three acceptable alternatives to handle the increased workload. Onealternative is to reassign present staff members, the second is to hire and train two new work-ers, and the third is to redesign current practice so that workers can readily collect theinformation with little additional effort. An unknown factor is the caseload for the coming yearwhen the new data will be collected on a trial basis. The estimated costs for various optionsand caseloads are shown in the following table:

CASELOAD

Moderate High Very High

Reassign staff $50* 60 85New staff 60 60 60Redesign collection 40 50 90

*Cost in $ thousands.

The director of social services has decided that reasonable caseload probabilities are .10 formoderate, .30 for high, and .60 for very high.

a. Which alternative will yield the minimum expected cost?

b. Construct a decision tree for this problem. Indicate the expected costs for the three decisionbranches.

c. Determine the expected value of perfect information.

d. Suppose the director of social services has the option of hiring an additional staff memberif one staff member is hired initially and the caseload turns out to be high or very high.Under that plan, the first entry in row 2 of the cost table (see Problem 10) will be 40 insteadof 60, the second entry will be 75, and the last entry will be 80. Construct a decision treethat shows the sequential nature of this decision, and determine which alternative willminimize expected cost.

45(45)

99

40

50

30

1/3

1/3

1/3

40

50

1/2

1/2

.30

.20

.50

.30

.20

.50

600

90

40

44

60

1/3

1/3

1/3

Alternative A

Alternative B



11. A manager has compiled estimated profits for various capacity alternatives but is reluctant toassign probabilities to the states of nature. The payoff table is:

STATE OF NATURE

#1 #2

A $20* 140Alternative B 120 80

C 100 40*In $ thousands.

a. Plot the expected-value lines on a graph.

b. Is there any alternative that would never be appropriate in terms of maximizing expectedprofit? Explain on the basis of your graph.

c. For what range of P(#2) would alternative A be the best choice if the goal is to maximizeexpected profit?

d. For what range of P(#1) would alternative A be the best choice if the goal is to maximizeexpected profit?

12. Repeat all parts of Problem 11, assuming the values in the payoff table are estimated costs andthe goal is to minimize expected costs.

13. The research staff of a marketing agency has assembled the following payoff table of esti-mated profits for four proposals:

Receive Not Receive Contract Contract

#1 $10* �2

Proposal #2 8 3#3 5 5#4 0 7

*In $ thousands.

Relative to the probability of not receiving the contract, determine the range of probability forwhich each of the proposals would maximize expected profit.

14. Given this payoff table:

STATE OF NATURE

#1 #2

A $120* 20

Alternative B 60 40C 10 110D 90 90

*In $ thousands.

a. Determine the range of P(#2) for which each alternative would be best, treating the payoffsas profits.

b. Answer part a treating the payoffs as costs.

15. A law firm is representing a company which is being sued by a customer. The decision iswhether to go through litigation or settle the case out of court. The cost of losing the case isestimated to be $1 million, whereas the cost of settling the case is $200,000. Winning the casein court would result in no loss. What is the minimum chance of winning for which thecompany should contest the case?

16. An 18 year old boy has just arrived at a hospital complaining of abdominal pain. The medicalfindings are consistent with appendicitis but not totally typical of appendicitis. The lab and



Abt, R., et al. “ The Dangerous Quest for Certainty in MarketForecasting,” Long Range Planning (12), April 1979,pp. 52–62.

Bierman, Harold, Charles P. Bonini, and Warren H. Haus-man. Quantitative Analysis for Business Decisions. 8thed. Burr Ridge, IL: Richard D. Irwin, 1991.

Clemen, R. T., Making Hard Decisions, 2nd ed. PacificGrove, CA: Duxbury Press, 1996.

Eppen, G. D., F. J. Gould, C. P. Schmidt, Jeffrey H. Moore,and Larry R. Weatherford. Introductory ManagementScience. 5th ed. Upper Saddle River, NJ: Prentice Hall,1998.

Hammond, J. S., et al. Smart choices. Boston: HarvardBusiness School press, 1999.

Stevenson, William J., Introduction to Management Science.3rd ed. Burr Ridge, IL: Richard D. Irwin, 1998.

Taylor, Bernard W., Introduction to Management Science.6th ed. Dubuque, Iowa: William C. Brown, 1999.

Turban, Efraim, and Jack Meredith. Fundamentals ofManagement Science. New York: McGraw-Hill, 1998.

X-ray test results are not clear. The surgeon is wondering whether to operate or wait 12 hours(in case it is not appendicitis). In this case, if the pain does not recede, then it must be appen-dicitis and the surgeon will operate. The probability that it is appendicitis is 56 percent usingpast experience with similar cases. Also, from similar cases, the probability that the appen-dicitis will perforate after 12 hours of wait is 6 percent. The payoffs are measured in terms ofdeath rate. The death rate of operating when appendicitis is present is 0.09 percent, while whenit is not present is 0.04 percent. The death rate of operating a perforated appendicitis is 0.64percent. Draw the decision tree and determine the best course of action in this case.1


Oglethorpe Power is an electric generating and distribu-tion cooperative supplying 20 percent of the electricity

used in Georgia, U.S. Most of the remaining demand issupplied by Georgia Power Co. Oglethorpe and GeorgiaPower jointly own most of the transmission lines in Georgia,and supply their excess electricity to Florida.

In the late 1990s, Florida Power and Light (FPL) indicatedto Oglethorpe that they were interested in constructing anothermajor transmission line between Florida and Georgia.Oglethorpe invited the consulting company, Applied DecisionAnalysis, Inc., to assist it in making a decision.

A team was formed which investigated the decision vari-ables, random variables, and their payoffs. Brainstormingresulted in an influence diagram and identification of the threedecision variables: line (joint with Georgia Power, alone, or noline), nature of control (Oglethorpe, or FPL), and whether toupgrade the associated facilities (joint with Georgia Power,alone, or no upgrade). Five random variables were identified:

construction cost (low, medium, or high), competitive situa-tion in Florida (good, fair, or bad), Florida demand (low,medium, or high). Oglethorpe’s share of demand (very low,low, medium, high, or very high), and future spot price forelectricity (low, medium, or high). The payoffs were measuredin terms of net present value.

The team estimated the probability of random events (thestates of nature for each random variable) and estimated thepayoffs for each combination of decision alternatives andstates of nature. Using the DPL software, the alternative (inbrackets) for each decision variable with highest expectedmonetary value was: line (alone), control (Oglethorpe), andupgrade (no). Then the risk profile (the distribution of NPV) ofthis solution was determined by DPL. Because this showedconsiderable possible negative NPV, the team identified therandom variable which most affected the downside risk: thecompetitive situation in Florida. The next step was to collectfurther information on this random variable in order to reducethe range of values for the payoff. The team reported all itsfindings to top management who started negotiating with FPL.

Applied Decision Analysis, Inc. was purchased by Price-waterhouseCoopers in 1998.

Source: A. Borison, “Oglethorpe Power Corporation Decides aboutInvesting in a Major Transmission System,” Interfaces (25)2,March–April 1995, pp. 25–36.

R E A D I N G

Oglethorpe Power:Example of DecisionAnalysis

SELECTEDBIBLIOGRAPHY ANDFURTHER READING

1J.R. Clarke, “The Application of Decision Analysis to Clinical Medicine,” Interfaces (17)2, March–April 1987,pp. 27–34


supplement ch05

Documents

decision problems

decision analysis approach

decision criterion

decision trees

decision maker

decision making

example of decision

simple decision problem