Example of decision making under certainty

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, 218L05S.2 Name some causes of poor 5S.9 Sensitivity Analysis, 224decisions. 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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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.

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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.

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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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SOLUTION

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EXAMPLE 5S-3

SOLUTION

Regret (opportunity loss) The differencebetweena 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.