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Multiattribute Preference Analysis with PerformanceTargetsRobert F. Bordley, Craig W. Kirkwood,
To cite this article:Robert F. Bordley, Craig W. Kirkwood, (2004) Multiattribute Preference Analysis with Performance Targets. OperationsResearch 52(6):823-835. http://dx.doi.org/10.1287/opre.1030.0093
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This paper develops an approach based on performance targets to assess a preference function for a multiobjective decisionunder uncertainty. This approach yields preference functions that are strategically equivalent to conventional multiattributeutility functions, but the target-oriented approach is more natural for some classes of decisions. In some situations, thetarget-oriented preference conditions are analogous to reliability theory conditions for series or parallel failure modes ina system. In such cases, reinterpreting the conditions using reliability concepts can be useful in assessing the preferencefunction. The target-oriented approach is also a generalization of common forms of goal programming. The approach hasparticular applicability for resource allocation decisions where the outcome of the decision is significantly determined bythe actions of other stakeholders to the decision, such as new product development or decision making in a controversialregulated environment.
Area of review : Decision Analysis.History : Received July 2000; revisions received April 2001, January 2002, May 2002, November 2002, February 2003;accepted March 2003.
1. IntroductionThis paper develops an approach based on performancetargets to assess a preference function for a multiobjec-tive decision under uncertainty. This approach is shownto yield preference functions that are strategically equiva-lent to conventional multiattribute utility functions, but thetarget-oriented approach is more natural for some classes ofdecisions. Therefore, this approach provides new methodsto assess a preference function for use in certain multiob-jective decision analyses.The target-oriented approach is particularly applicable
for resource allocation decisions where multiple stakehold-ers to the decision impact the success that results from theselected allocation of resources. Examples of such deci-sions include (1) budget allocation for new product devel-opment where competitors are simultaneously developingcompetitive products, and (2) allocation of project fund-ing related to controversial activities in regulated environ-ments. For example, a customer’s purchase decision for anew product may involve comparing your product’s per-formance against competing products on such attributes ascost, quality, and features. Your budget allocation decisionfor new product development in such a situation shouldconsider two types of uncertainty: (1) the performanceof your new product with respect to the attributes, and
(2) the performance of competitors’ products on thesesame attributes. The second type of uncertainty can beconceptualized by saying that your competitors’ productperformance establishes targets that your product will becompared against when a purchase decision is made. If theperformance of your competitors’ future products is uncer-tain when you make your product development budgetingdecisions, the performance of your potential products mustbe compared against uncertain targets set by your competi-tors’ products on the various attributes.
2. BackgroundSubstantial empirical evidence indicates that the conven-tional concave single-attribute utility function often doesnot provide a good description of individual preferences.As a substitute, Kahneman and Tversky (1979) propose anS-shaped value function, and Heath et al. (1999) suggestthat the inflection point in this S-shaped value function canbe interpreted as a target. Developing this concept further,Castagnoli and Li Calzi (1996) present a target-orienteddecision-making approach for decisions under uncertaintywith a single evaluation attribute, and this type of decision-making approach is now discussed.
For notational convenience, designate an evaluation attri-bute by Z, and an arbitrary specific level of that evaluationattribute by z. An expected utility decision maker is definedto be target oriented for a single-attribute decision if thedecision-maker’s utility for an outcome depends only onwhether a target is achieved with respect to Z. Thus, atarget-oriented decision maker has only two different util-ity levels, and because a utility function is only specifiedto within a positive affine transformation, these two util-ity levels can be set to one (if the target is achieved) andzero (if the target is not achieved). With this scaling for theutility function, a target-oriented decision-maker’s expectedutility for alternative a is
E�u �a�=∫ �
z=−�
{p�z �a×1+�1−p�z �a�×0}f �z �adz
=∫ �
z=−�p�z �af �z �adz�
where p�z � a is the probability that the target is achievedgiven that the attribute is at level z and alternative a isselected, and f �z � a is the probability density function forz given that a is selected.If targets are probabilistically independent of alterna-
tives, once z is specified, this reduces to
E�u � a�=∫ �
z=−�p�zf �z � adz� (1)
where p�z is the probability that the target is achievedgiven that the attribute is at level z. Thus, for a target-oriented decision maker it is not necessary to assess a utilityfunction; instead, it is necessary to determine the probabil-ity function p�z. As we discuss below, in some decisioncontexts this may be a more intuitively appealing task thanassessing a utility function. A special case of (1) is wherethe target is known for certain, and hence p�z is eitherzero or one, depending on z. Therefore, the target-orientedapproach applies to decisions with certain targets as wellas decisions with uncertain targets.It is clear from (1) that there is always an equivalent stan-
dard (nontarget-oriented) utility formulation for any target-oriented formulation because the utility function can be setequal to p�z to create such a formulation. The converse isalso true: Because utility functions consistent with the Sav-age axioms are bounded (Fishburn 1970), the utility func-tion can be rescaled so that it lies between zero and one overthe range of levels for Z that is of interest. Then, settingp�z equal to the (possibly rescaled) utility function createsa strategically equivalent target-oriented formulation.
2.2. Descriptive Target-Oriented Formulation
The decision making described in the preceding section isnormative in the sense that it assumes the decision makerwishes to obey the axioms of rational choice (Von Neu-mann and Morgenstern 1947, Pratt et al. 1964) that yield
expected utility as the decision criterion. An alternative for-mulation that also leads to (1) uses a target-oriented for-mulation to model descriptively the behavior of a decisionmaker. This approach models a decision maker as some-one who intuitively ranks alternatives in accordance withtheir probability of achieving a possibly uncertain targeton Z, and hence selects the alternative with the greatestprobability of achieving this target. Such a model might beconsidered a natural variant on Simon’s theory of boundedrationality. With this approach, the right-hand side of (1)is interpreted as the probability that the target will beachieved, given that alternative a is selected. Of course,there is always a normative expected utility interpretationof (1) that is mathematically equivalent to this descriptiveinterpretation.
3. Target-Oriented MultiattributeDecisions
Here are several illustrative decision-making situationswhere a target-oriented multiattribute decision-making app-roach is natural.
Product Development. A target-oriented approach toproduct development resource allocation is natural in somesituations where the product is a complex system, for exam-ple, an automobile, a personal computer, a television, anew missile defense system, or any of a variety of othercomplex products that are developed or improved by com-panies in competitive markets. Typically such products aredeveloped by sizable teams, with a key decision being theresources (budget, personnel, office space, testing capacity,etc.) that will be allocated to the subteams working on eachsubsystem of the product. Most such products are devel-oped in a competitive environment against other companies(or other opponents such as foreign military powers), andthe success of the new or improved product will be deter-mined by how well it performs relative to the competingproducts. Because competing products are usually underdevelopment at the same time, they give rise to uncer-tain performance targets against which your product willbe compared when it is finally released. A target-orienteddecision-making approach is relevant if potential customersfor the product make purchase decisions based on whetheryour product outperforms the competition on various setsof attributes. Examples of such decision rules include:
1. The Pugh Rule. This widely used concept selectionprocess (Pugh 1991) starts with a benchmark concept andcompares each proposed concept against the benchmark onseveral criteria. The proposed concept that is superior to thebenchmark on the largest number of criteria is chosen forfurther development. The Pugh rule is commonly imple-mented along with Quality Function Deployment (Clausing1998).
2. Plurality Voting. The customer chooses the productthat is superior to other products on the greatest number ofcriteria.
3. Elimination by Aspects. The customer ranksattributes in order of importance, and starting with the mostimportant attribute eliminates products that are inferior onthat attribute, repeats this procedure with the next mostimportant attribute, and continues in this way successivelyeliminating products that are inferior on successively lessimportant attributes (Tversky 1972).On the other hand, if potential customers make decisions
based on weighted average performance across attributes,product development teams will work to develop a prod-uct whose weighted average performance exceeds theweighted average performance of competing products. Inthis case, a target-oriented approach with a single attribute(namely, weighted average performance across the perfor-mance attributes) is appropriate.
Regulated Environments. Much business decisionmaking is conducted in an environment subject to gov-ernment regulation or standards with substantial scrutinyby interested parties. Examples of this include construct-ing new facilities such as warehouses, stores, manufactur-ing plants, pipelines, or power plants; or conducting certaintypes of business, such as the use of genetically modifiedplants or animals; or manufacturing involving or produc-ing hazardous materials. In such decision-making environ-ments, the performance required with respect to variousperformance attributes by the different stakeholders can beuncertain, and hence decision making is done in the pres-ence of uncertain targets.Often each stakeholder group in such a situation focuses
almost exclusively on performance with respect to the eval-uation attribute of interest to them. Hence, failure to pro-vide adequate performance with respect to each attribute, orat least with respect to a sufficient number of attributes, willlead to failure of the selected alternative. In such situations,a target-oriented decision-making approach is appropriate.
Setting Performance Standards. It is common inoperations management to set performance targets, suchas monthly sales quotas for sales personnel; or quality,throughput, and safety standards for manufacturing plants.These are often set to provide concrete incentives for opera-tions personnel who may have difficulty relating their dailypersonal activities to higher-level, somewhat abstract, cor-porate targets involving market share, revenue growth, netincome, etc. These higher-level corporate targets can beuncertain because these in turn address the highest objec-tive of a business, which is usually to be profitable and stayin business. Thus, the setting of performance standards isoften decision making with uncertain targets, and hence atarget-oriented approach is appropriate.
Resource Allocation Under Uncertain Competition.Product development, regulated decision making, and set-ting performance standards are examples of decisions withthree characteristics: (1) a fixed resource must be allo-cated among competing uses to produce an uncertain finalresult, (2) there are (possibly uncertain) targets with respect
to multiple performance attributes, and (3) the final suc-cess of the decision outcome is measured by the extentto which the multiple performance targets are met, andnot by detailed performance with respect to each perfor-mance attribute. Whenever these characteristics are present,a target-oriented approach to multiattribute preference anal-ysis is appropriate.The examples above emphasize that performance targets
may be uncertain, but decisions where some or all of thetargets are known for certain are special cases that are alsocovered by the results developed below.
4. Target-Oriented MultiattributeFormulation
This section extends the target-oriented approach in §2 todecisions with multiple evaluation attributes. We first con-sider a normative formulation that is analogous to the nor-mative formulation presented in §2.1 for the single-attributesituation.
Definition 1. With n attributes X = �X1�X2� � � � �Xn, adecision maker is defined to be target oriented if his or herutility for an outcome x= �x1� x2� � � � � xn depends only onwhich targets are met by that outcome, where there is asingle target for each attribute.
For example, a decision maker allocating a productdevelopment budget to upgrade an existing product mighthave three evaluation attributes: cost, quality, and features.If the decision maker is target oriented, alternative budgetallocations will be ranked based only on which of the cost,quality, and features targets are met, and not on the spe-cific levels that are achieved for the three attributes. Theseperformance targets might be set relative to a competitor’sproject that is also currently under development, and hencemight be uncertain due to the uncertain performance of thecompetitor’s product.It follows from Definition 1 that the utility function for
a target-oriented decision maker is completely specifiedby 2n − 2 constants where these constants are the utilitiesof achieving specific combinations of the various targets.(There are 2n such constants, but because a utility functionis only specified to within a positive affine transformation,two of these can be specified arbitrarily.) Therefore, to cal-culate expected utilities it is necessary to know the prob-ability for each of the 2n different possible combinationsof target achievement as a function of the levels for the nattributes. Define I = �I1� I2� � � � � In as a set of indicatorvariables where Ii equals one if the target for Xi is achievedand zero otherwise. Let Iu be the set of all 2
n combinationsof possible levels of I , and let p�I � x be the probability ofI given x. Then, the expected utility for alternative a is
where f �x � a is the probability density function over Xgiven a, and u�I is the decision maker’s utility functionover I .
Definition 2. For notational convenience, define thetarget-oriented preference function uT �x by
uT �x≡∑I∈Iuu�Ip�I � x� (3)
Using this definition, (2) shows that a target-orienteddecision-maker’s expected utility equals the expected valueof his or her target-oriented preference function. Equation(3) can be rewritten as
uT �x=∑I∈IukIp�I � x� (4)
where kI = u�I, and hence the decision-maker’s utilityfunction is specified by a set of 2n constants, where theseare the utilities of achieving each unique combination oftargets.Note that there is a descriptive formulation that is equiv-
alent to (2), just as there is a descriptive formulation forthe single-attribute case that is equivalent to (1). Specifi-cally, if u�I is interpreted as the probability that a par-ticular set of target achievements I is “good enough” withrespect to the entire set of targets, then the right side of(2) gives the probability that a particular alternative willbe “good enough” for the decision maker to select thisalternative. Hence, if it is assumed that a decision makerwill select the alternative that has the greatest probabilityof being “good enough,” then (2) provides an approach todescriptively model decision-making behavior in multiob-jective decisions.
5. Special Cases
5.1. Independent Targets
We now examine conditions on a decision-maker’s prefer-ences that lead to specific functional forms of (4). Theseconditions are useful because they can simplify the assess-ment of the target-oriented preference function in practicalapplications.
Definition 3. If a decision-maker’s probability of achiev-ing the target on any attribute Xi depends only on the levelxi, then the decision maker is said to have independenttargets.
Independent targets seem intuitively reasonable for manytarget-oriented decisions, and with independent targets, theIi are probabilistically independent, given x. If we definepi�xi to be the probability that the target for attribute Xiis achieved, given the level xi of that attribute, then inde-pendent targets imply
p�I � x={
n∏i=1
i�Ii=1
pi�xi
}{n∏i=1
i�Ii=0
�1−pi�xi�}� (5)
where a product is defined to be one if there are no factors.Equation (5) combines with (4) to yield
uT �x=∑I∈IukI
{n∏i=1
i�Ii=1
pi�xi
}{n∏i=1
i�Ii=0
�1−pi�xi�}� (6)
For example, with two attributes this becomes
uT �x1�x2=k��1−p1�x1��1−p2�x2�+k1p1�x1�1−p2�x2�+k2�1−p1�x1�p2�x2+k12p1�x1p2�x2� (7)
where k� = u�0�0, k1 = u�1�0, k2 = u�0�1, and k12 =u�1�1.Note that uT �x in (6) is fully determined by the n func-
tions pi�xi and the 2n constants kI . Thus, the amount
of information needed to determine uT �x when there areindependent targets is the same as that needed to deter-mine a multilinear multiattribute utility function (Keeneyand Raiffa 1976, §6.4; Kirkwood 1997, Theorem 9.41).In fact, Theorem 2 below shows that it is always possi-ble to find a multilinear utility function that is strategicallyequivalent to uT �x in (6), where two preference functionsfor decisions under uncertainty are said to be strategicallyequivalent if they give the same rank ordering for any setof alternatives and hence are positive affine transformationsof each other (Keeney and Raiffa 1976, Theorem 4.1; Kirk-wood 1997, Theorem 9.25).
5.2. Additive Target Preferences
Following standard terminology, a decision maker is saidto have additive independent preferences if the decision-maker’s rank ordering for any set of alternatives dependsonly on the marginal probability distributions over theattributes for each alternative (Keeney and Raiffa 1976,§6.5; Kirkwood 1997, Definition 9.31).
Theorem 1. The target-oriented preference function for atarget-oriented decision maker with independent targetsand additive independent preferences is strategically equiv-alent to
uT �x=n∑i=1Kipi�xi (8)
for some constants Ki. If an alternative that achieves all thetargets of a second alternative and also achieves additionaltargets is not less preferred than the second alternative,then Ki � 0 for all i.
The proofs for this theorem and the others below arein the appendix. The form of uT in (8) is similar to theform for the standard additive utility function, and Theo-rem 3 below shows that it is always possible to determine
an additive utility function that is strategically equivalentto an additive uT .
5.3. Reliability-Structured Target Preferences
While the additive target-oriented preference functionshown above has a similar form to the standard additivemultiattribute utility function, we now consider a type oftarget-oriented preference structure that is different fromthe structures usually assumed for multiattribute utilityfunctions.
Definition 4. A target-oriented decision maker with inde-pendent targets is said to have a reliability target structureif u�I can take on only two different levels. An outcomewith the higher utility level is called a success, and an out-come with the lower utility level is called a failure.
Because a utility function is only defined to within apositive affine transformation, it is always possible with areliability target structure to scale u�I so that the utilityfor a success is one and the utility for a failure is zero.This scaling convention is used for the remainder of thispaper.The potential usefulness of a reliability target structure
is illustrated by the new product development example dis-cussed above where there are three evaluation attributes:cost, quality, and features. One way to analyze this deci-sion is as follows: Suppose that potential customers for thisnew product will compare it against the competitors’ sim-ilar products, and hence a target-oriented decision analy-sis approach may be appropriate. Because these competingproducts are not yet on the market, the targets on the threeattributes that the new product must achieve to be a successare uncertain. A reliability target structure may be appro-priate if the decision is viewed as selecting the alternativewith the highest likelihood of acceptance by potential cus-tomers. For example, a potential customer may purchaseour product only if it is of higher quality than the com-petitors’ products and is also either less costly or has morefeatures. Assuming that the targets for the attributes areindependent and u�I is scaled to be either zero or one asdiscussed above, then
uT �xc� xq� xf = pq�xq{1− �1−pc�xc��1−pf �xf �
}�
where the subscripts c, q, and f refer to cost, quality, andfeatures, respectively.This is a special case of (6), but the kI do not need to
be directly assessed because they are determined by speci-fying the combinations of targets needed for success. Thisshows the origin of the term reliability target structure,because uT �x is determined using analogies with paral-lel and series elements in reliability. (The cost and fea-tures targets are analogous to parallel elements in reliabilityanalysis, and the quality target is analogous to an elementin series with these.) Theorem 4 below shows that thereis an equivalent multiplicative utility function for manyreliability-structured target-oriented preference functions.
6. Assessment Procedure with anApplication
6.1. Assessment Procedure
Assessing a target-oriented preference function of the formof (6) requires determining the n target probability func-tions pi�xi and the 2
n constants kI . While there are notheoretical restrictions on the shape of the pi�xi, in manyapplications these will be monotonic in xi with greater lev-els of xi either always having a higher probability or alwayshaving a lower probability. For example, in the new productdevelopment example discussed above, it seems reasonablethat pc�xc would monotonically decrease with increases inxc. That is, lower costs will always lead to a higher proba-bility of meeting the cost target.In a decision with monotonic pi�xi, these functions
might be determined by assessing the probability distribu-tion for the minimum level of Xi that will meet the target.Thus, for the new product development example, probabil-ity distributions might be assessed for the level of perfor-mance that the competitors will achieve in their productswith respect to cost, quality, and features. Then, either thecumulative distribution or the complementary cumulativedistribution for this uncertain quantity would be used aspi�xi, depending on whether more or less of an attributeis preferred.Assessment of the kI could be time consuming because
there are 2n of these. However, in analogy to applica-tions of multiattribute utility theory (Corner and Kirkwood1991), we might assume that additive independence holdsso that only the n constants in (8) must be assessed. Forexample, some variation on the SMART assessment pro-cedure (Edwards 1977, Edwards and Barron 1994) mightbe appropriate to assess these constants. This assessmentis simplified even further if a reliability target structure isassumed. Then, it is only necessary to determine the appro-priate series and parallel target structure, as illustrated bythe new product development example discussed above. Inthis case, no weights have to be determined, but instead theparallel/series structure is specified.
Keeney and Lilien (1987) consider a decision where a com-pany wanted to assess how prospective customers wouldevaluate a proposed new tester for very large-scale inte-grated circuits. They identified four categories of evaluationcriteria (technical, economic, software, and vendor support)with a total of 17 evaluation attributes, as shown in thefirst column of Table 1. The preference monotonicity foreach evaluation attribute is shown in the second column ofTable 1, and the scores on each of the evaluation attributesare shown in the third through fifth columns of the tablefor the OR 9000, which was the proposed new tester, andits two competitors, the J941 and the Sentry 50.
Keeney and Lilien (1987) assessed the measurable valuefunction for a lead user at a primary customer companyfor this testing equipment. This lead user first assessed aminimum acceptability level and a maximum desirabilitylevel for each attribute. Keeney and Lilien then confirmedthat the user’s preferences were describable by an addi-tive measurable value function, and they assessed a single-dimensional value function and an importance weight foreach attribute. The assessed additive measurable valuefunction was then used to evaluate the OR 9000 againstthe J941 and Sentry 50, and the results served as input todetermine that the proposed new tester was not competitiveenough to market.For this decision, it is natural to think in terms of perfor-
mance targets because the explicit purpose of the analysiswas to determine whether the OR 9000 was attractive whenjudged against the J941 and the Sentry 50. Thus, the per-formance of these two testers sets targets against which theOR 9000 is judged. Table 2 illustrates a possible target-oriented preference analysis for this decision. There is nouncertainty about the performance of the three testers, andpreferences are monotonic with respect to each evaluationattribute. Therefore, the target will be achieved for an evalu-ation attribute only if performance meets or exceeds a targetperformance level on that evaluation attribute. The targetperformance levels might be set using different criteria foreach evaluation attribute. For example, there might be someevaluation attributes where it would be judged necessaryonly to meet some minimal level of performance, while forother evaluation attributes it might be judged necessary toexceed the performance of both of the competitors by somethreshold amount (for example, 10%).
To illustrate a possible target-oriented analysis, Table 2assumes that the performance target for each evaluationattribute is equal to the best performance of either of thetwo competitors. For example, because higher levels of “pincapacity” are more preferable and the Sentry 50 has thehighest level for this evaluation attribute, the Sentry 50level (which is 256) is the performance target for this eval-uation attribute. The targets for each evaluation attribute areshown in the second column of Table 2.Keeney and Lilien (1987) used an additive measurable
value function, and if the additive independence conditionsin Theorem 1 hold, then the target-oriented preferencefunction will have the weighted-additive form (8). Keeneyand Lilien used a two-stage process to assess weights fora measurable value function. First, the relative weights forthe evaluation attributes within each of the four evaluationcategories were assessed, and then weights were assessedfor each of the four categories so that the overall weightfor each evaluation attribute is the product of its cate-gory weight and its within-category weight. To illustratethe target-oriented analysis procedure, Table 2 assumes thatthe weights used by Keeney and Lilien can be applied.Both the within-category weights and the category weights(which are 0.52, 0.14, 0.32, and 0.02) are shown in thethird column of Table 2.Using the performance data in Table 1 and the target
and weight information in Columns 2 and 3 of Table 2,the fourth through sixth columns of Table 2 show whethereach tester achieves the target on each evaluation attribute.For example, for “data rate,” both the OR 9000 and Sentry50 have a data rate of 50, which is the target level, andtherefore they achieve the target, and hence have pi�xi= 1
for this evaluation attribute. On the other hand, the J941does not achieve the target with respect to this evaluationattribute, and hence has pi�xi = 0. The weighted com-parisons in the rightmost three columns of Table 2 showthe weighted evaluation for each alternative within eachof the four evaluation categories and the overall valuefor each alternative using the weights in the third col-umn. This evaluation finds the OR 9000 to be the leastpreferred of the three alternatives, with an overall valueof 0.514.Both the Keeney and Lilien evaluation and this target-
oriented analysis assume that there is no uncertainty aboutthe performance of the alternatives. The target-orientedanalysis extends to the case with uncertainty in a straight-forward manner. Suppose first that the only uncertainty iswith respect to the performance of the proposed OR 9000,and there is no uncertainty about the performance of theJ941 and Sentry 50 because these are already in production.With an additive target-oriented preference function of theform of (8), only the marginal probability distributions forthe evaluation attributes impact the ranking of alternatives.Hence, the probability analysis can be done one attribute ata time.As an illustration, consider the “data rate” evaluation
attribute, and represent this by Y . Let yJ941 and ySentry 50 rep-resent the data rates for the J941 and Sentry 50. Then, analternative achieves the target for Y if y is at least as greatas max�yJ941� ySentry 50 = 50. Hence, the conditional proba-bility of achieving the target for Y as a function of y is
p�I � y={1� y � 50�
0� otherwise�
If fOR 9000�y and FOR 9000�y represent the probability densityfunction and cumulative distribution function, respectively,for yOR 9000, then the expected value of the target-orientedpreference function for data rate for the OR 9000 is
E�udata rate �OR 9000�=∫Yp�I � yfOR 9000�ydy
= 1− FOR 9000�50�which is known once FOR 9000�y is assessed using any ofthe standard methods.To illustrate a more complex case, suppose that the data
rates for the J941 and Sentry 50 are also uncertain. Then,p�I � y= Prob�y �max�yJ941� ySentry 50= Prob�y � yJ941 ∩y � ySentry 50. If the data rates for these two testers are prob-abilistically independent, then p�I � y= Prob�y � yJ941×Prob�y � ySentry 50, and the expected value for the OR 9000of the target preference function for data rate is
where the probability density functions and cumulative dis-tribution functions for yJ941 and ySentry 50 are representedanalogously to those for yOR 9000. It is straightforwardto evaluate this integral using a standard approximationmethod such as the extended Pearson-Tukey approxima-tion. (If the data rates for the J941 and Sentry 50 are not
independent, then the probability model will be more com-plex, as would also be true in a realistic conventional mul-tiattribute utility analysis for this situation.)The Keeney and Lilien (1987) measurable value anal-
ysis requires ranges for all the evaluation attributes andmidvalues for those ranges, while the target-oriented anal-ysis requires that a target be specified for each attribute.Both methods require that attribute weights be assessed.The competitive shortcomings of the OR 9000 can be iden-tified with either analysis approach, but these shortcom-ings are quickly apparent and easily understood from thetarget-oriented analysis results in Table 2. It is immediatelyclear from this table that the Sentry 50 has better techni-cal performance than the OR 9000, and technical perfor-mance has a weight of 0.52, which is much greater thanany other evaluation category. While either of the two anal-ysis approaches can be a valid way to analyze this decision,the target-oriented approach seems easier to explain to anontechnical audience, and it clearly emphasizes the differ-ences among the alternatives.
7. Comparison to MultiattributeUtility Analysis
This section shows that common multiattribute utility func-tion forms are strategically equivalent to various formsof target-oriented preference functions. However, in gen-eral there can be many target-oriented preference func-tions of a specified form that are strategically equivalentto a specified multiattribute utility function. For example,consider an additive multiattribute utility function u�x =∑ni=1 �iui�xi. Let x
oi represent the least-preferred xi and x
∗i
represent the most preferred xi in the domain of interest,and assume that the ui�xi are scaled so that ui�x
oi = 0 and
ui�x∗i = 1. Then, the additive multiattribute utility function
can always be translated into a valid strategically equiv-alent additive target-oriented preference function uT �x =∑ni=1Kipi�xi by setting pi�xi=moi +�m∗
i −moi ui�xi andKi = �i/�m∗
i − moi for any constants 0 � moi < m∗i � 1,
where moi = pi�xoi and m∗i = pi�x∗i .
For example, consider the two-attribute additive utilityfunction u�x1� x2= 0�4x1+ 0�6�1− x2, where the domainfor each Xi is 0� xi � 1. One strategically equivalent target-oriented preference function is found by setting moi = 0and m∗
i = 1, i= 1�2, to yielduT �x1� x2= 0�4x1+ 0�6�1− x2� (9)
and another strategically equivalent target-oriented prefer-ence function is found by setting mo1 = 0�4 and m∗
In (9), the probability of achieving the target for X1 whenx1 = 0 is zero, and the probability when x1 = 1 is one.However, in (10) the probability of achieving the target forX1 when x1 = 0 is 0.4, and the probability when x1 = 1 is0.9. Thus, the interpretation of (10) is different from (9),because in (9) there is a level of X1 for which the target on
X1 is certain to be achieved and another level for which itis certain not to be achieved, but in (10) the probability ofachieving the target on X1 is never less than 0.4 or greaterthan 0.9. However, in (10) the utility K1 of achieving thetarget for X1 is twice as great relative to the utility K2 ofachieving the target for X2 as it is in (9) (0.4 versus 0.6 in(9), but 0.8 versus 0.6 in (10)).Theorems are now presented which demonstrate that
target-oriented preference functions of the various typesreviewed above can often be converted into strategicallyequivalent multiattribute utility functions, and vice versa.In these theorems, the notation defined above for xoiand x∗i is used. Similarly, x
o = �xo1� xo2� � � � � xon and x∗ =�x∗1� x
∗2� � � � � x
∗n. In presentations of multiattribute utility
theory (Keeney and Raiffa 1976, Chapter 6; Kirkwood1997, Chapter 9), the utility function is usually scaled sothat u�xo = 0 and u�x∗ = 1, and for each xi the single-attribute utility function ui�xi is scaled so that ui�x
oi = 0
and ui�x∗i = 1, and these scaling conventions are used here.
Scaling constants for the single-attribute utility functionsare designated by �i and the multiplicative utility functionconstant by �.
Theorem 2. There always exists a multilinear utilityfunction
u�x=n∑i=1�iui�xi+
n∑i=1
∑j>i
�ijui�xiuj�xj
+n∑i=1
∑j>i
∑l>j
�ijlui�xiuj�xjul�xl
+ · · ·+�123···nu1�x1u2�x2 · · ·un�xnthat is strategically equivalent to any target-oriented pref-erence function of form (6), and vice versa.
Theorem 3. There always exists an additive utility func-tion u�x=∑n
i=1 �iui�xi that is strategically equivalent toany additive target-oriented preference function (8), andvice versa.
Theorem 4. There always exists:(1) A multiplicative utility function 1 + �u�x =∏ni=1�1 + ��iui�xi�, where 1 + � = ∏n
i=1�1 + ��i and� > 0, that is strategically equivalent to any reliability-structured target-oriented preference function with seriestargets, provided that pi�x
oi > 0 for all i,
(2) A reliability-structured target-oriented preferencefunction with series targets and pi�x
oi > 0 for all i that is
strategically equivalent to any multiplicative utility functionwith �> 0,
(3) A multiplicative utility function 1 + �u�x =∏ni=1�1 + ��iui�xi� with 1 + � = ∏n
i=1�1 + ��i, where−1 < � < 0, that is strategically equivalent to anyreliability-structured target-oriented preference functionwith parallel targets, provided that pi�x
oi < 1 for all i, and
(4) A reliability-structured target-oriented preferencefunction with parallel targets and pi�x
strategically equivalent to any multiplicative utility functionwith −1<�< 0.Theorem 4 shows that a multiplicative utility function
with substitutable attributes (−1 < � < 0) corresponds toa target-oriented preference function analogous to a paral-lel system in reliability theory, and a multiplicative pref-erence function with complementary attributes (0 < �)corresponds to a target-oriented preference function anal-ogous to a series system in reliability theory. Becauseseries and parallel systems are fundamental building blocksof reliability models and multiplicative utility functionsare common in multiattribute utility applications, Theo-rem 4 shows that there is a close mathematical relationshipbetween multiattribute utility theory and reliability theory.
8. Generalization: “Degree ofAchievement” of Targets
This section generalizes the development in preceding sec-tions to consider the “degree of achievement” of targets ina target-oriented preference function.
8.1. Single-Attribute Decisions
For a decision with a single evaluation attribute Z, let ztbe the possibly uncertain target level for Z and za be thepossibly uncertain actual performance for alternative a, andassume that utility is specified as a function u�zt� za. Forexample,
u�zt� za={0� za < zt�
1� otherwise�(11)
corresponds to a special case of §2.1 where preferences areincreasing, and if u�zt� za is a function only of za, thisformulation corresponds to conventional utility analysis.Note that u�xt� za is not necessarily monotonic with
respect to za in all practical decisions. For example,suppose
u�zt� za={−a�zt − za� za < zt�
b− c�za− zt� otherwise�(12)
for constants a� 0� b � 0, and c. (Equation (11) is a spe-cial case of (12) with a = 0, b = 1, and c = 0.) In (12),if a > 0 there is added loss of value for missing the targetzt on the low side (za < zt) by greater amounts, and eitheradded value, no change in value, or added loss for exceed-ing the target zt by greater amounts depending on whetherc < 0� c = 0, or c > 0. For example, if b = 0, a > 0, andc > 0, then the most preferred level is za = zt , and greaterdeviations from zt in either direction are increasingly lesspreferred. Examples where this might hold include manu-facturing processes where there is an “ideal” level for somecharacteristic of the product, materials management with atarget inventory level, or medical conditions with an ideallevel for a medical indicator, such as blood pressure.
General Formulation. If the probability density func-tion for zt and za, given a, is designated by ft�a�xt� xa � a,then
E�u � a�=∫ �
za=−�
∫ �
zt=−�u�zt� zaft�a�zt� za � adzt dza� (13)
If zt and za are probabilistically independent, given a, thenft�a�zt� za � a= ft�zt � a× fa�za � a, and if zt is also notdependent on a so that ft�a�zt� za � a= ft�zt× fa�za � a,then (13) becomes
E�u �a�=∫ �
za=−�fa�za �a
∫ �
zt=−�u�zt�zaft�ztdztdza� (14)
If p�za ≡∫ �zt=−� u�zt� zaft�zt dzt in (14), then this
equation is made equivalent to (1). (Because u�zt� za canbe rescaled by any positive affine transformation with-out changing the decision, it is always possible to specifyu�zt� za so that p�za gives valid probabilities.) Hence,there is always a target-oriented formulation (1) that isequivalent to (14), although that formulation may not havea natural interpretation in terms of the real-world decision.As discussed in §2.1, there is always a standard single-
attribute utility formulation that is strategically equivalentto any target-oriented specification of the form of (1).Because the preceding paragraph demonstrates that there isalways a specification of the form of (1) that is strategi-cally equivalent to any specification of the form of (14),therefore there is always a standard utility formulation thatis strategically equivalent to (14).
8.2. Multiattribute Decisions
The approach in the preceding section can be general-ized to multiattribute preferences, but utility independenceconcepts must be applied to develop a preference func-tion form that is practical for applications. Designate thetarget level for attribute Xi by xit , and the actual perfor-mance for Xi, given alternative a, by xia, and define xt ≡�x1t� x2t� � � � � xnt and xa ≡ �x1a� x2a� � � � � xna. Analogouslyto the single-attribute case in §8.1, assume that utility is afunction u�xt& xa. Designate the probability density func-tion over xt and xa, given a, by ft�a�xt& xa � a so that themultiattribute extension of (13) is
E�u � a�=∫ �
−�· · ·
∫ �
−�u�xt& xa ft�a�xt& xa � adxt dxa� (15)
Reducing the complexity of (15) so that it can beapplied requires simplifying both u�xt& xa and ft�a�xt& xa.To illustrate how this can be done, assume that the pairs�Xit�Xia� i= 1�2� � � � � n, are each additive independent ofthe remaining Xit and Xia. Then, u�xt& xa reduces to thestandard weighted-sum form
u�xt& xa=n∑i=1kiui�xit� xia (16)
for some constants ki and two-attribute utility functionsui�xit� xia. If, in addition, Xit is probabilistically indepen-dent of Xia, and Xit is probabilistically independent of a,
where ft�xt is the probability density function over the Xitand fa�xa � a is the conditional probability density functionover Xia, given a.Substituting (16) and (17) into (15) and integrating yields
E�u � a�=n∑i=1ki
∫ �
xia=−�fia�xia � a
·∫ �
xit=−�u�xit� xiafit�xit dxit dxia� (18)
Because each term in (18) is analogous to (14), the argu-ment that was applied to (14) also demonstrates that thereis a strategically equivalent target-oriented formulation for(18) of the form of (8). As with (13), a form such as (12)could be used for the u�xit� xia in (18) to represent dif-fering preferences for different degrees of achievement ofthe targets. Equation (18) assumes additive independenceamong the pairs �Xit�Xia, but other preference conditionscould also be applied, such as the utility independence andpairwise preferential independence conditions that lead toa multiplicative decomposition.
8.3. Relationship to Goal Programming
This section demonstrates that the basic weighted goal pro-gramming formulation is a special case of the model in§8.2. The basic weighted goal program is (Tamiz and Jones1996)
miny�'
n∑i=1�w−
i '−i +w+
i '+i (19)
subject to fi�y+ '−i − '+i = bi� i = 1� � � � � n, and y ∈ Cy .In this formulation, bi� i = 1� � � � � n, are the targets, ' rep-resents the set of all '−i � 0 and '+i � 0, which are thenegative or positive deviation, respectively, from each tar-get level, and w−
i � 0 and w+i � 0 are the weights for
these deviations in the optimization. (At most, one of '−iand '+i will be nonzero in any optimal solution.) Cy isan optional set of constraints on the decision variablesy = �y1� y2� � � � � ym. The solution to this goal program isthe feasible y that minimizes the weighted sum of the devi-ations between fi�y and bi.To show that this is a special case of generalized target-
oriented preference analysis, assume (16) holds and (12)holds with a� 0, b = 0, and c � 0 for each ui�xit� xia sothat
ui�xit� xia={−ai�xit − xia� xia < xit�
−ci�xia− xit� otherwise�(20)
For this case, the target-oriented decision with no uncer-tainty can be represented by
maxxa∈C
n∑i=1kiui�xit� xia� (21)
where C is the set of feasible xa = �x1a� x2a� � � � � xna.By defining '−i ≡max�xit − xia�0 to represent deviationsbelow the target levels xit in (20) and '
+i ≡max�xia−xit�0
to represent deviations above the target levels in that equa-tion, we can rewrite (20) as ui�xit� xia = −a'−i − c'+i .Substituting this representation for ui into (21) yields
min'�xa
n∑i=1�kiai'
−i + kici'+i (22)
subject to '−i − '+i = xit − xia, '+i � 0, '−i � 0, i =
1�2� � � � � n, and xa ∈ C. If xia is a function of a set ofdecision variables y, so that xia = fi�y with y ∈ Cy , andbi ≡ xit , we can rewrite '−i −'+i = xit−xia as fi�y+'−i −'+i = bi. Hence, (22), which we have just shown is equiva-lent to (20) and (21), becomes equivalent to (19) if we setw−i ≡ kiai and w+
i ≡ kici. Because (16) and (20) are specialcases of u�xt& xa, the weighted goal programming formu-lation in (19) is a special case of target-oriented preferenceanalysis.Stochastic goal programming (Ballestero 2001) treats
target-oriented decisions under uncertainty by replacingthe uncertain evaluation attribute levels with their corre-sponding expected utilities and then minimizing (19), anapproach that is not fully consistent with utility theory. Incontrast, target-oriented preference analysis under uncer-tainty, as presented in §8.2, extends the goal programmingformulation in (19) to decision making under uncertaintyin a way that is fully consistent with utility theory.
8.4. Further Generalization to Target Ranges
The formulation in §§8.1 and 8.2 can be generalized tomultiple target levels for each evaluation attribute, whichaddresses decisions with target ranges. Specifically, sup-pose there are two target levels xlit < x
uit for each Xi. If
xit ≡ �xlit� xuit, then with the same utility and probabilisticindependence conditions on Xit and Xia as assumed in §8.2,(18) will be a valid representation for this situation. As anexample, consider
ui�xlit� x
uit� xia=
−ai�xlit − xia� xia < x
lit�
0� xlit � xia � xuit�
−ci�xia− xuit� otherwise�
(23)
where ai > 0 and ci > 0, which assumes there is a rangeof levels xlit � xia � x
uit that are all equally preferred and
deviations in either direction from that range are less pre-ferred. (An example is a manufacturing process where anydimension for a manufactured component within a toler-ance range is equally acceptable.) For this case, by anal-ogous reasoning to that yielding (21), the target-orienteddecision with no uncertainty can be represented as
maxxa∈C
n∑i=1kiui�x
lit� x
uit� xia� (24)
where C is the set of feasible xa = �x1a� x2a� � � � � xna.
An analogous process to that in §8.3 develops a gener-alized version of (19) that is equivalent to (23) and (24).Define 'l−i ≡max�xlit − xia�0 and 'l+i ≡max�xia − xlit�0to represent deviations from the lower target levels xlit in(23), and 'u−i ≡ max�xuit − xia�0 and 'u+i ≡ max�xia −xuit�0 to represent deviations from the upper target levels.Then, an equivalent formulation to (23) and (24) can bewritten as
min'�xa
n∑i=1�kiai'
l−i + kici'u+i (25)
subject to 'l−i − 'l+i = xlit − xia, 'u−i − 'u+i = xuit − xia,'l+i � 0, 'l−i � 0, 'u+i � 0, 'u−i � 0, i = 1�2� � � � � n, andxa ∈C, where at most one of the deviations will be nonzerofor a specified i in any optimal solution. If xia is a functionof a set of decision variables y, so that xia = fi�y withy ∈Cy , and bi ≡ xit , w−
i ≡ kiai, and w+i ≡ kici, then (25) is
made equivalent to a generalized form of (19). Because (23)and (24) are special cases of u�xt& xa with target ranges,the goal programming formulation in (25) is a special caseof the generalized target-oriented preference formulation inthis section.
9. Concluding CommentsThis paper presents methods to model preference structuresinvolving targets on multiple evaluation attributes. Thisapproach can simplify the development of a multiattributepreference function for some decisions, and it appears tohave particular applicability for situations where the out-come of the decision is significantly determined by theactions of other stakeholders to the decision. Examples ofthese types of decisions include new product developmentand decision making in a highly regulated environment.
AppendixFollowing the notation of Keeney and Raiffa (1976), Xistands for all the attributes except Xi, x
′i stands for an arbi-
trary specified level of Xi, x′ stands for arbitrary speci-
fied levels of all the attributes, and �xi� x′i stands for any
level of Xi combined with any specified levels for the otherattributes.
Proof of Theorem 1. Consider two alternatives, a1,which has a probability 1/n of yielding x and a probability�n− 1/n of yielding x′; and a2 which has a probability1/n of yielding �xi� x
′i� i= 1�2� � � � � n. Because a1 and a2
have the same marginal probability distributions for eachxi, if the conditions of the theorem hold, then these twoalternatives must be equally preferred and hence have equalexpected values for uT .The expected value of uT for a1 is given by �1/n ·
uT �x+ ��n−1/n�uT �x′, and the expected value of uT fora2 is given by �1/n
∑ni=1 uT �xi� x
′i. Because the decision
maker has independent targets, then uT �x has the form of(6), and from inspection of (6)
uT �x= ai�xipi�xi+ bi�xi�1−pi�xi�= bi�xi+ ci�xipi�xi� (A-1)
for i = 1�2� � � � � n, for some functions ai�xi, bi�xi, andci�xi ≡ ai�xi − bi�xi. (Note that ai�xi and bi�xi arenot arbitrary because they are implicitly defined by (6).)Thus, the expected value of uT �x for a2 is equal to�1/n
∑ni=1�bi�x
′i+ ci�x′ipi�xi�. Equating this expression
to the expression for the expected value of uT for a1 andrearranging terms leads to
uT �x=−�n− 1uT �x′+n∑i=1bi�x
′i+
n∑i=1ci�x
′ipi�xi�
Defining Ki = ci�x′i leads to (8) except for the constant−�n−1uT �x′+∑ni=1 bi�x
′i. However, two target-oriented
preference functions that differ only by a constant arestrategically equivalent. Therefore, this constant can bedropped from uT �x, and hence (8) follows except that wemust establish that the Ki are nonnegative.The nonnegativity of the Ki under the conditions of the
theorem will be proved if ci�x′i= ai�x′i− bi�x′i� 0� i =
1�2� � � � � n. Compare ai�x′i and bi�x
′i in (A-1) with (6).
From (6) it follows that ai�x′i and bi�x
′i are made up of
terms that are pairwise identical except for the constantskI . Each pair of corresponding terms represents an outcomewhere the same targets are met except that for the term inai�x
′i, the target for xi is met in addition to the targets that
are met for the corresponding term in bi�x′i. Hence, from
the statement of the theorem, the outcome for the term inai�x
′i must not be less preferred than the outcome for the
corresponding term in bi�x′i, and therefore the kI for each
term included in ai�x′i must be at least as great as the
kI for the corresponding term included in bi�x′i. Hence,
Ki = ci�x′i= ai�x′i− bi�x′i� 0. �
For the following three proofs, it must be true that u�xand ui�xi are all scaled to lie between zero and one, butthere is no such requirement on uT �x or the pi�xi. How-ever, these can be rescaled to yield functions that are scaledbetween zero and one as follows:
u�x≡ �uT �x− uT �xo�/�uT �x∗− uT �xo�� (A-2)
ui�xi≡ �pi�xi−pi�xoi �/�pi�x∗i −pi�xoi �� (A-3)
Because u�x as defined by (A-2) is a positive affine trans-formation of uT �x, then u�x and uT �x are strategicallyequivalent.
Proof of Theorem 2. Keeney and Raiffa (1976, §6.5)have shown that if
then u�x has the multilinear form in Theorem 2. (Equation(A-4) is the same as (6.27) in Keeney and Raiffa (1976).Note that in this equation u�x is scaled so that u�xo= 0and u�x∗= 1 for some xo and x∗, and also ui�xoi = 0 andui�x
∗i = 1.) Therefore, showing that uT �x in (6) is strate-
gically equivalent to u�x in (A-4) will prove the theorem.From (A-1) in the proof of Theorem 1, we know that
uT �x = bi�xi + ci�xipi�xi� i = 1�2� � � � � n, for somefunctions bi�xi and ci�xi. Solve (A-2) and (A-3) for uT �xand ui�xi, respectively, and substitute the resulting expres-sions into this equation. Rearranging terms leads to
o�. Substituting first x = �x∗i � xi and then x =�xoi � xi into (A-5) yields the equations u�x
∗i � xi= b′i�xi+
c′i�xiui�x∗i = b′i�xi + c′i�xi and u�xoi � xi = b′i�xi +
c′i�xiui�xoi = b′i�xi. Solving these for b′i�xi and c′i�xi
and rearranging terms yields (A-4), and hence the result isproved.Because a key step in this proof is not intuitively
reversible, we will directly prove the converse of thistheorem. (The step that is not intuitively reversible isgoing from (A-1) to (6).) We will proceed by induc-tively constructing a strategically equivalent target-orientedpreference function (6) starting from any specified mul-tilinear utility function. This proof requires some newnotation. Specifically, define yk = �x1� x2� � � � � xk, yk =�xk+1� xk+2� � � � � xn, and Ik = �I1� I2� � � � � Ik, where eachIi is a zero-one indicator variable. Then, I
ku is defined to
be the set of all 2k combinations of possible levels of Ik.Finally, yI
k
k designates levels of x1� x2� � � � � xk as follows: IfIki = 1 then xi = x∗i , and if Iki = 0 then xi = xoi . For exam-ple, y�0�1�13 = �xo1� x∗2� x∗3.The induction proceeds as follows: Assume that for a
specific value of k
u�x= ∑I∈Ikuu(yI
k
k � yk){ k∏
i=1i�Iki =1
ui�xi
}
·{
k∏i=1
i�Iki =0
�1− ui�xi�}� (A-6)
(This is true from (A-4) for k= 1.) Then, apply (A-4) withi= k+ 1 to expand u�yIkk � yk in (A-6). The result isu�x= uk+1�xk+1
∑I∈Ikuu(yI
k
k � x∗k+1� yk+1
)
·{
k+1∏i=1
i�Iki =1
ui�xi
}{k+1∏i=1
i�Iki =0
�1− ui�xi�}
+ �1− uk+1�xk+1�∑I∈Ikuu(yI
k
k � xok+1� yk+1
)
·{
k+1∏i=1
i�Iki =1
ui�xi
}{k+1∏i=1
i�Iki =0
�1− ui�xi�}�
However, this can be rewritten as
u�x= ∑I∈Ik+1u
u(yI
k+1k+1 � yk+1
)
·{
k+1∏i=1
i�Ik+1i =1
ui�xi
}{k+1∏i=1
i�Ik+1i =0
�1− ui�xi�}�
and this is the same as (A-6) with k replaced by k + 1.Hence, if (A-6) holds for k it also holds for k+ 1. Whenk = n, (A-6) is the same as (6) if we set u�x = uT �x,u�yI
n
n � yn+1= kI , and pi�xi= ui�xi in (A-6). (When k=n, yn+1 is null, and hence u�yI
n
n � yn+1 is equal to a con-stant.) Because these substitutions result in valid values foruT �x, kI , and pi�xi, we have constructed a strategicallyequivalent target-oriented preference function for the multi-linear utility function, and thus the converse of the theoremis proved. �
Proof of Theorem 3. To show that there is always anadditive utility function that is strategically equivalent to(8), solve (A-2) and (A-3) for uT �x and pi�xi, respec-tively, and substitute into (8). This yields
uT �xo+ �uT �x∗− uT �xo�u�x
=n∑i=1Ki*pi�x
oi + �pi�x∗i −pi�xoi �ui�xi+�
Because uT �xo=∑n
i=1Kipi�xoi , this reduces to
�uT �x∗− uT �xo�u�x=
n∑i=1Ki�pi�x
∗i −pi�xoi �ui�xi�
Define �i =Ki�pi�x∗i −pi�xoi �/�uT �x∗−uT �xo� and theresult follows. If the �i do not sum to one, then multiply bythe appropriate (positive) constant so this is true. (Multipli-cation by a positive constant always yields a strategicallyequivalent utility function.)To show that there is always an additive target-oriented
preference function (8) that is strategically equivalent toany additive utility function, substitute into the additive util-ity function u�x=∑n
i=1 �iui�xi as follows: uT �x= u�x,pi�xi= ui�xi, and Ki = �i. The result is a valid additivetarget-oriented preference function. �
Proof of Theorem 4. Each part of this theorem is provedin order. For Part (1) of the theorem, a reliability-structuredtarget-oriented preference function with series targets hasthe form uT �x=
∏ni=1 pi�xi. Solving (A-2) and (A-3) for
uT �x and ui�xi, respectively, and substituting into thisequation yields
uT �xo+ �uT �x∗− uT �xo�u�x
=n∏i=1*pi�x
oi + �pi�x∗i −pi�xoi �ui�xi+�
Define � = �uT �x∗ − uT �x
o�/uT �xo and ��i =
�pi�x∗i −pi�xoi �/pi�xoi . By the conditions of the theorem
both � and �i must be greater than zero because uT �x∗ >
uT �xo and pi�x
∗i > pi�x
oi . Substitute into the equation
above to yield
uT �xo�1+�u�x�
={ n∏i=1pi�x
oi
}{ n∏i=1�1+��iui�xi�
}�
But because uT �xo =∏n
i=1 pi�xoi , therefore 1+ �u�x =∏n
i=1�1 + ��iui�xi�, which proves the result in Part (1).Each step of this proof is reversible, and so the con-verse of Part (1) is true, which establishes Part (2) of thetheorem.For Part (3) of the theorem, a reliability-structured target-
oriented preference function with parallel targets has theform 1−uT �x=
∏ni=1�1−pi�xi�. Solving (A-2) and (A-3)
for uT �x and ui�xi, respectively, and substituting into thisequation yields
1− uT �xo− �uT �x∗− uT �xo�u�x
=n∏i=1*1−pi�xoi − �pi�x∗i −pi�xoi �ui�xi+�
Define � = −�uT �x∗ − uT �xo�/�1 − uT �xo� and ��i =−�pi�x∗i − pi�xoi �/�1− pi�xoi �. By the conditions of thetheorem, pi�x
oi < 1, and therefore 1− uT �xo=
∏ni=1�1−
pi�xoi � > 0. Because uT �x
∗ > uT �xo, pi�xoi < 1, andpi�x
∗i > pi�x
oi , then −1 < � < 0 and 0 < �i. Substitute
into the equation above to yield
�1− uT �xo��1+�u�x�
={ n∏i=1�1−pi�xoi �
}{ n∏i=1�1+��iui�xi�
}�
However, because 1−uT �xo=∏ni=1�1−pi�xoi �, therefore
1+ �u�x =∏ni=1�1+ ��iui�xi�, which proves the result
in Part (3). Each step of this proof is reversible, and so theconverse of Part (3) is true, which establishes Part (4) ofthe theorem. �
AcknowledgmentsThe reviewers and editors provided helpful comments onboth the content and presentation of this paper, which
has been substantially improved as a result. The authorsparticularly thank the area editor, James S. Dyer, for point-ing out connections between their work and goal pro-gramming, and especially for the ideas underlying theformulation in §8.4.
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