Fuzzy Weirstrass theorem and convex fuzzy mappings

g e y w o r d s F u z z y numbers, Convexity, Continuity, Convex fuzzy mappings, Linear ordering, Fuzzy Weirstrass theorem, Fuzzy optimization. 1. I N T R O D U C T I O N Let R n denote the n-dimensional Eucl idean space. The suppor t , supp(#) , of a fuzzy set # : R ~ I = [0, 1] is defined as supp(/~) = {x • R n I #(x) > 0}. A fuzzy set # : R n --~ I is called fuzzy convex if #(Ax + (1 A)y) _> min{#(x) , #(y)}, for all x, y E supp(#) , and A c [0,1]. A fuzzy set # : R ~ ~ I is said to be normal if there exists a point x E R '~ such tha t #(x) = 1. A fuzzy number we t r ea t in this s tudy is a fuzzy set # : R 1 --* I which is normal , fuzzy convex, upper semicontinuous and with bounded suppor t . Supported by the National Science Council of the Republic of China under contract NSC 88-2213-E-155-021. This work was carried out while the first author was visiting the Department of Industrial and Manufacturing Systems Engineering, Kansas State University. *Author to whom all correspondence should be addressed. 0898-1221/06/$ see front matter (~ 2006 Elsevier Ltd. All rights reserved. Typeset by ,4j~4S-TEX doi: 10.1016/j .camwa. 2006.02.005 1742 Y.-R. SYAU AND E. S. LEE Let j r denote the set of all fuzzy numbers. A mapping from any nonempty set into j r will be called a fuzzy mapping. It is clear that each r • R 1 can be considered as a fuzzy number. Hence, each real-valued function can be considered as a fuzzy mapping. The concept of convexity for fuzzy mappings has been considered by many authors in fuzzy optimization. For example, in [1-5], the concept of convex fuzzy mappings defined through the "fuzzy-max" order was investigated. However, the "fuzzy-max" order is a partial ordering on the set of fuzzy numbers. In [6], Goetschel and Voxman proposed a linear ordering ___ on jr. For each fuzzy mapping f : R 1 ~ jr, based on the linear ordering _, they introduced a real-valued function T/ on the domain of the fuzzy mapping f . In [7], two concepts of convexity and quasiconvexity for a fuzzy mapping f are defined through the real-valued function T/ in t roduced in [6]. In this paper, we introduce the concept of convex fuzzy mappings directly through the linear ordering proposed in [6]. We define a ranking value function T on j r and the concept of monotonicity for a fuzzy mapping g : j r --* jr. Based on the ranking value function ~-, the concept of quasiconvex fuzzy mappings is also introduced. The continuity of fuzzy mappings through a metric on j r is studied, and the Weirstrass theorem is extended from real-valued functions to fuzzy mappings. The local-global minimum property of real-valued convex functions is extended to convex fuzzy mappings. As for real-valued convex functions, nonnegative linear combinations of convex fuzzy mappings are convex. Characterizations for convex fuzzy mappings and quasiconvex fuzzy mappings are also given. In addition, it is proved that every strict local minimizer of a quasiconvex fuzzy mapping is a global minimizer. 2. P R E L I M I N A R I E S In this section, for convenience, several definitions and results without proof from [3,6,8] are summarized below. It can be easily checked that the a-level set of a fuzzy number # E j r is a closed and bounded interval f { x • R I I ~ ( x ) > ~ } , i f 0 < ~ < l ; c l ( s u p p ( , ) ) , i f a = o, where cl(supp(#)) denotes the closure of supp(#). It was shown in [3] that a fuzzy set # : R 1 --* I is a fuzzy number if and only if (i) [#]~ is a closed and bounded interval for each a • [0, 1], and (ii) [#]1 ~ {~. Thus, we can identify a fuzzy number p with the parameterized triples, { ( a ( a ) , b ( a ) , a ) 1 0 < a < 1}, where a(a) and b(a) denote the leftand right-hand endpoints of [/z]~, respectively. For fuzzy numbers #, u E Y represented parametrically by {(a(a), b(a), a) [ 0 _< a _< 1} and {(c(a), d(a), a) I 0 < a < 1}, respectively, and each nonnegative real number r, we define the addition # + ~ and nonnegative scalar multiplication r# as follows, / ~ + u = {(a (a) +c(a) ,b(a)+d(a) ,~)10 < < 1}, r# = {(ra(a),rb(a),a) [0 < a < 1}. (2.1) (2.2) It is known that the addition and nonnegative scalar multiplication on j r defined by (2.1) and (2.2) are equivalent to those derived from the usual extension principle, and that j r is closed under the addition and nonnegative scalar multiplication. According to Goetschel and Voxman [6], we metricize 9 r by the metric, D({(a(a),b(a),a)l O < a < l } , { ( c ( a ) , d ( a ) , ~ ) 1 0 < a < l } ) = sup{max {la (a) c ( a ) l , Ib(a) d ( ~ ) l } I 0 < ~ < 1}, and define the following ordering, _, for jr. Fuzzy Weirstrass Theorem and Convex Fuzzy Mappings 1743 DEFINITION 2.1. (See [6, Definition 2.5].) Suppose that # = { ( a ( a ) , b ( a ) , a ) [ 0 <_ a < 1} and = { (c (~) , d(~), ~) I o < ~ < 1} ~ e f u ~ y numbers. Then ~ precedes ~ (~ ~ ~) i~ /o /o 1 la[a(a) + b(a)] da <_ a[c(a) + d(a)] da. (2.3) REMARK 2.1. (See [6].) The ordering ___ is reflexive and transitive; moreover, any two elements of ~" are comparable under the ordering ~, i.e., _ is a linear ordering for ~. 3. D E F I N I T I O N S A N D B A S I C R E S U L T S In this section, based on the linear ordering ~ on 3 r , we define a ranking value function and a strict ordering -< of __ on 5 r. Then, the concept of monotonicity for a fuzzy mapping g : ~" -~ ~" is proposed. We also introduce the concepts of convexity and continuity for fuzzy mappings based on the ordering _ and the metric D, respectively, introduced in the preceding section. First, motivated by the notion of the linear ordering _-< on 9 r , we define a ranking value function T : 9 v ~ R 1 as follows. DEFINITION 3.1. Let T : 2~ --~ R 1 be defined by /o 1 T (#) ---a [a (a) + b (a)] da, (3.1) for each # = { ( a ( a ) , b ( a ) , a ) l O < a < 1} e ~-. LEMMA 3.1. For ~, I~ E ~ , and k > O, 1. T(~ + U) = T( , ) + , (~) , 2. T(klz) = kr(lz). PROOF. Let {(a(a), b(a), a) I 0 < a < 1} and {(c(a), d(a), a) [ 0 _< a < 1} be the parametric representations of # and v, respectively. Then, for each a E [0, 1], [# + v]~ ----[a(a) + c(a), b(a) + d(a)] (3.2)