Competitiveness And Compensation In Decision Making A Fuzzy Set Based Interpretation

COMl’ETITIVENESS AND COMPENSATION IN DECISION MAKING: A FUZZY SET BASED INTERPRETATION RONALD R. YAGERt Iona College, New Rochelk, NY 10801,U.S.A. pllrport_Many problemsof vital concern to decision makers involve multipk criteria decision making. A problem of considerable interest in multiple-objectivedecision making involves the formulation of an overall decisionfunction from the variousindividualobjectives.The purposeof this artick is to bring togetherin a general framework two different approachesto this formubtion problem.The first approachis that suggestedby Zeleny[2]. The second approach is based upon the idea of fuzzy subsetswhich was developedby Zadeh[7.8]. An uptodate summaryof the state of the art of fuzzy set tbcory can be found in Gupta, Ragade, andYager, Advances in Fuzzy Set Theory and Applications, publisbedby North-Holland.

seopcd

Abatraet-The multiple objective decisionprobkm is investigated-par&ularly the questionof the form of the decisionfunction. Different modelsare investigated.The relationshipof competitivenessand compensation to the form of the models is studied. DitTering methods of including distinctions bchveen objectives are discussed. INTRODUCTION

In this paper we are interested in investigating some aspects of the multiple objective decision problem. We first discuss a framework in which to investigate the situation. In particular we are interested in analyzing two polar concepts. Competitive and non-competitive decisions. Compensatory and non-compensatory models. We define and discuss these ideas. We next investigate these ideas using fuzzy sets as a methodology for building decision models. We then discuss a vector space approach to multi-objective decision making. We are particularly interested in Zeleny’s method of the displaced ideal [t-3]. We then relate these vector models to the fuzzy set models. We discuss possible interpretations of the various vector models. We then investigate the situation when the objectives are not equivalent. We see how to incorporate this information in both the competitive and non-competitive models. Finally, we make some comments on the idea of tradeoff in muhiobjective decision models and relate these to the models discussed. A FRAMEWORK

FOR

MULTI-OBJECTIVE

DECISIONS

The multiple objective decision problem can be considered in the following framework from which certain interesting observations can be made. One has a set X of alternative solutions and a set of criteria which the decision maker desires to satisfy. Furthermore, one has some evaluation which measures the degree of satisfaction each alternative gives to each objective. That is, if X = {x,, x2,. . . ,x.} are the is a measure of the satisfaction which alternatives and A,, AZ,. . . A, are the criteria then, Ai alternative Xi gives to objective Ai. There has been considerable work done on the question of how one obtains these evaluations[M]. We shall not concern ourselves in this paper with this problem, but assume that this information is available in the form numbers in the unit interval. The larger the number the more satisfaction. As Zeleny states in [ I] we are faced with a decision problem only if the same alternative is not best for all the objectives. Under this dissonant condition, where there is no one clear cut best alternative, some means must be chosen to select an al&native. In general, the procedures used involved the combining of the multiple objectives to form an overall decision function D, a selection is then made by choosing the alternative which best satisfies this decision function. A number of different techniques or forms have been suggested for combining multiple objectives [6,12,25]. We shall be interested in investigating some of these forms and seeing tRonald R Yager received his BEE from the City Colkge of New York and his PhD. from the Po&ta%ic I&it& of Brooklyn in Systems Science. He is an associate professor of Management science at the Gmduate School of Business of Iona Colkge. Fuzzy sets, multiple criteria decision making and art&ial intelligence are his cummt subjects of principal interest. 285

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what implied relationships between the objectives are necessary to either justify or interpret that form. The crucial question is: how should the decision function be formed from the individual multiple objectives? The manner in which the individual objectives are combined implies some relationships between them. That is, given a set of objectives, if we have some additional information relating them it can help decide how we connect the objectives to form an overall decision function. Usually the information available for relating the objectives is soft. That is, there is some vagueness in the specification of the relationships. The objectives are usually related via some concept of tradeoff between the objectives. We shall not, in this paper, be especially interested in the question of tradeoffs, but be more interested in certain more general relationships between multi-objectives. We shall be interested in two polar concepts which relate objectives. The first pair is compensation-non-compensation. The second is competitive-non-competitive. The concepts of compensation-non-compensation involves the question of whether, for a given alternative a low value in one objective can be offset by a high value in a second objective[4,11,12,26]. In its most general sense this concerns itself with the question described as follows: Assume x and y are two alternatives and A and B are objectives, assume a
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under the threat of otherwise obtaining no solution. In this approach a solution is very dependent upon what each alternative gets. In this case there is a uniqueness of objectives. We shall first use the medium of fuzzy sets [7-l I] to help analyze these concepts as applied to multi-objective decision making. This theory will give us some insights into the quantification of the above ideas. A FUZZY

SET THEORETIC

INTERPRETATION

Assume X is a set of elements, a fuzzy subset A of X is a subset of X which has a characteristic function which takes values in the unit interval, A:

X+[O, 11.

If x E X then A(x) can be interpreted as the degree to which x satisfies the condition specified by A. An alternative interpretation of A(x) is the truthfulness of the statement x is A. If X is a set of alternatives and A is some objective in a decision then we can express A as a fuzzy subset of X. In this representation A(x) would be a measure of the satisfaction which alternative x gives to objective A. Furthermore, if we had a set of q objectives we could represent each by a fuzzy subset of X. This would give us q fuzzy sets A,, A*, . . . A,. We shall investigate various ways of combining these objectives to form a decision function D. In particular, we shall investigate the implications of the polar concepts competitivelnon-competitive (utilitarian) and compensatory/non-compensatory on the formulation of D. When the objectives come together in a competitive manner, they all want to be satisfied, thus any measure of satisfaction, which is what the overall decision function 1) is, must reflect this fact. In particular for any x E X, D should measure the degree to which x satisfies Al and A2 and A3.. . and A,,. Thus the “and” operation for combining fuzzy sets is consistent with the concept of competitiveness [8-10,131. A good alternative satisfies all the objectives. On the other hand, if the objectives come together in a non-competitive or utilitarian manner than the decision functions can choose which objectives to satisfy in order to optimize its satisfaction. That is, the interchangeability implied by this approach indicates that a good decision function should satisfy A, or A, or A3 or . . . A,. The distinction between the competitive and the utilitarian joining of the objectives lies in the use of the connectives “and” and “or” for the joining of the objectives. The definitions of the “and” and “or” connectives for fuzzy sets have been intensively studied. These studies have been both theoretical [I I, 14,15] and experimental [I6-181. In Bellman and Zadeh[ I I] and Bellman and Gertz[ 14) they show that the only way to define “and” on fuzzy sets so that the condition of non-compensation is satisfied is via the min operation. That is, for any x E X D(x) = Min [A,(x), A,(x), . . . . . . , A,(x)]. We shall denote D when using this method of connection as: D = A, rl A2 f-l A3. . . . n A,. An alternative derivation of this form of connection has been discussed by Yager[lO]. In[lO] it is suggested that when the objectives come together in the competitive situation they are bargaining with each other for the selection of the alternative, a situation which can be described in a game theoretic formulation. In this formulation we have q players, the q objectives. Each objective has n strategies, the selection of an alternative. The payoff matrix consists of q-tuples, each element corresponding to the satisfaction of an objective. Furthermore, this matrix has the property that the q-tuples have zero value unless all the objectives pick the same solution. That is, the objectives must all agree on one solution for anybody to get any satisfaction. The only non-zero payoff values lie on the “main diagonal” of the payoff matrix. It is known that one possible solution to this game is the minimax solution. It is shown in [ IO] that this minimax solution corresponds to the non-compensatory “and” described above.

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A second possible solution to this bargaining problem is the Nash[19] solution, where D(x) = [A,(x). A,(x). A,(x). . . . . . AP(x and the alternative selected is the x which maximizes D. In Bellman and Zadeh[ 1l] they have shown that this multiplicate joining of fuzzy sets also satisfies the properties of “anding”. However, this formulation has the property of partial compensation. Thus we shall denote the multiplicate form as the partially compensatory “and” and denote it as: D=A,.A2.

. . . . . . . .A,

An alternative “and” is obtained by using a form related to the bounded sum[22], where we define A(x)*B(x) = Max [0, A(x) + B(x) - 11, we denote “and” as AyAf . . . . . . . . . . . *A, Having chosen definitions for “and” the corresponding definitions for “or” naturally follow. A particularly interesting derivation has been suggested by Oden[17] and Goguen[20]. If A and B are two fuzzy-sets of X then DeMorgans Law requires (A(x) and B(x)) = -A(x) or B(x), where A(x) is not A defined as I- A(x). If we assume without loss of generality that A(x) 2 B(x), then for the non-compensatory “and” we get A(x) and B(x)-- = Min [A(x), B(x)] = B(x) and not (A(x) and B(x)) = 1 -B(x). Since (A(x) and B(x)) = A(x) or B(x), then A(x) orB(x) = 1 -B(x). Furthermore, since 1 -B(x) 2 A(x) this implies that “or” is the max operation. Thus for the non-compensatory utilitarian model we get AJx)] denoted D(x) = Al U A2 U A3 . . . . . . A,. Similarily for D(x) = MaxMI(X), &(x), compensatory case we get using the second of DeMorgans Laws A(x) or B(x) = A(x) and-) -

-

A(x) or B(x) = A(x) and B(x) = (I- A(x)). (I- B(x)) = 1 - B(x) - A(x) + A(x)B(x),

A(x) or B(x) = A(x) + B(x) - A(x) *B(x). Thus in the case compensatory non-competitive we get D = A, . A2.. . . . A, Since A,(x). A,(x) = A,(x) + A2(x) - A,(x). A,(x) then in general

D(x)= ii=l A(x)-

2 2 Ai(x)*Aj(x)+ K$+, j$+i 2 Ai(x)Aj(x)Adx) + * * *

j=i+l ill

For the other compensatory case A(x) and B(x) = Max IO,A(x) + B(x) - 11 then not (A(x) and B(x)) = 1 - Max [0, A(x) + B(x) - l]

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from this we see that A(x) or B(x) = Min [I, A(x) + B(x)]. Since not A(x) or not B(x) = Min [l, 1 -A(x) + IB(x)] = Min [l, 2 - A(x) - B(x)]. This is easily seen to be not (A(x) and B(x)). Thus we have developed, using the medium of fuzzy sets, a typology for decision functions. A VECTOR SPACE APPROACH

TO DECISION

MAKING

In this section we shall describe a vector space approach to solving multiple objective decision functions based upon the method of the displaced ideal suggested by Zeleny [ l-31. Assume, as before, we have a set of alternatives X = {x1, x2.. . . xn} and a set of q objectives, each expressed as a fuzzy subset over X. That is, for each objective we have a membership function Ai which associates with every x E X a number is the unit interval indicating how well x satisfies objective Ai. Let us reformulate this information. That is, instead of looking at the information in terms of objectives, we shall concentrate on alternatives. In particular, we can associate with each alternative a vector, whose elements are its satisfactions to the various objectives. That is, if Vi is the vector associated with the ith alternative then

The jth element of vector Vi is the grade of membership of xi in the jth fuzzy set, if we denoted this Vii then OS V, S 1. Vi can be called the satisfaction vector for ith alternative. If there exists some alternative xk such that Vki2 V, for ah j and ah i then xk is the optimal alternative. In general, this condition is not satisfied. Zeleny[l-31 has proposed a methodology for finding the best alternative in cases when this condition is not satisfied. This method is called the method of the displaced ideal. Using this method one defines the ideal alternative as the alternative which has one in all the components of its vector V. That is, the vector I, which consists of all ones would be the decision makers preference because it satisfies each objective perfectly. One could then use as a criteria for selecting the best among the available alternatives the one which has a satisfaction vector nearest to I. In this methodology he has suggested using the LP metric to measure the closeness of an alternative to the ideal. For each alternative he has suggested calculating

Then calculating i* such that Lp(i*) = Min Lp(xi)

xi E X

This he calls the compromise alternative with respect to p. It should be noted that i* depends upon p. In his papers, Zeleny emphasis the fact that “no specific choice of p is made or recommended.” An approach related to the method of displaced ideal can also be described. Let 0 be the alternative with zero satisfaction to each object. Then one could consider as a criteria for selecting the best available alternative the selection of the alternative which is furthest from 0. That is, we aspire to get as far away from the worst condition as possible. In this case we can also use the Mp metric to measure the distance from 0. We let Mp(i)=($(K,)p)l’P CAOR Vol. 7. No. 4-E

p=l,2,...

R.R.YAGER

290 measure the distance from the 0 solution. Then we calculate x* s.t.

Mp(x*) = max A@(x)

x E X.

This would be the solution furthest from the 0 alternative with respect to P. In the following we shall relate the vector approach to the fuzzy set approach discussed in the previous section. In particular, we shall be interested in seeing the relationship between the two vector approaches and the value of p to the ideas of compensation and competitiveness. COMPARISONOFFUZZYANDVECTORMETHODS We shall first look in detail at the solution using the fuzzy set approach for the noncompensatory competitive model. If we have q objectives A,, A*, A3,. . . A,, D=A, II A2 rl A3... fl A, and D(Xi) = min [AI( A&Xi) . . . Aq(xi)]. D is a fuzzy subset where for each Xi E X, D(Xi) is the value of the least satisfaction xi gives to any objective. Finally, the optimal alternative xOPis selected s.t.

D(xop)= FM$D(xi)t That is, xOPis the minimax solution. That is, the procedure consists of selecting for each alternative the min satisfaction over all the objectives, this gives us D(x), than taking the maximum member of D. Theorem 1. The optimal solution to the competitive non-compensatory fuzzy model is the alternative which is closest to the ideal alternative when using p = 03(the sup metric in R”). Proof: Consider some arbitrary alternative xi with its associated satisfaction vector

Then the distance from I,$ to I is Q(i) = d( V, I) =

(,z(1- Vij/‘)“‘s

By definition L,(i) = Max (1- Viji(. icjrq

Since the optimal solution is the i* s.t. L,(i*) = lim L,(i) and L,(i) = m;x (1 - V& = 1 - min {vi,} i then L,(i*) = rnp {I - min ) Vijll = max min (Vjl, i i

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which is the exact same solution as in the competitive non~om~nsato~ case. We now look at the non-competitive non-compensatory case. In this case we have D=A, U A2 U A3... U A, implying that for each alternative x E X, D(x) = Max [A,(x), A&c), A,(x)], for each x, D(x) is the value of the most satisfaction x gives to any objective. Finally, in this case, x,, is then again selected as D(x,,,) = Max D(x). XEX

In essence, we select i* s.t. Vi*j= Max V;, over al1i and J Theorem 2. The optimal solution to the noncompetitive non~ompensatory case is the alternative which is furthest from the 0 (zero) alternative using p = CQ. Proof. Recalling

MM) = d(

Vi9

a) =

[)J( VijY]"'

As in the previous case M,(i) = max 1Viii. The optimal solution is the i* s.t. I

Mp(i*) = Max M,(i).

This procedure selects the alternative which has the highest satisfaction to any objective. Thus we see that in both cases when p = m we get a non-compensatory type solution. However, the method of the displaced ideal selects the competitive optimal while the other method selects the non-competitive optimal. In the case when p = 1, we note that the method of the displaced ideal works as follows:

If we let K = 2 V+ the sum of the satisfactions given by alternative xi, then j=l

L,(i) = 1- 6.

The optimal solution in the case is Xi*s.t. L,(i*) =

Min (1 - F).

This implies L,(i*) = Max 77. That is, xi* is the alternative which has the greatest sum of I satisfactions. Let us ROWlook at the compensatory competitive models in fuzzy sets. In this case D=ATA$... where Al(x)*A&)

*%

= Max (0, A(x) + B(x) - 1) Then xoPis selected such that

We shall first prove a lemma Lemma: AI(X) * A*(X) * A&X) * * * * * A&)

I= Max 10,( E i=l

Ai

-(n - l))].

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R. R. YM.%R

Proof: We shall use induction. The result is evident for n = X and n = 2. Assume that it is true for some fixed n an&consider n + 1. Let

There are twocases. First if

then

and

Thus the result holds in this CEXLOn the other hand if

either

or

which concludes the proof. Based upon the preceding lemma tha following theorem follows by inspection. TBeama 3..~~~~~~

This theorem then implies that %LII optimal sofution in this compensatory competitivit)model wiil

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be the alternative with the greatest sum of membership just as in the displaced ideal case when D= 1. In the case where p = 1 the method of furthest distance from the 0 vector works as follows: M (i) = 4

vi,@)=

aA&Xi)

and the optimal alternative is the alternative which, maximizes M,(i). In this case, it is the alternative with the largest total satisfaction. Let us look at the compensatory non-competitive models in fuzzy sets. In this case

where A(x) 0 B(x) = Min [l, A(x) + B(x)],

the optimal solution is selected as x* p.t. 0(x*) = pz; D(x)

We can prove the following theorem Theorem 4. Assume x and y are such that Ai( if D = Al 0 A2 0 ***0 2 Ai(Yl>,$ ‘w

i=l ~f~~~.

A, then D(y) > D(X).

Let us define

E=A,*Al*As**+*

A,

and ,?!?=~,*&*.&*~~~A,,

where A=(l-A).

Recalling the results of three we show that if A<(y) then E(X) =ZE(y), S:&‘x)>~ i=l

i=1

Therefore if

we note that if

2 (I-

i=l

43)) > 2 (1-MY))

than &9

2 @Y).

However, since our hypothesis is that

then this is true. Thus g(x) z g(y). Recallingthe definition of the 0 which implies that 0 and * satisfy DeMorgans law that is,

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This implies that g(y) = 1 -D(y)

and

B(x) = 1 -D(x)

therefore D(y) = 1 - J??(Y) and

D(x) = 1 - E(x).

This implies that if ,??(x)~B(y) then D(y)rD(x). This theorem then implies that optimal solutions to the compensatory non-competitive model are also the solution or the vector model based on the furthest distance from the origin. Let us look at the other compensatory-type competitive model.

In this case D(x) = A,(x) * A2 - (x) * A,(x) * - . A&) if we take the log of D(x) we get In D(x) = In A,(x) + In A*(X) + - * * In A,(x). Since In is a monotonic transformation In D(x) and D(x) have the maximum at the same value. Nothing that the expansion (23) of In A(x) about the point A(x) = 1 is

lnA(x)=fF j=l

(A(x)-1).=(A(x)-1)-~(A(x)-1)‘+~(A(x)-1)’+

and since 0 s A(x) I 1 we get

Thus In D(X)=-

2 (1 -A,(x)+; 1

2 (1-&x),Z+~

2 (1-Ai(x))3+*

*’

In D(x) = - P$ 2 f (1- Ai(X)Y’* Thus this model involves a summation of terms similar to those found in the L,, metric used in the method of the displaced ideal. Thus we see that the multiplication “and” is in a sense a weighted average of the Lp metric “ands.” ANINTERPRETATIONOFTHEVECTORMODELS

We would like to use the ,results of the previous section to help give some possible interpretation to the vector models. The first vector model, Zeleny’s displaced ideal model, measures for each alternative its distance from the ideal solution by L,(i) = [,$ (1 - Aj(Xi))‘]“’

p = 1, 2, 3,. . .

The best solution for a given p is that Xi which minimizes L,(i). The second vector model, measures for each alternative its distance from the anti-ideal Cpas M,(i) = [A (Ai@)Y]“‘, i=I

JJ= 1,2,3,.

. . .

Competitiveness and compensation in decision making It then Xi E

selects

as the best solution

295

for a given p the Xi which maximizes M,(i) over all

X.

Two questions arise. The difference between the two models and the meaning of p. As noted before, different p’s will give different optimal solutions. In the previous section we proved four theorems. L,et us summarize these theorems which relate fuzzy set based models to the vector based models. Theorem 1 said that the solution to the non-compensatory competitive model is the same as the solution to the L, model. ‘Z’korem 2 said that the solution to the non-compensatoryutilitarianmodel is the solution to the

M, model. Theorem 3 said that the solution to the compensatory competitive model is the same as the solution to the L, model. Theorem 4 said that the solution to the compensatory utilitarian model is the same as the solution to the M, model. Based upon these theorems we conjecture the following understanding of the vector models. Fist, we note that the competitive solutions apply to the L type model. Whereas the non-competitive solutions apply to the M type model, Thus, we suggest that if the objectives are strung together via “and,” that is, they are truly competitive then we use the L type model. However, if the objectives are strung together via an “or” operation, that is, if the objectives are not competitive but interchangeable, we use the M type model. Next we note that p = 0~corresponded to the cases when there was no compensation between the objectives. Whereas p = 1 corresponded to the cases where there was absolute compensation. Thus, we Suggest that p is inversely related to the degree of compensation between the objectives. That is, if we define r E (0,l) as the degree of compensation between the objectives, then p = (l/r). In order to reinforce this interpretation of compensation let us consider MJx) = [,E, A!(x)]“~. The idea of compensation involves the ability of a small value for A,(x) to ‘S counteract the effects of a big Al(x). By considering the partial derivative of Mp with respect to A’s we can get some understanding of the effect.

aMp aAj(x) =

aM,, aAk(x)

(Add >p-’. Ai

Thus as p increases, decreasing the compensation, this goes to zero and diminishes the ability of i to compensate for K. NON

EQUIVALENT

OBJECTIVES

Our discussion so far has implicitly assumed that each of the objectives are equivalent in their effects on the overall decision function. This is not usually the case in most situations met in practice. There is usually some distinction between the objectives with respect to their effects on the decision function. In the vector type models the usual approach has been to associate some weight to each objective and then multiply the components in the metric by there appropriate weights. This is what is done by Zeleny[l-31. In Zeleny[l], however, mentions the possibility of incorporating the weights in a number of different manners. In Yager [ 10,241 he has suggested an approach based upon fuzzy sets for incorporating these differences in the competitive case. In this section we shall extend this approach to include the non-competitive case and the vector type models. As a prelimiary we shall introduce another operation on fuzzy sets, the raising of a set to a power. Assume A is a fuzzy subset of X, with grade of membership A(x) for each x E X. Let a be a non-negative real number. The fuzzy subset A” is defined as a fuzzy subset of X in which the membership grade for each x E X is [A(x)

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That is, each grade of membership is raised to the a power. When A(x) E (0,l) if a > 1, A”(x) A(x). If A(x) E (0, l} raising to a power has no effect. Zadeh[22] has suggested the appropriateness of using this type of operation for linguistic hedges, for example he suggests a = 2 corresponds to “very.” It should be noted that the requirements of being able to do this type of operation on fuzzy sets implies that Zadeh’s fuzzy sets have more structure than the lattices discussed by Goguen[20]. In order to see how to incorporate the effects of non-equivalence of objectives, we must distinguish between its effects in the competitive and non-competitive models, in that the non-equivalence of objectives have to be interpreted differently in these cases. Recalling, that in the competitive case each of the objectives is solely concerned with its own satisfaction and a solution is obtained after a “struggle” between the objectives to get their preferred alternative accepted. In this case, it appears that the most natural way for any distinction between objectives to enter the model is via some measure of there individual “strengths” in the struggle for selection of the accepted alternative. A common name used for this distinction is the importance of the objective. Therefore, we shall mean by the importance of an objective the strength of the objective in the competitive type struggle for alternative selection. In the non-competitive or utilitarian case the situation is different, in that the overall satisfaction is all that counts. Thus, the objectives are not competing but are conduits through which the overall decision function can obtain and measure satisfaction, hence the overall satisfaction is a result of the satisfactions to the various objectives. In this case a natural way for differences between alternatives to enter the model is via some measure for the overall decision function preference for the type of satisfaction one objective gives rather than the other. Something along the lines of a units of satisfaction to objective A is equivalent to b units of satisfaction to object B, that is the structure has a certain bias towards various objectives. We shall use the term contribution equivalence to measure this concept. In the sense, that the more preferred the objective, the less its contribution equivalence. If objective A is more preferred than objective B then a lower satisfaction to A is an equivalent contribution to the decision function than some higher value of B. If we use the idea of preference to denote the distinction between objectives then contribution equivalence is inversely related to preference. It should be noted that our definitions of preference and importance are very distinct and non-interchangeable concepts. For there is no struggle involved in the utilitarian case to measure the importance. While in the competitiveness case no objective is willing to forego its sovereignty to some higher power to impose its preferences. In the competitive case one should expect that as the importance (the strength in the struggle) of an objective increases the influence of that objective in determining the optimal alternative increases. In Yager [lo, 241he has suggested a model for including importances in the competitive case. This model is based upon the fuzzy set approach to multiple objective decision making. He suggests that we associate with each objective a number a indicative of its importance, then we use A? instead of Ai in the decision model. Thus for the non-compensatory “and” we get D = A?’

n A;’

rl AT'. . . A=q 4 *

Where ai > al if Ai is more important than Ah Furthermore, since as ai increases AT’decreases, then as ai increases the more likely for A:‘(x) to be the min [AT/(x)] and hence more influential in I

determining the decision. Using the multiplicative “and” we get

D(x)= fi &TX). Recalling that if K= fi Di then L-l

$= I

fi ai, i-l i#j

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then (&3&Q is largest when uj is smallest. This implies that the smallest term in the product is the most significant in determining the product. Since as cujincreases A?(x) decreases the more important an objective becomes the more it effects the decision function. An alternative interpretation of importance in the multiplicative case can be also seen. Assume we have equal objectives. Then as before

If we interpret the idea of more important as being more powerful in the struggle for alternative selection then an increase in importance can be expressed as an increase in the number of allies or “clones” in the struggle, as an objective becomes more powerful it gets help in the struggle. In particular, this help is in the form of clones of itself. Thus D(x) = A,(x)A,(n).

. . A,(x). AZ(x). . . A,(x) 1* . A4(x). **A,(x) ‘ni R2 %

where ni is the number of clones. The more powerful the Ai the greater the n, Therefore D(x) = AT’(x) . A;*(x). . . A;$x). So that nj increases as an objective becomes more important. Let us now investigate the noncompetitive or utili~ri~ case. In this case as the p~~e~~ce for the satisfaction from an objective increases then alternatives which are high in that objective should be more likely to be selected. We suggest the following procedure for the introduction of preference information into the non-competitive decision function. Associate with each objective a preference value pi, a non-negative number. The more preferred, the higher the pi. Then the cont~bution equivalent is Ci = (I/p&. Thus the larger the pi the smaller the Ci. For each objective calculate A?, that is, raise each objective to the reciprocal of its preference. Then use these new fuzzy sets in non-competitive model. Thus and

D(x)=Af’

U A2” U Af?..

U A$

A(x) = Max [A?(x)].

We noted that as pi increases then A?(x) increases and therefore, the satisfaction of n to the ith alternative increases. Furthermore, consider two objectives A and B. As suggested previously the concept of preference in utilitarian decision making implies that if A is preferred to B then less satisfaction in A is equivalent to the higher equivalent of B. In order to unders~nd this let us consider a two objective case A and I?. Assume pa >pb this implies Ca < Cb and D = Ace U BCb For tangibility assume A is the criteria a cheap car and B a luxurious car. The fact that Pa > Pb implies that the decision function gets more satisfaction out of a cheap car than a luxurious car. Assume in particular that p. = 2pb = l/2, cb = 2 and Ca = l/2, then

Using Zadeh’s concept of linguistic hedges then, D = moderately cheap car or a very luxurious car. If x is an alternative, A”2(~) is the degree that x satisfies the condition of being a moderately ’ cheap car and A(x) is the degree x satisfies the condition of being a cheap car and Aln(x) L A(x). B*(x) is the degree to which x satisfies the co~ition of being a very htxurious car and B(x) is the membership for a luxurious car and B*(x) 5 B(x). We see that the more preferred objectives have their membership grades increased and thereby making them more infhrential in the formulation of D. This linguistic interpretation brings home the idea that if Pa > Pb then less A is needed to satisfy D than B. In the multiplicative case, for mathematical simphcity we shah look at the two objective

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case. D=A.B D(x) = A(x) + B(x) - A(x) *B(x). If we introduce preferences C, < C,, then D(x) = A’“(x) + Bcb(x) - A’“(x) *Bcb(x). If we let AC”(x)= a and Bcb(x)= b we get. D= a t b-ab. Furthermore (dD/&z)= 1-b and (dD/ab) = 1 - a, are the rate of change of D(x) with respect to A and B respectively. We notice that as A becomes more preferred pa increases, Cu decreases, and a increases. As a increases, 1 -a decreases and, therefore, the rate of change of D with respect to a increases over that with respect to b. Let us now investigate the vector type models with inclusion of these operations for non-equality. In the competitive case we have

for the case of equivalent objectives. If we consider the non-equivalent objective case and use the methodology just developed we get L,(x) = Ii!, [l - Api(~))p]“~ where ai is the importance associated with the ith objective. It is suggested in[lO] that the average of the ai be 1. Denoting A?(X) as Ui,we get L(p) = (~$,>p= [ $ (1 - ai)‘] and g(P, I

= -p(l - ai)‘-‘.

First we note that (aL@)/aUi) is always non-positive. That is, as Ui increases L(p) decreases. This is as it should be, since recalling in this case the best alternative is the one with the smallest Lp’ Thus let US concern ourselves with I(aL(p)/aai)l. AS ai increases Ui decreases and (I - Ui) increases, then I(aL(p)/aai)l increases making the ith term more significant in the overall function. That is as ai increases the term becomes more influential in the determination of L(P). In the cooperative case

is the decision function for non-equivalent objective situation and

As objective i becomes preferred, Ci decreases, causing AC+’ and hence (JM(p)/aCi) to increase and therefore making the ith objective more influential in the decision model. TRADEOFFS

IN MULTIOBJECTIVE

MODELS

In many instances the relationship between the objectives in multiple objective decision making is not purely competitive or cooperative, but some combination. Many decision situations involve a set of criteria or objective functions plus some information on the decision makers willingness to make tradeoffs betweeen these objectives. For example, if the decision maker has four objectives A,, AZ, As, A4 he desires a solution which satisfies these four objectives. However, he may be willing to make tradeoffs between the objectives in the sense that if some alternative is very Al then it could be moderately AZ. Using the ideas of linguistic hedges as discussed in[ll, 221 we can incorporate these types of tradeoffs into a decision

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model. That is, as suggested in Zadeh [ 11,221 and Lakoff (271linguistic hedges such as very, sort of, moderately, etc. can be associated with some real or fuzzy positive number a such that for example, “very” A = A”, for the appropriate value of a. Therefore, assume in the context of a decision the decision maker wants an x (alternative) that is A, and A2 and A3 and A, or if x is “very A,” then it can be “moderately AZ” or if it is “very, very A3” then it can be sort of A,, etc. Using the above ideas we can express this type of decision as follows: Let T,, = A, fl A2 fl A3 f~ Al. Let each tradeoff be donoted as T . Then D = T,, or T, or T2 or T3. . . . or T.. That is, in essence the tradeoffs are connected to each other via an “or” operation as indicated by the statement of the decision maker. Furthermore we can use linguistic hedges to represent the individual tradeoffs. That is,

Where &ii is the power associated with hedge associated with the jth objective in the ith tradeoff. Using this approach we can formulate very complicated models using the fuzzy set structure. The difficulty associated with applying this method to decisions centers around the problem of obtaining the (I’, associated with various hedges. The current state of knowledge has not yet developed a relationship between hedges and powers. It may be possible to find these experimentally. Nevertheless, the structure suggested by this approach allows one to see the connection between the concepts of importance, preference and tradeoffs. In particular, if we approximate D as D, = AT1 fl Af2 rl A;3 fl A,“‘, this is the model we used as the competitive model with a signifying the importance. Thus we see that these importances are in essence a reflection of the tradeoffs the decision maker is willing to make. In Yager[24] we discusses a method for obtaining these values via a pairwise comparison of the objectives. A second possible approximation to D may be Q = Afl U A? U A? U A?, in this case the C’s are associated with the contribution equivalence. Again these may be a reflection of the tradeoffs. D2 would be a good approximation in the case where each tradeoff, T, T = A?’ fl A92 n A?3 fl A>’ is such that for j E (1, 2, 3, 4} aij+ (Yikfor all kf j. That is for example, if j = 1 then air % aiz, ai3, ai), which occurs when Ti = very Al and sort A2 and moderately A3 and sort of Al. For example: then since “and” is the min operation and high a’s reduce the grades of membership T = “very A,” = APit. Then if each Ti is of this form we would get Q as a good approximation. It is also possible to get approximation involving combinations at D, and Q type models. The idea of using these approximations is that it is easier for a decision maker to supply information with respect to his feelings for importance and preference than it is to obtain the linguistic hedge values. REFERENCES I. M. Zeleny, The theory of the displaced ideal. In Proc. of Multiple Criteria Decision Making, Kyoto, pp. 53-206. Springer-Verlag. New York (1976). 2. M. Zeleny, A conceptof compromisesolutionsand the methodof the displacedideal. Comput. & Ops. Res. 1,479-4% (1974). 3. M. Zeleny, The attribute dynamic attitude model. Mgmt.Sci. 23, (1),12-26 (1976). 4. R. L. Keeney and H. RaitTa,Lkcisions with Multiple Objectives. Wiley, New York (1976). 5. E. G. Petrov and V. G. Zotov. A selftheoreticalmethod for estimation of the efficiency of complex systems. Soviet Automatic Control 9.5745 (1976). 6. J. L. Cocharaneand M. Zeleny, Multipk Criteria Decision Making. The University of South Carolina Press, Columbia, s. c. (1973). 7. L. A. Zadeh. Outline of a new approachto the analysesof complex systemsand decisionprocesses.IEEE Trans. Syst. Man. Cvbem. 2.2844 (1973).

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8. R. E. Bellman and L. A. Zadeh, Decision making in a fuzzy environment. Mgmr. Sci. 17, 141-164(1970). 9. R. R. Yager and D. Basson, Decision making with fuzzy sets. Dee. Sci. 6,560-600 (1975). 10. R. R. Yager, Fuzzy decision making including unequal objectives. Fuzzy Sets and systek l(2) (1978). Il. R. E. Bellman and L. A. Zadeh, Local and fuzzy lo&s. In Modem Uses of Multiualued Logic (Ed. D. Epstein) D. Reidel Dordrecht (1976). 12. P. C. Fishbum, A survey multiattribute of multicriterion evaluation theories. In MultipIe Ctiteriu Problem Soloing (Ed. S. Zionts). DD. 181-224.Soringer-Verlag. New York (1978). 13. J. Rawls, A Theory of justice. Belknap Press Harvard University Cambridge, Mass (1971). 14. R. E. Bellman and M. Giertz, On the analytic formalism of the theory of fuzzy sets. Inform. Sci. 149-156(1973). IS. B. R. Gaines, Foundations of fuzzy reasoning. Int. 1. Man. Machine Studies 6,623-668 (1976). 16. H. M. Hersh and A. Caramazza, A fuzzy set approach to modifiers and vagueness in natural language. /. Experimental Psych. 105, 254-276(1976). 17. G. Oden, Integration of fuzzy logical information. i of Exp. Psychology 3,565-575 (1977). 18. H. J.Zimmerman,Resultsofempiricalstudiesinfuzzysetstheory. AppliedGeneralSystemsResearch,RecentDeuelopments and Trends (Ed. G. Klii), pp. 302-312.Plenum Press, New York (1977). 20. J. A. Goguen, The logic of inexact concepts. Synthese 19,325-373 (1%9). 21. G. Bachman and L. Narici, Functional Analysis. Academic Press, New York (1966). 22. L. A. Zadeh. A theory of approximate reasoning. Memo MN/58. Electronics Research Lab., University of California, Berkeley (1977). 23. W. L. Hart, Analytic Geometry and Calculus. D. C. Heath Boston (1957). 24. R. R. Yager, Multiple objective decision making using fuzzy sets. Int. /. of Man-Machine Studies 9,375-382 (1977). 25. S. Zionts, (Ed.), Mu/tip/e Criteria Problem Soloing. Springer-Verlag, New York (1978). 26. P. C. Fishburn, Noncompensatory preferences. Synthese 33,393403 (1976). 27. G. Lakoff, Hedges A study in meaning criteria and the logic of fuzzy concepts. J. Philo. bgic 4,458-508 (1973). I,

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