A Characterization of Convex Fuzzy Mappings

A Characterization of Convex Fuzzy Mappings

Chih-Yuan Chen (Nanya Institute of Technology, Taiwan), Cheng-Pin Wang (Yuan-Ze University, Taiwan) and Tetz C. Huang (Yuan-Ze University, Taiwan)
Copyright: © 2010 |Pages: 5
DOI: 10.4018/jalr.2010070104
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Abstract

As any real-valued functions can be regarded as a fuzzy mapping, the research on fuzzy mappings extends the research on real-valued functions. In this paper, a characterization of convex fuzzy mappings is obtained. Supposeis a fuzzy mapping, whereis a non-empty convex subset ofandis the set of all fuzzy numbers. With respect to the fuzzy-max order, is convex if and only if it is both quasi-convex and intermediate-point convex.
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Preliminaries

Letdenote the -dimensional Euclidean space. A fuzzy set ofis defined to be a function frominto. The support,, of a fuzzy setis defined by:

.

For, the -level setof a fuzzy setis defined by:

wheredenotes the closure of. A fuzzy number considered in this paper is a fuzzy setwith the following properties:

  • (1)is normal, i.e.,,

  • (2)is fuzzy convex, i.e.,, for anyand for any,

  • (3)is upper semicontinuous, i.e., for any,is a closed subset of, and

  • (4)is bounded.

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