- Email: [email protected]
(Φ1,Φ2)-Convex fuzzy mappings
Fuzzy Sets and Systems 138 (2003) 617 – 625 www.elsevier.com/locate/fss
(1; 2)-Convex fuzzy mappings Yu-Ru Syau∗ Department of Industrial Engineering, Da Yeh University, Chang-Hwa 51505, Taiwan, ROC Received 7 March 2000; received in revised form 7 November 2002; accepted 14 November 2002
Abstract The concepts of (1 ; 2 )-convexity, 1 -B-vexity and 1 -convexity for fuzzy mappings are introduced through the so-called “fuzzy max” order among fuzzy numbers. We show that the class of (1 ; 2 )-convex fuzzy mappings, 1 -B-vex fuzzy mappings and 1 -convex fuzzy mappings includes many well-known classes of fuzzy mappings such as convex fuzzy mappings and preinvex fuzzy mappings as its subclasses. We also give a characterization for 1 -B-vex fuzzy mappings by proposing the concept of 1 -B-vex sets. In addition some basic properties of fuzzy mappings from the standpoint of convex analysis are discussed. c 2002 Elsevier B.V. All rights reserved. Keywords: Fuzzy numbers; Convexity; (1 ; 2 )-convex fuzzy mappings
1. Introduction The concept of convexity is important for quantitative and qualitative studies in operations research or applied mathematics. It has also been considered by many authors in fuzzy optimization. Like a paper by Ammar and Metz [1], a general fuzzy nonlinear programming problem is formulated with application of the concept of convexity. Di?erent types of convexity and generalized convexity of fuzzy mappings were studied by several authors, including [2,4–6], aiming at applications to fuzzy nonlinear programming. Recently, Pini and Singh [7] considered a class of functions called (1 ; 2 )-convex functions from a uniAed point of view: the function 1 describes a generalized convex combination of arguments, and 2 determines generalized convex combinations of values. In this way, a large number of well known and new convexity conditions can be included.
The work was partially supported by the National Science Council of Republic of China under contract NSC 90-2218-E-212-013. ∗ Tel.: +886-4-852-8468; fax: +886-4-851-1270. E-mail address: [email protected] (Y.-R. Syau). c 2002 Elsevier B.V. All rights reserved. 0165-0114/03/$ - see front matter doi:10.1016/S0165-0114(02)00527-4
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Y.-R. Syau / Fuzzy Sets and Systems 138 (2003) 617 – 625
Motivated both by earlier research works and by the importance of the concept of convexity, we propose in this paper the concepts of (1 ; 2 )-convexity, 1 -B-vexity and 1 -convexity for fuzzy mappings. We show that the class of (1 ; 2 )-convex fuzzy mappings, 1 -B-vex fuzzy mappings and 1 -convex fuzzy mappings includes many well-known classes of fuzzy mappings such as convex fuzzy mappings and preinvex fuzzy mappings as its subclasses. We also give a characterization for 1 -B-vex fuzzy mappings by proposing the concept of 1 -B-vex sets. In addition some basic properties of fuzzy mappings from the standpoint of convex analysis are discussed. 2. Preliminaries This section will present some basic knowledge of fuzzy numbers and fuzzy mappings from the standpoint of convex analysis. For the convenience of reading, several deAnitions and results without proof from [3,5,7–9] will be listed below. A fuzzy number we treat in this paper is a fuzzy set u : R1 → [0; 1] which is normal, fuzzy convex, upper semicontinuous, and has bounded support. The family of fuzzy numbers will be denoted by F0 . It is obvious that the -level set of a fuzzy number u is a closed and bounded interval {x ∈ R1 |u(x) ¿ } if 0 ¡ 6 1; [a(); b()] = [u] = cl(supp u) if = 0; where cl(supp u) denotes the closure of the support, {x ∈ R1 |u(x)¿0}, of u. It is also obvious that each r ∈ R1 can be considered as a fuzzy number and thus R1 can be embedded in F0 . It is easily veriAed that a fuzzy set u : R1 → [0; 1] is a fuzzy number if and only if (i) [u] is a closed and bounded interval for each ∈ [0; 1], and (ii) [u]1 = ∅. Thus we can identify a fuzzy number u with the parameterized triples {(a(); b(); )|0 6 6 1}; where a() and b() denote the left- and right-hand endpoints of [u] , respectively. Denition 2.1. For u; v ∈ F0 , we say that u 4 v if for every ∈ [0; 1], a() 6 c()
and
b() 6 d();
where [u] = [a(); b()] and [v] = [c(); d()]. We say that u ≺ v if u 4 v and there exists 0 ∈ [0; 1] such that a(0 ) ¡ c(0 )
or
b(0 ) ¡ d(0 ):
We see that u = v, if u 4 v and v 4 u. It is often convenient to write v ¡ u (resp. v u) in place of u 4 v (u ≺ v). A subset S ∗ of F0 is said to be bounded above if there exists a fuzzy number u ∈ F0 , called an upper bound of S ∗ , such that v 4 u for every v ∈ S ∗ . Further, a fuzzy number u0 ∈ F0 is called
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the least upper bound (sup in short) for S ∗ if (i) u0 is an upper bound of S ∗ , and (ii) u0 4 u for every upper bound u of S ∗ . A lower bound and the greatest lower bound (inf in short) are deAned similarly. Let addition and nonnegative scalar multiplication on F0 be deAned by the usual extension principle (for details, see [5,9]). It is easily veriAed that addition and nonnegative scalar multiplication preserve the order on F0 , and that F0 is closed under these operations. It is also known [9] that sup{u; v} and inf {u; v} exist in F0 for any pair {u; v} ⊂ F0 . Let D ⊆ Rn be nonempty. A mapping f : D → F0 is a fuzzy mapping. The epigraph of f, denoted by epi(f), is deAned as epi(f) = {(x; u) : x ∈ D; u ∈ F0 ; f(x) u}: Denition 2.2. Let C be a nonempty convex subset of Rn . A fuzzy mapping f : C → F0 is said to be: (1) convex if for every ∈ [0; 1] and x; y ∈ C, f(x + (1 − )y) f(x) + (1 − )f(y); (2) quasi-convex if for every ∈ [0; 1] and x; y ∈ C, f(x + (1 − )y) sup{f(x); f(y)}: Recall that, by deAnition, a set K ⊆ Rn is said to be an invex set with respect to a function : K × K → Rn if x; y ∈ K implies that y + (x; y) ∈ K for all ∈ [0; 1]. Denition 2.3. Let K ⊆ Rn be a nonempty invex set with respect to a given function . A fuzzy mapping f : K → F0 is said to be preinvex if for every ∈ [0; 1] and x; y ∈ K, f(y + (x; y)) f(x) + (1 − )f(y):
3. Denitions and basic results Let S ⊆ Rn be nonempty, and let F be the family of fuzzy mappings deAned on S. Assume that 1 : S × S × [0; 1] → Rn with 1 (x; y; 0) = y; 1 (x; x; ) = x for all x; y ∈ S and ∈ [0; 1], and that 2 : S × S × [0; 1] × F → F0 with 2 (x; y; 0; f) = f(y); 2 (x; x; ; f) = f(x); 2 (x; y; ; f) 4 sup {f(x); f(y)} for all x; y ∈ S; ∈ [0; 1] and f ∈ F. A set X ⊆ S is 1 -convex if 1 (x; y; ) ∈ X for all x; y ∈ X; ∈ [0; 1]. From now on, S will be a 1 convex set. In what follows, let R∗ be the set of nonnegative real numbers. Let b : S × S × [0; 1] → R∗ , with b(x; y; ) ∈ [0; 1] for all x; y ∈ S and ∈ [0; 1]. For the sake of brevity, we shall omit the argument of b unless it is needed for speciAcation.
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Denition 3.1. A fuzzy mapping f : S → F0 is said to be: (1) (1 ; 2 )-convex if for every ∈ [0; 1] and x; y ∈ S, f(1 (x; y; )) 2 (x; y; ; f): (2) 1 -B-vex if for every ∈ [0; 1] and x; y ∈ S, f(1 (x; y; )) b(x; y; )f(x) + (1 − b(x; y; ))f(y): (3) strictly 1 -B-vex if for all x; y ∈ S; x = y; ∈ (0; 1), f(1 (x; y; )) ≺ b(x; y; )f(x) + (1 − b(x; y; ))f(y): (4) 1 -convex if for every ∈ [0; 1] and x; y ∈ S, f(1 (x; y; )) f(x) + (1 − )f(y): (5) 1 -quasiconvex if for every ∈ [0; 1] and x; y ∈ S, f(1 (x; y; )) sup{f(x); f(y)}: It is clear that every convex (resp. preinvex with (x; x) = O, where O being the origin of Rn ) fuzzy mapping is 1 -convex where 1 (x; y; ) = x + (1 − )y (resp. 1 (x; y; ) = y + (x; y)). Theorem 3.1. For a fuzzy mapping f, 1 -convex ⇒ 1 -B-vex ⇒ (1 ; 2 )-convex ⇒ 1 -quasiconvex: The following result which gives a constructive description of 1 -convex fuzzy mappings and 1 -B-vex fuzzy mappings is an immediate consequence of DeAnitions 2.1 and 3.1. Theorem 3.2. Let f be a fuzzy mapping de4ned on a 1 -convex set S, and for each x ∈ S, let {(c(; x); d(; x); ) | 0661} be the parametric representation of f(x). Then f is a 1 -convex (resp. 1 -B-vex) fuzzy mapping if and only if for each ∈ [0; 1]; c(; x) and, d(; x) are 1 -convex (resp. 1 -B-vex) functions of x ∈ S. In view of the deAnition of 1 -B-vex fuzzy mappings and the fact that addition and nonnegative scalar multiplication preserve the order on F0 , we obtain: Theorem 3.3. Let fj : S → F0 ; j = 1; 2; : : : ; N be 1 -B-vex fuzzy mappings with respect to the same b. Then the fuzzy mapping f : S → F0 de4ned by f(x) =
N
kj fj (x);
kj ¿ 0
j=1
is a 1 -B-vex fuzzy mapping on S.
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Corollary 3.1. Let fj : S → F0 ; j = 1; 2; : : : ; N be 1 -convex fuzzy mappings. Then the fuzzy mapping f : S → F0 de4ned by f(x) =
N
kj fj (x);
kj ¿ 0
j=1
is a 1 -convex fuzzy mapping on S. Denition 3.2. Given T ⊆ Rn × F0 ; T is said to be a 1 - B-vex set with respect to b if (x; u); (y; v) ∈ T implies that (1 (x; y; ); bu + (1 − b)v) ∈ T;
0 6 6 1:
It is obvious that each r ∈ R1 can be considered as a fuzzy number and thus it does make sense to speak of 1 -B-vex subset of Rn × R1 . We now give a characterization of 1 -B-vex fuzzy mappings in terms of their epigraphs. Theorem 3.4. A fuzzy mapping f : S → F0 is 1 -B-vex with respect to b if and only if its epigraph epi(f) is a 1 -B-vex set in Rn × F0 with respect to b. Proof. Let f : S → F0 be a 1 -B-vex fuzzy mapping and (x; u); (y; v) ∈ epi(f) with x; y ∈ S and u; v ∈ F0 . Then f(x) 4 u; f(y) 4 v. Since f is 1 -B-vex on S, and since addition and nonnegative scalar multiplication preserve the order on F0 , we have f(1 (x; y; )) bf(x) + (1 − b)f(y)
bu + (1 − b)v; 0 6 6 1; which implies that (1 (x; y; ); bu + (1 − b)v) ∈ epi(f);
0 6 6 1:
Thus, epi(f) is a 1 -B-vex set with respect to b. Conversely, assume that epi(f) is a 1 -B-vex set and x; y ∈ S. Then (x; f(x)); (y; f(y)) ∈ epi(f). Since epi(f) is a 1 -B-vex set with respect to b, it follows that (1 (x; y; ); bf(x) + (1 − b)f(y)) ∈ epi(f);
0 6 6 1;
which implies that f(1 (x; y; )) bf(x) + (1 − b)f(y);
0 6 6 1:
Hence, f is a 1 -B-vex fuzzy mapping with respect to b. This completes the proof. Theorem 3.5.If {Ti }i∈I is a family of 1 -B-vex sets in Rn × F0 with respect to b, then their intersection i∈I Ti is a 1 -B-vex set with respect to the same b.
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Proof. Assume that (x; u); (y; v) ∈ i∈I Ti and 0661. Then for each i ∈ I , we have (x; u); (y; v) ∈ Ti . Since each Ti is a 1 -B-vex set with respect b, it follows that (1 (x; y; ); bu + (1 − b)v) ∈ Ti ; which shows that
i ∈I
i ∈I
Ti is a 1 -B-vex set with respect to b.
Theorem 3.6. Let {fi }i∈I be an arbitrary nonempty collection of 1 -B-vex fuzzy mappings de4ned on S with respect to the same b such that for every x ∈ S; sup{fi (x) : i ∈ I } exists in F0 . Then the function f : S → F0 de4ned by f(x) = sup{fi (x) : i ∈ I } is a 1 -B-vex fuzzy mapping on S with respect to b. Proof. Since each fi is a 1 -B-vex fuzzy mapping with respect to b, its epigraph epi(fi ) = {(x; u) : x ∈ S; u ∈ F0 ; fi (x) u} is a 1 -B-vex set in Rn × F0 with respect to b, by Theorem 3.4. Hence their intersection epi(fi ) = {(x; u) : x ∈ S; u ∈ F0 ; fi (x) u; i ∈ I } i ∈I
is also a 1 -B-vex set in Rn × F0 with respect to b, by Theorem 3.5. It can be easily checked that this intersection is the epigraph of f. Hence, by Theorem 3.4, f is a 1 -B-vex fuzzy mapping on S with respect to b. Corollary 3.2. Let {fi }i∈I be an arbitrary nonempty collection of 1 -convex fuzzy mappings de4ned on S such that for every x ∈ S; sup{fi (x) : i ∈ I } exists in F0 . Then the function f : S → F0 de4ned by f(x) = sup{fi (x) : i ∈ I } is a 1 -convex fuzzy mapping on S. 4. Main results In this section, we present some properties of 1 -quasiconvex fuzzy mappings, 1 -B-vex fuzzy mappings and 1 -convex fuzzy mappings and discuss some applications to optimization. Theorem 4.1. A fuzzy mapping f : S → F0 is 1 -quasiconvex if and only if the set Lu = {x ∈ S : f(x) 4 u} is a 1 -convex (crisp) set for each u ∈ F0 . Proof. Let f : S → F0 be 1 -quasiconvex, and u ∈ F0 . If Lu is a singleton or an empty set then it is obvious a 1 -convex set. Assume that x; y ∈ Lu , i.e., f(x) 4 u and f(y) 4 u. So u is an upper
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bound of {f(x); f(y)}, it follows that sup{f(x); f(y)} 4 u. Since f is 1 -quasiconvex, we have for all ∈ [0; 1], f(1 (x; y; )) sup{f(x); f(y)} u: It follows that 1 (x; y; ) ∈ Lu for all ∈ [0; 1]. Thus Lu is a 1 -convex set. Conversely, let Lu be a 1 -convex set for every fuzzy number u. Let x; y ∈ S and u = sup{f(x); f(y)}. Then x; y ∈ Lu , and by the 1 -convexity of Lu , it follows that 1 (x; y; ) ∈ Lu for all ∈ [0; 1]. Hence, f(1 (x; y; )) u = sup{f(x); f(y)}: Thus, f : S → F0 is 1 -quasiconvex. This completes the proof. Theorem 4.2. Let f : S → F0 be 1 -convex. If g : F0 → F0 is a nondecreasing convex fuzzy mapping, then the mapping x → g(f(x)) is 1 -convex on S. Proof. Let x; y ∈ S, and ∈ [0; 1]. Since f : S → F0 is 1 -convex on S, we have f(1 (x; y; )) f(x) + (1 − )f(y): Since g : F0 → F0 is non-decreasing and convex, it follows that g(f(1 (x; y; ))) g(f(x) + (1 − )f(y))
g(f(x)) + (1 − )g(f(y)); and hence x → g(f(x)) is 1 -convex on S. This completes the proof. We now discuss some applications of 1 -B-vex fuzzy mappings to fuzzy optimization theory. Theorem 4.3. Let f be a fuzzy mapping de4ned on S such that inf {f(x) : x ∈ S} exists in F0 . Let # = inf {f(x) : x ∈ S}. (1) If f : S → F0 is 1 -B-vex, then the set $ = {x ∈ S : f(x) = #} is 1 -convex. (2) If f is strictly 1 -B-vex, then $ is singleton or empty. That is, if f is strictly 1 -B-vex, then f has at most one global minimizer. Proof. To prove part (1), let f : S → F0 be 1 -B-vex on S. If $ is a singleton or an empty set then it is obvious a 1 -convex set. Assume that x; y ∈ $, i.e., f(x) = f(y) = #. Since f is a 1 -B-vex fuzzy mapping on S, f(1 (x; y; )) bf(x) + (1 − b)f(y) = b# + (1 − b)# =# for all ∈ [0; 1]. Thus all points of the form 1 (x; y; ); ∈ [0; 1] belong to $. Hence $ is 1 -convex.
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Y.-R. Syau / Fuzzy Sets and Systems 138 (2003) 617 – 625
For the second part, the proof will be by contradiction: assume that there exist distinct points x; y ∈ S such that f(x) = f(y) = #. Since S is 1 -convex, then for ∈ (0; 1); 1 (x; y; ) ∈ S. Further since f is strictly 1 -B-vex on S, f(1 (x; y; )) ≺ bf(x) + (1 − b)f(y) = b# + (1 − b)# = #: This is contradicts that # = inf {f(x) : x ∈ S} and hence the result follows. Theorem 4.4. If f : S → F0 is 1 -B-vex with respect to b; lim→0+ 1 (x; y; ) = y for all x; y ∈ S, and b(x; y; )¿0 for all x; y ∈ S; 0¡¡d¡1 for some 4xed d, then every local minimizer of f is also a global minimizer of f over S. Proof. The proof is by contradiction. Let x˜ be a local minimizer of f, and suppose, by contradiction, that it is not a global minimizer of f over S. Then there exists some point x ∈ S satisfying f(x) ≺ f(x): ˜ By the 1 -B-vexity of f, for all ∈ [0; 1], ˜ )) bf(x) + (1 − b)f(x): ˜ f(1 (x; x; Since f(x) ≺ f(x), ˜ we have bf(x) + (1 − b)f(x) ˜ = b(f(x) − f(x)) ˜ + f(x) ˜ ≺ f(x) ˜ for all ∈ (0; d). Thus, for all ∈ (0; d), ˜ )) ≺ f(x): ˜ f(1 (x; x; Hence by the 1 -convexity of S, and since ˜ ) = x˜ lim 1 (x; x;
→0+
˜ ) ∈ S for all 0661, and 1 (x; x; ˜ ) are arbitrarily close to x˜ for suLciently it follows that 1 (x; x; small ¿0. This contradicts the deAnition of a local minimizer. Hence x˜ must be a global maximizer.
Corollary 4.1. Let f : S → F0 be 1 -convex, lim→0+ 1 (x; y; ) = y for all x; y ∈ S, and let x˜ ∈ S be a local minimizer of f. Then x˜ is also a global minimizer of f over S. 5. Discussions Motivated by earlier research works [2,3–7,9], we introduced in this paper the concept of (1 ; 2 )convexity, 1 -B-vexity and 1 -convexity into the class of fuzzy mappings through the so-called “fuzzy max” order among fuzzy numbers. It is shown that the class of (1 ; 2 )-convex fuzzy mappings, 1 -B-vex fuzzy mappings and 1 -convex fuzzy mappings includes many well-known classes of fuzzy mappings such as convex fuzzy mappings and preinvex fuzzy mappings as its subclasses,
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and that the class of 1 -B-vex fuzzy mappings and 1 -convex fuzzy mappings shares many properties with the class of convex functions. As pointed out in [2], we cannot give the notion of a di?erentiability of these fuzzy mappings in the usual manner, since the space of fuzzy numbers does not constitute any linear space. Goetschel and Voxman [3] have given the notion of a derivative for fuzzy mappings of one variable in a manner di?erent from the usual one. In an earlier paper [10], we have given the notion of a derivative for fuzzy mappings of several variables in ways that parallel the deAnition, proposed by Goetschel and Voxman [3], for a fuzzy mapping of one variable. Obviously, new fuzzy optimization approaches can be developed based on the concept of (1 ; 2 )-convexity, 1 -B-vexity and 1 -convexity. Furthermore, because of the use of these basic concepts, these new optimization approaches can be formulated based on sound convexity principles. References [1] E. Ammar, J. Metz, On fuzzy convexity and parametric fuzzy optimization, Fuzzy Sets and Systems 49 (1992) 135–141. [2] N. Furukawa, Convexity and local Lipschitz continuity of fuzzy-valued mappings, Fuzzy Sets and Systems 93 (1998) 113–119. [3] R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems 18 (1986) 31–43. [4] D. Li, Properties of B-vex fuzzy mappings and applications to fuzzy optimization, Fuzzy Sets and Systems 94 (1998) 253–260. [5] S. Nanda, K. Kar, Convex fuzzy mappings, Fuzzy Sets and Systems 48 (1992) 129–132. [6] M.A. Noor, Fuzzy preinvex functions, Fuzzy Sets and Systems 79 (1994) 95–104. [7] R. Pini, C. Singh, (1 ; 2 )-convexity, Optimization 40 (1997) 103–120. [8] Y.R. Syau, Preinvex fuzzy mappings, Comput. Math. Appl. 37 (1999) 31–39. [9] Y.R. Syau, On convex and concave fuzzy mappings, Fuzzy Sets and Systems 103 (1999) 163–168. [10] Y.R. Syau, Di?erentiability and convexity of fuzzy mappings, Comput. Math. Appl. 41 (2001) 73–81.
(1; 2)-Convex fuzzy mappings Yu-Ru Syau∗ Department of Industrial Engineering, Da Yeh University, Chang-Hwa 51505, Taiwan, ROC Received 7 March 2000; received in revised form 7 November 2002; accepted 14 November 2002
Abstract The concepts of (1 ; 2 )-convexity, 1 -B-vexity and 1 -convexity for fuzzy mappings are introduced through the so-called “fuzzy max” order among fuzzy numbers. We show that the class of (1 ; 2 )-convex fuzzy mappings, 1 -B-vex fuzzy mappings and 1 -convex fuzzy mappings includes many well-known classes of fuzzy mappings such as convex fuzzy mappings and preinvex fuzzy mappings as its subclasses. We also give a characterization for 1 -B-vex fuzzy mappings by proposing the concept of 1 -B-vex sets. In addition some basic properties of fuzzy mappings from the standpoint of convex analysis are discussed. c 2002 Elsevier B.V. All rights reserved. Keywords: Fuzzy numbers; Convexity; (1 ; 2 )-convex fuzzy mappings
1. Introduction The concept of convexity is important for quantitative and qualitative studies in operations research or applied mathematics. It has also been considered by many authors in fuzzy optimization. Like a paper by Ammar and Metz [1], a general fuzzy nonlinear programming problem is formulated with application of the concept of convexity. Di?erent types of convexity and generalized convexity of fuzzy mappings were studied by several authors, including [2,4–6], aiming at applications to fuzzy nonlinear programming. Recently, Pini and Singh [7] considered a class of functions called (1 ; 2 )-convex functions from a uniAed point of view: the function 1 describes a generalized convex combination of arguments, and 2 determines generalized convex combinations of values. In this way, a large number of well known and new convexity conditions can be included.
The work was partially supported by the National Science Council of Republic of China under contract NSC 90-2218-E-212-013. ∗ Tel.: +886-4-852-8468; fax: +886-4-851-1270. E-mail address: [email protected] (Y.-R. Syau). c 2002 Elsevier B.V. All rights reserved. 0165-0114/03/$ - see front matter doi:10.1016/S0165-0114(02)00527-4
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Y.-R. Syau / Fuzzy Sets and Systems 138 (2003) 617 – 625
Motivated both by earlier research works and by the importance of the concept of convexity, we propose in this paper the concepts of (1 ; 2 )-convexity, 1 -B-vexity and 1 -convexity for fuzzy mappings. We show that the class of (1 ; 2 )-convex fuzzy mappings, 1 -B-vex fuzzy mappings and 1 -convex fuzzy mappings includes many well-known classes of fuzzy mappings such as convex fuzzy mappings and preinvex fuzzy mappings as its subclasses. We also give a characterization for 1 -B-vex fuzzy mappings by proposing the concept of 1 -B-vex sets. In addition some basic properties of fuzzy mappings from the standpoint of convex analysis are discussed. 2. Preliminaries This section will present some basic knowledge of fuzzy numbers and fuzzy mappings from the standpoint of convex analysis. For the convenience of reading, several deAnitions and results without proof from [3,5,7–9] will be listed below. A fuzzy number we treat in this paper is a fuzzy set u : R1 → [0; 1] which is normal, fuzzy convex, upper semicontinuous, and has bounded support. The family of fuzzy numbers will be denoted by F0 . It is obvious that the -level set of a fuzzy number u is a closed and bounded interval {x ∈ R1 |u(x) ¿ } if 0 ¡ 6 1; [a(); b()] = [u] = cl(supp u) if = 0; where cl(supp u) denotes the closure of the support, {x ∈ R1 |u(x)¿0}, of u. It is also obvious that each r ∈ R1 can be considered as a fuzzy number and thus R1 can be embedded in F0 . It is easily veriAed that a fuzzy set u : R1 → [0; 1] is a fuzzy number if and only if (i) [u] is a closed and bounded interval for each ∈ [0; 1], and (ii) [u]1 = ∅. Thus we can identify a fuzzy number u with the parameterized triples {(a(); b(); )|0 6 6 1}; where a() and b() denote the left- and right-hand endpoints of [u] , respectively. Denition 2.1. For u; v ∈ F0 , we say that u 4 v if for every ∈ [0; 1], a() 6 c()
and
b() 6 d();
where [u] = [a(); b()] and [v] = [c(); d()]. We say that u ≺ v if u 4 v and there exists 0 ∈ [0; 1] such that a(0 ) ¡ c(0 )
or
b(0 ) ¡ d(0 ):
We see that u = v, if u 4 v and v 4 u. It is often convenient to write v ¡ u (resp. v u) in place of u 4 v (u ≺ v). A subset S ∗ of F0 is said to be bounded above if there exists a fuzzy number u ∈ F0 , called an upper bound of S ∗ , such that v 4 u for every v ∈ S ∗ . Further, a fuzzy number u0 ∈ F0 is called
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the least upper bound (sup in short) for S ∗ if (i) u0 is an upper bound of S ∗ , and (ii) u0 4 u for every upper bound u of S ∗ . A lower bound and the greatest lower bound (inf in short) are deAned similarly. Let addition and nonnegative scalar multiplication on F0 be deAned by the usual extension principle (for details, see [5,9]). It is easily veriAed that addition and nonnegative scalar multiplication preserve the order on F0 , and that F0 is closed under these operations. It is also known [9] that sup{u; v} and inf {u; v} exist in F0 for any pair {u; v} ⊂ F0 . Let D ⊆ Rn be nonempty. A mapping f : D → F0 is a fuzzy mapping. The epigraph of f, denoted by epi(f), is deAned as epi(f) = {(x; u) : x ∈ D; u ∈ F0 ; f(x) u}: Denition 2.2. Let C be a nonempty convex subset of Rn . A fuzzy mapping f : C → F0 is said to be: (1) convex if for every ∈ [0; 1] and x; y ∈ C, f(x + (1 − )y) f(x) + (1 − )f(y); (2) quasi-convex if for every ∈ [0; 1] and x; y ∈ C, f(x + (1 − )y) sup{f(x); f(y)}: Recall that, by deAnition, a set K ⊆ Rn is said to be an invex set with respect to a function : K × K → Rn if x; y ∈ K implies that y + (x; y) ∈ K for all ∈ [0; 1]. Denition 2.3. Let K ⊆ Rn be a nonempty invex set with respect to a given function . A fuzzy mapping f : K → F0 is said to be preinvex if for every ∈ [0; 1] and x; y ∈ K, f(y + (x; y)) f(x) + (1 − )f(y):
3. Denitions and basic results Let S ⊆ Rn be nonempty, and let F be the family of fuzzy mappings deAned on S. Assume that 1 : S × S × [0; 1] → Rn with 1 (x; y; 0) = y; 1 (x; x; ) = x for all x; y ∈ S and ∈ [0; 1], and that 2 : S × S × [0; 1] × F → F0 with 2 (x; y; 0; f) = f(y); 2 (x; x; ; f) = f(x); 2 (x; y; ; f) 4 sup {f(x); f(y)} for all x; y ∈ S; ∈ [0; 1] and f ∈ F. A set X ⊆ S is 1 -convex if 1 (x; y; ) ∈ X for all x; y ∈ X; ∈ [0; 1]. From now on, S will be a 1 convex set. In what follows, let R∗ be the set of nonnegative real numbers. Let b : S × S × [0; 1] → R∗ , with b(x; y; ) ∈ [0; 1] for all x; y ∈ S and ∈ [0; 1]. For the sake of brevity, we shall omit the argument of b unless it is needed for speciAcation.
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Denition 3.1. A fuzzy mapping f : S → F0 is said to be: (1) (1 ; 2 )-convex if for every ∈ [0; 1] and x; y ∈ S, f(1 (x; y; )) 2 (x; y; ; f): (2) 1 -B-vex if for every ∈ [0; 1] and x; y ∈ S, f(1 (x; y; )) b(x; y; )f(x) + (1 − b(x; y; ))f(y): (3) strictly 1 -B-vex if for all x; y ∈ S; x = y; ∈ (0; 1), f(1 (x; y; )) ≺ b(x; y; )f(x) + (1 − b(x; y; ))f(y): (4) 1 -convex if for every ∈ [0; 1] and x; y ∈ S, f(1 (x; y; )) f(x) + (1 − )f(y): (5) 1 -quasiconvex if for every ∈ [0; 1] and x; y ∈ S, f(1 (x; y; )) sup{f(x); f(y)}: It is clear that every convex (resp. preinvex with (x; x) = O, where O being the origin of Rn ) fuzzy mapping is 1 -convex where 1 (x; y; ) = x + (1 − )y (resp. 1 (x; y; ) = y + (x; y)). Theorem 3.1. For a fuzzy mapping f, 1 -convex ⇒ 1 -B-vex ⇒ (1 ; 2 )-convex ⇒ 1 -quasiconvex: The following result which gives a constructive description of 1 -convex fuzzy mappings and 1 -B-vex fuzzy mappings is an immediate consequence of DeAnitions 2.1 and 3.1. Theorem 3.2. Let f be a fuzzy mapping de4ned on a 1 -convex set S, and for each x ∈ S, let {(c(; x); d(; x); ) | 0661} be the parametric representation of f(x). Then f is a 1 -convex (resp. 1 -B-vex) fuzzy mapping if and only if for each ∈ [0; 1]; c(; x) and, d(; x) are 1 -convex (resp. 1 -B-vex) functions of x ∈ S. In view of the deAnition of 1 -B-vex fuzzy mappings and the fact that addition and nonnegative scalar multiplication preserve the order on F0 , we obtain: Theorem 3.3. Let fj : S → F0 ; j = 1; 2; : : : ; N be 1 -B-vex fuzzy mappings with respect to the same b. Then the fuzzy mapping f : S → F0 de4ned by f(x) =
N
kj fj (x);
kj ¿ 0
j=1
is a 1 -B-vex fuzzy mapping on S.
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Corollary 3.1. Let fj : S → F0 ; j = 1; 2; : : : ; N be 1 -convex fuzzy mappings. Then the fuzzy mapping f : S → F0 de4ned by f(x) =
N
kj fj (x);
kj ¿ 0
j=1
is a 1 -convex fuzzy mapping on S. Denition 3.2. Given T ⊆ Rn × F0 ; T is said to be a 1 - B-vex set with respect to b if (x; u); (y; v) ∈ T implies that (1 (x; y; ); bu + (1 − b)v) ∈ T;
0 6 6 1:
It is obvious that each r ∈ R1 can be considered as a fuzzy number and thus it does make sense to speak of 1 -B-vex subset of Rn × R1 . We now give a characterization of 1 -B-vex fuzzy mappings in terms of their epigraphs. Theorem 3.4. A fuzzy mapping f : S → F0 is 1 -B-vex with respect to b if and only if its epigraph epi(f) is a 1 -B-vex set in Rn × F0 with respect to b. Proof. Let f : S → F0 be a 1 -B-vex fuzzy mapping and (x; u); (y; v) ∈ epi(f) with x; y ∈ S and u; v ∈ F0 . Then f(x) 4 u; f(y) 4 v. Since f is 1 -B-vex on S, and since addition and nonnegative scalar multiplication preserve the order on F0 , we have f(1 (x; y; )) bf(x) + (1 − b)f(y)
bu + (1 − b)v; 0 6 6 1; which implies that (1 (x; y; ); bu + (1 − b)v) ∈ epi(f);
0 6 6 1:
Thus, epi(f) is a 1 -B-vex set with respect to b. Conversely, assume that epi(f) is a 1 -B-vex set and x; y ∈ S. Then (x; f(x)); (y; f(y)) ∈ epi(f). Since epi(f) is a 1 -B-vex set with respect to b, it follows that (1 (x; y; ); bf(x) + (1 − b)f(y)) ∈ epi(f);
0 6 6 1;
which implies that f(1 (x; y; )) bf(x) + (1 − b)f(y);
0 6 6 1:
Hence, f is a 1 -B-vex fuzzy mapping with respect to b. This completes the proof. Theorem 3.5.If {Ti }i∈I is a family of 1 -B-vex sets in Rn × F0 with respect to b, then their intersection i∈I Ti is a 1 -B-vex set with respect to the same b.
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Proof. Assume that (x; u); (y; v) ∈ i∈I Ti and 0661. Then for each i ∈ I , we have (x; u); (y; v) ∈ Ti . Since each Ti is a 1 -B-vex set with respect b, it follows that (1 (x; y; ); bu + (1 − b)v) ∈ Ti ; which shows that
i ∈I
i ∈I
Ti is a 1 -B-vex set with respect to b.
Theorem 3.6. Let {fi }i∈I be an arbitrary nonempty collection of 1 -B-vex fuzzy mappings de4ned on S with respect to the same b such that for every x ∈ S; sup{fi (x) : i ∈ I } exists in F0 . Then the function f : S → F0 de4ned by f(x) = sup{fi (x) : i ∈ I } is a 1 -B-vex fuzzy mapping on S with respect to b. Proof. Since each fi is a 1 -B-vex fuzzy mapping with respect to b, its epigraph epi(fi ) = {(x; u) : x ∈ S; u ∈ F0 ; fi (x) u} is a 1 -B-vex set in Rn × F0 with respect to b, by Theorem 3.4. Hence their intersection epi(fi ) = {(x; u) : x ∈ S; u ∈ F0 ; fi (x) u; i ∈ I } i ∈I
is also a 1 -B-vex set in Rn × F0 with respect to b, by Theorem 3.5. It can be easily checked that this intersection is the epigraph of f. Hence, by Theorem 3.4, f is a 1 -B-vex fuzzy mapping on S with respect to b. Corollary 3.2. Let {fi }i∈I be an arbitrary nonempty collection of 1 -convex fuzzy mappings de4ned on S such that for every x ∈ S; sup{fi (x) : i ∈ I } exists in F0 . Then the function f : S → F0 de4ned by f(x) = sup{fi (x) : i ∈ I } is a 1 -convex fuzzy mapping on S. 4. Main results In this section, we present some properties of 1 -quasiconvex fuzzy mappings, 1 -B-vex fuzzy mappings and 1 -convex fuzzy mappings and discuss some applications to optimization. Theorem 4.1. A fuzzy mapping f : S → F0 is 1 -quasiconvex if and only if the set Lu = {x ∈ S : f(x) 4 u} is a 1 -convex (crisp) set for each u ∈ F0 . Proof. Let f : S → F0 be 1 -quasiconvex, and u ∈ F0 . If Lu is a singleton or an empty set then it is obvious a 1 -convex set. Assume that x; y ∈ Lu , i.e., f(x) 4 u and f(y) 4 u. So u is an upper
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bound of {f(x); f(y)}, it follows that sup{f(x); f(y)} 4 u. Since f is 1 -quasiconvex, we have for all ∈ [0; 1], f(1 (x; y; )) sup{f(x); f(y)} u: It follows that 1 (x; y; ) ∈ Lu for all ∈ [0; 1]. Thus Lu is a 1 -convex set. Conversely, let Lu be a 1 -convex set for every fuzzy number u. Let x; y ∈ S and u = sup{f(x); f(y)}. Then x; y ∈ Lu , and by the 1 -convexity of Lu , it follows that 1 (x; y; ) ∈ Lu for all ∈ [0; 1]. Hence, f(1 (x; y; )) u = sup{f(x); f(y)}: Thus, f : S → F0 is 1 -quasiconvex. This completes the proof. Theorem 4.2. Let f : S → F0 be 1 -convex. If g : F0 → F0 is a nondecreasing convex fuzzy mapping, then the mapping x → g(f(x)) is 1 -convex on S. Proof. Let x; y ∈ S, and ∈ [0; 1]. Since f : S → F0 is 1 -convex on S, we have f(1 (x; y; )) f(x) + (1 − )f(y): Since g : F0 → F0 is non-decreasing and convex, it follows that g(f(1 (x; y; ))) g(f(x) + (1 − )f(y))
g(f(x)) + (1 − )g(f(y)); and hence x → g(f(x)) is 1 -convex on S. This completes the proof. We now discuss some applications of 1 -B-vex fuzzy mappings to fuzzy optimization theory. Theorem 4.3. Let f be a fuzzy mapping de4ned on S such that inf {f(x) : x ∈ S} exists in F0 . Let # = inf {f(x) : x ∈ S}. (1) If f : S → F0 is 1 -B-vex, then the set $ = {x ∈ S : f(x) = #} is 1 -convex. (2) If f is strictly 1 -B-vex, then $ is singleton or empty. That is, if f is strictly 1 -B-vex, then f has at most one global minimizer. Proof. To prove part (1), let f : S → F0 be 1 -B-vex on S. If $ is a singleton or an empty set then it is obvious a 1 -convex set. Assume that x; y ∈ $, i.e., f(x) = f(y) = #. Since f is a 1 -B-vex fuzzy mapping on S, f(1 (x; y; )) bf(x) + (1 − b)f(y) = b# + (1 − b)# =# for all ∈ [0; 1]. Thus all points of the form 1 (x; y; ); ∈ [0; 1] belong to $. Hence $ is 1 -convex.
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For the second part, the proof will be by contradiction: assume that there exist distinct points x; y ∈ S such that f(x) = f(y) = #. Since S is 1 -convex, then for ∈ (0; 1); 1 (x; y; ) ∈ S. Further since f is strictly 1 -B-vex on S, f(1 (x; y; )) ≺ bf(x) + (1 − b)f(y) = b# + (1 − b)# = #: This is contradicts that # = inf {f(x) : x ∈ S} and hence the result follows. Theorem 4.4. If f : S → F0 is 1 -B-vex with respect to b; lim→0+ 1 (x; y; ) = y for all x; y ∈ S, and b(x; y; )¿0 for all x; y ∈ S; 0¡¡d¡1 for some 4xed d, then every local minimizer of f is also a global minimizer of f over S. Proof. The proof is by contradiction. Let x˜ be a local minimizer of f, and suppose, by contradiction, that it is not a global minimizer of f over S. Then there exists some point x ∈ S satisfying f(x) ≺ f(x): ˜ By the 1 -B-vexity of f, for all ∈ [0; 1], ˜ )) bf(x) + (1 − b)f(x): ˜ f(1 (x; x; Since f(x) ≺ f(x), ˜ we have bf(x) + (1 − b)f(x) ˜ = b(f(x) − f(x)) ˜ + f(x) ˜ ≺ f(x) ˜ for all ∈ (0; d). Thus, for all ∈ (0; d), ˜ )) ≺ f(x): ˜ f(1 (x; x; Hence by the 1 -convexity of S, and since ˜ ) = x˜ lim 1 (x; x;
→0+
˜ ) ∈ S for all 0661, and 1 (x; x; ˜ ) are arbitrarily close to x˜ for suLciently it follows that 1 (x; x; small ¿0. This contradicts the deAnition of a local minimizer. Hence x˜ must be a global maximizer.
Corollary 4.1. Let f : S → F0 be 1 -convex, lim→0+ 1 (x; y; ) = y for all x; y ∈ S, and let x˜ ∈ S be a local minimizer of f. Then x˜ is also a global minimizer of f over S. 5. Discussions Motivated by earlier research works [2,3–7,9], we introduced in this paper the concept of (1 ; 2 )convexity, 1 -B-vexity and 1 -convexity into the class of fuzzy mappings through the so-called “fuzzy max” order among fuzzy numbers. It is shown that the class of (1 ; 2 )-convex fuzzy mappings, 1 -B-vex fuzzy mappings and 1 -convex fuzzy mappings includes many well-known classes of fuzzy mappings such as convex fuzzy mappings and preinvex fuzzy mappings as its subclasses,
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and that the class of 1 -B-vex fuzzy mappings and 1 -convex fuzzy mappings shares many properties with the class of convex functions. As pointed out in [2], we cannot give the notion of a di?erentiability of these fuzzy mappings in the usual manner, since the space of fuzzy numbers does not constitute any linear space. Goetschel and Voxman [3] have given the notion of a derivative for fuzzy mappings of one variable in a manner di?erent from the usual one. In an earlier paper [10], we have given the notion of a derivative for fuzzy mappings of several variables in ways that parallel the deAnition, proposed by Goetschel and Voxman [3], for a fuzzy mapping of one variable. Obviously, new fuzzy optimization approaches can be developed based on the concept of (1 ; 2 )-convexity, 1 -B-vexity and 1 -convexity. Furthermore, because of the use of these basic concepts, these new optimization approaches can be formulated based on sound convexity principles. References [1] E. Ammar, J. Metz, On fuzzy convexity and parametric fuzzy optimization, Fuzzy Sets and Systems 49 (1992) 135–141. [2] N. Furukawa, Convexity and local Lipschitz continuity of fuzzy-valued mappings, Fuzzy Sets and Systems 93 (1998) 113–119. [3] R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems 18 (1986) 31–43. [4] D. Li, Properties of B-vex fuzzy mappings and applications to fuzzy optimization, Fuzzy Sets and Systems 94 (1998) 253–260. [5] S. Nanda, K. Kar, Convex fuzzy mappings, Fuzzy Sets and Systems 48 (1992) 129–132. [6] M.A. Noor, Fuzzy preinvex functions, Fuzzy Sets and Systems 79 (1994) 95–104. [7] R. Pini, C. Singh, (1 ; 2 )-convexity, Optimization 40 (1997) 103–120. [8] Y.R. Syau, Preinvex fuzzy mappings, Comput. Math. Appl. 37 (1999) 31–39. [9] Y.R. Syau, On convex and concave fuzzy mappings, Fuzzy Sets and Systems 103 (1999) 163–168. [10] Y.R. Syau, Di?erentiability and convexity of fuzzy mappings, Comput. Math. Appl. 41 (2001) 73–81.