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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 EngineeringEconomic 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 200yearold 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 postmortem 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. Postmortem phase Analysis: (a) Determine value of eliminating uncertainty in crucial state
variables (b) Develop most economical informationgathering 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. PostMorten: Phase The postmortem 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
informationgathering program, if,_in fact, it would be proﬁtable to gather more
information. The informationgathering 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 informationgathering programs. Of course, once the informationgathering 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 informationgathering
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 postmortem 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 decisionanalysis 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 decisionanalysis
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 decisionanalysis 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 decisionmaking
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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 Decision Analysis, Decision Making, RONALD A. HOWARD

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