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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 are reflected in risk and time preferences. These developments are now on the horizon and in some cases in sight; for example, although we have tended'to think of the utility theory as an academic pursuit, one of our major companies was recently faced with the question, ” Is 10 million dollars of profit sufficient to incur one chance in 1 mil- lion of losing 1 billion dollars? " Although the loss is staggering, it is realistic for the company concerned. Should such a large company be risk-indifferent and make decisions on an expected value basis? Are stockholders responsible for diversifying their risk externally to the company or should the company be risk-averting on their behalf? For the first time the company faced these ques- tions in a formal way rather than deciding the particular question oh its own merits and this we must regard as a step forward. Decision analysis has had its critics, of course. One said, “In the final analysis, aren't decisions politically based? " The best answer to that came from a high official in the executive branch of our government who said, “ The better the logical basis for a decision, the more difficult it is for extraneous political factors to hold sway.” It may be discouraging in the short run to see logic over- ridden by the tactical situation, but one must expect to lose battles to win the war. Another criticism is, “If this is such a good idea, why haven’t I heard of it before?” One very practical reason is that the operations we conduct in the course of a decision analysis would be expensive to carry out without using computers. To this extent decision analysis is a product of our technology. There are other answers, however. One is that the idea of probability as a state of mind and not of things is only now regaining its proper place in the world of thought. The opposing heresy lay heavy on the race for the better part of a century. We should note that most of the operations research performed in World War II required mathematical and probabilistic concepts that were readily available to Napoleon. One wonders about how the introduction of formal methods for decision making at that time might have affected the course of history. 104 DECISION ANALYSIS: APPLIED DECISION THEORY 3. THE PRINCIPLES OF THE DECISION ANALYST Next we turn to the principles of the decision analyst, the professional who embarks on preparing a decision analysis. His first principle is to identify and isolate the components of the decision—the uncertainty, risk aversion, time preference, and problem structure. Often arguments over which is the best decision arise because the participants do not realize that they are arguing on different grounds. Thus it is possible for A to think that a certain alternative is riskier than it is in B’s opinion, either because A assigns different probabilities to the outcomesthan B but both are equally risk—averting, or because A and B assign the same probabilities to the outcomes but differ in their risk aversion. If we are to make progress in resolving the argument, we must identify the nature of the difficulty and bring it into the open. Similar clarifications may be made in the areas of time preference or in the measurement of the value of outcomes. One aid in reducing the problem to its fundamental components is restricting the vocabulary that can be used in discussing the problem. Thus we carry on the discussion in terms ofevents, random variables, probabilities, density functions, expectations, outcomes, and alternatives. We do not allow fuzzy thinking about the nature of these terms. Thus “The density function of the probability” and “The confidence in the probability estimate” must be nipped in the bud. We speak of “ assigning,” not “ estimating," the probabilities of events and think of this assignment as based on our “state of information.” These conventions eliminate statements like the one recently made on a TV panel of doctors who were discussing the right of a patient to participate in decision making on his treatment. One doctor asserted that the patient should be told of “some kind of a chance of a likelihood of a bad result." I am sure that the doctor was a victim of the pressures of the program and would agree with us that telling the patient the probability the doctor would assign to a bad result would be preferable. One principle that is vital to the decision analyst is professional detachment in selecting alternatives. The analyst must not become involved in the heated political controversies that often surround decisions except to reduce them to a common basis. He must demonstrate his willingness to change the recommended alternative in the face of new information if he is to earn the respect of all con- cerned. This professional detachment may, in fact, be the analyst’s single most valuable characteristic. Logic is often severely strained when we are personally involved. The detachment of the analyst has another positive benefit. As an observer he may be able to suggest alternatives that may have escaped those who are intimately involved with the problem. He may suggest delaying action, buying insurance, or performing a test, depending on the nature of the decision. Of course, the comprehensive knowledge of the properties of the existing alternatives that the decision analyst must gain is a major aid in formulating new alternatives. Since it is a rare decision that does not imply other present and future decisions, the decision analyst must establish a scope for the analysis that is 105 RONALD A. HOWARD broad enough to provide meaningful answers but not broad enough to impose impractical computational requirements. Perhaps the fundamental question in establishing scope is how much to spend on decision analysis. Because the approach could be applied both to selecting a meal from a restaurant menu and to allocating the federal budget, the analyst needs some guidelines to determine when the analysis is worthwhile. The question of how much decision analysis is an economic problem sus- ceptible to a simpler decision analysis, but rather than pursue that road let us pose an arbitrary and reasonable but indefensible rule of thumb: spend at least 1 percent of the resources to be allocated on the question of how they should be allocated. Thus, if we were going to buy a 2000-.dollar automobile, the rule indicates a 20-dollar analysis, whereas for a 20,000-dollar house it would specify a ZOO-dollar analysis. A 1-million-dollar decision would justify 10,000 dollars’ worth of analysis or, let us say, about three man—months. The initial reaction to this guideline has been that it is conservative in the sense of not spending much on analysis; yet, when we apply it to many decisions now made by business and government, the reaction is that the actual expenditures on analysis are only one-tenth or one-hundredth as large as the rule would prescribe. Of course, we can all construct situations in which a much smaller or larger expenditure than given by the rule would be appropriate, and each organization can set its own rule, perhaps making the amount spent on analysis nonlinear in the re- sources to be allocated. Nevertheless, the 1 percent figure has served well to illustrate where decision analysis can be expected to have the highest payoff. The professional nature of the decision analyst becomes apparent when he balances realism in the various parts of the decision-analysis model. Here he can be guided only by what used to be called engineering judgment. One principle he should follow is to’ avoid sophistication in any part of the problem when that sophistication would not affect the result. We can describe this informally by saying that he should strive for a constant “wince ” level as he surveys all parts of the analysis. One indication that he has achieved this state is that he would be torn among many possibilities for improvement if we allowed him to devote more time and resources to the decision model. 4. THE ENCODING OF SUBJECTIVE INFORMATION One unique feature of decision analysis is the encoding of subjective infor- mation, both in the form of risk aversion and in the assignment of probabilities. 4.1. Risk Aversion and Time Preference Since we are dealing in most cases with enterprises rather than individuals, the appropriate risk aversion and time preference should be that of the enter- prise. The problem of establishing such norms is beyond our present scope. It is easy, however, to demonstrate to managers, or to anyone else for that. matter, that the phenomenon of risk aversion exists and that it varies widely from individual to individual. One question useful in doing this is, ” How much would you have to be paid to call a coin, double or nothing, for next year's 106 DECISION ANALYSIS: APPLIED DECISION THEORY salary?" Regardless of the salary level of the individuals involved, this is a provocative question. We point out that only a rare individual would play such a game for a payment of zero and that virtually everyone would play for a payment equal to next year’s salary, since then there would be nothing to lose. Thereafter we are merely haggling over the price. Payments in the range of 60 percent to 99 percent of next year's salary seem to satisfy the vast majority of professional individuals. The steps required to go from a realization of personal risk aversion and time preference to corporate counterparts and finally to a reward system for managers that will encourage them to make decisions consistent with corporate risk aversion and time preference remain a fascinating area of research. 4.2. Encoding of Uncertainty When we begin the probabilistic phase of the decision analysis, We face the problem of encoding the uncertainty in each of the crucial state variables. We shall want to have the prior probability distributions assigned by the people within the enterprise who are most knowledgeable about each state variable. Thus the priors on engineering variables will typically be assigned by the engineering department; on marketing variables, by the marketing department, and so on. However, since we are in each case attempting to encode a probability distribution that reflects a state of mind and since most individuals have real difficulty in thinking about uncertainty, the method we use to extract the priors is extremely important. As people participate in the prior-gathering process, their attitudes are indicated successively by, “This is ridiculous,” “ It can't be done," “ I have told you what you want to know but it doesn’t mean anything," “ Yes, it seems to reflect the way I feel," and “ \Vhy doesn’t everybody do this? ” In gathering the information we must be careful to overcome the defenses the individual develops as a result of being asked for estimates that are often a combination of targets, wishful thinking, and expectations. The biggest diffi— culty is in conveying to the man that you are interested in his state of knowledge and not in measuring him or setting a goal for him. If the subject has some experience with probability, he often attempts to make all his priors look like normal distributions, a characteristic we may designate as “bellshaped” thinking. Although normal distributions are appro- priate priors in some circumstances, we must avoid making them a foregone conclusion. Experience has shown certain procedures to be effective in this almost psychoanalytic process of prior measurement. The first procedure is to make the measurement in a private interview to eliminate group pressure and to over- come the vague notions that most people exhibit about matters probabilistic. Sending around forms on which the subjects are supposed to draw their priors has been worse than useless, unless the subjects were already experienced in decision analysis. Next we ask questions of the form, " What are the chances that x will exceed 10,” because people seem much more comfortable in assigning probabilities to events than they are in sketching a density function. As these questions are 107 RONALD A. HOWARD asked, we skip around, asking the probability that x will be “greater than 50, less than 10, greater than 30,” often asking the same question again later in the interview. The replies are recorded out of the view of the subject in order to frustrate any attempt at forced consistency on his part. As the interview pro- ceeds, the subject often considers the questions with greater and greater care, so that his answers toward the end of the interview may represent his feelings much better than his initial answers. We can change the form of the questions by asking the subject to dine the domain of the random variable into n mutually exclusive regions with equal probability. (Of course, we would never put the question to him that way.) We can use the answers to all these questions to draw the complementary cumulative distribution for the variable, a form of representation that seems easiest to convey to people without formal prob- abilistic training. The result of this interview is a prior that the subject is willing to live with, regardless of whether we are going to use it to govern a lottery on who buys coffee or on the disposal of his life savings. \Ve can test it by comparing the prior with known probabilistic mechanisms; for example, if he says that a is the median of the distribution of x, then he should be indifferent about whether we pay him one hundred dollars if x exceeds a or if he can call the toss of a coin correctly. If he is not indifferent, then we must require him to change (1 until he is. The end result of such questions is to producea prior that the subject is not tempted to change in any way, and we have thus achieved our final goal. The prior—gathering process is not cheap, but we perform it only on the crucial state variables. In cases in which the interview procedure is not appropriate, the analyst can often obtain a satisfactory prior by drawing one himself and then letting the subject change it until the subject is satisfied. This technique may also be useful as an educational device in preparation for the interview. If two or more variables are dependent, we must gather priors on conditional as well as marginal distributions. The procedure is generally the same but somewhat more involved. However, we have the benefit of being able to apply some checks on our results. Thus, if we have two dependent variables x and y, we can obtain the joint distribution by measuring the prior on x and the con— ditional on y, given x, or, alternatively, by measuring the prior onyand the con- ditional on 9:, given y. If we follow both routes, we have a consistency check on the joint distribution. Since the treating ofjoint variables is a source of expense, we should formulate the problem to avoid them whenever possible. To illustrate the nature of prior gathering we present the example shown in Figure 3. The decision in a major problem was thought to depend primarily on the average lifetime of a new material. Since the material had never been made and test results would not be available until three years after the decision was required, it was necessary to encode the knowledge the company now had concerning the life of the material. This knowledge resided in three professional metallurgists who were experts in that field of technology. These men were interviewed separately according to the principles we have described. They produced the points labeled “ Subjects 1, 2, and 3 ” in Figure 3. These results have several interesting features. We note, for example, that for l: 17 Subject 108 DECISION ANALYSIS: APPLIED DECISION THEORY 1.0 + Subject 1 0 Subject 2 E. 0.8 0 Subject 3 g 0 6 E ' E 1.; o 4 Consensus E :3 0.2 o 5 10 15 20 I Figure 3. Priors on lifetime of material. 2 assigned probability 0.2 and 0.25 at various points in the interview. On the whole, however, the subjects were remarkably consistent in their assignments. We observe that Subject 3 was more pessimistic than Subject 1. At the conclusion of the three interviews the three subjects were brought together and shown the results. At this point a vigorous discussion took place. Subjects 1 and 3, in particular, brought forth information of which the other two members of the group were unaware. As the result of this information exchange, the three group members drew the consensus curve—each subject said that this curve represented the state of information about the material life at the end of the meeting. It has been suggested that the proper way to reconcile divergent priors is to assign weights to each, multiply, and add, but this experiment is convincing evidence that any such mechanistic procedure misses the point. Divergent priors are an excellent indicator of divergent states of information. The ex- perience just described not only produced the company's present encoding of uncertainty about the lifetime of the material but at the same time encouraged the exchange of information within the group. 5. A DECISION-ANALYSIS EXAMPLE To illustrate the flavor of application let us consider a recent decision analysis in the area of product introduction. Although the problem was really from another industry, let us suppose that it was concerned with the development and production of a new type of aircraft. There were two major alternatives: to develop and sell a new aircraft (A2) or to continue manufacturing and selling 109 RONALD A. HOWARD Deterministic business model Sensitivity analysis Market priors Physical Operating “3:123:23” Profit priors cost function model lottery Production Capital priors cost function Figure 4. Decision analysis for new product introduction. the present product (A1). The decision was to be based on the present value of future expected profits at a discounting rate of 10 percent per year. Initially, the decision was supposed to rest on the lifetime of the material for which we obtained the priors in Figure 3; however, a complete decision analysis was desired. Since several hundred million dollars in present value of profit were at stake, the decision analysis was well justified. The general scheme of the analysis appears in Figure 4. The first step was to construct a model of the business, a model that was primarily a model of the market. The profit associated with each alternative was described in terms of the price of the product, its operating capital costs, the behavior of its competi- tors, and the national characteristics of customers. The actual profit and dis- counted profit were computed over a 22-year time period. A suspicion grew that this model did not adequately capture the regional nature of demand. Consequently a new model was constructed that included the market charac- teristics, region by region and customer by customer. Moving to the more detailed basis affected the predictions so much that the additional refinement was clearly justified. Other attempts at refinement, however, did not affect the results sufficiently to justify a still more refined model. Now, the sensitivity analysis was performed to determine the crucial state variables, which turned out to be the operating cost, capital cost, and a few market parameters. Because of the complexity of the original business model, an approximate business model essentially quadratic in form was constructed to show how profit depended on these crucial state variables in the domain of interest. The coefficients of the approximate business model were established by runs on the complete business model. The market priors were directly assigned with little trouble. However, because the operating and capital costs were the two most important variables llO DECISION ANALYSIS: APPLIED DECISION THEORY in the problem, these priors were assigned according to a more detailed pro- cedure. First, the operating cost was related to various physical features of the design by the engineering department. This relationship was called the oper- ating-cost function. One of the many input physical variables was the average lifetime of the material whose priors appear in Figure 3. All but two of the 12 physical input variables were independent. The priors on the whole set of input variables were gathered and used with the operating—cost function in a Monte Carlo simulation that produced a prior for the operating cost of the product. The capital-cost function was again developed by engineering but was much simpler in form. The input certainties were the production costs for various parts of the product. Again, a Monte Carlo analysis produced a prior on capital cost. . Once we had established priors on all inputs to the approximate business model, we could determine the profit lottery for each alternative, in this case by using numerical analysis. The present-value profit lotteries for the two alternatives looked very much like those shown in Figure 1. The new product alternative A2 sto- chastically dominated the alternative A1 of continuing to manufacture the present product. The result showed two interesting facets of the problem. First, it had been expected that the profit lottery for the new product alternative would be considerably broader than it was for the old product. The image was that of a profitable and risky new venture compared with a less profitable but less risky standard venture. In fact, the results showed that the uncertainties in profit were about the same for both alternatives, thus showing how initial concepts may be misleading. The second interesting facet was that the average lifetime of the material whose priors appear in Figure 3 was actually of little consequence in the de- cision. It was true enough that profits were critically dependent on this lifetime if the design were fixed, but if the design were left flexible to accommodate to different average material lifetimes profits would be little affected. Furthermore, leaving the design flexible was not an expensive alternative; therefore another initial conception had to be modified. However, the problem did not yield so easily. Figure 5 shows the present value of profits through each number of years t for each alternative. Note that if we ignore returns beyond year 7 the new product has a higher present value but that if we consider returns over the entire 22-year period the relationship reverses, as we have already noted. When management saw these results, they were considerably disturbed. The division in question had been under heavy pressure to show a profit in the near future—alternative A2 would not meet that requirement. 'Thus the question of time preference that had been quickly passed off as one of present value at 10 percent per year became the central issue in the decision. The question was whether the division was interested in the quick kill or the long pull. At last report the division was still trying to convince the company to extend its profit horizon. This problem clearly illustrates the use of decision analysis in clarifying the ill RONALD A. HOWARD 500 § Alternative A2 § ‘8” ._. O 0 Expected present value of profit through year t in millions of dollars 0 2 4 6 81012141618202224 Year: Figure 5. Expected present value of profit. issues surrounding a decision. A decision that might have been made on the basis of a material lifetime was shown to depend more fundamentally on’ the question of time preference for profit. The nine man—months of effort devoted to this analysis were considered well spent by the company. The review com- mittee for the decision commented, “ We have never had such a realistic analysis of a new business venture before." The company is now interested in insti- tuting decision-analysis procedures at several organizational levels. ll2 DECISION ANALYSIS: APPLIED DECISION THEORY 6. CONCLUSION Decision analysis offers operations research a second chance at top manage- ment. By foregoing statistical reproducibility we can begin to analyze the one-of—a-kind problems that managers have previously had to handle without assistance. Experience indicates that the higher up the chain of management we progress the more readily the concepts we have outlined are accepted. A typical reaction is, “ I have been doing this all along, but now I see how to reduce my ideas to-numbers." Decision analysis is no more than a procedure for applying logic. The ultimate limitation to its applicability lies not in its ability to cope with problems but in man’s desire to be logical. ANALYSE DES DECISIONS: THEORIE APPLIQUEE DES DECISIONS RESUME Au cours de ces derniéres anne'es, la the’orie de decision a été de plus en plus acceptée en tant que cadre conceptuel pour la prise de décision. Ccpendant, cette the’orie a surtout affecté les statisticiens plutét que les personnes qui en ont le plus bcsoin: les responsables dc decisions. Cctte étude décrit un proce’dé qui permet de replaccr dcs problemes de decision réels dans la structure de la the'orie de décision. Le proeédé d’analyse de décision englobe chaque e'tape, du mesurage des choix de risques et des jugemcnts portant sur des facteurs critiques par l'établissement de structures des facteurs rclatifs a la technique, au marche’, a la rivalitc’ commercials et a l'cnvironnement, jusqu’au mesurage des preferences subjectives et dc la valeur dc la prediction. L’analyse de décision met en perspective les nombrcux instruments de simulation, d’analyse nu- mérique, et de transformations de probabilitc’s qui deviennent de plus en plus commodcs depuis le développement des systemes d’ordinateurs électroniques dont les difl'e'rentes “stations” dependent d'une “centrale” unique. Le procédé est applique a un probleme de de'cision re'ellc qui s’e'tend sur dcs dizaines d'anne’s et dont la valeur actuelle est de plusieurs centaines de millions de dollars. Cette étude analyse le probleme de la determination des dépenses consacrées a l’analyse de decisions. L’une des plus importantes propriétés de ce proce'dé tient au nombre des bénéficcs auxiliaircs créés au cours de l’élabor— ation de ce genre d'étude. L’expérience montre que ces bénéfices peuvant excéder en valeur le cofit des de’penses consacrées a l’e'laboration dc la decision. H3 ...
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This note was uploaded on 09/28/2010 for the course MS&E 252 taught by Professor Howard during the Fall '08 term at Stanford.

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