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HO2_Applied+Decision+Theory - DECISION ANALYSIS APPLIED...

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Unformatted text preview: DECISION ANALYSIS: APPLIED DECISION THEORY Analyse des Decisions: The’orz'e Applique’e ’ des De’cz'sions RONALD A. HOWARD Institute in Engineering-Economic Systems Stanford University, Cali orm'a United States of America I. INTRODUCTION Decision theory in the modern sense has existed for more than a decade. Most of the effort among the present developers of the theory has been devoted to Bayesian analysis of problems formerly treated by classical statistics. Many practical management decision problems, hoWever, can be handled by formal structures that are far from novel theoretically. The world of top management decision making is not often structured by simple Bernoulli, Poisson, or normal models. Indeed, Bayes's theorem itself may not be so important. A statistician for a major company wrote a report in which he commented that for all the talk about the Bayesian revolution he did not know of a single application in the company in which Bayes’s theorem was actually used. The observation was probably quite correct—but what it shows by implication is that the most sig- nificant part of the revolution is not Bayes's theorem or conjugate distributions but rather the concept of probability as a state of mind, a 200-year-old concept. Thus the real promise of decision theory lies in its ability to provide a broad logical basis for decision making in the face of uncertainty rather than in any specific models. The purpose of this article is to outline a formal procedure for the analysis of decision problems, a procedure that I call “decision analysis." We shall also discuss several of the practical problems that arise when we attempt to apply the decision analysis formalism. 2. DECISION ANALYSIS To describe decision analysis it is first necessary to define a decision. A decision is an irrevocable allocation of resources, irrevocable in the sense that it is im- possible or extremely costly to change back to the situation that existed before making the decision. Thus for our purposes a decision is not a mental commit— ment to follow a course of action but rather the actual pursuit of that course of action. This definition often serves to identify the real decision maker within a loosely structured organization. Finding the exact nature of the decision to be 97 RONALD A. HO\VA RD made, however, and who will make it, remains one of the fundamental problems of the decision analyst. Having defined a decision, let us clarify the concept by drawing a necessary distinction between a good decision and a good outcome. A good decision is a logical decision—one based on the uncertainties, values, and preferences of the decision maker. A good outcome is one that is profitable or otherwise highly valued. In short, a good outcome is one that we wish would happen. Hopefully, by making good decisions in all the situations that face us we shall ensure as high a percentage as possible of good outcomes. We may be disappointed to find that a good decision has produced a bad outcome or dismayed to learn that someone who has made what we consider to be a bad decision has enjoyed a good outcome. Yet, pending the invention of the true clairvoyant, we find no better alternative in the pursuit of good outcomes than to make good decisions. Decision analysis is a logical procedure for the balancing of the factors that influence a decision. The procedure incorporates uncertainties, values, and preferences in a basic structure that models the decision. Typically, it includes technical, marketing, competitive, and environmental factors. The essence of the procedure is the construction of a structural model of the decision in a form suitable for computation and manipulation; the realization of this model is often a set of computer programs. 2.1. The Decision Analysis Procedure Table 1 lists the three phases of a decision analysis that are worth distinction: the deterministic, probabilistic, and post-mortem phases. TABLE 1 The Decision Analysis Procedure I. Deterministic phase Define the decision Identify the alternatives Assign values to outcomes Select state variables Establish relationship at state variables . Specify time preference amewwr An lysis: (a) Determine dominance to eliminate alternatives (b) Measure sensitivity to identify cnicial state variables II. Probabilistic phase 1. Encode uncertainty on crucial state variables Analysis: Develop profit lottery 2. Encode risk preference Analysis: Select best alternative III. Post-mortem phase Analysis: (a) Determine value of eliminating uncertainty in crucial state variables (b) Develop most economical information-gathering program 98 DECISION ANALYSIS: APPLIED DECISION THEORY 2.1.1. The Deterministic Phase The first step in the deterministic 'phase is to answer the question, ” What decision must be made?” Strange as it may seem, many people with what appear to be decision problems have never asked themselves that question. \Ve must distinguish between situations in which there is a decision to be made and situations in which we are simply worried about a bad outcome. If we have resources to allocate, we have a decision problem, but if we are only hand wringing about circumstances beyond our control no formal analysis will help. The difference is that between selecting a surgeon to operate on a member of your family and waiting for the result of the operation. We may be in a state of anguish throughout, but decision analysis can help only with the first question. The next step is to identify the alternatives that are available, to answer the question, “ What courses of action are open to us? " Alternative generation is the most creative part of the decision analysis procedure. Often the introduction of a new alternative eliminates the need for further formal analysis. Although the synthesis of new alternatives necessarily does not fall within the province of the decision analysis procedure, the procedure does evaluate alternatives and thereby suggests the defects in present alternatives that new alternatives might remedy. Thus the existence of an analytic procedure is the first step toward synthesis. We continue the deterministic phase by assigning values to the various outcomes that might be produced by each alternative. We thus answer the question, “How are you going to determine which outcomes are good and which are bad? " In business problems this will typically be a measure of profit. Military and governmental applications should also consider profit, measured perhaps with more difficulty, because these decision makers are also allocating the economic resources of the nation. Even when we agree on the measure of profit to be assigned to each outcome, it may be difficult to make the assignment until the values of a number of variables associated with each outcome are specified. We call these variables the state variables of the decision. Their selection is the next step in the deterministic phase. A typical problem will have state variables of many kinds: costs of manu- facture, prices charged by competitors, the failure rate of the product, etc. We select them by asking the question, “If you had a crystal ball, what numerical questions would you ask it about the outcome in order to specify your profit measure?" At the same time that we select these variables we should assign both nominal values for them and the range over which they might vary for future reference. Next we establish how the state variables are related to each other and to the measure of performance. We construct, in essence, a profit function that shows how profit is related to the factors that underlie the decision. The con- struction of this profit function requires considerable judgment to avoid the twin difficulties of excessive complexity and unreal simplicity. If the results of the decision extend over a long time period, it will be neces- sary to have the decision maker specify his time preference for profit. We must 99 RONALD A. HOWARD ask, ” How does profit received in the future compare in value to profit received today? ” or an equivalent question. In cases in which we can assume a perfect financial environment the present value of future profit at some rate of interest will be the answer. In many large decision problems, however, the nature of the undertaking has an effect on the basic financial structure of the enterprise. In these cases a much more realistic modeling of the time preference for profit is necessary. Now that we have completed the steps in the deterministic phase we have a deterministic model of the decision problem. We next perform two closely related analyses. We perform them by setting the state variables to their nominal values and then sweeping each through its range of values, individually and jointly, as judgment dictates. Throughout this process we observe which alternative would be best and how much value would be associated with each alternative. We often observe that regardless of the values the state variables take on in their ranges one alternative is always superior to another, a condition we describe by saying that the first alternative dominates the second. The principle of dominance may often permit a major reduction in the number of alternatives that need be considered. As a result of this procedure we have performed a sensitivity analysis on the state variables. \Ve know how much a 10 percent change in one of the variables will affect profit, hence the optimum alternative. Similarly, we know how changes in state variables may interact to affect the decision. This sensi- tivity analysis shows us where uncertainty is important. We identify those state variables to which the outcome is sensitive as “crucial” state variables. Deter- mining how uncertainties in the crucial state variable influence the decision is the concern of the probabilistic phase of the decision analysis. 2.1.2. Probabilistic Phase The probabilistic phase begins by encoding uncertainties on each of the crucial state variables; that is, gathering priors on them. A subset of the crucial state variables will usually be independent—for these only a single probability distribution is necessary. The remainder will have to be treated by collecting conditional as well as marginal distributions. We have more to say on this process later. The next step is to find the uncertainty in profit for each alternative implied by the functional relationship of profit to the crucial state variables and the probability distribution on those crucial state variables for the alternative. We call this derived probability distribution of profit the profit lottery of the alternative. In a few cases the profit lottery can be derived analytically and in many by numerical analysis procedures. In any case it may be approximated by a Monte Carlo simulation. Regardless of the procedure used, the result is a probability distribution on profit (or perhaps on discounted profit) for each of the alternatives that remain in the problem. Now we must consider how to choose between two alternatives with different profit lotteries. In one case the choice is easy. Suppose that we plot the profit lottery for each alternative in complementary cumulative form; that is, plot the 100 DECISION ANALYSIS: APPLIED DECISION THEORY Alternative .41 Profit lottery (density function) Profit Alternative A2 Profit lottery (probability of profit exceeding x) Figure 1. Stochastic dominance. probability of profit exceeding .7: for any given x. Suppose further, as shown in Figure 1, that the complementary cumulative for alternative A2 always lies above that for alternative A1. This means that for any number .1: there is a higher probability of profit exceeding that number with alternative A2 than with alternative A1. In this case we would prefer alternative A2 to alternative A1 , provided only that we liked more profit better than less profit. We describe this situation by saying that the profit from alternative A2 is stochastically greater than the profit from alternative A1 or equivalently by saying that alter- native A2 stochastically dominates alternative A1. Stochastic dominance is a concept that appeals intuitively to management; it applies in a surprising number of cases. Alternative A1 Alternative A2 Profit lottery (density function) Profit Alternative A; Alternative A2 Profit lottery (probability of profit exceeding x) Figure 2. Lack of stochastic dominance. lOl RONALD A. HOWARD Figure 2, however, illustrates a case in which stochastic dominance does not apply. When faced with a situation like this, we must either abandon formal methods and leave the selection of the best alternative to judgment or delve into the measurement of risk preference. If we choose to measure risk preference, we begin the second step of the probabilistic phase. We must construct a utility function for the decision niaker that will tell us whether or not, for example, he would prefer a certain 4 million dollars profit to equal chances of earning zero or 10 million dollars. Although these questions are quite foreign to management, they are being asked increasingly often with promising results. Of course, when risk preference is established in the form of a utility function, the best alternative is the one whose profit lottery has the highest utility. 2.1.3. Post-Morten: Phase The post-mortem phase of the procedure is composed entirely of analysis. This phase begins when the best alternative has been selected as the result of the probabilistic phase. Here we use the concepts of the clairvoyant lottery to establish a dollar value of eliminating uncertainty in each of the state variables individually and jointly. Being able to show the impact of uncertainties on profit is one of the most important features of decision analysis. It leads directly to the next step of the post—mortem, which is finding the most economical information-gathering program, if,_in fact, it would be profitable to gather more information. The information-gathering program may be physical research, a marketing survey, or the hiring of a consultant. Perhaps in no other area of its operations is an enterprise in such need of substantiating analysis as it is in the justification of information-gathering programs. Of course, once the information-gathering scheme, if any, is completed, its information modifies the probability distributions on the crucial state variables and consequently afi'ects the decision. Indeed, if the information-gathering program were not expected to modify the probability distributions on the crucial state variables it would not be conducted. We then repeat the proba- bilistic phase by using the new probability distributions to find the profit lotteries and then enter the post-mortem phase once more to determine whether further information gathering is worthwhile. Thus the decision analysis is a vital structure that lets us compare at any time the values of such alternatives as acting, postponing action and buying information, or refusing to consider the problem further. We must remember that the analysis is always based on the current state of knowledge. Overnight there can arrive a piece of infor- mation that changes the nature of the Conclusions entirely. Of course, having captured the basic structure of the problem, we are in an excellent position to incorporate any such information. Finally, as the result of the analysis the decision maker embarks on a course of action. At this point he may be interested in the behavior of several of the state variables for planning purposes; for example, having decided to introduce a new product, he may want to examine the probability distributions for its sales in future years to make subsidiary decisions on distribution facilities or 102 DECISION ANALYSIS: APPLIED DECISION THEORY on the size of the sales force. The decision-analysis model readily provides such planning information. 2.2. The Advantages of Decision Analysis Decision analysis has many advantages, of which we have described just a few, such as its comprehensiveness and vitality as a model of the decision and its ability to place a dollar value on uncertainty. We should point out further that the procedure is relevant to both one of a kind and repetitive decisions. Decision analysis offers the operations research profession the opportunity to extend its scope beyond its traditional primary concern with repetitively verifiable operations. One of the most important advantages of decision analysis lies in the way it encourages meaningful communication among the members of the enterprise because it provides a common language in which to discuss decision problems. Thus engineers and marketing planners with quite different jargons can appreci- ated one another’s contributions to a decision. Both can use the decision-analysis language to convey their feelings to management quickly and effectively. A phenomenon that seems to be the result of the decision—analysis language is the successive structuring of staff groups to provide reports that are useful in decision-analysis terms. Thus, if the decision problem being analyzed starts in an engineering group, that group ultimately seeks inputs from marketing, product planning, the legal staff, and so on, that are compatible with the proba- bilistic analysis. Soon these groups begin to think in probabilistic terms and to emphasize probabilistic thinking in their reports. The process seems irrever- sible in that,once the staff of an organization becomes comfortable in dealing with probabilistic phenomena they are never again satisfied with deterministic or expected value approaches to problems. Thus the existence of decision- analysis concepts as a language for communication may be its most important advantage. 2.3. The Hierarchy of Decision Analysis It is informative to place decision analysis in the hierarchy of techniques that have been developed to treat decision problems. “'e see that a decision analysis requires two supporting activities. One is a lower order activity that we call alternative evaluation; the second, a higher order activity that we call goal setting. Performing a decision analysis requires evaluating alternatives according to the goals thathave been set for the decision. The practitioners of operations research are quite experienced in alternative evaluation in both industrial and military contexts. In fact, in spite of the lip service paid to objective functions, only rare operations researchers have had the scope necessary to consider the goal—setting problems. All mankind seems inexpcrt at goal setting, although it is the most important problem we face. Perhaps the role of decision analysis is to allow the discussion of decisions to be carried on at a level that shows the explicit need for goals or criteria for selection of the best alternative. We need to make goals explicit only 103 RONALD A. HOWARD if the decision maker is going to delegate the making of the decision or if he is unsure of his ability to be consistent in selecting the best alternative. \Ve shall not comment on whether there is a trend toward more or less delegation of decision making. However, it is becoming clear to those with decision-making responsibilities that the increasing complexity of the operations under their control requires correspondingly more formal approaches to the problem of organizing the information that bears on a decision if inconsistent decisions are to be avoided. The history of the analysis of the procurement of military weapons systems points this out. Recent years have shown the progression of procurement thinking from effectiveness to cost effectiveness. In this respect the military authorities have been able to catch up in their decision—making apparatus to what industry had been doing in its simpler problems for years. Other agencies of government are now in the process of making the same transition. Now all must move on to the inclusion of uncertainty, to the establishment of goals that a...
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