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On formalization of a general fuzzy mathematical problem
Fuzzy Sets and Systems 3 (1980) 311--321. © North-Holland Publishing Company
ON FORMALIZATION OF A GENERAL MATHEMATICAL PROBLEM
FUZZY
S. A. ORLOVSKY Computing Center of the U.S.S.R. Academy of Sciences, Vavilova, 40, Moscow, U.S.S.R. Received February 1978 Revised November 1978 The problem considered in this paper concerns decision-making on the basis of information presented in a fuzzy form. It differs from the usual mathematical programming problems of maximizing a given function over a given set of alternatives mainly in that the values of such function are fuzzy and are represented by fuzzy sets. The question is how to choose alternatives from the given set which give in some sense "the best" fuzzy value of this function. The fuzzy-valued function in question may be thought of as a fuzzy utility function specifying fuzzy utility estimates to the alternatives, or as a performance function of a system with reaction to choices of alternatives (controls) allowing only fuzzy description. The values of this function have the form of fuzzy subsets of some universal set of estimates (or reactions) and a binary preference relation (generally fuzzy) is assumed to be specified in this set. In the paper this relation is extended from the elements of the universal set onto fuzzy subsets of this set. Some properties of this induced relation are studied. This relation is used to extract a fuzzy subset of nondominated alternatives and maximal elements of this subset are suggested as rational choices. It is shown that under some assumptions alternatives exist which are in fact unfuzzily nondominated thus serving as unfuzzy solutions to a fuzzily formulated problem.
Key Words: Fuzzy, Utility function, Preference relation, Nondominated alternatives.
1. ln~oduction Decision-making is probably the most important and popular aspect of application of mathematical methods in various fields of human activity. In real-world situations decisions are nearly always made on the basis of information which is at least in part of fuzzy nature. In some cases fuzzy information is used as an approximation of probably more rigorous information available and in this form appears to be quite convenient and sufficient for making good enough decisions. In other cases fuzzy information is the only form of information at hand. Th~ first step in mathematically attacking a practical decision-making problem consists in formulating a suitable mathematical model of a system or a situation analysed. And if we intend to make reasonably adequate mathematical models of such situations to help practicians in searching for rational decisions, we should be able to introduce fuzziness into our models and to suggest means of processing fuzzy information. 311
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S.A. Orlovsky
The first and fruitful momentum towards mathematical formalization of fuzziness pioneered by L. A. Zadeh is in its progress now, when numerous attempts are being made to explore the ability of the fuzzy sets theory to become a useful tool for adequate mathematical analysis of real-world situations. This paper should be considered as just one of *..ese attempts. We analyse in it a problem of making decisions when fuzzy information available is presented in the form of a fuzzy-valued utility function (performance function). But before going into details we should like to outline a number of related to each other classes of mathematical models of decision-making situations. Two elements can be extracted from a description of a decision-making problem (DM problem). The first is a set of feasible choices (decisions, alternatives) which may be described either in a fuzzy or in an unfuzzy form. The second element is information (fuzzy or unfuzzy) available about preferences between alternatives. The particular form of a DM problem depends largely on the forn~t in which this infl)rmation is presented. In the general case this information has the form of a fuzzy preference relation specified in a given set of alternatives by means of its membership function. This kind of a DM problem which we shall refer to as P1 has been considered in Orlovsky [5]. In some cases preferences between alternatives may be described by a so called utility function. This function maps a given set of alternatives into a set of estimates of alternatives and a preference relation is specified in this last set. This function therefore allows to compare alternatives with each other by their estimates. If the estimates are numbers, then the DM problem is referred to as mathematical programming problem. In the fuzzy case utility functions may have various forms. The most general form has the function which maps the set of alternatiw;s into a class of fuzzy subsets of a set of estimates. In other words for every air, :native this function specifies a fuzzy estimate in the form of a fuzzy set of estimates. As an illustration to this a utility function may be thought of as a performance function of a system under control. Fuzzy values of this function are then fuzzily described reactions of the system to controls. If we want to rationally control the system we shall be able to compare fuzzy reactions with each other, to decide which of them are more satisfactory. Mathematically this problem involves the necessity of extending a generally fuzzy preference relation from elements of a universal set of reactions onto fuzzy subsets of this set. This kind of a DM problem which may be referred to as a general fuzzy mathematical programming problem (P2) is treated in this paper, and we show how to reduce it to the problem P1 and to use the results obtained in Orlovsky [5] to rationally choose alternatives. Probably more related to practical situations is another class of DM problems in which information about preferences between alternatives is supplied in the form of an unfuzzily specified utility function containing parameters the values of which are fuzzy. The set of alternatives may also be described with unfuzzily specified functions containing fuzzy parameters. This model seems to be quite typical for practical situations when the values of parameters are obtained from experts. But even in cases when these parameters are the results of measurements they are
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intervals and not just numbers. This type of a DM problem which we refer to as that of mathematical programming with fuzzy parameters can be reduced to the problem of the type P2 and then to P1, and we hope to show in the next paper how to treat it using the results obtained in this paper and in Orlovsky [5].
2. Formulation of the problem A general fuzzy mathematical programming problem (FMP) considered in this paper is formulated as follows. Two sets are specified: X - - t h e set of alternatives (feasible choices) and the universal set Y of estimates (utility values for alternatives from X). Let ~ be the class of all fuzzy sets (FS) in Y, that is the class of all functions u" Y---~[0, 1]. A fuzzy ,,':::",,,,.,,,yfunction (performance function)~" X---~ ~7 is specified, assigning a fuzzy estimate for every alternative x from X. We have already mentioned in Section 1 a possible interpretation of such function. The mapping • is more conveniently described with the function ¢ : X × Y---~ [0, 1], in which case for x°~ X fixed the function q~(x°, y)~ ff gives the fuzzy estimate (a membership function of the estimate) for the alternative x °. To complete the formulation of the general FMP problem we assume that a fuzzy nonstrict preference relation (FPR) /x" Y× Y---,[0, 1] is specified in the universal set of estimates Y. so that the value tx(z. y), z, y c Y is a degree to which the preference z ~ y is true. The question arises now how to rationally choose an alternative in the set X on the basis of information in the form of the function ¢ and the FPR tz?, or rather how might the rationality of choice be understood in this context? In this paper a notion of a rational choice is suggested which seems both natural and in agreement with what is understood as rationality in unfuzzy problems. Various types of FMP problems and approaches to their treatment have bccn analysed by Bellman and Zadeh [1], Negoita et al. [2, 3], S. Orlovsky [4] and other authors. In those papers either fuzzy goal sets were considered, or different approaches to the problem of minimization of a given unfuzzy function over a fuzzy set have been studied. As we shall see in the sequel the FMP problem with a fuzzy goal function considered here requires a different approach. In this case alternatives in the set X should be compared with each other on the basis of their fuzzy estimates given by the function ~0 and the F P R / x in the universal set Y. That necessitates a search for a preference relation in the set X which agrees with the function ¢ and the FPR /z. Having found this preference relation (which is generally fuzzy) one can choose alternatives in the set X using a fuzzy subset of nondominated alternatives of the set X, which have been introduced in Orlovsky [5].
3. Pre|immary definitions We thought it worthwhile to present here some definitions from Orlovsky [5] which shall be needed in the sequel.
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A fuzzy preference relation (FPR) in a set X is a fuzzy subset of the product set X x X with the membership function r l : X x X---~[0, 1]. It is assumed that a F P R is reflexive, i.e. rl(x, x ) = ] for any x from X. FPR 7/ is called weakly linear if ~l(x, y ) = 0 = = > r l ( y , x ) > 0 and strongly linear if for any xt, x2 from X at least one of the equations ri(xl, x2) = 1, Tl(x2, x 0 = 1 holds. For a given FPR rI two corresponding fuzzy relations may be introduced: (1) Fuzzy indifference (equivalence) relation rIe(x, y) = rain {~(x, y), rI(y, x)}
V x,y E X.
(2) Fuzzy strict preference relation rI~(x, y ) =
~Tl(x, y ) - r I ( y , x~ if ~(x, y)>~ rI(y, x), / 0 otherwise.
It is easily ascertained that a strongly linear F P R possesses the property riS(x, y ) = 1 - r i ( y , x)
for any x,y s X.
For a given set X with a F P R rl in it we introduce the subset of nondominated (ND) alternatives, having the membership function of the form rIND(X) = inf [1 -- ri~(y, x)]= 1 - s u p rl~(y, x) = 1 - s u p [ r I ( y , x ) - r I ( x , y)]
anyx~X.
yaX
The value rIND(x) is understood as a degree to which the alternative x is dominated by no one of the elements of the set X.
4. Extension of a tuzzy preference relation to classes of fuzzy sets Prior to analysing the general k-~P problem formulated in Section 2 we consider in this section an auxiliary problem of extending a given F P R from the elements of the set in which this FPR is defined onto fuzzy subsets of this set. Formally this problem may be stated as follows: if Ix: Y × Y---~[0, 1] is a F P R in a given set Y and ~ is a class of fuzzy subsets of Y, what is the form of a F P R induced by Ix in the class i f ? To obtain the desired F P R we shall reason in the following manner. If v ~ ~, then as it is easily understood the value ri(v, y) = sup min {v(z), ~z(z, y)} zEy
may be taken as a degree to which the FS v is preferred to the element y e Y. Similarly the value ri(y, v) = sup min {v(z), Ix(y, z)} z~Y
is a degree to which the reverse preference is true.
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N o w if v~ and Vz are any two FS's in Y, then the value ~(v~, v2) = s u p min {v~(y), sup min {v2(z), tt(y, z)}} ycY
z~Y
= sup min {vl(y), M2(Z), ~[£(y, ,~)} y,z~Y
is a degree of the preference v~ ~ v2. O n e can easily see that the degree of the reverse preference (v~ <~ v2) is given by the equation rl(v2, v l ) - sup min {v1(y), v2(z),/z(z, y)}. y,z~Y
T h e F F R rl obtained is defined in the class o~ and is induced by the F P R Ix specified in the universal set Y. Now we shall consider some usefull properties of the F P R ~). T H E O R E M 4.1. I f a FPR ix in Y is reJtexive, then the induced FPR rl is reflective in the class of all normal FS's in Y.
ProoL If v is normal and Ix is reflexive, then we have rl(v, v ) = sup min {v(y), v(z), Ix(y, z)} y,z~Y
>~sup rain {v(y), v(y)} = 1, y~Y
i.e. rl(v, v)= 1. T H E O R E M 4.2. If FPR t~ in Y is strongly linear, then the induced FPR r~ is strongly linear in the class of all normal FS's in Y. Proof. As ~t is assumed to be strongly linear it suffices to show that for any two normal FS's v~, v2 in Y at least one of the equations holds: T~(Vl, l J 2 ) = 1
or
lt](v2, v j ) = 1.
Assume the contrary, i.e. that for some two normal FS's v t and v2 the inequalities ~(v~, rE) = sup min{v~(z), v2(y), Ix(z, y)} = p~ < 1,
(4.1)
• l(vl, v2) = sup min{v~(z), rE(y), IX(y, z)} = P2 < I
(4.2)
~,z~Y ¥,z~Y
hold simultaneously. Now for any n u m b e r e from the interval 0 < e < m i n { ( l - p l ) , introduce two sets Y:=/Ylv,(Y)>~supv,(y)-e=l-e I.
~,~Y
1, J
( l - p 2 ) } we
i = l , 2.
A s / ~ is strongly linear then for any two elements y~ ~ Y~ and Y2 a y2 one of the
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S.A. Orlovsky
equations (a) Ix(y,, y2) = 1 or (b) IX(Y2, Y~)= 1 holds. In case (a) we obtain ~(vt, v2) = sup min{v~(z), rE(y), Ix(z, y)} y.z~Y
~min{vt(yl), vz(Y2), Ix(Y1, Y2)}~ 1 - e > Pl. Similarly in case (b) we obtain "q(v2, vl)>p2. Thence we conclude that the inequalities (4.1) and (4.2) can not hold simultaneously. Theorem 4.3. If FPR Ix in Y is weakly linear, then the induced FPR rl is weakly linear in the class of all nonempty FS's in Y. The proof of this theorem is much similar to that of Theorem 4.2 and is not presented here. Note that the transitivity of IX in Y does not generally imply the transitivity of r/, an example to this being presented at the end of this section. If an uifuzzy preference relation R is specified in the set ~; then using the above definition we obtain for the induced FPR: n(v~, v2) = sup min{vl(Z),
/"2(Y)}-
(z, v ) ~ R
It is easily verified that a linear unfuzzy relation R is strongly linear as defined in Section 3. Then from Theorem 4.2 we conclude that the FPR r/ induced by an unfuzzy linear relation R is strongly linear in the class of all normal FS's in Y. From the other hand as it has been noted in Section 3 for a strongly linear FPR r/ the equation rl"(v~, v2) = 1-rl(v2, v~) holds for any FS's v~ and v2, where ~ is the corresponding fuzzy strict preference relation. From here we conclude that for any FPR induced by an unfuzzy linear relation R the equation rt"(vl, v2~ = 1 -
sup min{vt(z),
v2(y)}
(V.z)~R
holds for any normal FS's u~ and v2 in Y. In a more particular case when Y is the number axis and R(>I) is the natural order in Y the induced FPR r / m a y be expressed in a simpler form provided the class ~ is less wide. In doing this we shall make use of the following definition: A fuzzy set v in Y is called convex if all the sets Y k : = { y l y e Y , v(y)>~k}
0~
are convex. 1 Now we shall prove the following Theorem 4.4. The natural order (>I) in the number axis Y induces a FPR rl in the ~It is understood of course that Y is closed with respect to the operations of addition and multiplication by a number defined in this set.
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class of all FS's in Y possessing the property: for any two normal convex FS's vl and v2 in Y one of the equations holds 1](//1, //2)= 1,
~(~'1, V2)=sup min{vl(y), v2(Y)}.
Proof. Denote A = s u p min{vl(y), v2(y)}. y~y" As the natural order is linear then according to Theorem 4.1 it suffices to show that under the assumptions of Theorem 4.4 the inequalities rl(v~, v2)> A and n(v2, v l ) > A can not hold simultaneously. Assume the contrary, i.e. that the inequalities sup min{v~(z),/"2(Y)} > A, Z~y sup min{vl(z), v2(y)} > A hold simultaneously. That imply the existence of elements z °, y°, z 1, y ~ y with the properties: (a) z ° > y°; z ~> y~ and (b) vl(z°)>A,
v2(y°) >A,
vl(zl)>A,
v2(yl)>A.
One can easily ascertain that if (a) is valid, then the set I = con(z °, z l ) N c o n ( y °, yl) (con=convex hull) is not empty. As vl and v2 are convex we conclude from (b) that for any ~el:/:~v~(fp)>A and v 2 ( ~ ) > A and min{v,(~), v2(~)} > A = s u p min{vl(y), //2(Y)}, which is impossible. The following example shows that Theorem 4.4 is not generally valid without the convexity assumption.
Example 4.1.
Consider two FS's v~ and V 2 in the number axis with the membership functions shown on Fig. 1. As it is easily seen in this example ~l(v~, v2) = 1
/4,
#~
Fig. 1.
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S.A. Orlovsky
but sup min{v~(y), v2(y)}< 1. v~Y
Theorem 4.4 and the properties of strongly linear FPR's imply that for any two normal convex FS's of the number axis the following equations hold rle(vl, v2)= sup min{vl(y), v2(y)}, V~-Y
ns(vl, v2)=
1 - s u p min{v~(y), rE(y)} ~v 0
if ~l(v~, v2) = 1, otherwise.
To complete this section we shall make a few comments on some properties of the FPR introduced. Let us consider three fuzzy subsets v~, v2 and v3 of the number axis as shown on Fig. 2. Using (4.3) we see that in this case -qe(v~, v2)= tie(v2, v~)= 1, rl~(v~, ~y3)=0. Thus we conclude that v~ and v2 are equivalent to a degree 1. This seems unnatural as the FS v2 is located more to the right with respect to Vl, or in other words v2 is "shifted" to the region with greater values of y. Let us, however, give these FS's the following interpretation° Let points on the axis Y represent values of length and assume that vi (~ = 1, 2, 3) represents the result of measurement of the length of object i, "width" (ai, bi) of v~ reflecting precision of the measurement. It is obvious that within the given range of precision one has no justification to state that object 2 is longer than object 1 (and of course it is not reasonable to state the reverse). Thus within the given precision objects 1 and 2 are indistinguishable from each other by their lengths, and it is this fact that is reflected by the equation rle(v~, v2)= 1. From the other hand the precision in this case is sufficient to state that object 3 is longer than object 1, i.e. rl'(v3, vl)= 1. This example also illustrates nontransitivity of FPR ~1, as rl~(v~, v3)=0, then ~l~(v~, v2)= 1, rl~(v2, v3) = 1 do not imply r~(v~, v3)= 1. In what follows the extension procedure introduced in this section is applied t o obtain a solution for the general FMP problem.
"/l
Q~
3 I ! !
I ! I
I I I I I
I I I I I
Fig. 2.
i I ! I I I I
i I ! I
I I ! I I
6~
y
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5. Fuzzy set of nondominated alternatives for a general FMP problem As it has been introduced in Section 2 a general FMP problem is described in the following terms: a set of alternatives X, a universal set of estimates Y, a fuzzy goal function , : X x Y-->[0, 1] and a FPR g : Y x Y-->[0, 1]. In treating this problem here we shall rely upon the results and reasoning from Section 4 to introduce corresponding to t¢ and/x FPR ~i in the set X and then we shall specify the FS of nondominated alternatives in the fuzzily ordered set (X, TI) as it has been suggested in Orlovsky [5]. For every alternative x e X the function ¢ gives the corresponding utility value , ( x , y) in the form of a FS in the universal set Y. From this fact and using the extension of FPR t~ to classes of FS's in Y we obtain the FPR rl induced in X in the form TI(X1, X2) = sup min{,~o(xt, z), go(x2, y), /x(z, y)}. ~',ZE Y
After having obtained this FPR we can formulate the general FMP problem considered as that of rationally choosing alternatives in the fuzzily ordered set
(x, ~). To find a solution to this problem we shall specify the fuzzy subset of nondominated alternatives in the form lIND(x)
=
1 - s u p rl~(L x ) = 1--sup[rl(.L x)-rl(x, k)], ~zX
xEX
with rF being the corresponding to rl FR of strict preference. For the general problem considered the FS ,1Nn has the form 71ND(x) = 1--sup/su p min{q~(~, z), q~(x, y), g(z, y)} YccXtvc~Y; zc:Y
- s u p min{¢(~, z), ~(x, y), ~(y, z)}|.
(5.1~
J
y~:Y; z~Y
The formula (5.1} gives a means of processing membership values of the FS's under consideration to obtain the FS of nondominated alternatives. Alternatives having the maximal membership value in ~lN" may be naturally considered as rational choices for the FMP problem analysed. In the case when the universal set Y is the naturally ordered number axis ana the function q~(x, y) possesses the property supv~vq~(x, y ) = 1 for any x from X, we obtain the followin, expression for rl N O . *I~D(x) = inf ~upmin{¢(~, y), ~o(x, y)}, Vx e X i~N(~
J~ 'f
with
N ( x ) = { ~ l ~ i , , n(~, x)= l}. Note that the reflexivit3¢ of rl implies x ~ N(x), V x ~ X and therefore N(x) # ~,
Vx~X.
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320
In the context of decision-making problems alternatives having the membership value 1 in the set rIND i.e. alternatives which are strictly dominated to a positive degree by no one of alternatives of the set X are of special interest. We call them unfuzzy nondominated alternatives (UND alternatives). In the next section we suggest some sufficient conditions for the existence of U N D alternatives in a general FMP problem.
6. Existence of U N D alternatives We shall formulate here relatively simple sufficient conditions in the form of the following theorem. Theorera 6.1. If the sets X ,and Y are bicompact, Y being a subset o[ the number axis, the function ~ : X × Y--->[0, 1] is continuous c , X × Y, supy~y ~(x, y ) = 1 [or any x e X and 7):X× X--->[0, 1] is the F I R induce,~ in X by the natural order in Y and by the [unction q~, then there exists at least one UND alternative in the [uzzily ordered set (X, 7)). Proof. (1) We introduce the set Y,, = L.Jx~x YO:') with
Y(x)={ylyeY,
~0(x, y) = max q~(x, 9)}.
Under the assumptions of the theorem we obviously have Y(x)#l~, V x e X, therefore Ym # t~. We shall show now that Ym is closed in Y. Consider any y°e Y~,, where Y~, denotes the closure of Ym in Y. Consider a sequence {y~}= Ym, {Yi}--->Y°, and a corresponding sequeace {xi}c X, such that y~ e Y(x~), i = 1, 2, . . . . Denote x ° a limit point of x~. As X is bicompact we have x°~ X. Assume now that y°e Y(x°), i.e. that
,p(x °, y ° ) - m a x ¢(x °, y ) < 0 .
(6.1)
As q~ is continuous on X x Y and consequently the function re(x)= maxy~v q~(x, y) is lower semicontinuous on X, we can find neighbourhoods U(y°). U(x °) of y° and x ° in Y and X respectively in such a way that the inequality
q~(x, y ) - m ( x ) < O holds for any y e U(y °) and x e U(x°). As it can be easily seen we can choose elements 9 and ~ of the above sequences satisfying 9 e U(y°), ~ e U(x °) and 9 e Y(~). Then from (6.1) we obtain q~(£, 9 ) - r n a x 02, y ) < 0 y~Y
which is in contradiction with the above definition of Y(x). Thus we conclude that y° e Ym, which implies t~at Ym is closed in Y (therefore Y,, is bicompact in Y). (2) As ~he natural order in Y is linear and Ym is bicompact there exists y" e ym with the property y" I> y, '¢ y e Ym.
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(3) Consider now any x" ~ X for which y" ~ Y ( x " ) . We shall show that x n is a U N D alternative in the set (X, rl). If we assume the existence of ~ ~ X which is strictly to a positive degree dominates x", i.e. rl'02, x n) > 0, then we have • 102, x " ) = 1,
(6.2)
rl(x", J2) = sup min{q~(x", y), q~(~, z ) } < 1. y~z
Using the assumption supz~v q:(~, z ) = 1 and the fact that y " ~ Y,,, is the maximal element in the set ('i'm, t>) we may write sup min{cp(x", y), q~(~, z)}~ > sup{~0(x", y"), q~(~, z)} = sup q~(~, z ) = 1 and consequently ~(x", ~ ) = 1, which contradicts (6.2). Therefore ~X~, x " ) > 0 is impossible and thus ~ ~ ( ~ , x " ) = 0 for any ~ X which means that x" is U N D alternative in (X, ~).
7. Summary The most characteristic of a general FMP problem is a fuzzy representation of utility values for alternatives from a given set. Analysis of fuzzy prefercnce relations enables to describe this problem in formal terms and to introduce a set of nondominated alternatives which is fuzzy reflecting the fu,zy nature of the problem. The rational choice of alternatives may be made on the basis of maximization of the membership function of this fuzzy set. In some cases the given set of alternatives contains elements which are dominated by no one of the other alternatives to a degree 1. These elements (alternatives) are in fact definitely nondominated and represent what may be referred to as unfuzzy solution to a fuzzily formulated decision-making problem.
Acknowledgement I am thankful to the referees for their comments which have helped to make the paper as I hope more readable.
References [1] R.E. Bellman and L. A. Zadeh, Decision-making in a fuzzy environment, Management Sci. 17 (1970) 8141-8164. [2] C.V. Negoita and M. Sularia, On fuzzy mathematical programming and tolerances in planning. Econ. Comput. Econ. Cybern. Stud. Res. 1 (19761 3-14. [3] C.V. Negoita, S. Minoiu and E. Stan, On considering imprecision in dynamic linear programming, Econom. Comp. Econom. Cybernet. Stud. Res. 3 (1976) 83-95. [4] S.A. Orlovsky, On programming with fuzzy constraint sets, Kybernetes 6 (1977) ! -201. [5] S.A. Odovsky, Decision-making with a fuzzy preference relation, Int. J. Fuzzy Sets Systems | t3) (1978) 155-167.
ON FORMALIZATION OF A GENERAL MATHEMATICAL PROBLEM
FUZZY
S. A. ORLOVSKY Computing Center of the U.S.S.R. Academy of Sciences, Vavilova, 40, Moscow, U.S.S.R. Received February 1978 Revised November 1978 The problem considered in this paper concerns decision-making on the basis of information presented in a fuzzy form. It differs from the usual mathematical programming problems of maximizing a given function over a given set of alternatives mainly in that the values of such function are fuzzy and are represented by fuzzy sets. The question is how to choose alternatives from the given set which give in some sense "the best" fuzzy value of this function. The fuzzy-valued function in question may be thought of as a fuzzy utility function specifying fuzzy utility estimates to the alternatives, or as a performance function of a system with reaction to choices of alternatives (controls) allowing only fuzzy description. The values of this function have the form of fuzzy subsets of some universal set of estimates (or reactions) and a binary preference relation (generally fuzzy) is assumed to be specified in this set. In the paper this relation is extended from the elements of the universal set onto fuzzy subsets of this set. Some properties of this induced relation are studied. This relation is used to extract a fuzzy subset of nondominated alternatives and maximal elements of this subset are suggested as rational choices. It is shown that under some assumptions alternatives exist which are in fact unfuzzily nondominated thus serving as unfuzzy solutions to a fuzzily formulated problem.
Key Words: Fuzzy, Utility function, Preference relation, Nondominated alternatives.
1. ln~oduction Decision-making is probably the most important and popular aspect of application of mathematical methods in various fields of human activity. In real-world situations decisions are nearly always made on the basis of information which is at least in part of fuzzy nature. In some cases fuzzy information is used as an approximation of probably more rigorous information available and in this form appears to be quite convenient and sufficient for making good enough decisions. In other cases fuzzy information is the only form of information at hand. Th~ first step in mathematically attacking a practical decision-making problem consists in formulating a suitable mathematical model of a system or a situation analysed. And if we intend to make reasonably adequate mathematical models of such situations to help practicians in searching for rational decisions, we should be able to introduce fuzziness into our models and to suggest means of processing fuzzy information. 311
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The first and fruitful momentum towards mathematical formalization of fuzziness pioneered by L. A. Zadeh is in its progress now, when numerous attempts are being made to explore the ability of the fuzzy sets theory to become a useful tool for adequate mathematical analysis of real-world situations. This paper should be considered as just one of *..ese attempts. We analyse in it a problem of making decisions when fuzzy information available is presented in the form of a fuzzy-valued utility function (performance function). But before going into details we should like to outline a number of related to each other classes of mathematical models of decision-making situations. Two elements can be extracted from a description of a decision-making problem (DM problem). The first is a set of feasible choices (decisions, alternatives) which may be described either in a fuzzy or in an unfuzzy form. The second element is information (fuzzy or unfuzzy) available about preferences between alternatives. The particular form of a DM problem depends largely on the forn~t in which this infl)rmation is presented. In the general case this information has the form of a fuzzy preference relation specified in a given set of alternatives by means of its membership function. This kind of a DM problem which we shall refer to as P1 has been considered in Orlovsky [5]. In some cases preferences between alternatives may be described by a so called utility function. This function maps a given set of alternatives into a set of estimates of alternatives and a preference relation is specified in this last set. This function therefore allows to compare alternatives with each other by their estimates. If the estimates are numbers, then the DM problem is referred to as mathematical programming problem. In the fuzzy case utility functions may have various forms. The most general form has the function which maps the set of alternatiw;s into a class of fuzzy subsets of a set of estimates. In other words for every air, :native this function specifies a fuzzy estimate in the form of a fuzzy set of estimates. As an illustration to this a utility function may be thought of as a performance function of a system under control. Fuzzy values of this function are then fuzzily described reactions of the system to controls. If we want to rationally control the system we shall be able to compare fuzzy reactions with each other, to decide which of them are more satisfactory. Mathematically this problem involves the necessity of extending a generally fuzzy preference relation from elements of a universal set of reactions onto fuzzy subsets of this set. This kind of a DM problem which may be referred to as a general fuzzy mathematical programming problem (P2) is treated in this paper, and we show how to reduce it to the problem P1 and to use the results obtained in Orlovsky [5] to rationally choose alternatives. Probably more related to practical situations is another class of DM problems in which information about preferences between alternatives is supplied in the form of an unfuzzily specified utility function containing parameters the values of which are fuzzy. The set of alternatives may also be described with unfuzzily specified functions containing fuzzy parameters. This model seems to be quite typical for practical situations when the values of parameters are obtained from experts. But even in cases when these parameters are the results of measurements they are
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intervals and not just numbers. This type of a DM problem which we refer to as that of mathematical programming with fuzzy parameters can be reduced to the problem of the type P2 and then to P1, and we hope to show in the next paper how to treat it using the results obtained in this paper and in Orlovsky [5].
2. Formulation of the problem A general fuzzy mathematical programming problem (FMP) considered in this paper is formulated as follows. Two sets are specified: X - - t h e set of alternatives (feasible choices) and the universal set Y of estimates (utility values for alternatives from X). Let ~ be the class of all fuzzy sets (FS) in Y, that is the class of all functions u" Y---~[0, 1]. A fuzzy ,,':::",,,,.,,,yfunction (performance function)~" X---~ ~7 is specified, assigning a fuzzy estimate for every alternative x from X. We have already mentioned in Section 1 a possible interpretation of such function. The mapping • is more conveniently described with the function ¢ : X × Y---~ [0, 1], in which case for x°~ X fixed the function q~(x°, y)~ ff gives the fuzzy estimate (a membership function of the estimate) for the alternative x °. To complete the formulation of the general FMP problem we assume that a fuzzy nonstrict preference relation (FPR) /x" Y× Y---,[0, 1] is specified in the universal set of estimates Y. so that the value tx(z. y), z, y c Y is a degree to which the preference z ~ y is true. The question arises now how to rationally choose an alternative in the set X on the basis of information in the form of the function ¢ and the FPR tz?, or rather how might the rationality of choice be understood in this context? In this paper a notion of a rational choice is suggested which seems both natural and in agreement with what is understood as rationality in unfuzzy problems. Various types of FMP problems and approaches to their treatment have bccn analysed by Bellman and Zadeh [1], Negoita et al. [2, 3], S. Orlovsky [4] and other authors. In those papers either fuzzy goal sets were considered, or different approaches to the problem of minimization of a given unfuzzy function over a fuzzy set have been studied. As we shall see in the sequel the FMP problem with a fuzzy goal function considered here requires a different approach. In this case alternatives in the set X should be compared with each other on the basis of their fuzzy estimates given by the function ~0 and the F P R / x in the universal set Y. That necessitates a search for a preference relation in the set X which agrees with the function ¢ and the FPR /z. Having found this preference relation (which is generally fuzzy) one can choose alternatives in the set X using a fuzzy subset of nondominated alternatives of the set X, which have been introduced in Orlovsky [5].
3. Pre|immary definitions We thought it worthwhile to present here some definitions from Orlovsky [5] which shall be needed in the sequel.
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A fuzzy preference relation (FPR) in a set X is a fuzzy subset of the product set X x X with the membership function r l : X x X---~[0, 1]. It is assumed that a F P R is reflexive, i.e. rl(x, x ) = ] for any x from X. FPR 7/ is called weakly linear if ~l(x, y ) = 0 = = > r l ( y , x ) > 0 and strongly linear if for any xt, x2 from X at least one of the equations ri(xl, x2) = 1, Tl(x2, x 0 = 1 holds. For a given FPR rI two corresponding fuzzy relations may be introduced: (1) Fuzzy indifference (equivalence) relation rIe(x, y) = rain {~(x, y), rI(y, x)}
V x,y E X.
(2) Fuzzy strict preference relation rI~(x, y ) =
~Tl(x, y ) - r I ( y , x~ if ~(x, y)>~ rI(y, x), / 0 otherwise.
It is easily ascertained that a strongly linear F P R possesses the property riS(x, y ) = 1 - r i ( y , x)
for any x,y s X.
For a given set X with a F P R rl in it we introduce the subset of nondominated (ND) alternatives, having the membership function of the form rIND(X) = inf [1 -- ri~(y, x)]= 1 - s u p rl~(y, x) = 1 - s u p [ r I ( y , x ) - r I ( x , y)]
anyx~X.
yaX
The value rIND(x) is understood as a degree to which the alternative x is dominated by no one of the elements of the set X.
4. Extension of a tuzzy preference relation to classes of fuzzy sets Prior to analysing the general k-~P problem formulated in Section 2 we consider in this section an auxiliary problem of extending a given F P R from the elements of the set in which this FPR is defined onto fuzzy subsets of this set. Formally this problem may be stated as follows: if Ix: Y × Y---~[0, 1] is a F P R in a given set Y and ~ is a class of fuzzy subsets of Y, what is the form of a F P R induced by Ix in the class i f ? To obtain the desired F P R we shall reason in the following manner. If v ~ ~, then as it is easily understood the value ri(v, y) = sup min {v(z), ~z(z, y)} zEy
may be taken as a degree to which the FS v is preferred to the element y e Y. Similarly the value ri(y, v) = sup min {v(z), Ix(y, z)} z~Y
is a degree to which the reverse preference is true.
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N o w if v~ and Vz are any two FS's in Y, then the value ~(v~, v2) = s u p min {v~(y), sup min {v2(z), tt(y, z)}} ycY
z~Y
= sup min {vl(y), M2(Z), ~[£(y, ,~)} y,z~Y
is a degree of the preference v~ ~ v2. O n e can easily see that the degree of the reverse preference (v~ <~ v2) is given by the equation rl(v2, v l ) - sup min {v1(y), v2(z),/z(z, y)}. y,z~Y
T h e F F R rl obtained is defined in the class o~ and is induced by the F P R Ix specified in the universal set Y. Now we shall consider some usefull properties of the F P R ~). T H E O R E M 4.1. I f a FPR ix in Y is reJtexive, then the induced FPR rl is reflective in the class of all normal FS's in Y.
ProoL If v is normal and Ix is reflexive, then we have rl(v, v ) = sup min {v(y), v(z), Ix(y, z)} y,z~Y
>~sup rain {v(y), v(y)} = 1, y~Y
i.e. rl(v, v)= 1. T H E O R E M 4.2. If FPR t~ in Y is strongly linear, then the induced FPR r~ is strongly linear in the class of all normal FS's in Y. Proof. As ~t is assumed to be strongly linear it suffices to show that for any two normal FS's v~, v2 in Y at least one of the equations holds: T~(Vl, l J 2 ) = 1
or
lt](v2, v j ) = 1.
Assume the contrary, i.e. that for some two normal FS's v t and v2 the inequalities ~(v~, rE) = sup min{v~(z), v2(y), Ix(z, y)} = p~ < 1,
(4.1)
• l(vl, v2) = sup min{v~(z), rE(y), IX(y, z)} = P2 < I
(4.2)
~,z~Y ¥,z~Y
hold simultaneously. Now for any n u m b e r e from the interval 0 < e < m i n { ( l - p l ) , introduce two sets Y:=/Ylv,(Y)>~supv,(y)-e=l-e I.
~,~Y
1, J
( l - p 2 ) } we
i = l , 2.
A s / ~ is strongly linear then for any two elements y~ ~ Y~ and Y2 a y2 one of the
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equations (a) Ix(y,, y2) = 1 or (b) IX(Y2, Y~)= 1 holds. In case (a) we obtain ~(vt, v2) = sup min{v~(z), rE(y), Ix(z, y)} y.z~Y
~min{vt(yl), vz(Y2), Ix(Y1, Y2)}~ 1 - e > Pl. Similarly in case (b) we obtain "q(v2, vl)>p2. Thence we conclude that the inequalities (4.1) and (4.2) can not hold simultaneously. Theorem 4.3. If FPR Ix in Y is weakly linear, then the induced FPR rl is weakly linear in the class of all nonempty FS's in Y. The proof of this theorem is much similar to that of Theorem 4.2 and is not presented here. Note that the transitivity of IX in Y does not generally imply the transitivity of r/, an example to this being presented at the end of this section. If an uifuzzy preference relation R is specified in the set ~; then using the above definition we obtain for the induced FPR: n(v~, v2) = sup min{vl(Z),
/"2(Y)}-
(z, v ) ~ R
It is easily verified that a linear unfuzzy relation R is strongly linear as defined in Section 3. Then from Theorem 4.2 we conclude that the FPR r/ induced by an unfuzzy linear relation R is strongly linear in the class of all normal FS's in Y. From the other hand as it has been noted in Section 3 for a strongly linear FPR r/ the equation rl"(v~, v2) = 1-rl(v2, v~) holds for any FS's v~ and v2, where ~ is the corresponding fuzzy strict preference relation. From here we conclude that for any FPR induced by an unfuzzy linear relation R the equation rt"(vl, v2~ = 1 -
sup min{vt(z),
v2(y)}
(V.z)~R
holds for any normal FS's u~ and v2 in Y. In a more particular case when Y is the number axis and R(>I) is the natural order in Y the induced FPR r / m a y be expressed in a simpler form provided the class ~ is less wide. In doing this we shall make use of the following definition: A fuzzy set v in Y is called convex if all the sets Y k : = { y l y e Y , v(y)>~k}
0~
are convex. 1 Now we shall prove the following Theorem 4.4. The natural order (>I) in the number axis Y induces a FPR rl in the ~It is understood of course that Y is closed with respect to the operations of addition and multiplication by a number defined in this set.
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class of all FS's in Y possessing the property: for any two normal convex FS's vl and v2 in Y one of the equations holds 1](//1, //2)= 1,
~(~'1, V2)=sup min{vl(y), v2(Y)}.
Proof. Denote A = s u p min{vl(y), v2(y)}. y~y" As the natural order is linear then according to Theorem 4.1 it suffices to show that under the assumptions of Theorem 4.4 the inequalities rl(v~, v2)> A and n(v2, v l ) > A can not hold simultaneously. Assume the contrary, i.e. that the inequalities sup min{v~(z),/"2(Y)} > A, Z~y sup min{vl(z), v2(y)} > A hold simultaneously. That imply the existence of elements z °, y°, z 1, y ~ y with the properties: (a) z ° > y°; z ~> y~ and (b) vl(z°)>A,
v2(y°) >A,
vl(zl)>A,
v2(yl)>A.
One can easily ascertain that if (a) is valid, then the set I = con(z °, z l ) N c o n ( y °, yl) (con=convex hull) is not empty. As vl and v2 are convex we conclude from (b) that for any ~el:/:~v~(fp)>A and v 2 ( ~ ) > A and min{v,(~), v2(~)} > A = s u p min{vl(y), //2(Y)}, which is impossible. The following example shows that Theorem 4.4 is not generally valid without the convexity assumption.
Example 4.1.
Consider two FS's v~ and V 2 in the number axis with the membership functions shown on Fig. 1. As it is easily seen in this example ~l(v~, v2) = 1
/4,
#~
Fig. 1.
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S.A. Orlovsky
but sup min{v~(y), v2(y)}< 1. v~Y
Theorem 4.4 and the properties of strongly linear FPR's imply that for any two normal convex FS's of the number axis the following equations hold rle(vl, v2)= sup min{vl(y), v2(y)}, V~-Y
ns(vl, v2)=
1 - s u p min{v~(y), rE(y)} ~v 0
if ~l(v~, v2) = 1, otherwise.
To complete this section we shall make a few comments on some properties of the FPR introduced. Let us consider three fuzzy subsets v~, v2 and v3 of the number axis as shown on Fig. 2. Using (4.3) we see that in this case -qe(v~, v2)= tie(v2, v~)= 1, rl~(v~, ~y3)=0. Thus we conclude that v~ and v2 are equivalent to a degree 1. This seems unnatural as the FS v2 is located more to the right with respect to Vl, or in other words v2 is "shifted" to the region with greater values of y. Let us, however, give these FS's the following interpretation° Let points on the axis Y represent values of length and assume that vi (~ = 1, 2, 3) represents the result of measurement of the length of object i, "width" (ai, bi) of v~ reflecting precision of the measurement. It is obvious that within the given range of precision one has no justification to state that object 2 is longer than object 1 (and of course it is not reasonable to state the reverse). Thus within the given precision objects 1 and 2 are indistinguishable from each other by their lengths, and it is this fact that is reflected by the equation rle(v~, v2)= 1. From the other hand the precision in this case is sufficient to state that object 3 is longer than object 1, i.e. rl'(v3, vl)= 1. This example also illustrates nontransitivity of FPR ~1, as rl~(v~, v3)=0, then ~l~(v~, v2)= 1, rl~(v2, v3) = 1 do not imply r~(v~, v3)= 1. In what follows the extension procedure introduced in this section is applied t o obtain a solution for the general FMP problem.
"/l
Q~
3 I ! !
I ! I
I I I I I
I I I I I
Fig. 2.
i I ! I I I I
i I ! I
I I ! I I
6~
y
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5. Fuzzy set of nondominated alternatives for a general FMP problem As it has been introduced in Section 2 a general FMP problem is described in the following terms: a set of alternatives X, a universal set of estimates Y, a fuzzy goal function , : X x Y-->[0, 1] and a FPR g : Y x Y-->[0, 1]. In treating this problem here we shall rely upon the results and reasoning from Section 4 to introduce corresponding to t¢ and/x FPR ~i in the set X and then we shall specify the FS of nondominated alternatives in the fuzzily ordered set (X, TI) as it has been suggested in Orlovsky [5]. For every alternative x e X the function ¢ gives the corresponding utility value , ( x , y) in the form of a FS in the universal set Y. From this fact and using the extension of FPR t~ to classes of FS's in Y we obtain the FPR rl induced in X in the form TI(X1, X2) = sup min{,~o(xt, z), go(x2, y), /x(z, y)}. ~',ZE Y
After having obtained this FPR we can formulate the general FMP problem considered as that of rationally choosing alternatives in the fuzzily ordered set
(x, ~). To find a solution to this problem we shall specify the fuzzy subset of nondominated alternatives in the form lIND(x)
=
1 - s u p rl~(L x ) = 1--sup[rl(.L x)-rl(x, k)], ~zX
xEX
with rF being the corresponding to rl FR of strict preference. For the general problem considered the FS ,1Nn has the form 71ND(x) = 1--sup/su p min{q~(~, z), q~(x, y), g(z, y)} YccXtvc~Y; zc:Y
- s u p min{¢(~, z), ~(x, y), ~(y, z)}|.
(5.1~
J
y~:Y; z~Y
The formula (5.1} gives a means of processing membership values of the FS's under consideration to obtain the FS of nondominated alternatives. Alternatives having the maximal membership value in ~lN" may be naturally considered as rational choices for the FMP problem analysed. In the case when the universal set Y is the naturally ordered number axis ana the function q~(x, y) possesses the property supv~vq~(x, y ) = 1 for any x from X, we obtain the followin, expression for rl N O . *I~D(x) = inf ~upmin{¢(~, y), ~o(x, y)}, Vx e X i~N(~
J~ 'f
with
N ( x ) = { ~ l ~ i , , n(~, x)= l}. Note that the reflexivit3¢ of rl implies x ~ N(x), V x ~ X and therefore N(x) # ~,
Vx~X.
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In the context of decision-making problems alternatives having the membership value 1 in the set rIND i.e. alternatives which are strictly dominated to a positive degree by no one of alternatives of the set X are of special interest. We call them unfuzzy nondominated alternatives (UND alternatives). In the next section we suggest some sufficient conditions for the existence of U N D alternatives in a general FMP problem.
6. Existence of U N D alternatives We shall formulate here relatively simple sufficient conditions in the form of the following theorem. Theorera 6.1. If the sets X ,and Y are bicompact, Y being a subset o[ the number axis, the function ~ : X × Y--->[0, 1] is continuous c , X × Y, supy~y ~(x, y ) = 1 [or any x e X and 7):X× X--->[0, 1] is the F I R induce,~ in X by the natural order in Y and by the [unction q~, then there exists at least one UND alternative in the [uzzily ordered set (X, 7)). Proof. (1) We introduce the set Y,, = L.Jx~x YO:') with
Y(x)={ylyeY,
~0(x, y) = max q~(x, 9)}.
Under the assumptions of the theorem we obviously have Y(x)#l~, V x e X, therefore Ym # t~. We shall show now that Ym is closed in Y. Consider any y°e Y~,, where Y~, denotes the closure of Ym in Y. Consider a sequence {y~}= Ym, {Yi}--->Y°, and a corresponding sequeace {xi}c X, such that y~ e Y(x~), i = 1, 2, . . . . Denote x ° a limit point of x~. As X is bicompact we have x°~ X. Assume now that y°e Y(x°), i.e. that
,p(x °, y ° ) - m a x ¢(x °, y ) < 0 .
(6.1)
As q~ is continuous on X x Y and consequently the function re(x)= maxy~v q~(x, y) is lower semicontinuous on X, we can find neighbourhoods U(y°). U(x °) of y° and x ° in Y and X respectively in such a way that the inequality
q~(x, y ) - m ( x ) < O holds for any y e U(y °) and x e U(x°). As it can be easily seen we can choose elements 9 and ~ of the above sequences satisfying 9 e U(y°), ~ e U(x °) and 9 e Y(~). Then from (6.1) we obtain q~(£, 9 ) - r n a x 02, y ) < 0 y~Y
which is in contradiction with the above definition of Y(x). Thus we conclude that y° e Ym, which implies t~at Ym is closed in Y (therefore Y,, is bicompact in Y). (2) As ~he natural order in Y is linear and Ym is bicompact there exists y" e ym with the property y" I> y, '¢ y e Ym.
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(3) Consider now any x" ~ X for which y" ~ Y ( x " ) . We shall show that x n is a U N D alternative in the set (X, rl). If we assume the existence of ~ ~ X which is strictly to a positive degree dominates x", i.e. rl'02, x n) > 0, then we have • 102, x " ) = 1,
(6.2)
rl(x", J2) = sup min{q~(x", y), q~(~, z ) } < 1. y~z
Using the assumption supz~v q:(~, z ) = 1 and the fact that y " ~ Y,,, is the maximal element in the set ('i'm, t>) we may write sup min{cp(x", y), q~(~, z)}~ > sup{~0(x", y"), q~(~, z)} = sup q~(~, z ) = 1 and consequently ~(x", ~ ) = 1, which contradicts (6.2). Therefore ~X~, x " ) > 0 is impossible and thus ~ ~ ( ~ , x " ) = 0 for any ~ X which means that x" is U N D alternative in (X, ~).
7. Summary The most characteristic of a general FMP problem is a fuzzy representation of utility values for alternatives from a given set. Analysis of fuzzy prefercnce relations enables to describe this problem in formal terms and to introduce a set of nondominated alternatives which is fuzzy reflecting the fu,zy nature of the problem. The rational choice of alternatives may be made on the basis of maximization of the membership function of this fuzzy set. In some cases the given set of alternatives contains elements which are dominated by no one of the other alternatives to a degree 1. These elements (alternatives) are in fact definitely nondominated and represent what may be referred to as unfuzzy solution to a fuzzily formulated decision-making problem.
Acknowledgement I am thankful to the referees for their comments which have helped to make the paper as I hope more readable.
References [1] R.E. Bellman and L. A. Zadeh, Decision-making in a fuzzy environment, Management Sci. 17 (1970) 8141-8164. [2] C.V. Negoita and M. Sularia, On fuzzy mathematical programming and tolerances in planning. Econ. Comput. Econ. Cybern. Stud. Res. 1 (19761 3-14. [3] C.V. Negoita, S. Minoiu and E. Stan, On considering imprecision in dynamic linear programming, Econom. Comp. Econom. Cybernet. Stud. Res. 3 (1976) 83-95. [4] S.A. Orlovsky, On programming with fuzzy constraint sets, Kybernetes 6 (1977) ! -201. [5] S.A. Odovsky, Decision-making with a fuzzy preference relation, Int. J. Fuzzy Sets Systems | t3) (1978) 155-167.