- Email: [email protected]
On generalized convexity of fuzzy variables maps
Applied Mathematics and Computation 157 (2004) 65–75 www.elsevier.com/locate/amc
On generalized convexity of fuzzy variables maps E.E. Ammar Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt
Abstract In this paper, we investigate the convexity and quasiconvexity of fuzzy functions by use the interval analysis. Some properties of convex and quasiconvex fuzzy functions are introduced. Applications of fuzzy convexity and quasiconvexity to optimality are discussed. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Fuzzy variables; Fuzzy functions; Convexity; a-level sets; Interval analysis; Optimality
1. Introduction The concept of convexity is important for quantitative and qualitative studies in operation research and helps to find optimal solutions. This was considered by many authors also in fuzzy optimization. The convexity of fuzzy sets was introduced by many authors like Zadeh [15], Brown [2], Drewniak [3], Lowen [5], Liu [4] and Ammar [1]. For the convexity of fuzzy functions there are a little interested papers. Puri and Ralescu [8] define the concept of the differential of fuzzy function which extends the differential of a set-valued function. Mario and David [6] introduce the notion of the differentiability of fuzzy continuous mappings defined an fuzzy topological vector space. Nada [7] introduces the concept of convexity and logarithmic convexity with some properties of fuzzy mappings. Syau [10– 12] gives the definition of fuzzy convexity and concavity of fuzzy mappings which corresponding to usual cases, some of fuzzy convexity and weakly E-mail address: [email protected] (E.E. Ammar). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.08.027
66
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
convexity are presented. Hong and Xu [13] introduced some properties of convex fuzzy optimization and characteristics of the convex and quasiconvex fuzzy mappings. Vijayaraju and Marudai [14] give some results on fixed point theorems for contractive fuzzy mappings. Sense the convexity of sets and functions [9] plays a central rule to characterized the optimally criteria in operation research. So, in this paper, we study the convexity and quasiconvexity of fuzzy function. In Section 2 we give a preliminaries of some basic notions. In Section 3, the definition and some properties of convex fuzzy functions are considered. In Section 4, quasiconvexity and strictly quasiconvexity are defined. Also qualitative analysis of the above notions in sense of interval analysis are given. In Section 5, an application of the convexity and generalized convexity of fuzzy functions to optimality are presented. In this paper, we use ^ and _ denoted to minimum (infimum) and maximum (superimum), respectively. 2. Preliminaries Assume that F ðRÞ denoted the set of all compact (bounded and closed) fuzzy numbers in the real line, i.e., if ~x 2 F ðRÞ, then ~x satisfies (i) ft: t 2 R; ~xðtÞ ¼ 1g 6¼ £. (ii) For every a 2 ½0; 1 ; ~xa ¼ ft: t 2 R; ~xðtÞ P ag ¼ ½xLa ; xU a is a finite closed on R, where R ¼ ð 1; 1Þ. Definition 1. A fuzzy point ~xa (fuzzy variables) in F ðRÞ is a membership function ~xa : R ! ½0; 1 such that t 2 R, a > 0 and a if t ¼ x; ~xa ¼ 0 else;
1 ~x(t)
α
~x L
t
~x U
R
where 0 < a 6 1. The point t is called the support of xa and a its membership degree [6]. The fuzzy point ~xa is said to be contained in, or to belong to, a fuzzy ~ within ~xa 2 A ~ iff ~xa ðtÞ 6 AðtÞ. ~ set A
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
67
Definition 2. Let ~x, ~y 2 F ðRÞ. We define that ~xa 6 ~ya iff ~xa ðtÞ 6 ~ya ðtÞ for every U a 2 ð0; 1 , i.e., xLa ðtÞ 6 yaL ðtÞ and xU a ðtÞ 6 ya ðtÞ Definition 3. For any two fuzzy numbers ~x0a ¼ ½~x0L x0U x00a ¼ ½~x00L x00U a ;~ a , ~ a ;~ a follows: ~x0a þ ~x00a ¼ ½~x0L x00L x0U x00U a þ~ a ;~ a þ~ a ;
ð1Þ
~x0a
ð2Þ
~x00a
¼
½~x0L a
~x00L x0U a ;~ a
~x00U a ;
k~x0a ¼ ½k~x0L x0U a ; k~ a ;
ð3Þ
~x0a ~x00a ¼ ½~x0L x0U x00L x00U a ;~ a ½~ a ;~ a h ¼ ~x0L x00L x0L x00U x0U x00L x0U x00U a ~ a ^~ a ~ a ^~ a ~ a ^~ a ~ a ; i 00L 0L 00U 0U 00L 0U 00U ~x0L ~ ~ ~ ~ ~ ~ ~ x _ x x _ x x _ x x : a a a a a a a a
ð4Þ
e ! F ðRÞð~x ! f ð~xÞÞ is said to be fuzzy real valuedDefinition 4. The map f : D e where D e is said to be the domain of definition for f ð~xÞ, function defined on D, and ~x is the independent fuzzy variable. Example 1. Let ~ a 2 F ðRÞ be a given fuzzy number; then e f1 ð~xÞ ¼ ~ a þ ~x for any ~x 2 D;
ð5Þ
e f2 ð~xÞ ¼ ~ a ~x for any ~x 2 D;
ð6Þ
e f3 ð~xÞ ¼ ~ a ~x for any ~x 2 D
ð7Þ
e are fuzzy real variable-valued functions defined on D.
3. Convex fuzzy functions e ! F ðRÞ is said to be convex fuzzy Definition 5. The fuzzy function f : D function if and only if f ðk~x0a þ ð1 kÞ~x00a Þ 6 kf ð~x0a Þ þ ð1 kÞf ð~x00a Þ;
ð8Þ
e 0 6 k 6 1 and a 2 ð0; 1 . where ~x0a , ~x00a 2 D, e ! F ðRÞ is said to be strictly convex fuzzy function if and The function f : D only if f ðk~x0a þ ð1 kÞ~x00a Þ < kf ð~x0a Þ þ ð1 kÞf ð~x00a Þ;
ð9Þ
e 0 < k < 1. where ~x0a , ~x00a 2 D, The fuzzy function is said to be concave (strictly concave) if f is convex (strictly convex).
68
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
e ! F ðRÞ is convex fuzzy function, ~x 2 D. e If we Lemma 1. Suppose that f : D define f ð~xÞ ¼ sup af ð~xa Þ; a>0
1. f ð~xÞ 2 F ðRÞ, 2. for any a 2 ð0; 1 , ðf ð~xÞÞa ¼ f ð~xa Þ ¼ ½f ðxLa Þ; f ðxU a Þ . Proof 1. Let ~xðx0 Þ ¼ 1 and y0 ¼ f ðx0 Þ, x0 2 Rn ; then y0 2 f ð~x1 Þ (i.e., at a ¼ 1), f ð~x1 Þ ¼ ff ðxÞ j x 2 ~x1 ¼ ½~xL1 ; ~xU 1 g then f ð~xÞy0 ¼ sup a f ð~xa Þðy0 Þ ¼ sup a ¼ 1: a>0
y0 2f ð~xa Þ
2. Since for any a 2 ð0; 1 follows: f ð~xa Þ ¼ ff ðxÞjx 2 ~xa ¼ ½~xLa ; ~xU a g and since f is convex on a convex a-level set ~xa , there exist f ðxLa Þ ¼
^x2~x f ðxÞ a
and
f ðxU a Þ ¼
_x2~x f ðxÞ a
such that ðf ð~xÞÞa ¼ f ð~xa Þ ¼ f ðxLa Þ; f ðxU a Þ :
e ! F ðRÞ is convex iff Theorem 1. A fuzzy function f : D ^ L ^ U bf ð^xLa Þ; f ð^xU a Þc 6 bf ðxa Þ; f ðxa Þc; where 00L U 00U ^xLa ¼ kx0L xa ¼ kx0U a þ ð1 kÞxa ; ^ a þ ð1 kÞxa ; 00L f^ ð~xLa Þ ¼ kf ðx0L a Þ þ ð1 kÞf ðxa Þ; 0U 00U f^ ð~xU a Þ ¼ kf ðxa Þ þ ð1 kÞf ðxa Þ
and
e ~x0a ; ~x00a 2 D:
e ! F ðRÞ is convex if Proof. From Definition 5 the function f : D f ðk~x0a þ ð1 kÞ~x00a Þ 6 kf ðx0a Þ þ ð1 kÞf ðx00a Þ; e 0 6 k 6 1. where ~x0 ,~x00 2 D,
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
69
The first side, using Lemma 1 and from (1) and (3) 0U 00L 00U f ðk~x0a þ ð1 kÞ~x00a Þ ¼ f ðk½x0L a ; xa þ ð1 kÞ½xa ; xa Þ 00L 0U 00U ¼ f ð½kx0L a þ ð1 kÞxa ; kxa þ ð1 kÞxa Þ
xLa Þ; f ð^xU ¼ f ð½^xLa ; ^xU a Þ; ¼ ½f ð^ a Þ ;
ð10Þ
00L U 00U where ^xLa ¼ kx0L xa ¼ kx0U a þ ð1 kÞxa ; ^ a þ ð1 kÞxa . The second side, using Lemma 1 0U 00L 00U kf ð~x0a Þ þ ð1 kÞf ð~x00a Þ ¼ kf ½x0L a ; xa þ ð1 kÞf ½xa ; xa 0U 00L 00U ¼ k½f ðx0L a Þ; f ðxa Þ þ ð1 kÞ½f ðxa Þ; f ðxa Þ 0U 00L 00U ¼ ½kf ðx0L a Þ; kf ðxa Þ þ ½ð1 kÞf ðxa Þ; ð1 kÞf ðxa Þ 00L 0U 00U ¼ ½kf ðx0L a Þ þ ð1 kÞf ðxa Þ; kf ðxa Þ þ ð1 kÞf ðxa Þ ð11Þ ¼ ½f^ ðxL Þ; f^ ðxU Þ : a
a
From (8), (10) and (11) the result follows. h e ! F ðRÞ is convex iff its fuzzy epigraph Theorem 2. A fuzzy function f : D e f ¼ fð~xa ; dÞ: ~x 2 D; e d 2 F ðRÞ; f ð~xa Þ 6 dg G e F ðRÞ. is convex fuzzy subset in D e f and ð~x00 ; d2 Þ 2 G ef. Proof. Let f be convex fuzzy function. Assume ð~x0a ; d1 Þ 2 G a Since f is convex fuzzy function, for each k 2 I ¼ ½0; 1 follows: f ðk~x0a þ ð1 kÞ~x00a Þ 6 kf ð~x0a Þ þ ð1 kÞf ð~x00a Þ 6 kd1 þ ð1 kÞd2 : Hence ef ðk~x0a þ ð1 kÞ~x00a ; kd1 þ ð1 kÞd2 Þ 2 G or ef: kð~x0a ; d1 Þ þ ð1 kÞð~x00a ; d2 Þ 2 G e f is convex. Thus G e f be convex. Assume ~x0 ; ~x00 2 D. e Then Conversely, let G a a ef ð~x0a ; f ð~x0a ÞÞ ¼ ð~x0 ; d1 Þ 2 G
and
ef: ð~x00a ; f ð~x00a ÞÞ ¼ ð~x00 ; d2 Þ 2 G
e f is convex, for each k 2 I follows Since G ef ðk~x0a ; ð1 kÞ~x00a ; kf ð~x0a Þ þ ð1 kÞf ð~x00a ÞÞ 2 G or f ðk~x0a ; ð1 kÞ~x00a Þ 6 kf ð~x0a Þ þ ð1 kÞf ð~x00a Þ: Hence f is convex fuzzy and this complete the proof.
h
70
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
Theorem 3. Let ðfj Þj2J be a family (finite or infinite) of convex and bounded fuzzy e then f ð~xa Þ ¼ _ fj ð~xa Þ is convex fuzzy function on D. e function on D, j2J
e their fuzzy epigraphs Proof. Since each is a convex fuzzy functions on D, e fj ¼ fð~xa ; dÞ j~x 2 D; e d 2 F ðRÞ; fj ð~xa Þ 6 dg G
ð12Þ
are convex fuzzy subset in Rnþ1 by Theorem 2, and hence their intersection \ e fj ¼ fð~xa ; dÞj~x 2 D; e fj ð~xa Þ 6 d; j 2 J g G j2J
is fuzzy convex set (since the intersection of any convex fuzzy sets is fuzzy convex). But this convex intersection is the epigraphs of f . Hence f is convex e by Theorem 2. h function on D e ! F ðRÞ be convex fuzzy functions. Theorem 4. Let f : F ðRÞ ! F ðRÞ and g : D If f is non-decreasing, then the function ~x ! f ðgð~xÞÞ is convex fuzzy function. e and k 2 ½0; 1 . Since g : D e ! F ðRÞ is convex, we have Proof. Let ~x0 , ~x00 2 D gðk~x0a þ ð1 kÞ~x00a Þ 6 kgð~x0a Þ þ ð1 kÞgð~x00a Þ: Since f : F ðRÞ ! F ðRÞ is non-decreasing and convex, it follows that f ðgðk~x0a þ ð1 kÞ~x00a ÞÞ 6 f ðkgð~x0a Þ þ ð1 kÞgð~x00a ÞÞ 6 kf ðgð~x0a ÞÞ þ ð1 kÞf ðgð~x00a ÞÞ: Hence ~x ! f ðgð~xÞÞ is convex.
h
4. Quasiconvex fuzzy functions e ! F ðRÞ is said to be quasiconvex fuzzy Definition 6. The fuzzy function f : D function if and only if f ðk~x0a þ ð1 kÞ~x00a Þ 6 f ð~x0a Þ _ f ð~x00a Þ
ð13Þ
e k 2 I and a 2 ð0; 1 . for any ~x0a ; ~x00a 2 D, The fuzzy function is said to be strictly quasiconvex if and only if relation (13) holds only with (<) for ~x0a 6¼ ~x00a , k 2 I 0 ¼ ð0; 1Þ. e ! F ðRÞ is said to be a fuzzy closed Remark 1. The fuzzy function f : D 1 function if and only if for a 2 I; f ½a; 1 is closed [6].
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
71
e ! F ðRÞ is quasiconvex if and only if the Theorem 5. A fuzzy function f : D fuzzy set e e f ð~xd Þ 6 ag 6 D e L fa ¼ f~x 2 D:
ð14Þ
is convex for each d; a 2 F ðRÞ. e Let ~x0 ; ~x00 any two fuzzy Proof. Let f be a quasiconvex fuzzy function on D. d d e e variables on L fa D a , a 2 ð0; 1 . Then for k 2 I, f ðk~x0d þ ð1 kÞ~x00d 6 f ð~x0d Þ _ f ð~x00d Þ 6 a; then k~x0d þ ð1 kÞ~x00d 2 e L fa . Hence e L fa is convex set. e Conversely, let L fa be convex fuzzy set. Then for any ~x0d ; x00d 2 e L fa and k 2 I, 0 k~xd þ ð1 kÞ~x00d 2 e L fa . Let ~x0d ; ~x00d 2 e L fa be such that f ð~x0d Þ 6 a and f ð~x00d Þ 6 f ð~x0d Þ 6 a. Now f ðk~x0d þ ð1 kÞ~x00d Þ 6 a ¼ f ð~x0d Þ _ f ð~x00d Þ. Thus f is a quasiconvex fuzzy function and this completes the proof. h e ! F ðRÞ be a fuzzy closed function. Then f is a quasiTheorem 6. Let f : D e then there exist convex fuzzy function if and only if for all ~x0d ; ~x00d 2 D, 0 k 2 I ¼ ð0; 1Þ; d 2 ð0; 1 , such that f ðk~x0d þ ð1 kÞ~x00d Þ 6 f ð~x0d Þ _ f ð~x00d Þ: Proof. From Theorem 5 it is sufficient to show that f 1 ½k; 1 is a convex set for all k 2 I ¼ ½0; 1 . In fact if that is not true, then there exists another c ¼ d 2 I, c < k, such that f 1 ½c; 1 is not convex set, that is there exist ~x0c ; x00c 2 f 1 ½c; 1 and k0 2 I such that ~x^k0 ¼ k0~x0c þ ð1 k0 Þ~x00c 2 f 1 ½c; 1 : Assume for c < k that ~ ¼ f 1 ½c; 1 ^ ½~x0 ; ~x^ ; A k0 ~ 0 ¼ f 1 ½c; 1 \ ½~x0L þ ~x^L ; ~x0U þ ~x^U A c c k0 c k0 and ~ 00 ¼ f 1 ½c; 1 \ ½~x00L þ ~x^L ; ~x00U þ ~x^U ; A c c k0 c k0 where b~x0L x^L x0U x^U x0c þ ð1 kÞ~x^k0 ; k 2 Ig c þ~ k0 ; ~ c þ~ k0 c ¼ fk~ and b~xLk0 þ ~x00L x^U x00U x^k0 þ ð1 kÞ~x00c ; k 2 Ig: c ;~ k0 þ ~ c c ¼ fk~
72
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
~ 0 and A ~ 00 are bounded e from [8] it follows A Since f is fuzzy closed function D, c c ~ 0 , ~x^ 2 A ~ 00 . Thus there exists ~x0 2 A ~ 0 and ~x00 2 A ~ 00 closed fuzzy sets and ~x^k0 62 A c k0 c c c such that min kð~xLc ðtÞ ~x^L xU x^U k0 ðtÞÞ; ð~ c ðtÞ ~ k0 ðtÞÞk ~ 0c t2A
x^L x0U x^U ¼ kð~x0L c ðtÞ ~ k0 ðtÞÞ; ð~ c ðtÞ ~ k0 ðtÞÞk; min00 kð~xLc ðtÞ ~x^L xU x^U k0 ðtÞÞ; ð~ c ðtÞ ~ k0 ðtÞÞk ~ t2A c
¼ kð~x00L x^L x00U x^U c ðtÞ ~ k0 ðtÞÞ; ð~ c ðtÞ ~ k0 ðtÞÞk: Hence x^L x0U x^U f 1 ½c; 1 \ ð~x0L c þ~ k0 ; ~ c þ~ k0 ¼ ; and f 1 ½c; 1 \ ½~x^L x00L x^U x00U k0 þ ~ c ;~ k0 þ ~ c Þ ¼ ;; that is f 1 ½c; 1 \ ½~x0L x00L x0U x00U k0 þ ~ c ;~ k0 þ ~ c Þ ¼ ;:
ð15Þ
On the other hand, by the assumption of the theorem, there exists an k 2 I 0 f ðk~x0d þ ð1 kÞ~x00d Þ 6 f ð~x0d Þ _ f ð~x00d Þ: Since ~x0c 2 f 1 ½c; 1 and ~x00c 2 f 1 ½c; 1 , we have f ðk~x0c þ ð1 kÞ~x00c Þ P c, that is k~x0c þ ð1 kÞ~x00c 2 f 1 ½c; 1 . This contradicts (15). Thus f 1 ½c; 1 is convex fuzzy set for all k ¼ c 2 I and this complete the proof. h e ! F ðRÞ be a fuzzy closed function and suppose there exists Theorem 7. Let f : D e an k 2 I and a 2 ð0; 1 such that f ðk~x0a þ ð1 kÞ~x00a Þ < f ð~x0a Þ for all ~x0a ; ~x00a 2 D 0 00 satisfying f ð~xa Þ > f ð~xa Þ. Then f is a quasiconvex fuzzy function. e there exists a Proof. By Theorem 5, it is sufficient to show that for all ~x0a ; ~x00a 2 D, 0 real number c 2 I ¼ ð0; 1Þ such that f ðc~x0a þ ð1 cÞ~x00a Þ 6 f ð~x0a Þ _ f ð~x00a Þ: e such that for all c 2 I 0 Suppose, on the contrary, that there exist ~x0a ; ~x00a 2 D f ðc~x0a þ ð1 cÞ~x00a Þ > f ð~x0a Þ _ f ð~x00a Þ:
ð16Þ
If f ð~x0a Þ 6¼ f ð~x00a Þ, without loss of generality, we can suppose that f ð~x0a Þ > f ð~x00a Þ. By the assumption of the theorem, we have f ðk~x0a þ ð1 kÞ~x00a Þ < f ð~x0a Þ: This leads to a contradiction.
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
73
If f ð~x0a Þ ¼ f ð~x00a Þ, by inequality (16) and the hypothesis of the theorem, we obtain f ð~x00a Þ < f fðkc þ 1 kÞ~x0a þ kð1 cÞ~x00a g ¼ f fk½c~x0a þ ð1 cÞ~x00a þ ð1 kÞ~x0a g < f fc~x0a þ ð1 cÞ~x00a g
ð17Þ
for all c 2 I 0 . By the conditions of the theorem again, we have f fk½ðkc þ 1 kÞ~x0a þ kð1 cÞ~x00a þ ð1 kÞ~x00a g > f fðkc þ 1 kÞ~x0a þ kð1 cÞ~x00a g > f fc~x0a þ ð1 cÞ~x00a g ðbyð16ÞÞ
ðbyð17ÞÞ
for all c 2 I 0 . Let c ¼ c ¼ k=ð1 þ kÞ 2 I 0 . Then the above inequality becomes f fðc~x0a þ ð1 cÞ~x00a Þg > f fc~x0a þ ð1 cÞ~x00a g; this leads to a contradiction. Thus inequality (16) is not true. Then f is quasiconvex fuzzy function and this completes the proof. h e ! F ðRÞ be a quasiconvex fuzzy function. If there exists Theorem 8. Let f : D e implies that k 2 I. For every pair of distinct fuzzy variables ~x0 , ~x00 2 D 0 00 0 00 f ðk~xa þ ð1 kÞ~xa Þ < f ð~xa Þ _ f ð~xa Þ. Thus f is strictly quasiconvex fuzzy function e on D. e Then Proof. Suppose that f is not strictly quasiconvex fuzzy function on D. e ~x0 ¼ there exists ~x0 , ~x00 2 D, 6 ~x00 , c 2 I and a 2 ð0; 1 such that f ðc~x0a þ ð1 cÞ~x00a Þ P f ð~x0a Þ _ f ð~x00a Þ: Let ~za ¼ c~x0a þ ð1 cÞ~x00a , the above inequality implies f ð~za Þ > f ð~x0a Þ _ f ð~x00a Þ:
ð18Þ
Since f is a quasiconvex function, we have f ð~za Þ 6 f ð~x0a Þ _ f ð~x00a Þ:
ð19Þ
Eqs. (18) and (19) imply f ð~za Þ ¼ f ð~x0a Þ _ f ð~x00a Þ:
ð20Þ
Choose b satisfying 0 < b < c < k < 1 such that c ¼ kb þ ð1 kÞb: Let ~x0a ¼ b þ ð1 bÞ~x00a ; ~x00a ¼ b~x0a þ ð1 bÞ~x00a :
ð21Þ
74
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
Thus k~x0a þ ð1 kÞ~x00a ¼ k½b~x0a þ ð1 bÞ~x00a þ ð1 kÞ½b~x0a þ ð1 bÞ~x00a ¼ k½b~x0a þ ð1 bÞ~x00a þ b~x0a þ ð1 bÞ~x00a k½b~x0a þ ð1 bÞ~x00a ¼ b~x0a þ ð1 bÞ~x00a ¼ ~za :
ð22Þ
Again since f is a quasiconvex fuzzy function, this implies f ð~x0a Þ 6 f ð~x0a Þ _ f ð~x00a Þ;
ð23Þ
f ð~x00a Þ 6 f ð~x0a Þ
ð24Þ
_
f ð~x00a Þ:
According to the assumption of the theorem and (21)–(24), we have f ð~za Þ < f ð~x0a Þ _ f ð~x00a Þ 6 ½f ð~x0a Þ _ f ð~x00a Þ _ ½f ð~x0a Þ _ f ð~x00a Þ ¼ f ð~x0a Þ _ f ð~x00a Þ; which contradicts (20). h e ! F ðRÞ be (strictly) convex fuzzy function; then f is Theorem 9. Let f : D e (strictly) quasiconvex on D. e ! F ðRÞ be (strictly) convex fuzzy function. Let ~x0 ; ~x00 2 D, e Proof. Let f : D a a ðk 2 ð0; 1ÞÞ k 2 ½0; 1 and a 2 ð0; 1 ; then from Definition 5 follows: f ðk~x0a þ ð1 kÞx00a Þð<Þ 6 kf ð~x0a Þ þ ð1 kÞf ð~x00a Þ 6 f ð~x0a Þ _ f ð~x00a Þ: Thus f is (strictly) quasiconvex fuzzy function.
h
5. Application to optimality We discuss some applications of convexity and quasiconvexity fuzzy functions to optimality theory. For that purpose we assume that X is a normed e X is a compact fuzzy set. linear space over the field of fuzzy numbers and D e X . Let Theorem 10. Let f be a convex fuzzy function defined on D e e e ~ L a ¼ min~x2D~ f ð~xa Þ; then the set W a ¼ f~xa 2 D: f ð~xa Þ ¼ La g is a convex subset of e If f is strictly convex fuzzy, then W e has only one fuzzy point. D. e and 0 6 k 6 1. Then since f is convex fuzzy Proof (First). Let ~x0 ; ~x00 2 W function f ðk~x0a þ ð1 kÞ~x00a Þ 6 kf ð~x0a Þ þ ð1 kÞf ð~x00a Þ ¼ k e L a þ ð1 kÞ e La ¼ e La: ea and hence W e a is convex set. Thus ~x0a ¼ k~x0a þ ð1 kÞ~x00a 2 W
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
75
(Second). Assume to the contrary, that f ð~x0 Þ ¼ f ð~x00 Þ ¼ e L. Since f is strictly convex fuzzy function L a þ ð1 kÞ e La ¼ e La f ðk~x0a þ ð1 kÞ~x00a Þ < kf ð~x0a Þ þ ð1 kÞf ð~x00a Þ ¼ k e and since e L a ¼ min~x2D~ f ð~xa Þ and hence the result follows. h e ! F ðRÞ be a strictly quasiconvex fuzzy function. If ~x is a Theorem 11. Let f : D local minimum fuzzy point of f , then it is also a global minimum fuzzy point of f e on D. e Then there Proof. Assume that ~x is a local minmum fuzzy point of f in D. ea D e a of ~x 2 B e a such that exists a fuzzy neighbourhood B a f ð~xa Þ 6 f ð~xa Þ
e a ; ~xa 6¼ ~x : for all ~xa 2 B a
ð25Þ
e If ~xa is not a global minimum fuzzy point, then there is some ~x a 2 B a such that xa Þ: f ð~x a Þ < f ð~ e a for some k, 0 < k < 1. From (25) e is convex, ~xa ¼ k~x þ ð1 kÞ~x 2 B Since D a a f ð~xa Þ 6 f ð~xa Þ. Since f is strictly quasiconvex (Definition 6), f ð~xa Þ < f ð~xa Þ and this is a contradiction. h
References [1] E.E. Ammar, Some properties of convex fuzzy sets and convex fuzzy cones, Fuzzy Sets Syst. 106 (1999) 381–386. [2] J.G. Brown, A note on fuzzy sets, Inform. Control 18 (1971) 32–39. [3] J. Drewniak, Convex and strongly convex fuzzy sets, J. Math. Anal. Appl. 126 (1987) 292–300. [4] Y.M. Liu, Some properties of convex fuzzy sets, J. Math. Anal. Appl. 111 (1985) 119–129. [5] R. Lowen, Convex sets, Fuzzy Sets Syst. 3 (1980) 291–310. [6] F. Mario, H.F. David, Differentiation of fuzzy continuous mappings on fuzzy topological vector spaces, J. Math. Anal. Appl. 121 (1987) 579–601. [7] S. Nada, Convex fuzzy mappings, Fuzzy Sets Syst. 48 (1992) 129–132. [8] M.L. Puri, D.A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl. 91 (1983) 552– 558. [9] R. Rockafeller, Convex Analysis, Princeton University Press, Princeton, NJ, 1970. [10] Y.-R. Syau, Some properties of convex fuzzy mappings, J. Fuzzy Math. 7 (1999) 151–160. [11] Y.-R. Syau, Invex and generalized convex fuzzy mappings, Fuzzy Sets Syst. 115 (2000) 455– 461. [12] Y.-R. Syau, Some properties of weakly convex fuzzy mappings, Fuzzy Sets Syst. 123 (2001) 203–207. [13] Y. Hong, J. Xu, A class of convex fuzzy mappings, Fuzzy Sets Syst. 129 (2002) 47–56. [14] P. Vijayaraju, M. Marudai, Fixed point theorems for fuzzy mappings, Fuzzy Sets Syst. 135 (2003) 401–408. [15] L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353.
On generalized convexity of fuzzy variables maps E.E. Ammar Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt
Abstract In this paper, we investigate the convexity and quasiconvexity of fuzzy functions by use the interval analysis. Some properties of convex and quasiconvex fuzzy functions are introduced. Applications of fuzzy convexity and quasiconvexity to optimality are discussed. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Fuzzy variables; Fuzzy functions; Convexity; a-level sets; Interval analysis; Optimality
1. Introduction The concept of convexity is important for quantitative and qualitative studies in operation research and helps to find optimal solutions. This was considered by many authors also in fuzzy optimization. The convexity of fuzzy sets was introduced by many authors like Zadeh [15], Brown [2], Drewniak [3], Lowen [5], Liu [4] and Ammar [1]. For the convexity of fuzzy functions there are a little interested papers. Puri and Ralescu [8] define the concept of the differential of fuzzy function which extends the differential of a set-valued function. Mario and David [6] introduce the notion of the differentiability of fuzzy continuous mappings defined an fuzzy topological vector space. Nada [7] introduces the concept of convexity and logarithmic convexity with some properties of fuzzy mappings. Syau [10– 12] gives the definition of fuzzy convexity and concavity of fuzzy mappings which corresponding to usual cases, some of fuzzy convexity and weakly E-mail address: [email protected] (E.E. Ammar). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.08.027
66
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
convexity are presented. Hong and Xu [13] introduced some properties of convex fuzzy optimization and characteristics of the convex and quasiconvex fuzzy mappings. Vijayaraju and Marudai [14] give some results on fixed point theorems for contractive fuzzy mappings. Sense the convexity of sets and functions [9] plays a central rule to characterized the optimally criteria in operation research. So, in this paper, we study the convexity and quasiconvexity of fuzzy function. In Section 2 we give a preliminaries of some basic notions. In Section 3, the definition and some properties of convex fuzzy functions are considered. In Section 4, quasiconvexity and strictly quasiconvexity are defined. Also qualitative analysis of the above notions in sense of interval analysis are given. In Section 5, an application of the convexity and generalized convexity of fuzzy functions to optimality are presented. In this paper, we use ^ and _ denoted to minimum (infimum) and maximum (superimum), respectively. 2. Preliminaries Assume that F ðRÞ denoted the set of all compact (bounded and closed) fuzzy numbers in the real line, i.e., if ~x 2 F ðRÞ, then ~x satisfies (i) ft: t 2 R; ~xðtÞ ¼ 1g 6¼ £. (ii) For every a 2 ½0; 1 ; ~xa ¼ ft: t 2 R; ~xðtÞ P ag ¼ ½xLa ; xU a is a finite closed on R, where R ¼ ð 1; 1Þ. Definition 1. A fuzzy point ~xa (fuzzy variables) in F ðRÞ is a membership function ~xa : R ! ½0; 1 such that t 2 R, a > 0 and a if t ¼ x; ~xa ¼ 0 else;
1 ~x(t)
α
~x L
t
~x U
R
where 0 < a 6 1. The point t is called the support of xa and a its membership degree [6]. The fuzzy point ~xa is said to be contained in, or to belong to, a fuzzy ~ within ~xa 2 A ~ iff ~xa ðtÞ 6 AðtÞ. ~ set A
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
67
Definition 2. Let ~x, ~y 2 F ðRÞ. We define that ~xa 6 ~ya iff ~xa ðtÞ 6 ~ya ðtÞ for every U a 2 ð0; 1 , i.e., xLa ðtÞ 6 yaL ðtÞ and xU a ðtÞ 6 ya ðtÞ Definition 3. For any two fuzzy numbers ~x0a ¼ ½~x0L x0U x00a ¼ ½~x00L x00U a ;~ a , ~ a ;~ a follows: ~x0a þ ~x00a ¼ ½~x0L x00L x0U x00U a þ~ a ;~ a þ~ a ;
ð1Þ
~x0a
ð2Þ
~x00a
¼
½~x0L a
~x00L x0U a ;~ a
~x00U a ;
k~x0a ¼ ½k~x0L x0U a ; k~ a ;
ð3Þ
~x0a ~x00a ¼ ½~x0L x0U x00L x00U a ;~ a ½~ a ;~ a h ¼ ~x0L x00L x0L x00U x0U x00L x0U x00U a ~ a ^~ a ~ a ^~ a ~ a ^~ a ~ a ; i 00L 0L 00U 0U 00L 0U 00U ~x0L ~ ~ ~ ~ ~ ~ ~ x _ x x _ x x _ x x : a a a a a a a a
ð4Þ
e ! F ðRÞð~x ! f ð~xÞÞ is said to be fuzzy real valuedDefinition 4. The map f : D e where D e is said to be the domain of definition for f ð~xÞ, function defined on D, and ~x is the independent fuzzy variable. Example 1. Let ~ a 2 F ðRÞ be a given fuzzy number; then e f1 ð~xÞ ¼ ~ a þ ~x for any ~x 2 D;
ð5Þ
e f2 ð~xÞ ¼ ~ a ~x for any ~x 2 D;
ð6Þ
e f3 ð~xÞ ¼ ~ a ~x for any ~x 2 D
ð7Þ
e are fuzzy real variable-valued functions defined on D.
3. Convex fuzzy functions e ! F ðRÞ is said to be convex fuzzy Definition 5. The fuzzy function f : D function if and only if f ðk~x0a þ ð1 kÞ~x00a Þ 6 kf ð~x0a Þ þ ð1 kÞf ð~x00a Þ;
ð8Þ
e 0 6 k 6 1 and a 2 ð0; 1 . where ~x0a , ~x00a 2 D, e ! F ðRÞ is said to be strictly convex fuzzy function if and The function f : D only if f ðk~x0a þ ð1 kÞ~x00a Þ < kf ð~x0a Þ þ ð1 kÞf ð~x00a Þ;
ð9Þ
e 0 < k < 1. where ~x0a , ~x00a 2 D, The fuzzy function is said to be concave (strictly concave) if f is convex (strictly convex).
68
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
e ! F ðRÞ is convex fuzzy function, ~x 2 D. e If we Lemma 1. Suppose that f : D define f ð~xÞ ¼ sup af ð~xa Þ; a>0
1. f ð~xÞ 2 F ðRÞ, 2. for any a 2 ð0; 1 , ðf ð~xÞÞa ¼ f ð~xa Þ ¼ ½f ðxLa Þ; f ðxU a Þ . Proof 1. Let ~xðx0 Þ ¼ 1 and y0 ¼ f ðx0 Þ, x0 2 Rn ; then y0 2 f ð~x1 Þ (i.e., at a ¼ 1), f ð~x1 Þ ¼ ff ðxÞ j x 2 ~x1 ¼ ½~xL1 ; ~xU 1 g then f ð~xÞy0 ¼ sup a f ð~xa Þðy0 Þ ¼ sup a ¼ 1: a>0
y0 2f ð~xa Þ
2. Since for any a 2 ð0; 1 follows: f ð~xa Þ ¼ ff ðxÞjx 2 ~xa ¼ ½~xLa ; ~xU a g and since f is convex on a convex a-level set ~xa , there exist f ðxLa Þ ¼
^x2~x f ðxÞ a
and
f ðxU a Þ ¼
_x2~x f ðxÞ a
such that ðf ð~xÞÞa ¼ f ð~xa Þ ¼ f ðxLa Þ; f ðxU a Þ :
e ! F ðRÞ is convex iff Theorem 1. A fuzzy function f : D ^ L ^ U bf ð^xLa Þ; f ð^xU a Þc 6 bf ðxa Þ; f ðxa Þc; where 00L U 00U ^xLa ¼ kx0L xa ¼ kx0U a þ ð1 kÞxa ; ^ a þ ð1 kÞxa ; 00L f^ ð~xLa Þ ¼ kf ðx0L a Þ þ ð1 kÞf ðxa Þ; 0U 00U f^ ð~xU a Þ ¼ kf ðxa Þ þ ð1 kÞf ðxa Þ
and
e ~x0a ; ~x00a 2 D:
e ! F ðRÞ is convex if Proof. From Definition 5 the function f : D f ðk~x0a þ ð1 kÞ~x00a Þ 6 kf ðx0a Þ þ ð1 kÞf ðx00a Þ; e 0 6 k 6 1. where ~x0 ,~x00 2 D,
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
69
The first side, using Lemma 1 and from (1) and (3) 0U 00L 00U f ðk~x0a þ ð1 kÞ~x00a Þ ¼ f ðk½x0L a ; xa þ ð1 kÞ½xa ; xa Þ 00L 0U 00U ¼ f ð½kx0L a þ ð1 kÞxa ; kxa þ ð1 kÞxa Þ
xLa Þ; f ð^xU ¼ f ð½^xLa ; ^xU a Þ; ¼ ½f ð^ a Þ ;
ð10Þ
00L U 00U where ^xLa ¼ kx0L xa ¼ kx0U a þ ð1 kÞxa ; ^ a þ ð1 kÞxa . The second side, using Lemma 1 0U 00L 00U kf ð~x0a Þ þ ð1 kÞf ð~x00a Þ ¼ kf ½x0L a ; xa þ ð1 kÞf ½xa ; xa 0U 00L 00U ¼ k½f ðx0L a Þ; f ðxa Þ þ ð1 kÞ½f ðxa Þ; f ðxa Þ 0U 00L 00U ¼ ½kf ðx0L a Þ; kf ðxa Þ þ ½ð1 kÞf ðxa Þ; ð1 kÞf ðxa Þ 00L 0U 00U ¼ ½kf ðx0L a Þ þ ð1 kÞf ðxa Þ; kf ðxa Þ þ ð1 kÞf ðxa Þ ð11Þ ¼ ½f^ ðxL Þ; f^ ðxU Þ : a
a
From (8), (10) and (11) the result follows. h e ! F ðRÞ is convex iff its fuzzy epigraph Theorem 2. A fuzzy function f : D e f ¼ fð~xa ; dÞ: ~x 2 D; e d 2 F ðRÞ; f ð~xa Þ 6 dg G e F ðRÞ. is convex fuzzy subset in D e f and ð~x00 ; d2 Þ 2 G ef. Proof. Let f be convex fuzzy function. Assume ð~x0a ; d1 Þ 2 G a Since f is convex fuzzy function, for each k 2 I ¼ ½0; 1 follows: f ðk~x0a þ ð1 kÞ~x00a Þ 6 kf ð~x0a Þ þ ð1 kÞf ð~x00a Þ 6 kd1 þ ð1 kÞd2 : Hence ef ðk~x0a þ ð1 kÞ~x00a ; kd1 þ ð1 kÞd2 Þ 2 G or ef: kð~x0a ; d1 Þ þ ð1 kÞð~x00a ; d2 Þ 2 G e f is convex. Thus G e f be convex. Assume ~x0 ; ~x00 2 D. e Then Conversely, let G a a ef ð~x0a ; f ð~x0a ÞÞ ¼ ð~x0 ; d1 Þ 2 G
and
ef: ð~x00a ; f ð~x00a ÞÞ ¼ ð~x00 ; d2 Þ 2 G
e f is convex, for each k 2 I follows Since G ef ðk~x0a ; ð1 kÞ~x00a ; kf ð~x0a Þ þ ð1 kÞf ð~x00a ÞÞ 2 G or f ðk~x0a ; ð1 kÞ~x00a Þ 6 kf ð~x0a Þ þ ð1 kÞf ð~x00a Þ: Hence f is convex fuzzy and this complete the proof.
h
70
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
Theorem 3. Let ðfj Þj2J be a family (finite or infinite) of convex and bounded fuzzy e then f ð~xa Þ ¼ _ fj ð~xa Þ is convex fuzzy function on D. e function on D, j2J
e their fuzzy epigraphs Proof. Since each is a convex fuzzy functions on D, e fj ¼ fð~xa ; dÞ j~x 2 D; e d 2 F ðRÞ; fj ð~xa Þ 6 dg G
ð12Þ
are convex fuzzy subset in Rnþ1 by Theorem 2, and hence their intersection \ e fj ¼ fð~xa ; dÞj~x 2 D; e fj ð~xa Þ 6 d; j 2 J g G j2J
is fuzzy convex set (since the intersection of any convex fuzzy sets is fuzzy convex). But this convex intersection is the epigraphs of f . Hence f is convex e by Theorem 2. h function on D e ! F ðRÞ be convex fuzzy functions. Theorem 4. Let f : F ðRÞ ! F ðRÞ and g : D If f is non-decreasing, then the function ~x ! f ðgð~xÞÞ is convex fuzzy function. e and k 2 ½0; 1 . Since g : D e ! F ðRÞ is convex, we have Proof. Let ~x0 , ~x00 2 D gðk~x0a þ ð1 kÞ~x00a Þ 6 kgð~x0a Þ þ ð1 kÞgð~x00a Þ: Since f : F ðRÞ ! F ðRÞ is non-decreasing and convex, it follows that f ðgðk~x0a þ ð1 kÞ~x00a ÞÞ 6 f ðkgð~x0a Þ þ ð1 kÞgð~x00a ÞÞ 6 kf ðgð~x0a ÞÞ þ ð1 kÞf ðgð~x00a ÞÞ: Hence ~x ! f ðgð~xÞÞ is convex.
h
4. Quasiconvex fuzzy functions e ! F ðRÞ is said to be quasiconvex fuzzy Definition 6. The fuzzy function f : D function if and only if f ðk~x0a þ ð1 kÞ~x00a Þ 6 f ð~x0a Þ _ f ð~x00a Þ
ð13Þ
e k 2 I and a 2 ð0; 1 . for any ~x0a ; ~x00a 2 D, The fuzzy function is said to be strictly quasiconvex if and only if relation (13) holds only with (<) for ~x0a 6¼ ~x00a , k 2 I 0 ¼ ð0; 1Þ. e ! F ðRÞ is said to be a fuzzy closed Remark 1. The fuzzy function f : D 1 function if and only if for a 2 I; f ½a; 1 is closed [6].
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
71
e ! F ðRÞ is quasiconvex if and only if the Theorem 5. A fuzzy function f : D fuzzy set e e f ð~xd Þ 6 ag 6 D e L fa ¼ f~x 2 D:
ð14Þ
is convex for each d; a 2 F ðRÞ. e Let ~x0 ; ~x00 any two fuzzy Proof. Let f be a quasiconvex fuzzy function on D. d d e e variables on L fa D a , a 2 ð0; 1 . Then for k 2 I, f ðk~x0d þ ð1 kÞ~x00d 6 f ð~x0d Þ _ f ð~x00d Þ 6 a; then k~x0d þ ð1 kÞ~x00d 2 e L fa . Hence e L fa is convex set. e Conversely, let L fa be convex fuzzy set. Then for any ~x0d ; x00d 2 e L fa and k 2 I, 0 k~xd þ ð1 kÞ~x00d 2 e L fa . Let ~x0d ; ~x00d 2 e L fa be such that f ð~x0d Þ 6 a and f ð~x00d Þ 6 f ð~x0d Þ 6 a. Now f ðk~x0d þ ð1 kÞ~x00d Þ 6 a ¼ f ð~x0d Þ _ f ð~x00d Þ. Thus f is a quasiconvex fuzzy function and this completes the proof. h e ! F ðRÞ be a fuzzy closed function. Then f is a quasiTheorem 6. Let f : D e then there exist convex fuzzy function if and only if for all ~x0d ; ~x00d 2 D, 0 k 2 I ¼ ð0; 1Þ; d 2 ð0; 1 , such that f ðk~x0d þ ð1 kÞ~x00d Þ 6 f ð~x0d Þ _ f ð~x00d Þ: Proof. From Theorem 5 it is sufficient to show that f 1 ½k; 1 is a convex set for all k 2 I ¼ ½0; 1 . In fact if that is not true, then there exists another c ¼ d 2 I, c < k, such that f 1 ½c; 1 is not convex set, that is there exist ~x0c ; x00c 2 f 1 ½c; 1 and k0 2 I such that ~x^k0 ¼ k0~x0c þ ð1 k0 Þ~x00c 2 f 1 ½c; 1 : Assume for c < k that ~ ¼ f 1 ½c; 1 ^ ½~x0 ; ~x^ ; A k0 ~ 0 ¼ f 1 ½c; 1 \ ½~x0L þ ~x^L ; ~x0U þ ~x^U A c c k0 c k0 and ~ 00 ¼ f 1 ½c; 1 \ ½~x00L þ ~x^L ; ~x00U þ ~x^U ; A c c k0 c k0 where b~x0L x^L x0U x^U x0c þ ð1 kÞ~x^k0 ; k 2 Ig c þ~ k0 ; ~ c þ~ k0 c ¼ fk~ and b~xLk0 þ ~x00L x^U x00U x^k0 þ ð1 kÞ~x00c ; k 2 Ig: c ;~ k0 þ ~ c c ¼ fk~
72
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
~ 0 and A ~ 00 are bounded e from [8] it follows A Since f is fuzzy closed function D, c c ~ 0 , ~x^ 2 A ~ 00 . Thus there exists ~x0 2 A ~ 0 and ~x00 2 A ~ 00 closed fuzzy sets and ~x^k0 62 A c k0 c c c such that min kð~xLc ðtÞ ~x^L xU x^U k0 ðtÞÞ; ð~ c ðtÞ ~ k0 ðtÞÞk ~ 0c t2A
x^L x0U x^U ¼ kð~x0L c ðtÞ ~ k0 ðtÞÞ; ð~ c ðtÞ ~ k0 ðtÞÞk; min00 kð~xLc ðtÞ ~x^L xU x^U k0 ðtÞÞ; ð~ c ðtÞ ~ k0 ðtÞÞk ~ t2A c
¼ kð~x00L x^L x00U x^U c ðtÞ ~ k0 ðtÞÞ; ð~ c ðtÞ ~ k0 ðtÞÞk: Hence x^L x0U x^U f 1 ½c; 1 \ ð~x0L c þ~ k0 ; ~ c þ~ k0 ¼ ; and f 1 ½c; 1 \ ½~x^L x00L x^U x00U k0 þ ~ c ;~ k0 þ ~ c Þ ¼ ;; that is f 1 ½c; 1 \ ½~x0L x00L x0U x00U k0 þ ~ c ;~ k0 þ ~ c Þ ¼ ;:
ð15Þ
On the other hand, by the assumption of the theorem, there exists an k 2 I 0 f ðk~x0d þ ð1 kÞ~x00d Þ 6 f ð~x0d Þ _ f ð~x00d Þ: Since ~x0c 2 f 1 ½c; 1 and ~x00c 2 f 1 ½c; 1 , we have f ðk~x0c þ ð1 kÞ~x00c Þ P c, that is k~x0c þ ð1 kÞ~x00c 2 f 1 ½c; 1 . This contradicts (15). Thus f 1 ½c; 1 is convex fuzzy set for all k ¼ c 2 I and this complete the proof. h e ! F ðRÞ be a fuzzy closed function and suppose there exists Theorem 7. Let f : D e an k 2 I and a 2 ð0; 1 such that f ðk~x0a þ ð1 kÞ~x00a Þ < f ð~x0a Þ for all ~x0a ; ~x00a 2 D 0 00 satisfying f ð~xa Þ > f ð~xa Þ. Then f is a quasiconvex fuzzy function. e there exists a Proof. By Theorem 5, it is sufficient to show that for all ~x0a ; ~x00a 2 D, 0 real number c 2 I ¼ ð0; 1Þ such that f ðc~x0a þ ð1 cÞ~x00a Þ 6 f ð~x0a Þ _ f ð~x00a Þ: e such that for all c 2 I 0 Suppose, on the contrary, that there exist ~x0a ; ~x00a 2 D f ðc~x0a þ ð1 cÞ~x00a Þ > f ð~x0a Þ _ f ð~x00a Þ:
ð16Þ
If f ð~x0a Þ 6¼ f ð~x00a Þ, without loss of generality, we can suppose that f ð~x0a Þ > f ð~x00a Þ. By the assumption of the theorem, we have f ðk~x0a þ ð1 kÞ~x00a Þ < f ð~x0a Þ: This leads to a contradiction.
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
73
If f ð~x0a Þ ¼ f ð~x00a Þ, by inequality (16) and the hypothesis of the theorem, we obtain f ð~x00a Þ < f fðkc þ 1 kÞ~x0a þ kð1 cÞ~x00a g ¼ f fk½c~x0a þ ð1 cÞ~x00a þ ð1 kÞ~x0a g < f fc~x0a þ ð1 cÞ~x00a g
ð17Þ
for all c 2 I 0 . By the conditions of the theorem again, we have f fk½ðkc þ 1 kÞ~x0a þ kð1 cÞ~x00a þ ð1 kÞ~x00a g > f fðkc þ 1 kÞ~x0a þ kð1 cÞ~x00a g > f fc~x0a þ ð1 cÞ~x00a g ðbyð16ÞÞ
ðbyð17ÞÞ
for all c 2 I 0 . Let c ¼ c ¼ k=ð1 þ kÞ 2 I 0 . Then the above inequality becomes f fðc~x0a þ ð1 cÞ~x00a Þg > f fc~x0a þ ð1 cÞ~x00a g; this leads to a contradiction. Thus inequality (16) is not true. Then f is quasiconvex fuzzy function and this completes the proof. h e ! F ðRÞ be a quasiconvex fuzzy function. If there exists Theorem 8. Let f : D e implies that k 2 I. For every pair of distinct fuzzy variables ~x0 , ~x00 2 D 0 00 0 00 f ðk~xa þ ð1 kÞ~xa Þ < f ð~xa Þ _ f ð~xa Þ. Thus f is strictly quasiconvex fuzzy function e on D. e Then Proof. Suppose that f is not strictly quasiconvex fuzzy function on D. e ~x0 ¼ there exists ~x0 , ~x00 2 D, 6 ~x00 , c 2 I and a 2 ð0; 1 such that f ðc~x0a þ ð1 cÞ~x00a Þ P f ð~x0a Þ _ f ð~x00a Þ: Let ~za ¼ c~x0a þ ð1 cÞ~x00a , the above inequality implies f ð~za Þ > f ð~x0a Þ _ f ð~x00a Þ:
ð18Þ
Since f is a quasiconvex function, we have f ð~za Þ 6 f ð~x0a Þ _ f ð~x00a Þ:
ð19Þ
Eqs. (18) and (19) imply f ð~za Þ ¼ f ð~x0a Þ _ f ð~x00a Þ:
ð20Þ
Choose b satisfying 0 < b < c < k < 1 such that c ¼ kb þ ð1 kÞb: Let ~x0a ¼ b þ ð1 bÞ~x00a ; ~x00a ¼ b~x0a þ ð1 bÞ~x00a :
ð21Þ
74
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
Thus k~x0a þ ð1 kÞ~x00a ¼ k½b~x0a þ ð1 bÞ~x00a þ ð1 kÞ½b~x0a þ ð1 bÞ~x00a ¼ k½b~x0a þ ð1 bÞ~x00a þ b~x0a þ ð1 bÞ~x00a k½b~x0a þ ð1 bÞ~x00a ¼ b~x0a þ ð1 bÞ~x00a ¼ ~za :
ð22Þ
Again since f is a quasiconvex fuzzy function, this implies f ð~x0a Þ 6 f ð~x0a Þ _ f ð~x00a Þ;
ð23Þ
f ð~x00a Þ 6 f ð~x0a Þ
ð24Þ
_
f ð~x00a Þ:
According to the assumption of the theorem and (21)–(24), we have f ð~za Þ < f ð~x0a Þ _ f ð~x00a Þ 6 ½f ð~x0a Þ _ f ð~x00a Þ _ ½f ð~x0a Þ _ f ð~x00a Þ ¼ f ð~x0a Þ _ f ð~x00a Þ; which contradicts (20). h e ! F ðRÞ be (strictly) convex fuzzy function; then f is Theorem 9. Let f : D e (strictly) quasiconvex on D. e ! F ðRÞ be (strictly) convex fuzzy function. Let ~x0 ; ~x00 2 D, e Proof. Let f : D a a ðk 2 ð0; 1ÞÞ k 2 ½0; 1 and a 2 ð0; 1 ; then from Definition 5 follows: f ðk~x0a þ ð1 kÞx00a Þð<Þ 6 kf ð~x0a Þ þ ð1 kÞf ð~x00a Þ 6 f ð~x0a Þ _ f ð~x00a Þ: Thus f is (strictly) quasiconvex fuzzy function.
h
5. Application to optimality We discuss some applications of convexity and quasiconvexity fuzzy functions to optimality theory. For that purpose we assume that X is a normed e X is a compact fuzzy set. linear space over the field of fuzzy numbers and D e X . Let Theorem 10. Let f be a convex fuzzy function defined on D e e e ~ L a ¼ min~x2D~ f ð~xa Þ; then the set W a ¼ f~xa 2 D: f ð~xa Þ ¼ La g is a convex subset of e If f is strictly convex fuzzy, then W e has only one fuzzy point. D. e and 0 6 k 6 1. Then since f is convex fuzzy Proof (First). Let ~x0 ; ~x00 2 W function f ðk~x0a þ ð1 kÞ~x00a Þ 6 kf ð~x0a Þ þ ð1 kÞf ð~x00a Þ ¼ k e L a þ ð1 kÞ e La ¼ e La: ea and hence W e a is convex set. Thus ~x0a ¼ k~x0a þ ð1 kÞ~x00a 2 W
E.E. Ammar / Appl. Math. Comput. 157 (2004) 65–75
75
(Second). Assume to the contrary, that f ð~x0 Þ ¼ f ð~x00 Þ ¼ e L. Since f is strictly convex fuzzy function L a þ ð1 kÞ e La ¼ e La f ðk~x0a þ ð1 kÞ~x00a Þ < kf ð~x0a Þ þ ð1 kÞf ð~x00a Þ ¼ k e and since e L a ¼ min~x2D~ f ð~xa Þ and hence the result follows. h e ! F ðRÞ be a strictly quasiconvex fuzzy function. If ~x is a Theorem 11. Let f : D local minimum fuzzy point of f , then it is also a global minimum fuzzy point of f e on D. e Then there Proof. Assume that ~x is a local minmum fuzzy point of f in D. ea D e a of ~x 2 B e a such that exists a fuzzy neighbourhood B a f ð~xa Þ 6 f ð~xa Þ
e a ; ~xa 6¼ ~x : for all ~xa 2 B a
ð25Þ
e If ~xa is not a global minimum fuzzy point, then there is some ~x a 2 B a such that xa Þ: f ð~x a Þ < f ð~ e a for some k, 0 < k < 1. From (25) e is convex, ~xa ¼ k~x þ ð1 kÞ~x 2 B Since D a a f ð~xa Þ 6 f ð~xa Þ. Since f is strictly quasiconvex (Definition 6), f ð~xa Þ < f ð~xa Þ and this is a contradiction. h
References [1] E.E. Ammar, Some properties of convex fuzzy sets and convex fuzzy cones, Fuzzy Sets Syst. 106 (1999) 381–386. [2] J.G. Brown, A note on fuzzy sets, Inform. Control 18 (1971) 32–39. [3] J. Drewniak, Convex and strongly convex fuzzy sets, J. Math. Anal. Appl. 126 (1987) 292–300. [4] Y.M. Liu, Some properties of convex fuzzy sets, J. Math. Anal. Appl. 111 (1985) 119–129. [5] R. Lowen, Convex sets, Fuzzy Sets Syst. 3 (1980) 291–310. [6] F. Mario, H.F. David, Differentiation of fuzzy continuous mappings on fuzzy topological vector spaces, J. Math. Anal. Appl. 121 (1987) 579–601. [7] S. Nada, Convex fuzzy mappings, Fuzzy Sets Syst. 48 (1992) 129–132. [8] M.L. Puri, D.A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl. 91 (1983) 552– 558. [9] R. Rockafeller, Convex Analysis, Princeton University Press, Princeton, NJ, 1970. [10] Y.-R. Syau, Some properties of convex fuzzy mappings, J. Fuzzy Math. 7 (1999) 151–160. [11] Y.-R. Syau, Invex and generalized convex fuzzy mappings, Fuzzy Sets Syst. 115 (2000) 455– 461. [12] Y.-R. Syau, Some properties of weakly convex fuzzy mappings, Fuzzy Sets Syst. 123 (2001) 203–207. [13] Y. Hong, J. Xu, A class of convex fuzzy mappings, Fuzzy Sets Syst. 129 (2002) 47–56. [14] P. Vijayaraju, M. Marudai, Fixed point theorems for fuzzy mappings, Fuzzy Sets Syst. 135 (2003) 401–408. [15] L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353.