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Using fuzzy binary relations for identifying noninferior decision alternatives
Fuz~ Sets and Systems 2,5 (1988) 21-32 Nollh-Hoilend
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USD~IG F U Z Z Y BII~ARY g ~ L A T I O N $ F O R I D ~ ~ I ~ G NO~UOR D l g C ~ l O N .gLTE~'~IATIVES Serge V. OVCHINNIKOV* San F r c a c ~ State Uniwes~, San Franc/sco, CA 94132, USA
Vladimir M. OZERNOY CeJifomia State Unieers'mp, Hayward, C.~#i94542, USA Received, January 1986 Re~ April 1986 A ~ noninferiority relation is introduced. B~sed on this relation, fuzzy relations of preference, strict preference, indifference, and incomparability are specified, and their pmpe~ea are investigated. A f~gzy noninfefior sub~t is defined and iis basic ch~actefisfic prope~ea are established. An algorithm is presented for identifying a f~.y ~et of noninferior alternatives in decision-making situations that require the selection of a certain number of noninferior alternatives for more detailed study.
Keywords: Noninferiolity relation, Strict preference, Preference, Indifference, Incomparability, Noninferior subset.
L ~~u~on
In many decision-making problems the purpose of the analysis is to determine the 'best' decision alternative, in such problems a complete rank-order of the set of feasible alternatives is not needed. lns~¢a ~, d , the analysis of the problem usually starts with the identification of all nondominated alternatives. The most preferred alternative is then selected from among nondominated alternatives, while all dominated alternatives are rejected. ,,Another decision-making problem involves the selection of a relatively small number of preferred decision alternatives for more detailed study. The preferred alternatives are chosen from among nondominated alternatives or from among top rank-ordered alternatives. In this case the remaining alternatives are also rejected. Several methods have been proposed for the identification of nondominated alternatives. Oaft and Ozernoy [2] develop an algorithm that makes it possible to identify nondominated altern•ives independently of the specific way in which binary relations are introduced that would define concepts such as 'more preferred alternative' and 'set of nondominated alternatives'. The algorithm ex#icitly uses a concept of incomparability of alternatives. Methods developed for the selection of the 'best' alternative from among * Research sponsored by NSF Grant IST-8~3431. 0165-0114/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)
22
S.V. O v c ~ k o v ,
P.M. Ozemoy
nondominated alternatives as well as methods developed for the selection of a relatvely small number of preferred altematves use additonal information about file decision maker's preferences. For example, Kirkwood and Sarin [3] derive conditions to determine whether two multcriteria alternatives can be ranked the wei~t~,additive ev~uaton ~ ~ o n , They also develop ~ al that p ~ a l l y ~ k ~ r d e r s the set of alternatives based on the p~rwise r a n ~ g information. Ozernoy and GaR [6] present a maR|step procedure that permits at each step at least parti~ identification and eli~nation of dominated alternatives by obt~ning a sequence of ~fferent types of preference information from the decision maker. Zionts [13] points out that there may be some instances where the elimination of dominated alternatives is inexpedient. For example, a dominated alternative may be sufficiently dose to a nondominated altematve in the criteria space so that the decision maker might make a choice based on some secondary criteria not used in the analysis. In addition, attempts to precisely quantify alternatives would sometimes give an unrealistic view of the precision with which alternatives could be evaluated. In such situations, dominated aRernatives should not be excluded from further analysis. In this paper, we introduce a new approach for | d e n t i n g nondominated decision alternatives. This aproach is based on the use of a fuzzy noninferiority relation which allows to determine a fuzzy noninferior subset of a finite set. The value of the membership function is regarded as the degree to which an element is noninferior in the finite set. A procedure is deve|oped for the selection of noninferior elements. Thus, considering the finite set as a set of feasible decision alternatives in a decision-making problem, we prove that the 1-cut of a noninferior subset is a nonempty set of unfuuy undominated alternatives in the set of feasible alternatives, reestablishing the main result of Orlovsky [4]. The work in this article establishes an approach that allows one to identify different numbers of noninferior decision alternatives based on different v~ues of in constructing an ~ c u t of a noninferior subset. This is of particular interest since by the change of a value of ~ one can identify a required number of noninfer/or alternatives (or a number close to the required one) for more detailed
study.
This paper is organized as follows. Section 2 introduces a fuzzy noninferiority relation and its prope~es. Based on this relation, the fuzzy relations of strict preference, indifference, incomparability, and preference are defined. A theorem is proved which establishes some compatibility properties for these relations. ]t is also shown that the inverse image of a crisp nondominance relation under fuzzy correspondence is a fuzzy noninferiority relation- a result which is of particular interest in the area of ftgzy multi-criteria decision-making. In ~ t i o n 3 a ~ noninferior subset is defined. It is shown that two noninferior elements are either incomparable or equiva|ent. Basic characteristic properties of a fuzzy noninferior sub~et are specified. These are f u r y analogs of res~cfive p r o p e l s of crisp choice functions. Section 4 presents an algorithm for constructing an ~-cut of a noninferior subset and identf3~g noninferior elements.
Fuzzy binary relaaons
23
Let Q be a fuzzy binary relation on a finite set A. This relation is defined by its membership function Q(x, y), which maps a direct product A × A into the unit interval [0,1]. We regard this relation as an noninferiori~ relation: the value Q(x, y) represents the degree to which element x is not inferior to y. This interpretation imposes: certain restrictions~ on Q, which we introduce as properties of this relation. First of all, Q is a reflexive fuzzy binary relation, i.e.
Q(x, x) = 1 for all x cA, meaning that any element in A is certainly not inferior to itself. The second property is completeness (strong linearity in [4]) which is defined as follows: Q(x, y) v Q(y, x) = I for all x, y E A. (We define, as usual) a v b ~=max(a, b) and a ^ b = rain(a, b).) This property guarantees that, for any two given elements in A, certainly at least one of them is not inferior to another one. There are many ways to define an antisymmetric part of a fuzzy binary relations [4, 5, 7]. In this paper we adopt the following definition [4]: P(x, y) - (Q(x, y) - Q(y, x)) v 0 for all x, y ¢ A, which corresponds to the crisp definition given by P = Q \ Q-!. This relation is regarded as a strict preference associated with the noninferiofity relation Q. As usual, such a relation is supposed to be a transiave fuzzy binary relation, i.e. (PP)
P(x, y) ^ P(y, z) <<.P(x, z) for all x, y, z ¢ A.
Note that, by completeness of Q, P is a fuzzy complement of Q - I i.e.
P(x,y)---l-Q(y,x)
forallx, y a A .
In the crisp case, reflexive, complete binary relations with transitive antisymmetric parts are called quasi.transitive relations [8]. We use the same name for fuzzy binary relations satisfying all the introduced properties and, following [8], denote transitivity property of strict preference by (PP). Therefore, by definition, a fuzzy quasi-transitive relation is a reflexive compiete fuzzy binary relation satisfying the (PP) property. It is easy to verify that a reflexive ~ y binary relation Q is a fuzzy quasi-transitive relation if and only if the following property is sastisfied (QT)
Q(x,y) v Q(y, z) >~ Q(x, z),
for all x, y, z ~ A. A symmetric par~ S of a fuzzy binary relation Q is defined by
S(x, y) = Q(x, y) ^ Q(y, x) for all x, y ~ A. The value S(x, y) is considered to be a degree to which elements x and y are incomparable or indifferem. In order to separate incomparability and indifference subrelations of $ we
24
S.V. Ovchinnikov, V.Mo Ozernoy
&fine an/nd/fference relation I as a crisp equivalence relation contained in $. M e n l(x, y) is a function with range {0, 1}, such that
O) (2) O) (4)
Z(x, z ) = 1, ICx,y) ffi 1(y, x), z(x, y)^ z(y, z ) ~ z(y, z), x(x, y) ~ s(x, y),
for ~ x, y, z cA. We denote the transitivity property 0 ) of the indifference relation by (II). An incon~rability relation N is defined by its membership function as follows
N(x,y)ffiS(x,y)-l(x,y)
for all x, y cA.
It is e u y to veri~ that N is an an~flexive and symmetric ~ binary relation, i.e. N(x,x)~O and N(x,y)=N(y,x), for all x, y cA. We also have the decomposition $ m N U 1, such that N N 1 m ~. Since an indifference relation I is not defined by means of the relation Q, ~ r t ~ comp~bdity conditions must be imposed on I in order to guarantee properties similar to crisp ones (see [2]). We employ the following compatibility ¢on~tions:
(QI) (IQ)
Q(x. y) ^ l(y, z) ~ Q(x, z), I(x, y) ^ Q(y, z) ~ Q(x, z),
for all x, y, z ¢ A, which are similar to the conditions used in [8]. A fuzzy preference relation R is defined as the union P U I of a strict preference "''~ ce P an~ l"Ii ~neren relation L i.e.
R(x, y) = P(x, y) v1(x, y) for all x, yEA. Further, in this paper, we assume that the noninfefiofity relation Q and indifference relation I satisfy properties (QT), (lI), (QI) and (IQ). (Note that, in many important special cases, I is an identity relation; then conditions (]I), (QI), The following theorem establishes some compatibilRy properties of the strict preference relation P, the preference relation R, the indifference relation 1, and the incomparability relation N (of. [2, 8]). First, we prove a useful lemma.
2.1. 1(y, z) = 1 ~ t i e ~ Q(x, y) = Q(x, z) aria Q(.v, x) = O(z, x). ~ .
By (QI), we have
Q(x, y) = Q(x, y) ^ 1(y, z) ~ Q(x, z) auo Q(x, z) = Q(x, z) ^ l(z, y) ~ Q(x, y). Hence, Q(x, y) = Q(x, z). Similarly, by (IQ), Q(y, x) --- Q(Z, x). E! 2oL Conditions (PP), (II), (QI) and (IQ) togethe~ impgy if, I) (n,)
e(x, y) ^ ~(y, z j ~ p(x, z), Z(x, y) ^ P(y, z)<~ p(x, z),
Fuzzy bina~ relations
(ND :~N) (RR)
N(x, y) ^ 1(y, z ) ~ ~(x, z), 1(x, y) ^ U(y, z) ~ u(x, z), R(x, y) ^ R(y, z) ~ R(x, z),
for all x, y, z e A.
Proof. (PI) Obvious, if l(y, z)ffi 0. Otherwise, by Lamina 2.1, e(x, y) ffi(Q(x, y) - Q(y, z)) v o = (Q(x, z) - Q(z, x)) v o-- ~'(x,z).
(IP) Similar to (PI). (NI) Obvious, if l(y, z ) = 0 or N(x, y ) = O. Otherwise, l(y, z ) = 1 and I(x, y) -- 0. Therefore, by transitivity of I, l(z, x) = O. By l.emma 2.1, we have
N(x, y) = s(x, y) - 1(x, y) = Q(x, y) ^ Q(y, x) ,o Q(x, z) ^ ¢2(z, x) = s(x, z) - v,(x, z) = N(x, z).
(IN) Similar to (NI). (RR) Follows from
R(x, y) ^ n(y, z) --- (P(x, y) v I(x, y)) ^ (e(y, z) v I(y, z)) = (P(x, y) ^ e(y, z)) v (P(x, y) A l(y, z)) v if(x, y) ^ e(y, z))
v if(x,y) ^ I(y, z))~
Q(x, y)-- V F(x, u) ^ Q(u, v) ^ F(y, ~,)= ~ F(x, u) ^ F(y, v). U~U
Lamina 2.2. The inverse image Q of a crisp quasi-transiave relation Q on B under fuzzy mapping F is a reflexive and complete fuzzy binary relation on A.
Pn~f. (1) Q(X, x) -- VuOv F(x, u) ^ F(x, v) <~Vu F(x, u) = 1, sin~ Q is a reflexive binary relation and F(x) is a normal fuzzy subset.
$.V. O v c ~ v ,
V.M. Ozemoy
(2) Let uo¢ F~(x) and Vo~ Fdy). We have
~>IF(x, Uo) ^ Q(Uo, Vo) ^ F(y, Vo)] v [F(y, no) ^ Q(Vo, uo) ^ F(x, Uo)] = Q(Uo, Vo) v (}(vo, Uo)= 1, since Q is a complete binary relation.
[3
~ ~ 2.2. The inverse image of a crisp quasi-trans~v¢ binary relation Q under fuzzy correspondence F is a fuzzy quasi~ransitive relation Q. Proof. By Lemma 2.2, it su~ces to prove that Q satisfies the (QT) property
or that, for any given s and t such that sQt, there are u and q such that uQ~, and
sQq,
IF(y, u) ^ F(x, t)] v IF(z, s) ^ F(y, q)] ~ F(z, s) ^ F(x, t). Let us prove, first, that there are Uo and qo such that uoQt, sQqo, and F(y, Uo) v F(y, qo) = I. Suppose that F(y, u) < I and F(y, q) < I for all u and q such that uQt and sQq. Then, for any w ¢ Fj(y), wQt and sQw are both false, which implies both tPw and wPs. Therefore, tPs, which contradicts sQt. We have now
IF(y, Uo) ^ F(x, 01 v [F(z, s) ^ F(y, qo)] [F(y, uo) v F(z, s)] ^ [~(y, Uo) v F(y, qo)] ^ [F(x, 0 v F(z, s)] ^ IF(x, t) v ~(y, qo)] F(z, s) ^ [F(x, 0 v F(z, s)] ^ F(x, t) = F(z, s) ^ F(x, 0. n In practice, A is a set of decision alternatives, B is a subset of finite dimensional real space ('criteria' space), and Q is a vector nondominance relation. It is assumed that ~ y information about alternative a e A is given in the form of a normal ~ subset F(a) of B representing the value of a linguistlc va~ab|e. For example, this f u r y subset m i ~ t represent the following judgement about alternative a: "alternative a is more or less good with respect to criterion C~ and f~r ~ t h respect to criterion C2". A fuzzy set F(a) is regarded then as a fuzzy ~ a g e of alternative a e A under ~kzzy correspondence F between A and B. In accordance ~ Theorem 2.2, the inverse image Q of Q under F is a noninfefiority relation on the set of alternatives a. Therefore, fuzzy information
Fuzzy binary relations
27
about alternatives generates a noninfer~ofity relation by means of the inverse image construction. (Note that, in the one-dimensional case, the so defined noninferiofity relation is the dominance relation R8 introduced in [10] by means of a different construction.) 3. Fuzzy n ~ e d o r
~but
Given noninfefiority relation Q on a finite set A we define a f u r y nom'nferior subset L(A) by means of its memben~hip function L(A)(x) as follows:
L(A)(x) = y~A Q(x, y) for x cA. The value L(A)(x) is regarded as a degree to which an element x is noninferior in a. Since Q is a complement of p - i a noninferior subset defined above coinci~es with the fuzzy set of nondominated elements introduced in [4]. (Note that, in the particular case when Q is generated as an inverse image of the vector nondominance relation (see previous section), the so defined noninferior subset can be regarded as a fuzzy Pareto set of nondominated alternatives.) The follo~ng theorem shows that a noninferior subset is, actually, a normal fuzzy subset of A (el. Theorem 6.4 in [4])
3.1. V , , , L(A)(x)
-
1.
~f. Suppose that ~,eA L(A)(x)< 1. Then for any x e A there is y e A such that Q(x, y ) < 1. Then, by finiteness of A, there is a cyc|e Xo, x ~ , . . . , x, = Xo, such that Q(x, x~+l) < 1, for 0 ~
1 > Q(Xo, a contradiction.
x,)
Q(xo, xo) = 1,
[21
In accordance with the theorem proved, the t-cut of a noninferior subset is a nonempty crisp set; the elements of this crisp set can be regarded as nonfuzzy undominated elements in A (el. [4]). A noninferior subset L(A) is compatible with the indifference relation 1, as the following theorem demonstrates. Theorem 3.2. l(x, y ) = l implies L(A)(x)= L(A)(y). ~oL
Follows immediately from Lemma 2.1.
[3
]n the crisp case two noninfe~or elements are either incomparable or equivalent [2]. It is also true in the framework developed in this paper, as the next theorem shows.
Theorem 3.3. L(A)(x) ^ L(A)(y)~ N(x, y) v l(x, y).
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$.V. O v c ~ v ,
~ .
V.M. Ozernoy
Since L(A)(u) <~Q(u, t) for all t e A, we have L(A)(x) ^ L(A)Lrj <- fl(x, Y) ^ Q(y, x) f N(x, y) v l(x, y).
El
Considering a noninferior subset L(A) as a function of A, where A is a finite subset of a certain universe set, one can regard L(A) as a fuzzy choice function [5]. Then the value L(A)(x) is a degree to which x is the 'best' element in the set A. The next three theorems establish basic characteristic prope~es of the: fuzzy choice function L(A).
T b o t e m 3,4. A' G A implies L(A °) ~ L(A) f3 A'.
~ .
For x E A', we have
z(A')(x) = , ~ Q(x, 0
;',~
Q(x, 0 = L(A)(x).
E!
The property established in the theorem proved in Sun's property (~r) [8]; it is called Heritage property (H) in [1].
T h e o ~ 3.5. L(A' U A") ~_L(A') N L(A"). ~f.
For x EA' OA",
= L ( A ' U A")(x).
Otherwise, L(A')(x) A L(A")(x)=0.
E!
The proved property is the Concordance property (C) in [1] or property (y) in
[9]. Finally, we establish a property which can be regarded as a fuzzy analog of the classical 'independence of irrelevant alternatives' property. Theorem 3.6. Let A ' be a subset of A such that L(A)(y) < L(A)(x) for all x e A' and y e A \A'. Then L(A') ffi L(A) N A '. Preef. From Theorem 3.4, L(A') ~ L(A) N A ~. Suppose that there is x E A' such that L(A')(x) > L(A)(x), i.e.
A O(x, 0 > , ~ Q(x, 0.
t~A ~
It means that there exists a y ~ A \ A ° such that, ~o~~~=l u ~ A',
Q(x, y) < Q(x, u).
(,)
since L(A)(y) < L(A)(x), there is a y, suchthat Q(y, y~) < L(A)(x). If y, ~A', there is a y= such that Q(y~, yz)< L(A)(x), and so on. Since A is a finite set and there is no cycle with the property Q(y~,y ~ . t ) < 1 (see the proof of Theorem 3.1),
Fuzzy b~.ry reigns we construct a sequence Yoffi Y, Y~,..., Yn such that y~ ¢ A \ A ' for i < n, y~ ¢ A', and Q(y~, y~+~)
Q(y, y,) V| Q(Y,,
<
L(A)(x) Q(x,t).
In particular,Q(y, y,,)< Q(x, y~), which implies,together with (,), Q(x, y) v 0(Y, Y,) < Q(x, y,) - a contradiction with the (QT) property, v! Recall that L(A)(x) is considered to be a degree to which an element x ~ the 'best' in A. Let us suppose that a threshold ¢ is given such that elements x with L ( A ) ( x ) < ¢ are deleted from consideration (irrelevant alternatives). ~ e n Theorem 3.6 states that the (fuzzy) 'best' elements in the remainder set A' are exactly the same as in the given set A. Actually, A' is an ~-eut of f e ~ y ~ t L(A). An algorithm constructing the ~-cuts of a noninferior subset L(A) is presented in the next section. In the conclusion of this section, we establish necessary and su~cient conditions on a fuzzy binary relation Q to generate normal noninfefior subsets. We say that a fuzzy binary relation P is an acydic relation if, for any finite sequence X o , . . . , x,, P(x~, xj+l) > 0, for all 0 ~ i ~ n - 1, implies P(x,, Xo) = O.
"l~¢o~m 3.7. Let Q be a fuzzy binary relation.A noninferiorsubset L(A) defined by its membership function L(A)(x)= A Q(x, y) yea
is a normal fuzzy subset for any finite set A if and only if (~ is a i~'~qexivecomplete fuzzy binary relation with acyclic antisymme~ic part P. Preefo Sufficiency is proved, essentially, by repeating the argument used in the proof of Theorem 3.1. To prove necessl"ty, we note first that, for any element x, 1 = l,((x})(x) ~ Q(x, x) which implies reflexivity of Q. Second, we have, for A -- {x, y},
I = L(A)(x) v L(A)(y) ffi(Q(x, x) ^ Q(s, y)) v (Q(y, x) ^ Q(y, y))
= Q(x, y) v O(y, x) which proves completeness of Q. Finally, let A -- {Xo,..., xn} and P(x~,x~+1)> O for all 0 ~ i ~ n - 1. Therefore, Q(x~+l, x~) < 1 for all 0 ~ i ~ n - 1. Hence, L(A)(xj) < 1 for all 0 ~ i <~ n - 1. Since L(A) is a normal fuzzy set, L(A)(x,) = 1, which implies Q(x,, Xo)= 1. Therefore, P(x,, xo)=0, i.e. P is an acyclic fuzzy binary relation. O
By definition, an ~-cut of a fuzzy set L(A) is a crisp set L~(A)= {x A: L(A)(x) >~~). Since L(A)(x) = A,~A Q(x, y), we have
L~(A ) = {x cA" (Vy e A ) xQ,~y },
30
8.V. ~ m k o v ,
where Q,, is an C-cut of ~
V.M. Ozemoy
binary relation Q, i.e. xQ~,y if and only if
We introduce a symmetric $,, and an antisymmetric P~, parts of Q,, as follows:
xS~y if and only if xQ~y and yQ,,x, and
xP,,y if and only if xQ~,y and not yQ,~c. Note, that P, is not an e-cut of fuzzy strict preference P, while $,,/s an ~-cut of $.
4.1. Q~ is a c~sp quasi.transiti've binary relation. Proof. Obviously, Q~ is a reflexive complete binary relation. Let us prove transitivity of P¢. By definition, xP~y if and only if Q(y, x ) < ~. Suppose that xP, y and yP~z, or, equivalently, Q(y, x) < o~, and Q(z, y) < ¢. Then, by the (QT) property, ~ > Q(z, y) v Q(y, x) ~ Q(z, x). Hence, xP~z. [3
4~. For any y ¢ A \L~(A), ~ere is an x ¢ L~(A) such that xP~y. ~ . Since Area Q(Y, t) ffi L(A)(y) < ~, there e~sts a Yl such that Q(y, yl) < o~. Hence, YtP~Y. If y~ f L~(A), then there exists a Y2such that YzP~Yt, and so on. Since P~ is a partial order and A is a finite set, there is an x ¢ L~(A) such that
xP~y. 0 Let B = A U {b}, b ¢ A, so that B has just one more element than A. Then, from Theorem 3.4, L~(A)~L~(B)NA. Therefore, any element in L~(B) is either b, or an element of L~(A). The following theorem establishes important relationships between sets L~(B) and L~(A) in terms of relation P~,. 4.L (i) If there ~ an x ¢ L~(A) such th~ xPab, then L~(B) = L~(A); (ii) otherwise, L~(B)=(L~(A)XY)O {b}, where Y = {x ¢ L,~(A): bPax}.
Tk~
Proof. (i) Since xP~b, bQ~t is false. Therefore, b ~ L~(B). Suppose that there is a t ¢ L,,(A) such that t ~ L~(B). Then bP~t, which implies, by transitivity of P~, xP~t. Hence tQ,~x is false, which contradicts t ¢ L~(A). (ii) In this case, for all xcL~(A) we have b Q ~ . Hence, L ~ ( A ) \ Y f { t ¢ L,~(A): b$~t}. It suffices to prove that (1) any element of Y does not belong to L (B), (2) any element of L (A)\ Y belongs to L,,(B), and 0 ) b belongs to Lffi(B). (1) Let t ¢ Y. Then bP~t. Therefore, ~,~b is false. Hence, t does not belong to
L.(a).
(2) Let t ¢ L~(A) \ Y. Then tQ~.v for all y ¢ A and tQ~b. Therefore, t belongs (3) Since b Q ~ for all x ¢L~(A), "it suffices to prove that bQ~y for all yeAkL~(A). Suppose that yP~b for some yeAkL~(A). Then, by Lemma 4.2, there is an x ¢ L,,(A) such that xP~y. By transitivity of P~, xP~,bwhich contradicts
bQ z.
o
Fuzzy binary rehuions
31
The dichotomy of Theorem 4.1 can he easily expressed in terms of the membership function ~(x, y). Indeed, xP~b if and only if ~(b, x) < ~; ~herefore, the two cases of Theorem ~.1 are (1) ~(b, x) < ~, and (2) Q(b, ~) ~>~. It follows from Theorem 4.1 that, in order to obtain L~(B) from L~(A), one needs to compare b w/th elements of L~(A) only. We introduce now an algorithm, based on the resu~ts of Theorem ~.1, which effectively constructs L,(A)(See also [2] where a sin~lar algo~thm is suggested in
the crisp case.) Suppose that A = { a t , . . . , a,}, and consider a nested family of subsets A t c A 2 c z ' " ~ A , - - A , where A~={at,...,a~}. Given a set L~(A~){ a t , , . . . , at.}, the following a|gorithm constructs L~(A~+~). Step 1. Initialize the counter i of elements in L(Ak) to 0: i = 0. Step 2. Increment the ~unter by 1: i :--- i ÷ 1. Step 3. Check if there are more elements in L~(Ak) to compare with ak÷l: ff i < m then go to Step 4, otherwise go to Step 7. Step 4. Check if a~P~ah÷~:if Q(ak÷~,a~) < ~ then go to Step 8, otherwise go to Step 5. Step 5. Check if ak,$~ak+~:if Q(ak, ak÷~)~~ then go to Step 2, other~se go to Step 6. Step 6. Delete element ak, from L~(Ak) and go to Step 2. Step 7. Add element ak+t to the remaining elements in L~(Ak). Stop. Step 8. L,(A~+t)- L~(At). Stop. Obviously, L~(A1) -- AI. Then the above a|gorithm is applied recursively to the nested family At = A 2 c ' ' "=An=A until the ~-cut of the noninferior set L(A) is constructed. The algorithm has several distinctive features. In particu|ar, the algorithm makes it possible to identify (1) noninferior decision alternatives independently of the specific way in which the fuzzy binary relation Q is given; (2) noninfe~or decisions alternatives when the set of feasible decision alternatives is specified as a list and when ~easib|e alternatives are generated in the course of so|ring the decision-making problem; (3) a required number of decision alternatives (or a numbe~ close to the required one) for more detailed study in situations where the e|imination of dominated alternatives is inexpedient. References [1] M.A. Aizonnan and A.V. MaHshcvs~, General theory of best variants choice: Some aspects, IEEE Trans. Automat. Control ~ (1981) 1030-1040. [2] M.G. Gaft and V.M. Ozernoy,Isolation of a set of noninfe~or solutions and their estimates in decision-making problems With a vector-valuod c~tc~on, Automat. Remote Control 34 (1974) 1787-1795. [3] C.W. Fdrkwood and R.K. Satin, Ranking with partial infornmtion: A method and application, Oper. Res. 33 (1985) 38-48.
32
$. V. ~ v ,
V.M. Ozernoy
S.A. ~ , D e c i ~ n - ~ g ~ t h a f u ~ p~femnce rela~on, Fuzzy Sets and Systems I (1~s) ~--167. $ ~ of ~ binary r~lations, Fuzzy Sets and Systems 6 (1981) 169-195. [5] S.V. ~ o v , [6] V.M. Ozemoy and M.G. Gaff, Mul~rRerion decision problems, in: D.E. Bell, R.L. Kceny, and H. Ra~, Eds., Conflicting Objectives in Decisiom (John Wiley, London, 1977) 17-39. and P. Vincke, Fury preferences in an optimization perspective (unpublished). [7] M. ~ choice and Social w e m ~ (Oliver & ~ y a , ~ b ~ h - L o n d o n , 1970). [8! ~.K. Sen, ~ ~ and revealed preference, Rev. EcOnom. Stnd, 38 (1971) 307-317. [9] A.K. Sen, ~ approach to decision m~n8 with fuz~ sets, IEEE [I0] R.M. Tong and P.P. Bonimone, A ~ c Trans. Systems Man Cybemet. 10 (1980) 716-723. [11] L.A. Z~eh, Optimality and non-sealmr valued performance criteria, IEEE Trans. Automat. Control (1963) 315-316. [hi L.A. Zadeh, The l i ~ t i ¢ approach and its applications to de¢iu'on analysis, in: Y.C. Ho a ~ S.K. Mitter, Eds,, Dize~ons in Lard-Scale Systems (Plenum Press, New York, 1976) 339--361. S. Zionts, Multiple criteria nmthemati~ programmi~: An overview lind several approaches, in: G. Fande| and $. Spronk, Eds., Decision Methods and AppH~tions (Springer, Berlin, 1985) 85-128.
21
USD~IG F U Z Z Y BII~ARY g ~ L A T I O N $ F O R I D ~ ~ I ~ G NO~UOR D l g C ~ l O N .gLTE~'~IATIVES Serge V. OVCHINNIKOV* San F r c a c ~ State Uniwes~, San Franc/sco, CA 94132, USA
Vladimir M. OZERNOY CeJifomia State Unieers'mp, Hayward, C.~#i94542, USA Received, January 1986 Re~ April 1986 A ~ noninferiority relation is introduced. B~sed on this relation, fuzzy relations of preference, strict preference, indifference, and incomparability are specified, and their pmpe~ea are investigated. A f~gzy noninfefior sub~t is defined and iis basic ch~actefisfic prope~ea are established. An algorithm is presented for identifying a f~.y ~et of noninferior alternatives in decision-making situations that require the selection of a certain number of noninferior alternatives for more detailed study.
Keywords: Noninferiolity relation, Strict preference, Preference, Indifference, Incomparability, Noninferior subset.
L ~~u~on
In many decision-making problems the purpose of the analysis is to determine the 'best' decision alternative, in such problems a complete rank-order of the set of feasible alternatives is not needed. lns~¢a ~, d , the analysis of the problem usually starts with the identification of all nondominated alternatives. The most preferred alternative is then selected from among nondominated alternatives, while all dominated alternatives are rejected. ,,Another decision-making problem involves the selection of a relatively small number of preferred decision alternatives for more detailed study. The preferred alternatives are chosen from among nondominated alternatives or from among top rank-ordered alternatives. In this case the remaining alternatives are also rejected. Several methods have been proposed for the identification of nondominated alternatives. Oaft and Ozernoy [2] develop an algorithm that makes it possible to identify nondominated altern•ives independently of the specific way in which binary relations are introduced that would define concepts such as 'more preferred alternative' and 'set of nondominated alternatives'. The algorithm ex#icitly uses a concept of incomparability of alternatives. Methods developed for the selection of the 'best' alternative from among * Research sponsored by NSF Grant IST-8~3431. 0165-0114/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)
22
S.V. O v c ~ k o v ,
P.M. Ozemoy
nondominated alternatives as well as methods developed for the selection of a relatvely small number of preferred altematves use additonal information about file decision maker's preferences. For example, Kirkwood and Sarin [3] derive conditions to determine whether two multcriteria alternatives can be ranked the wei~t~,additive ev~uaton ~ ~ o n , They also develop ~ al that p ~ a l l y ~ k ~ r d e r s the set of alternatives based on the p~rwise r a n ~ g information. Ozernoy and GaR [6] present a maR|step procedure that permits at each step at least parti~ identification and eli~nation of dominated alternatives by obt~ning a sequence of ~fferent types of preference information from the decision maker. Zionts [13] points out that there may be some instances where the elimination of dominated alternatives is inexpedient. For example, a dominated alternative may be sufficiently dose to a nondominated altematve in the criteria space so that the decision maker might make a choice based on some secondary criteria not used in the analysis. In addition, attempts to precisely quantify alternatives would sometimes give an unrealistic view of the precision with which alternatives could be evaluated. In such situations, dominated aRernatives should not be excluded from further analysis. In this paper, we introduce a new approach for | d e n t i n g nondominated decision alternatives. This aproach is based on the use of a fuzzy noninferiority relation which allows to determine a fuzzy noninferior subset of a finite set. The value of the membership function is regarded as the degree to which an element is noninferior in the finite set. A procedure is deve|oped for the selection of noninferior elements. Thus, considering the finite set as a set of feasible decision alternatives in a decision-making problem, we prove that the 1-cut of a noninferior subset is a nonempty set of unfuuy undominated alternatives in the set of feasible alternatives, reestablishing the main result of Orlovsky [4]. The work in this article establishes an approach that allows one to identify different numbers of noninferior decision alternatives based on different v~ues of in constructing an ~ c u t of a noninferior subset. This is of particular interest since by the change of a value of ~ one can identify a required number of noninfer/or alternatives (or a number close to the required one) for more detailed
study.
This paper is organized as follows. Section 2 introduces a fuzzy noninferiority relation and its prope~es. Based on this relation, the fuzzy relations of strict preference, indifference, incomparability, and preference are defined. A theorem is proved which establishes some compatibility properties for these relations. ]t is also shown that the inverse image of a crisp nondominance relation under fuzzy correspondence is a fuzzy noninferiority relation- a result which is of particular interest in the area of ftgzy multi-criteria decision-making. In ~ t i o n 3 a ~ noninferior subset is defined. It is shown that two noninferior elements are either incomparable or equiva|ent. Basic characteristic properties of a fuzzy noninferior sub~et are specified. These are f u r y analogs of res~cfive p r o p e l s of crisp choice functions. Section 4 presents an algorithm for constructing an ~-cut of a noninferior subset and identf3~g noninferior elements.
Fuzzy binary relaaons
23
Let Q be a fuzzy binary relation on a finite set A. This relation is defined by its membership function Q(x, y), which maps a direct product A × A into the unit interval [0,1]. We regard this relation as an noninferiori~ relation: the value Q(x, y) represents the degree to which element x is not inferior to y. This interpretation imposes: certain restrictions~ on Q, which we introduce as properties of this relation. First of all, Q is a reflexive fuzzy binary relation, i.e.
Q(x, x) = 1 for all x cA, meaning that any element in A is certainly not inferior to itself. The second property is completeness (strong linearity in [4]) which is defined as follows: Q(x, y) v Q(y, x) = I for all x, y E A. (We define, as usual) a v b ~=max(a, b) and a ^ b = rain(a, b).) This property guarantees that, for any two given elements in A, certainly at least one of them is not inferior to another one. There are many ways to define an antisymmetric part of a fuzzy binary relations [4, 5, 7]. In this paper we adopt the following definition [4]: P(x, y) - (Q(x, y) - Q(y, x)) v 0 for all x, y ¢ A, which corresponds to the crisp definition given by P = Q \ Q-!. This relation is regarded as a strict preference associated with the noninferiofity relation Q. As usual, such a relation is supposed to be a transiave fuzzy binary relation, i.e. (PP)
P(x, y) ^ P(y, z) <<.P(x, z) for all x, y, z ¢ A.
Note that, by completeness of Q, P is a fuzzy complement of Q - I i.e.
P(x,y)---l-Q(y,x)
forallx, y a A .
In the crisp case, reflexive, complete binary relations with transitive antisymmetric parts are called quasi.transitive relations [8]. We use the same name for fuzzy binary relations satisfying all the introduced properties and, following [8], denote transitivity property of strict preference by (PP). Therefore, by definition, a fuzzy quasi-transitive relation is a reflexive compiete fuzzy binary relation satisfying the (PP) property. It is easy to verify that a reflexive ~ y binary relation Q is a fuzzy quasi-transitive relation if and only if the following property is sastisfied (QT)
Q(x,y) v Q(y, z) >~ Q(x, z),
for all x, y, z ~ A. A symmetric par~ S of a fuzzy binary relation Q is defined by
S(x, y) = Q(x, y) ^ Q(y, x) for all x, y ~ A. The value S(x, y) is considered to be a degree to which elements x and y are incomparable or indifferem. In order to separate incomparability and indifference subrelations of $ we
24
S.V. Ovchinnikov, V.Mo Ozernoy
&fine an/nd/fference relation I as a crisp equivalence relation contained in $. M e n l(x, y) is a function with range {0, 1}, such that
O) (2) O) (4)
Z(x, z ) = 1, ICx,y) ffi 1(y, x), z(x, y)^ z(y, z ) ~ z(y, z), x(x, y) ~ s(x, y),
for ~ x, y, z cA. We denote the transitivity property 0 ) of the indifference relation by (II). An incon~rability relation N is defined by its membership function as follows
N(x,y)ffiS(x,y)-l(x,y)
for all x, y cA.
It is e u y to veri~ that N is an an~flexive and symmetric ~ binary relation, i.e. N(x,x)~O and N(x,y)=N(y,x), for all x, y cA. We also have the decomposition $ m N U 1, such that N N 1 m ~. Since an indifference relation I is not defined by means of the relation Q, ~ r t ~ comp~bdity conditions must be imposed on I in order to guarantee properties similar to crisp ones (see [2]). We employ the following compatibility ¢on~tions:
(QI) (IQ)
Q(x. y) ^ l(y, z) ~ Q(x, z), I(x, y) ^ Q(y, z) ~ Q(x, z),
for all x, y, z ¢ A, which are similar to the conditions used in [8]. A fuzzy preference relation R is defined as the union P U I of a strict preference "''~ ce P an~ l"Ii ~neren relation L i.e.
R(x, y) = P(x, y) v1(x, y) for all x, yEA. Further, in this paper, we assume that the noninfefiofity relation Q and indifference relation I satisfy properties (QT), (lI), (QI) and (IQ). (Note that, in many important special cases, I is an identity relation; then conditions (]I), (QI), The following theorem establishes some compatibilRy properties of the strict preference relation P, the preference relation R, the indifference relation 1, and the incomparability relation N (of. [2, 8]). First, we prove a useful lemma.
2.1. 1(y, z) = 1 ~ t i e ~ Q(x, y) = Q(x, z) aria Q(.v, x) = O(z, x). ~ .
By (QI), we have
Q(x, y) = Q(x, y) ^ 1(y, z) ~ Q(x, z) auo Q(x, z) = Q(x, z) ^ l(z, y) ~ Q(x, y). Hence, Q(x, y) = Q(x, z). Similarly, by (IQ), Q(y, x) --- Q(Z, x). E! 2oL Conditions (PP), (II), (QI) and (IQ) togethe~ impgy if, I) (n,)
e(x, y) ^ ~(y, z j ~ p(x, z), Z(x, y) ^ P(y, z)<~ p(x, z),
Fuzzy bina~ relations
(ND :~N) (RR)
N(x, y) ^ 1(y, z ) ~ ~(x, z), 1(x, y) ^ U(y, z) ~ u(x, z), R(x, y) ^ R(y, z) ~ R(x, z),
for all x, y, z e A.
Proof. (PI) Obvious, if l(y, z)ffi 0. Otherwise, by Lamina 2.1, e(x, y) ffi(Q(x, y) - Q(y, z)) v o = (Q(x, z) - Q(z, x)) v o-- ~'(x,z).
(IP) Similar to (PI). (NI) Obvious, if l(y, z ) = 0 or N(x, y ) = O. Otherwise, l(y, z ) = 1 and I(x, y) -- 0. Therefore, by transitivity of I, l(z, x) = O. By l.emma 2.1, we have
N(x, y) = s(x, y) - 1(x, y) = Q(x, y) ^ Q(y, x) ,o Q(x, z) ^ ¢2(z, x) = s(x, z) - v,(x, z) = N(x, z).
(IN) Similar to (NI). (RR) Follows from
R(x, y) ^ n(y, z) --- (P(x, y) v I(x, y)) ^ (e(y, z) v I(y, z)) = (P(x, y) ^ e(y, z)) v (P(x, y) A l(y, z)) v if(x, y) ^ e(y, z))
v if(x,y) ^ I(y, z))~
Q(x, y)-- V F(x, u) ^ Q(u, v) ^ F(y, ~,)= ~ F(x, u) ^ F(y, v). U~U
Lamina 2.2. The inverse image Q of a crisp quasi-transiave relation Q on B under fuzzy mapping F is a reflexive and complete fuzzy binary relation on A.
Pn~f. (1) Q(X, x) -- VuOv F(x, u) ^ F(x, v) <~Vu F(x, u) = 1, sin~ Q is a reflexive binary relation and F(x) is a normal fuzzy subset.
$.V. O v c ~ v ,
V.M. Ozemoy
(2) Let uo¢ F~(x) and Vo~ Fdy). We have
~>IF(x, Uo) ^ Q(Uo, Vo) ^ F(y, Vo)] v [F(y, no) ^ Q(Vo, uo) ^ F(x, Uo)] = Q(Uo, Vo) v (}(vo, Uo)= 1, since Q is a complete binary relation.
[3
~ ~ 2.2. The inverse image of a crisp quasi-trans~v¢ binary relation Q under fuzzy correspondence F is a fuzzy quasi~ransitive relation Q. Proof. By Lemma 2.2, it su~ces to prove that Q satisfies the (QT) property
or that, for any given s and t such that sQt, there are u and q such that uQ~, and
sQq,
IF(y, u) ^ F(x, t)] v IF(z, s) ^ F(y, q)] ~ F(z, s) ^ F(x, t). Let us prove, first, that there are Uo and qo such that uoQt, sQqo, and F(y, Uo) v F(y, qo) = I. Suppose that F(y, u) < I and F(y, q) < I for all u and q such that uQt and sQq. Then, for any w ¢ Fj(y), wQt and sQw are both false, which implies both tPw and wPs. Therefore, tPs, which contradicts sQt. We have now
IF(y, Uo) ^ F(x, 01 v [F(z, s) ^ F(y, qo)] [F(y, uo) v F(z, s)] ^ [~(y, Uo) v F(y, qo)] ^ [F(x, 0 v F(z, s)] ^ IF(x, t) v ~(y, qo)] F(z, s) ^ [F(x, 0 v F(z, s)] ^ F(x, t) = F(z, s) ^ F(x, 0. n In practice, A is a set of decision alternatives, B is a subset of finite dimensional real space ('criteria' space), and Q is a vector nondominance relation. It is assumed that ~ y information about alternative a e A is given in the form of a normal ~ subset F(a) of B representing the value of a linguistlc va~ab|e. For example, this f u r y subset m i ~ t represent the following judgement about alternative a: "alternative a is more or less good with respect to criterion C~ and f~r ~ t h respect to criterion C2". A fuzzy set F(a) is regarded then as a fuzzy ~ a g e of alternative a e A under ~kzzy correspondence F between A and B. In accordance ~ Theorem 2.2, the inverse image Q of Q under F is a noninfefiority relation on the set of alternatives a. Therefore, fuzzy information
Fuzzy binary relations
27
about alternatives generates a noninfer~ofity relation by means of the inverse image construction. (Note that, in the one-dimensional case, the so defined noninferiofity relation is the dominance relation R8 introduced in [10] by means of a different construction.) 3. Fuzzy n ~ e d o r
~but
Given noninfefiority relation Q on a finite set A we define a f u r y nom'nferior subset L(A) by means of its memben~hip function L(A)(x) as follows:
L(A)(x) = y~A Q(x, y) for x cA. The value L(A)(x) is regarded as a degree to which an element x is noninferior in a. Since Q is a complement of p - i a noninferior subset defined above coinci~es with the fuzzy set of nondominated elements introduced in [4]. (Note that, in the particular case when Q is generated as an inverse image of the vector nondominance relation (see previous section), the so defined noninferior subset can be regarded as a fuzzy Pareto set of nondominated alternatives.) The follo~ng theorem shows that a noninferior subset is, actually, a normal fuzzy subset of A (el. Theorem 6.4 in [4])
3.1. V , , , L(A)(x)
-
1.
~f. Suppose that ~,eA L(A)(x)< 1. Then for any x e A there is y e A such that Q(x, y ) < 1. Then, by finiteness of A, there is a cyc|e Xo, x ~ , . . . , x, = Xo, such that Q(x, x~+l) < 1, for 0 ~
1 > Q(Xo, a contradiction.
x,)
Q(xo, xo) = 1,
[21
In accordance with the theorem proved, the t-cut of a noninferior subset is a nonempty crisp set; the elements of this crisp set can be regarded as nonfuzzy undominated elements in A (el. [4]). A noninferior subset L(A) is compatible with the indifference relation 1, as the following theorem demonstrates. Theorem 3.2. l(x, y ) = l implies L(A)(x)= L(A)(y). ~oL
Follows immediately from Lemma 2.1.
[3
]n the crisp case two noninfe~or elements are either incomparable or equivalent [2]. It is also true in the framework developed in this paper, as the next theorem shows.
Theorem 3.3. L(A)(x) ^ L(A)(y)~ N(x, y) v l(x, y).
28
$.V. O v c ~ v ,
~ .
V.M. Ozernoy
Since L(A)(u) <~Q(u, t) for all t e A, we have L(A)(x) ^ L(A)Lrj <- fl(x, Y) ^ Q(y, x) f N(x, y) v l(x, y).
El
Considering a noninferior subset L(A) as a function of A, where A is a finite subset of a certain universe set, one can regard L(A) as a fuzzy choice function [5]. Then the value L(A)(x) is a degree to which x is the 'best' element in the set A. The next three theorems establish basic characteristic prope~es of the: fuzzy choice function L(A).
T b o t e m 3,4. A' G A implies L(A °) ~ L(A) f3 A'.
~ .
For x E A', we have
z(A')(x) = , ~ Q(x, 0
;',~
Q(x, 0 = L(A)(x).
E!
The property established in the theorem proved in Sun's property (~r) [8]; it is called Heritage property (H) in [1].
T h e o ~ 3.5. L(A' U A") ~_L(A') N L(A"). ~f.
For x EA' OA",
= L ( A ' U A")(x).
Otherwise, L(A')(x) A L(A")(x)=0.
E!
The proved property is the Concordance property (C) in [1] or property (y) in
[9]. Finally, we establish a property which can be regarded as a fuzzy analog of the classical 'independence of irrelevant alternatives' property. Theorem 3.6. Let A ' be a subset of A such that L(A)(y) < L(A)(x) for all x e A' and y e A \A'. Then L(A') ffi L(A) N A '. Preef. From Theorem 3.4, L(A') ~ L(A) N A ~. Suppose that there is x E A' such that L(A')(x) > L(A)(x), i.e.
A O(x, 0 > , ~ Q(x, 0.
t~A ~
It means that there exists a y ~ A \ A ° such that, ~o~~~=l u ~ A',
Q(x, y) < Q(x, u).
(,)
since L(A)(y) < L(A)(x), there is a y, suchthat Q(y, y~) < L(A)(x). If y, ~A', there is a y= such that Q(y~, yz)< L(A)(x), and so on. Since A is a finite set and there is no cycle with the property Q(y~,y ~ . t ) < 1 (see the proof of Theorem 3.1),
Fuzzy b~.ry reigns we construct a sequence Yoffi Y, Y~,..., Yn such that y~ ¢ A \ A ' for i < n, y~ ¢ A', and Q(y~, y~+~)
Q(y, y,) V| Q(Y,,
<
L(A)(x) Q(x,t).
In particular,Q(y, y,,)< Q(x, y~), which implies,together with (,), Q(x, y) v 0(Y, Y,) < Q(x, y,) - a contradiction with the (QT) property, v! Recall that L(A)(x) is considered to be a degree to which an element x ~ the 'best' in A. Let us suppose that a threshold ¢ is given such that elements x with L ( A ) ( x ) < ¢ are deleted from consideration (irrelevant alternatives). ~ e n Theorem 3.6 states that the (fuzzy) 'best' elements in the remainder set A' are exactly the same as in the given set A. Actually, A' is an ~-eut of f e ~ y ~ t L(A). An algorithm constructing the ~-cuts of a noninferior subset L(A) is presented in the next section. In the conclusion of this section, we establish necessary and su~cient conditions on a fuzzy binary relation Q to generate normal noninfefior subsets. We say that a fuzzy binary relation P is an acydic relation if, for any finite sequence X o , . . . , x,, P(x~, xj+l) > 0, for all 0 ~ i ~ n - 1, implies P(x,, Xo) = O.
"l~¢o~m 3.7. Let Q be a fuzzy binary relation.A noninferiorsubset L(A) defined by its membership function L(A)(x)= A Q(x, y) yea
is a normal fuzzy subset for any finite set A if and only if (~ is a i~'~qexivecomplete fuzzy binary relation with acyclic antisymme~ic part P. Preefo Sufficiency is proved, essentially, by repeating the argument used in the proof of Theorem 3.1. To prove necessl"ty, we note first that, for any element x, 1 = l,((x})(x) ~ Q(x, x) which implies reflexivity of Q. Second, we have, for A -- {x, y},
I = L(A)(x) v L(A)(y) ffi(Q(x, x) ^ Q(s, y)) v (Q(y, x) ^ Q(y, y))
= Q(x, y) v O(y, x) which proves completeness of Q. Finally, let A -- {Xo,..., xn} and P(x~,x~+1)> O for all 0 ~ i ~ n - 1. Therefore, Q(x~+l, x~) < 1 for all 0 ~ i ~ n - 1. Hence, L(A)(xj) < 1 for all 0 ~ i <~ n - 1. Since L(A) is a normal fuzzy set, L(A)(x,) = 1, which implies Q(x,, Xo)= 1. Therefore, P(x,, xo)=0, i.e. P is an acyclic fuzzy binary relation. O
By definition, an ~-cut of a fuzzy set L(A) is a crisp set L~(A)= {x A: L(A)(x) >~~). Since L(A)(x) = A,~A Q(x, y), we have
L~(A ) = {x cA" (Vy e A ) xQ,~y },
30
8.V. ~ m k o v ,
where Q,, is an C-cut of ~
V.M. Ozemoy
binary relation Q, i.e. xQ~,y if and only if
We introduce a symmetric $,, and an antisymmetric P~, parts of Q,, as follows:
xS~y if and only if xQ~y and yQ,,x, and
xP,,y if and only if xQ~,y and not yQ,~c. Note, that P, is not an e-cut of fuzzy strict preference P, while $,,/s an ~-cut of $.
4.1. Q~ is a c~sp quasi.transiti've binary relation. Proof. Obviously, Q~ is a reflexive complete binary relation. Let us prove transitivity of P¢. By definition, xP~y if and only if Q(y, x ) < ~. Suppose that xP, y and yP~z, or, equivalently, Q(y, x) < o~, and Q(z, y) < ¢. Then, by the (QT) property, ~ > Q(z, y) v Q(y, x) ~ Q(z, x). Hence, xP~z. [3
4~. For any y ¢ A \L~(A), ~ere is an x ¢ L~(A) such that xP~y. ~ . Since Area Q(Y, t) ffi L(A)(y) < ~, there e~sts a Yl such that Q(y, yl) < o~. Hence, YtP~Y. If y~ f L~(A), then there exists a Y2such that YzP~Yt, and so on. Since P~ is a partial order and A is a finite set, there is an x ¢ L~(A) such that
xP~y. 0 Let B = A U {b}, b ¢ A, so that B has just one more element than A. Then, from Theorem 3.4, L~(A)~L~(B)NA. Therefore, any element in L~(B) is either b, or an element of L~(A). The following theorem establishes important relationships between sets L~(B) and L~(A) in terms of relation P~,. 4.L (i) If there ~ an x ¢ L~(A) such th~ xPab, then L~(B) = L~(A); (ii) otherwise, L~(B)=(L~(A)XY)O {b}, where Y = {x ¢ L,~(A): bPax}.
Tk~
Proof. (i) Since xP~b, bQ~t is false. Therefore, b ~ L~(B). Suppose that there is a t ¢ L,,(A) such that t ~ L~(B). Then bP~t, which implies, by transitivity of P~, xP~t. Hence tQ,~x is false, which contradicts t ¢ L~(A). (ii) In this case, for all xcL~(A) we have b Q ~ . Hence, L ~ ( A ) \ Y f { t ¢ L,~(A): b$~t}. It suffices to prove that (1) any element of Y does not belong to L (B), (2) any element of L (A)\ Y belongs to L,,(B), and 0 ) b belongs to Lffi(B). (1) Let t ¢ Y. Then bP~t. Therefore, ~,~b is false. Hence, t does not belong to
L.(a).
(2) Let t ¢ L~(A) \ Y. Then tQ~.v for all y ¢ A and tQ~b. Therefore, t belongs (3) Since b Q ~ for all x ¢L~(A), "it suffices to prove that bQ~y for all yeAkL~(A). Suppose that yP~b for some yeAkL~(A). Then, by Lemma 4.2, there is an x ¢ L,,(A) such that xP~y. By transitivity of P~, xP~,bwhich contradicts
bQ z.
o
Fuzzy binary rehuions
31
The dichotomy of Theorem 4.1 can he easily expressed in terms of the membership function ~(x, y). Indeed, xP~b if and only if ~(b, x) < ~; ~herefore, the two cases of Theorem ~.1 are (1) ~(b, x) < ~, and (2) Q(b, ~) ~>~. It follows from Theorem 4.1 that, in order to obtain L~(B) from L~(A), one needs to compare b w/th elements of L~(A) only. We introduce now an algorithm, based on the resu~ts of Theorem ~.1, which effectively constructs L,(A)(See also [2] where a sin~lar algo~thm is suggested in
the crisp case.) Suppose that A = { a t , . . . , a,}, and consider a nested family of subsets A t c A 2 c z ' " ~ A , - - A , where A~={at,...,a~}. Given a set L~(A~){ a t , , . . . , at.}, the following a|gorithm constructs L~(A~+~). Step 1. Initialize the counter i of elements in L(Ak) to 0: i = 0. Step 2. Increment the ~unter by 1: i :--- i ÷ 1. Step 3. Check if there are more elements in L~(Ak) to compare with ak÷l: ff i < m then go to Step 4, otherwise go to Step 7. Step 4. Check if a~P~ah÷~:if Q(ak÷~,a~) < ~ then go to Step 8, otherwise go to Step 5. Step 5. Check if ak,$~ak+~:if Q(ak, ak÷~)~~ then go to Step 2, other~se go to Step 6. Step 6. Delete element ak, from L~(Ak) and go to Step 2. Step 7. Add element ak+t to the remaining elements in L~(Ak). Stop. Step 8. L,(A~+t)- L~(At). Stop. Obviously, L~(A1) -- AI. Then the above a|gorithm is applied recursively to the nested family At = A 2 c ' ' "=An=A until the ~-cut of the noninferior set L(A) is constructed. The algorithm has several distinctive features. In particu|ar, the algorithm makes it possible to identify (1) noninferior decision alternatives independently of the specific way in which the fuzzy binary relation Q is given; (2) noninfe~or decisions alternatives when the set of feasible decision alternatives is specified as a list and when ~easib|e alternatives are generated in the course of so|ring the decision-making problem; (3) a required number of decision alternatives (or a numbe~ close to the required one) for more detailed study in situations where the e|imination of dominated alternatives is inexpedient. References [1] M.A. Aizonnan and A.V. MaHshcvs~, General theory of best variants choice: Some aspects, IEEE Trans. Automat. Control ~ (1981) 1030-1040. [2] M.G. Gaft and V.M. Ozernoy,Isolation of a set of noninfe~or solutions and their estimates in decision-making problems With a vector-valuod c~tc~on, Automat. Remote Control 34 (1974) 1787-1795. [3] C.W. Fdrkwood and R.K. Satin, Ranking with partial infornmtion: A method and application, Oper. Res. 33 (1985) 38-48.
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$. V. ~ v ,
V.M. Ozernoy
S.A. ~ , D e c i ~ n - ~ g ~ t h a f u ~ p~femnce rela~on, Fuzzy Sets and Systems I (1~s) ~--167. $ ~ of ~ binary r~lations, Fuzzy Sets and Systems 6 (1981) 169-195. [5] S.V. ~ o v , [6] V.M. Ozemoy and M.G. Gaff, Mul~rRerion decision problems, in: D.E. Bell, R.L. Kceny, and H. Ra~, Eds., Conflicting Objectives in Decisiom (John Wiley, London, 1977) 17-39. and P. Vincke, Fury preferences in an optimization perspective (unpublished). [7] M. ~ choice and Social w e m ~ (Oliver & ~ y a , ~ b ~ h - L o n d o n , 1970). [8! ~.K. Sen, ~ ~ and revealed preference, Rev. EcOnom. Stnd, 38 (1971) 307-317. [9] A.K. Sen, ~ approach to decision m~n8 with fuz~ sets, IEEE [I0] R.M. Tong and P.P. Bonimone, A ~ c Trans. Systems Man Cybemet. 10 (1980) 716-723. [11] L.A. Z~eh, Optimality and non-sealmr valued performance criteria, IEEE Trans. Automat. Control (1963) 315-316. [hi L.A. Zadeh, The l i ~ t i ¢ approach and its applications to de¢iu'on analysis, in: Y.C. Ho a ~ S.K. Mitter, Eds,, Dize~ons in Lard-Scale Systems (Plenum Press, New York, 1976) 339--361. S. Zionts, Multiple criteria nmthemati~ programmi~: An overview lind several approaches, in: G. Fande| and $. Spronk, Eds., Decision Methods and AppH~tions (Springer, Berlin, 1985) 85-128.