Making Decisions Under Certainty, Uncertainty, and Risk

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Decision Theory LEARNING OBJECTIVES SUPPLEMENT OUTLINE 5S.5 Decision Making under Uncertainty, 219 After completing this supplement, you 5S.6 Decision Making under Risk, 220 5S.1 Introduction, 216 should be able to: 5S.7 Decision Trees, 2215S.2 The Decision Process and Causes of L05S.1 Outline the steps in the decision Poor Decisions, 217 5S.8 Expected Value of Perfect

process. Information, 2235S.3 Decision Environments, 218 L05S.2 Name some causes of poor 5S.9 Sensitivity Analysis, 224

decisions. 5S.4 Decision Making under Certainty, 218

L05S.3 Describe and use techniques that apply to decision making under uncertainty.

L05S.4 Describe and use the expected- value approach.

L05S.5 Construct a decision tree and use it to analyze a problem.

L05S.6 Compute the expected value of perfect information.

L05S.7 Conduct sensitivity analysis on a simple decision problem.

55.1 INTRODUCTION -

Decision theory represents a general approach to decision making. It is suitable for a wide range of operations management decisions. Among them are capacity planning, product and service design, equipment selection, and location planning. Decisions that lend themselves to a decision theory approach tend to be characterized by the following elements:

1. A set of possible future conditions that will have a bearing on the results of the decision.

2. 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; the competitor will or will not introduce a new product). These are called states of nature.

2. Develop a list of possible alternatives, one of which may be to do nothing. t

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

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216

217 Supplement to Chapter Five Decision Theory

If possible, estimate the likelihood of each possible future condition.

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

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

POSSIBLE FUTURE DEMAND

Alternatives Low Moderate High Small facility $10* $10 $10 Medium facility 7 12 12 Large facil ity (4) 2 16 'Present value in $ millions.

The payoffs are shown in the body of the table. In this instance, the payoffs are in terms of present values, which represent equivalent current dollar values of expected future income less costs. This is a convenient measure because it places all alternatives on a comparable basis. If a small facility is built, the payoff will be the same for all three possible states of nature. For a medium facility, low demand will have a present value of $7 million, whereas both moderate and high demand will have present values of $12 million. A large facility will have a loss of $4 million if demand is low, a present value of $2 million if demand is moder- ate, and a present value of $16 million if demand is high.

The problem for the decision maker is to select one of the alternatives, taking the present value into account.

Evaluation of the alternatives differs according to the degree of certainty associated with the possible future conditions.

5S.2 THE DECISION PROCESS AND CAUSES OF POOR DECISIONS

Despite the best efforts of a manager, a decision occasionally turns out poorly due to unfore- seeable circumstances. Luckily, such occurrences are not common. Often, failures can be traced to a combination of mistakes in the decision process, to bounded rationality, or to suboptimization.

The decision process consists of these steps:

1. Identify the problem.

2. Specify objectives and criteria for a solution.

3. Develop suitable alternatives.

4. Analyze and compare alternatives.

5. Select the best alternative.

6. Implement the solution.

7. Monitor to see that desired result is achieved.

In many cases, managers fail to appreciate the importance of each step in the decision- making process. They may skip a step or not devote enough effort to completing it before jumping to the next step. Sometimes this happens owing to a manager's style of making quick decisions or a failure to recognize the consequences of a poor decision. The manager's ego can be a factor. This sometimes happens when the manager has experienced a series of successes- important decisions that turned out right. Some managers then get the impression that they can do no wrong. But they soon run into trouble, which is usually enough to bring them back down

Payoff table Table showing the expected payoffs for each alternative in every possible state of nature.

L05S.1 Outline the steps in the decision process.

L05S.2 Name some causes of poor decisions.

••

218

Bounded rationality The limitations on decision making caused by costs, human abilities, time, technology, and availability of information.

Suboptimization The result of different departments each attempting to reach a solu- tion that is optimum for that department.

Certainty Environment in which relevant parameters have known values.

Risk Environment in which cer- tain future events have probable outcomes.

Uncertainty Environment in which it is impossible to assess the likelihood of various future events.

EXAMPLE 55-1

Supplement to Chapter Five Decision Theory

to earth. Other managers seem oblivious to negative results and continue the process they associate with their previous successes, not recognizing that some of that success may have been due more to luck than to any special abilities of their own. A part of the problem may be the manager's unwillingness. to admit a mistake. Yet other managers demonstrate an inability to make a decision; they stall long past the time when the decision should have been rendered .

Of course, not all managers fall into these traps-it seems safe to say that the majority do not. Even so, this does not necessarily mean that every decision works out as expected. Another factor with which managers must contend is bounded rationality, or the limits imposed on decision making by costs, human abilities, time, technology, and the availability of information. Because of these limitations, managers cannot always expect to reach deci- sions that are optimal in the sense of providing the best possible outcome (e.g., highest profit, least cost). Instead, they must often resort to achieving a satisfactory solution.

Still another cause of poor decisions is that organizations typically departmentalize deci- sions. Naturally, there is a great deal of justification for the use of departments in terms of overcoming span-of-control problems and human limitations. However, suboptimization can occur. This is a result of different departments' attempts to reach a solution that is optimum for each. Unfortunately, what is optimal for one department may not be optimal for the orga- nization as a whole. If you are familiar with the theory of constraints (see Chapter 16), subop- timization and local optima are conceptually the same, with the same negative consequences.

5S.3 DECISION ENVIRONMENTS Operations management decision environments are classified according to the degree of cer- tainty present. There are three basic categories: certainty, risk, and uncertainty.

. Certainty means that relevant parameters such as costs, capacity, and demand have known values.

Risk means that certain parameters have probabilistic outcomes.

Uncertainty means that it is impossible to assess the likelihood of various possible future events.

Consider these situations:

1. Profit per unit is $5. You have an order for 200 units. How much profit will you make? (This is an example of certainty since unit profits and total demand are known.)

2. Profit is $5 per unit. Based on previous experience, there is a 50 percent chance of an order for 100 units and a 50 percent chance of an order for 200 units. What is expected profit? (This is an example of risk since demand outcomes are probabilistic.)

3. Profit is $5 per unit. The probabilities of potential demands are unknown. (This is an example of uncertainty.)

The importance of these different decision environments is that they require different anal- ysis techniques. Some techniques are better suited for one category than for others.

5S.4 DECISION MAKING UNDER CERTAINTY When it is known for certain which of the possible future conditions will actually happen, the decision is usually relatively straightforward: Simply choose the alternative that has the best payoff under that state of nature. Example 5S-1 illustrates this.

Determine the best alternative in the payoff table on the previous page for each of-the cases: It is known with certainty that demand will be (a) low, (b) moderate, (c) high.

219

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Supplement to Chapter Five Decision Theory

Choose the alternative with the highest payoff. Thus, if we know demand will be low, we would elect to build the small facility and realize a payoff of : 10 million. If we know demand will be moderate, a medium factory would yield the highest payoff ($12 million versus either $10 million or $2 million). For high demand, a large facility would provide the highest payoff.

Although complete certainty is rare in such situations, this kind of exercise provides some perspective on the analysis. Moreover, in some instances, there may be an opportunity to con- sider allocation of funds to research efforts, which may reduce or remove some of the uncer- tainty surrounding the states of nature, converting uncertainty to risk or to certainty.

5S.5 DECISION MAKING UNDER UNCERTAINTY At the opposite extreme is complete uncertainty: 0 information is available on how likely the various states of nature are. Under those conditions, four possible decision criteria are maximin, maximax, Laplace, and minimax regret. These approaches can be defined as follows:

Maximin-Determine the worst possible payoff for each alternative, and choose the alternative that has the "best worst." The maximin approach is essentially a pessimistic one because it takes into account only the worst possible outcome for each alternative. The actual outcome may not be as bad as that, but this approach establishes a "guaranteed minimum."

Maximax-Determine the best possible payoff, and choose the alternative with that pay- off. The maximax approach is an optimistic, "go for it" strategy; it does not take into account any payoff other than the best.

Laplace-Determine the average payoff for each alternative, and choose the alternative with the best average. The Laplace approach treats the states of nature as equally likely.

Minimax regret-Determine the worst regret for each alternative, and choose the alter- native with the "best worst." This approach seeks to minimize the difference between the payoff that is realized and the best payoff for each state of nature.

The next two examples illustrate these decision criteria.

Referring to the payoff table on page 217, determine which alternative would be chosen under each of these strategies:

a. Maximin

b. Maximax

c. Laplace

a. Using maximin, the worst payoffs for the alternatives are as follows: Small facility: $10 million

Medium facility: 7 million

Large facility: -4 million

Hence, since $10 million is the best, choose to build the small facility using the maximin strategy.

b. Using maximax, the best payoffs are as follows: Small facility: $10 million

Medium facility: 12 million

Large facility: 16 million

The best overall payoff is the $16 million in the third row. Hence, the maximax criterion leads to building a large facility.

SOLUTION

L05S.3 Describe and use techniques that apply to deci- sion making under uncertainty.

Maximin Choose the alterna- tive with the best of the worst possible payoffs.

Maximax Choose the alter- native with the best possible payoff.

Laplace Choose the alternative with the best average payoff of any of the alternatives.

Minimax regret Choose the alternative that has the least of the worst regrets.

EXAMPLE 5S-2

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EXAMPLE 5S-3

SOLUTION

Regret (opportunity loss) The differencebetween a given payoff and the best payoff for a state of nature.

L05S.4 Describe and use the expected-value approach.

Expected monetary value (EMV) criterion The best expected value among the alternatives.

Supplement to Chapter Five Decision Theory

c. For the Laplace criterion, first find the row totals, and then divide each of those amounts by the number of states of nature (three in this case). Thus, we have

Row Total (in $ millions)

Row Average (in $ millions)

Small facility

Medium facility

Large facility

$30 31 14

$10.00 10.33 4.67

Because the medium facility has the highest average, it would be chosen under the Laplace criterion.

Determine which alternative would be chosen using a minimax regret approach to the capac- ity planning program.

The first step in this approach is to prepare a table of regrets (or opportunity losses). To do this, subtract every payoff in each column from the best payoff in that column. For instance, in the first column, the best payoff is 10, so each of the three numbers in that column must be subtracted from 10. Going down the column, the regrets will be 10 - 10 = 0, 10 - 7 = 3, and 10 - (-4) = 14. In the second column, the best payoff is 12. Subtracting each payoff from 12 yields 2, 0, and 10. In the third column, 16 is the best payoff. The regrets are 6, 4, and 0. These results are summarized in a regret table:

REGRETS (IN $ MILLIONS)

Alternatives Low Moderate High Worst

,Small facility $0 $2 $6 $6 Medium facility 3 0 4 4 Large facility 14 10 0 14

The second step is to identify the worst regret for each alternative. For the first alternative, the worst is 6; for the second, the worst is 4; and for the third, the worst is 14.

The best of these worst regrets would be chosen using minimax regret. The lowest regret is 4, which is for a medium facility. Hence, that alternative would be chosen.

Solved Problem 6 at the end of this supplement illustrates decision making under uncer- tainty when the payoffs represent costs.

The main weakness of these approaches (except for Laplace) is that they do not take into account all of the payoffs. Instead, they focus on the worst or best, and so they lose some information. Still, for a given set of circumstances, each has certain merits that can be helpful to a decision maker.

5S.6 DECISION MAKING UNDER RISK Between the two extremes of certainty and uncertainty lies the case of risk: The probability of occurrence for each state of nature is known. (Note that because the states are mutually exclusive and collectively exhaustive, these probabilities must add to 1.00.) A widely used approach under such circumstances is the expected monetary value criterion. The expected value is computed for each alternative, and the one with the best expected value is selected. The expected value is the sum of the payoffs for an alternative where each payoff is weighted by the probability for the relevant state of nature. Thus, the approach is

Expected monetary value (EMV) criterion-Determine the expected 'payoff or each alternative, and choose the alternative that has the best expected payoff.

Supplement to Chapter Five Decision Theory

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

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

= .30($10) = .30($7) = .30($-4)

EVsmall

EVmedium

EVlarge

.50($10)

.50($12)

.50($2)

+ .20($10) = $10 .20($12) = $10.5 .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 is nei- ther risk averse nor risk seeking, but is risk neutral. Typically, well-established organizations with numerous decisions of this nature tend to use expected value because it provides an indi- cation of the long-run, average payoff. That is, the expected-value amount (e.g., $10.5 million in the last example) is not an actual payoff but an expected or average amount that would be approximated if a large number of identical decisions were to be made. Hence, if a decision maker applies this criterion to a large number of similar decisions, the expected payoff for the total will approximate the sum of the individual expected payoffs.

55.7 DECISION TREES In health care the array of treatment options and medical costs makes tools such as decision trees particularly valuable in diagnosing and prescribing treatment plans. For example, if a 20-year-old and a 50-year-old both are brought into an emergency room complaining of chest pains, the attending physician, after asking each some questions on family history, patient his- tory, general health, and recent events and activities, will use a decision tree to sort through the options to arrive at the appropriate decision for each patient.

Decision trees are tools that have many practical applications, not only in health care but also in legal cases and a wide array of management decision making, including credit card fraud; loan, credit, and insurance risk analysis; decisions on new product or service develop- ment; and location analysis.

A decision tree is a schematic representation of the alternatives available to a decision maker and their possible consequences. The term gets its name from the treelike appearance of the diagram (see Figure 5S.1). Although tree diagrams can be used in place of a payoff

Payoff 1

Payoff 2 \>-,

~e °CJ",o

Payoff 3 Initial decision

Payoff 4 C-?

00 6'(9 ~ Payoff 5

Payoff 6

• Decision point o Chance event

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EXAMPLE 55-4

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l05S.5 Construct a decision tree and use it to analyze a problem.

Decision tree A schematic representation of the available alternatives and their possible consequences.

FIGURE 5S.1 Format of a decision tree

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EXAMPLE 55-5

SOLUTION

Supplement to Chapter Five Decision Theory

table, they are particularly useful for analyzing situations that involve sequential decisions. For instance, a manager may initially decide to build a small facility only to discover that demand is much higher than anticipated. In this case, the manager may then be called upon to make a subsequent decision on whether to expand or build an additional facility.

A decision tree is composed of a number of nodes that have branches emanating from them (see Figure 5S.1). Square nodes denote decision points, and circular nodes denote chance events. Read the tree from left to right. Branches leaving square nodes represent alternatives; branches leaving circular nodes represent chance events (i.e., the possible states of nature).

After the tree has been drawn, it is analyzed from right to left; that is, starting with the last decision that might be made. For each decision, choose the alternative that will yield the 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 has narrowed the choices to two: large or small. Information has been collected on payoffs, and a decision tree has been constructed. Analyze the decision tree and determine which initial alternative (build small or build large) should be chosen in order to maximize expected monetary value.

o \./;'\ $40 {\\'O~

N'l 0e

\..0

,,\~~ $40 ~0'V.

\)0 Overtime $50

E:'JrIJ Cillo

$55 ,,\~~ ($10)

~0'V. \)0

o \./;'\ {\\'O~

Reo. N'l 0e\..0 liCe

IJric $50es High

oerr, Cillo $70(.6)

S,.. "110-:ge

The dollar amounts at the branch ends indicate the estimated payoffs if the sequence of chance events and decisions that is traced back to the initial decision occurs. For example, if the initial decision is to build a small facility and it turns out that demand is low, the payoff will be $40 (thousand). Similarly, if a small facility is built, demand turns out high, and a later decision is made to expand, the payoff will be $55 (thousand). The figures in parentheses on branches leaving the chance nodes indicate the probabilities of those states of nature. Hence, the probability of low demand is .4, and the probability of high demand is .6. Payoffs in parentheses indicate losses.

Analyze the decisions from right to left:

1. Determine which alternative would be selected for each possible second decision. For a small facility with high demand, there are three choices: do nothing, work overtime, and expand. Because expand has the highest payoff, you would choose it. Indicate this by placing a double slash through each of the other alternatives. Similarly, for a large facility with low demand, there are two choices: do nothing and reduce prices'Xcsa would choose reduce prices because it has the higher expected value, so a double slash is placed on the other branch.

Supplement to Chapter Five Decision Theory 223 Determine the product of the chance probabilities and their respective payoffs for the remaining 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 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 expected value than the small facility .

.8 EXPECTED VALUE OF PERFECT INFORMATION

certain situations, it is possible to ascertain which state of nature will actually occur in the e. For instance, the choice of location for a restaurant may weigh heavily on whether a

- . highway will be constructed or whether a zoning permit will be issued. A decision maker have probabilities for these states of nature; however, it may be possible to delay a deci- until it is clear which state of nature will occur. This might involve taking an option to

_.; the land. If the state of nature is favorable, the option can be exercised; if it is unfavorable, option can be allowed to expire. The question to consider is whether the cost of the option

ill be less than the expected gain due to delaying the decision (i.e., the expected payoff ve the expected value). The expected gain is the expected value of perfect information,

EVPI. Other possible ways of obtaining perfect information depend somewhat on the nature of

- decision being made. Information about consumer preferences might come from market ch, additional information about a product could come from product testing, or legal

rts might be called on. There are two ways to determine the EVPL One is to compute the expected payoff under

enainty and subtract the expected payoff under risk. That is,

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

"':-singthe information from Example 5S-4, determine the expected value of perfect informa- n using Formula 5S-1.

-=- t, compute the expected payoff under certainty. To do this, identify the best payoff under h state of nature. Then combine these by weighting each payoff by the probability of that e of nature and adding the amounts. Thus, the best payoff under low demand is $10, the t under moderate demand is 12, and the best under high demand is $16. The expected

yoff under certainty is, then,

.30($10) + .50($12) + .20($16) = $12.2

L05S.6 Compute the expected value of perfect information.

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

EXAMPLE 5S-6

SOLUTION #

224

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EXAMPLE 5S-7

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SOLUTION

Ul5S.7 Conduct sensitivity analysis on a simple decision problem.

Sensitivity analysis Deter- mining the range of probability for which an alternative has the best expected payoff.

EXAMPLE 5S-8

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SOLUTION

Supplement to Chapter Five Decision Theory

The expected payoff under risk, as computed in Example 5S--4, is $10.5. The EVPI is the dif- ference between these:

EVPI = $12.2 - $10.5 = $1.7

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

A second approach is to use the regret table to compute the EVPI. To do this, find the expected regret for each alternative. The minimum expected regret is equal to the EVPI.

Determine the expected value of perfect information for the capacity-planning problem using the expected regret approach.

Using information from Examples 5S-2, 5S-3, and 5S--4, we can compute the expected regret for each alternative. Thus: '

Small facility Medium facility Large facility

+ .20(6) = 2.2 + .20(4) = 1.7 [minimum] + .20(0) = 9.2

.30(0)

.30(3)

.30(14)

+ .50(2) + .50(0) + .50(10)

The lowest expected regret is l.7, which is associated with the second alternative. Hence, the EVPI is $l.7 million, which agrees with the previous example using the other approach.

5S.9 SENSITIVITY ANALYSIS Generally speaking, both the payoffs and the probabilities in this kind of a decision problem are estimated values. Consequently, it can be useful for the decision maker to have some indication of how sensitive the choice of an alternative is to changes in one or more of these values. Unfortunately, it is impossible to consider all possible combinations of every variable in a typical problem. Nevertheless, there are certain things a decision maker can do to judge the sensitivity of probability estimates.

Sensitivity analysis provides a range of probability over which the choice of alternatives would remain the same. The approach illustrated here is useful when there are two states of nature. It involves constructing a graph and then using algebra to determine a range of prob- abilities for which a given solution is best. In effect, the graph provides a visual indication of the range of probability over which the various alternatives are optimal, and the algebra provides exact values of the endpoints of the ranges. Example 5S-8 illustrates the procedure .•.•. Given the following table, determine the range of probability for state of nature #2, that is, P(2), for which each alternative is optimal under the expected-value approach.

STATE OF NATURE

#1 #2

A 4 12 Alternative B 16 2

C 12 8

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

Supplement to Chapter Five Decision Theory 225

16 16

14

12

10

#2 8 Payoff

6

4

2

.8 1.0

14

12

10

#1 Payoff 8

6 I I

4 I I

2 I --B best--~C best-i~--

o .2 .4 .6 P(2)

The graph shows the range of values of P(2) over which each alternative is optimal. Thus, low values of P(2) [and thus high values of P(l), since P(l) + P(2) = 1.0], alternative B have the highest expected value; for intermediate values of P(2), alternative C is best; and

higher values of P(2), alternative A is best. To find exact values of the ranges, determine where the upper parts of the lines intersect. e that at the intersections, the two alternatives represented by the lines would be equivalent

terms of expected value. Hence, the decision maker would be indifferent between the two :nat point. To determine the intersections, you must obtain the equation of each line. This relatively simple to do. Because these are straight lines, they have the form y = a + bx, - e a is the y-intercept value at the left axis, b is the slope of the line, and x is P(2). Slope

fined as the change in y for a one-unit change in x. In this type of problem, the distance 'een the two vertical axes is 1.0. Consequently, the slope of each line is equal to the right- value minus the left-hand value. The slopes and equations are as follows:

#1 #2 Slope Equation 4 12 12 - 4 = + 8 4 + 81'1:2)

16 2 2 -16 = -14 16 - 141'1:2)

12 8 8-12=- 4 12-41'1:2)

From the graph, we can see that alternative B is best from P(2) 5 0 to the point where that ight line intersects the straight line of alternative C, and that begins the region where C is r. To find that point, solve for the value of P(2) at their intersection. This requires setting

two equations equal to each other and solving for P(2). Thus,

16 - 14P(2) = 12 - 4P(2)

Rearranging terms yields

4 = lOP(2) olving yields P(2) = AO. Thus, alternative B is best from P(2) = 0 up to P(2) = AO. B C are equivalent at P(2) = AO.

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

+ 8P(2) = 12 - 4P(2) Rearranging terms results in

12P(2) = 8

olving yields P(2) = .67. Thus, alternative C is best from P(2) > AO up to P(2) = .67, re A and C are equivalent. For values of P(2) greater than .67 up to P(2) = 1.0, A is best.

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

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SUMMARY

.. " KEY TERMS

SOLVED PROBLEMS

Problem 1

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Solution

Problem 2

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Supplement to Chapter Five Decision Theory

Decision making is an integral part of operations management. Decision theory is a general approach to decision making that is useful in many different aspects of

operations management. Decision theory provides a framework for the analysis of decisions. It includes a number of techniques that can be classified according to the degree of uncertainty associated with a particular decision problem. Two visual tools useful for analyzing some decision problems are decision trees and graphical sensitivity analysis.

bounded rationality, 218 Laplace, 219 risk, 218 certainty, 218 maximax, 219 sensitivity analysis, 224 decision tree, 221 maximin, 219 suboptimization, 218 expected monetary value minimax regret, 219 uncertainty, 218 (EMV) criterion, 220 payoff table, 217 expected value of perfect regret {opportunity information (EVPI), 223 loss), 220

The following solved problems refer to this payoff table:

New No Bridge New Built Bridge

Alternative capacity A 1 14 for new store B 2 10

C 4 6

where A = small, B = medium, and C = large.

Assume the payoffs represent profits. Determine the alternative that would be chosen under each of these decision criteria:

a. Maximin.

b. Maximax.

c. Laplace.

New No New Maximin Maximax Laplace Bridge Bridge (worst) (best) (average)

A 1 14 1 14 [best] 15 7 2 = 7.5 [best] B 2 10 2 10 1272=6 C 4 6 4 [best] 6 1072=5

Thus, the alternatives chosen would be C under maximin, A under maximax, and A under Laplace.

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

Plot a straight line for each alternative. Do this by plotting the payoff for new bridge on the left axis and the payoff for no new bridge on the right axis and then connecting the two points. Each line represents the expected profit for an alternative for the entire range of probability of no new bridge. 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 probability, alterna- tive C is best, and for higher values, alternative A is best (B is never the highest line, so it is never optimal).

Supplement to Chapter Five Decision Theory 227

Payoff for new bridge 14

Payoff for no new bridge

10

6

4

2 1

o .27 P(no new bridge)

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

Thus, for C you have 4 + (6 - 4)P, which is 4 + 2P For A, I + (14 - I)P, which is I + 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) are

A: .27 < P :s 1.00 B: never optimal C: o s P < .27

Using the information in the payoff table, develop a table of regrets, and then Problem 3 a. Determine the alternative that would be chosen under minimax regret. b. Determine the expected value of perfect information using the regret table, assuming that the eXcel

probability of a new bridge being built is .60. mhhe.com/stevenson12e

To obtain the regrets, subtract all payoffs in each column from the best payoff in the column. The Solution regrets are

New No New Bridge Bridge

A

B

C

3

2 o

o 4

8

a. Minimax regret involves finding the worst regret for each alternative and then choosing native that has the "best" worst. Thus, you would choose A:

the alter-

A

B

C

Worst

3 [best] 4

8

iI'

...•.

228 Supplement to Chapter Five Decision Theory

b. Once the regret table has been developed, you can compute the EVPI as the smallest expected regret. Since the probability of a new bridge is given as .60, we can deduce that the probability of no new bridge is l.00 - .60 = .40. The expected regrets are

A: .60(3) + .40(0) = l.80 B: .60(2) + .40(4) = 2.80 C: .60(0) + .40(8) = 3.20

Hence, the EVPI is l.80.

mhhe.com/stevenson12e

Using the probabilities of .60 for a new bridge and .40 for no new bridge,

a. Compute the expected value of each alternative in the payoff table, and identify the alternative that would be selected under the expected-value approach.

b. Construct a decision tree for the problem showing expected values.

Problem 4

eXcel

Solution a. A: .60(1) + .40(14) = 6.20 [best] B: .60(2) + .40(10) = 5.20 C: .60(4) + .40('6) = 4.80

b. Expected values

A

'O'i::J\ $1

e"'Q~llj, ~e-l'l

No /)ew 6 .

rid $14ge (.40)

e \.'O\)1 $2 'Q~\og~e-l'l

!\to /)ew

6riq, {Jer.

.40) $10

~o\)\ $4 '0ge .

'>J'I 'o~\ ~e

!\to /)ew

6riq,»:,40) $6

.60(1) = .60

$6.20 (best)

.40( 14) = 5.60

.60(2) = 1.20

$5.20B

c .40(10) = 4.00

.60(4) = 2.40

$4.80

.40(6) = 2.40

Problem 5 Compute the EVPI using the information from the previous problem.

Solution U sing Formula 5S-1, the EVPI is the expected payoff under certainty minus the maximum expected value. The expected payoff under certainty involves multiplying the best payoff i~ each column by

Supplement to Chapter Five Decision Theory

the c?lumn probability and then summing those amounts. The best payoff in the first column is 4, and the b'est in the second is 14. Thus,

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

Then

EVPI = 8.00 - 6.20 = 1.80

(This agrees with the result obtained in Solved Problem 3b.)

Excel solution:

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u: - s;'. &. ~ - 1i'.!11 ~ ~ 11 1M se (e1'cr

C Jl4 "', ~-! C D

Noles GA 8

1 Pavoff Table

2~ 3

H

~ K

II' bN Brid Noow u"" ro = 4 Probability = 0.& 0.4 Min Max Avg EMV 5 A 1 ,. 1 14 7.5 6.2 6 B 2 10 2 10 6 5.2 7 C 4 6 4 6 5 4.S :. 8

;~i 11

12 Opportunity Loss Table 13 New Bridae No Max EOl

:::ill A 3 0 3 U = EVPI 15. 8 2 4 4 2.S 16 C 0 8 S 3.2 17 16 19

20 21 Criteria Optimal Alternative Valli<

22! J Maximin .1 C 4 I23 Maximax A 142ti.... Laplace A SoNedPrnbem6.....".

7S .11·H ~ - ~ ~ Solved Problem 1 3 4a s.o..dProblem2 50Ned Ptoblem 4b REady IIIiEJI!!! tro. -~ a +

Placing the problem data in the cell positions shown, the expected monetary value (EMV) for each alternative is shown in column J.

Then, the overall EMV is obtained in column L as the maximum of the values in LS, L6, and L7. The EVPI is obtained using the Opportunity Loss Table as the minimum EOL value in Jl4, JlS,

and Jl6.

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

a. Determine the choice that you would make under each of these strategies: maximin, minimin, and Laplace. *

b. Develop the regret table, and identify the alternative chosen using minimax regret. Then find the EVPI if P(new bridge) = .60.

c. Using sensitivity analysis, determine the range of P(no new bridge) for which each alternative would be optimal.

d. If P(new bridge) .60 and P(no new bridge) .40, find the alternative chosen to minimize expected cost.

Problem 6

eXcel mhhe.com/stevenson12e

"Minimin is the reverse of maximax; for costs, minimin identifies the lowest (best) cost.

229

230 Supplement to Chapter Five Decision Theory

Solution a. New No New Maximin Minimin Laplace

Bridge Bridge (worst) (best) (average)

A 1 14 14 1 [best] 15.,. 2 = 7.5

- B 2 10 10 2 12.,. 2 = 6 C 4 6 6 [best] 4 10 .,. 2 = 5 [best]

b. Develop the regret table by subtracting the lowest cost in each column from each of the values in the column. (Note that none of the values is negative.)

New No New Bridge Bridge Worst.

A B

C

o 8 1 4 3 0

EVPI = .60(3) + .40(0) = 1.80

8

4

3 [best]

c. The graph is identical to that shown in Solved Problem 2. However, the lines now represent expected costs, so the best alternative for a given value of P(no new bridge) is the lowest line. Hence, for very low values of P(no new bridge), A is best; for intermediate values, B is best; and for high values, C is best. You can set the equations of A and B, and Band C, equal to 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

4

Cost with no new bridge

Cost with new bridge 14

10

6

2 1

o .20 .33 P(no new bridge)

d. 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 that were computed in Solved Problem 4. However, now you want the alternative that has the lowest expected payoff rather than the one with the highest payoff. Consequently, alternative C is the best because its expected payoff is the lowest of the three. Alternatively, the best choice (C) could be determined using the sensitivity ranges found in part c since .40 lies in the range .33 to 1.0.

Supplement to Chapter Five Decision Theory 231

1. What is the chief role of the operations manager?

2. Listjhe steps in the decision-making process.

3. Explain the term bounded rationality.

4. Explain the term suboptimization.

5. What are some of the reasons for poor decisions?

6. What information is contained in a payoff table?

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

8. Contrast maximax and maximin decision strategies. Under what circumstances is each appropriate?

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

10. Explain or define each of these terms: a. Laplace criterion. b. Minimax regret. c. Expected value. d. Expected value of perfect information.

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

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

13. Identify three potential unethical actions or inactions related to decision analysis and the ethical principle each violates (see Chapter 1).

1. A small building contractor has recently experienced two successive years in which work opportu- nities exceeded the firm's capacity. The contractor must now make a decision on capacity for next year. Estimated profits under each of the two possible states of nature are as shown in the table below. Which alternative should be selected if the decision criterion is a. Maximax? b. Maximin? c. Laplace? d. Minimax regret?

NEXT YEAR'S DEMAND

Alternative Low High Do nothing $50' $60 Expand 20 80 Subcontract 40 70 'Profit in $ thousands.

2. Refer to Problem 1. Suppose after a certain amount of discussion, the contractor is able to subjec- tively assess the probabilities of low and high demand: P(low) = .3 and P(high) = .7. a. Determine the expected profit of each alternative. Which alternative is best? Why? b. Analyze the problem using a decision tree. Show the expected profit of each alternative on the tree. c. Compute the expected value of perfect information. How could the contractor use this

knowledge?

3. Refer to Problems 1 and 2. Construct a graph that will enable you to perform sensitivity analy- sis on the problem. Over what range of P(high) would the alternative of doing nothing be best? Expand? Subcontract?

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

DISCUSSION AND REVIEW QUESTIONS

PROBLEMS

232 Supplement to Chapter Five Decision Theory

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 low demand is estimated to be AO.

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 of P

(demand low).

5. Determine the course of action that has the highest expected payoff for this decision tree.

~------------ $1.0*

...__----- $1.3

------ $1.3

...__----- $1.5

:==--------=:=.:..:."-.-----$1.6

-------- $1.8

otni(\9...-------- $0.7 00 (\

Other use #1:=--~=:....==-:.!~---$1.5 6 \..6.\ '3(\

\ 6e«\ ------$1.0 S«\'3\ Expand Medium demand (.5)I:-----------'----'----------- $1.6

ot\\\(\9...-------- $1.6 00 (\

Subcontractt:=---~=~~---- $1.5

-------- $1.7

otni(\9...__------ ($0.9) 00 (\

Other use #1:==----~=~=-"~--- $1.4

-------- $1.0

....-------$1:0

:=--~=:....==-:.!~---$1.1

-------- $0.9

* Net present value in millions '------------- $2.4

6. The lease of Theme Park, Inc., is about to expire. Management must decide whether to renew the lease for another 10 years or to relocate near the site of a proposed motel. The town planning board is currently debating the merits of granting approval to the motel. A consultant has estimated the net present value of Theme Park's two alternatives under each state of nature as shown on the fol- lowing page. •

Supplement to Chapter Five Decision Theory 233

What course of action would you recommend using? a. Maximax. b. Maximin. c. Laplace. d·. Minimax regret.

Motel Motel Options Approved Rejected

Renew s 500,000 $4,000,00 Relocate 5,000,000 100,00

7. Refer to Problem 6. Suppose that the management of Theme Park, Inc., has decided that there is a .35 probability that the motel's application will be approved. a. If management uses maximum expected monetary value as the decision criterion, which alter-

native 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 planning board

considers the motel's application, would you advise management to sign the lease? The lease will cost $24,000.

8. 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, management decides

that an estimate of approval is approximately .45. How sensitive is the solution to this revised estimate? Explain.

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

9. A firm must decide whether to construct a small, medium, or large stamping plant. A consultant's report indicates 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 value will be $42 million. If demand turns out to be high, the firm can either subcontract and realize the net present value of $42 million or expand greatly 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, its net present value is estimated at 22 million; if demand turns out to be high, the firm could do nothing and realize a net present value of $46 million, or it could expand and realize a net present 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. b. What is the maximin alternative? c. Compute the EVPI and interpret it. d. Perform sensitivity analysis on P(high).

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

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

The net present value associated with the purchase of two machines initially is $75,000 if demand 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, there are three options. One option is to do nothing, which would have a net present value of $90,000. A second option is to subcontract; that would have a net present value of $110,000. The third option is to purchase a second machine. This option would have a net present value of $100,000.

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

234 Supplement to Chapter Five Decision Theory

11. Determine the course of action that has the highest EMV for the accompanying tree diagram.

'13 0

'13 60

~ '13 90 .30

40

.50 44''-Ie p..

-a.\.\ p..\\.e~0 .20' 60

.30

'13~--'-"--(45) }--_-'-'/"-.3 _ 45

,-----,-'/-,,-3 - 99

'-------40

.50-r----------------50

~----30 .20 '12

~--40

12. A logistics provider plans to have a new warehouse built to handle increasing demands for its services. Although the company is unsure of how much demand there will be, it must decide now on the size (large or small) of the warehouse. Preliminary estimates are that if a small warehouse is built and demand is low, the monthly income will be $700,000. If demand is high, it will have to either expand the facility or lease additional space. Leasing will result in a monthly income of $100,000 while expanding will result in a monthly income of $500,000.

If a large warehouse is built and demand is low, monthly income will only be $40,000, while if demand is high, monthly income will be 2 million. a. Construct a tree diagram for this decision. b. Using your tree diagram, identify the choice that would be made using each of the four

approaches for decision making under uncertainty.

13. The director of social services of a county has learned that the state has mandated additional information requirements. This will place an additional burden on the agency. The director has identified three acceptable alternatives to handle the increased workload. One alternative is to reassign present staff members, the second is to hire and train two new workers, and the third is to redesign current practice so that workers can readily collect the information with little additional effort. An unknown factor is the caseload for the coming year when the new data will be collected on a trial basis. The estimated costs for various options and caseloads are shown in the following table:

CASELOAD

High Very HighModerate

$50' 60 40

Reassignstaff Newstaff Redesigncollection 'Costin$ thousands.

60 60 50

85 60 90

Assuming that past experience has shown the probabilities of various caseloads to be unreliable, what decision would be appropriate using each of the following criteria? a. Maximin. b. Maximax. c. Minimax regret. d. Laplace.

Supplement to Chapter Five Decision Theory 235

14. After contemplating the caseload question (see previous problem), the director of social services has decided that reasonable caseload probabilities are .10 for moderate, .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 decision

branches. c. Determine the expected value of perfect information using an opportunity loss table.

15. Given this payoff table:

STATE OF NATURE

#1 #2

A 20 $120' Alternative B 4060

C 11010 o 9090

'Payoff in $ thousands.

a. Determine the range of pel) for which each alternative would be best, treating the payoffs as profits.

b. Answer part a treating the payoffs as costs.

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

STATE OF NATURE

#1 #2

A 140 $20' Alternative B 80120

C 40100 'Profit 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 expected

profit? 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 maximize

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

expected profit?

17. Repeat all parts of Problem 16, assuming the values in the payoff table are estimated costs and the goal is to minimize expected costs.

18. The research staff of a marketing agency has assembled the following payoff table of estimated profits:

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

Receive Not Receive Contract Contract

#1 $10' 22 Proposal #2 8 3

#3 5 5 #4 o 7

'Cost in $ thousands. ... Anderson, David R., Dennis J. Sweeney, Thomas A.

Williams, Jeffrey D. Camm, and R. Kipp Martin. An Introduction to Management Science, 13th ed.

Stevenson, William J., and Ceyhun Ozgur. Introduction to Management Science with Spreadsheets. New York: McGraw-Hilllrwin, 2007.

SELECTED BIBLIOGRAPHY

Mason, OH: South-Western, 2010. Taylor, Bernard W. Introduction to Management Science. AND FURTHER 11th ed. Dubuque, lA: William C. Brown, 2013. READINGS

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