Preinvex Fuzzy-valued Function and Its Application in Fuzzy Optimization
Based on the ordering of fuzzy numbers proposed by Goetschel and Voxman, in this paper, the representations and characterizations of semi-E-preinvex fuzzy-valued function are defined and obtained. As an application, the conditions of strictly local optima
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Abstract Based on the ordering of fuzzy numbers proposed by Goetschel and Voxman, in this paper, the representations and characterizations of semi-E-preinvex fuzzy-valued function are defined and obtained. As an application, the conditions of strictly local optimal solution and global optimal solution in the mathematical programming problem are discussed. Keywords Fuzzy numbers · semi-E-preinvexity · fuzzy optimization
1 Introduction The concept of fuzzy set was introduced by Zadeh in [11]. Since then, many applications of fuzzy set have been widely developed. Just as many systems with parameter uncertainty, the optimization theory with parameter uncertainty such as in objective function, constraints, or both of objective function and constraints, is often dealt. It is well known that the classical theory of convex analysis and mathematical programming are closely linked each other. Some authors have discussed the convexity, quasi-convexity and B−convex of fuzzy mappings [6, 8]. In 1994, Noor [5] introduced the concept of preinvex fuzzy-valued functions over the field of real numbers R, and obtained some properties of preinvex fuzzy-valued functions. After that, the properties of preinvex fuzzy-valued functions have been developed and generalized by many authors [7–10] and applied in fuzzy optimization problem [3]. The essence Z. Gong (B) College of Mathematics and Statistics, Northwest Normal University, 730070 Lanzhou, China e-mail: [email protected] Y. Bai Department of Mathematics, Longdong University, 745000 Qingyang, China W. Pan School of Economics and Management, Tsinghua University, 100084 Beijing, China B.-Y. Cao and H. Nasseri (eds.), Fuzzy Information & Engineering and Operations Research & Management, Advances in Intelligent Systems and Computing 211, DOI: 10.1007/978-3-642-38667-1_4, © Springer-Verlag Berlin Heidelberg 2014
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of preinvex fuzzy-valued functions is investigated and some judge theorems and a characterization of preinvex fuzzy-valued functions are obtained by using the upper (lower) semi-continuity [4]. In this paper, some representations and characterizations of semi-E-preinvex fuzzy-valued functions are obtained. As an application, the conditions of strictly local optimal solution and global optimal solution in the mathematical programming problem are discussed.
2 Preliminaries and Definitions A fuzzy number is a mapping u: R → [0, 1], with the following properties: 1. 2. 3. 4.
u u u x
is normal, i.e., there exists x0 ∈ R with u(x0 ) = 1; is convex fuzzy set; is semicontinuous on R and ∈ R : u(x) > 0 is compact.
Let F be the set of all fuzzy numbers on R. For u ∈ F, we write [u]α = [u − (α), u + (α)], then the following conditions are satisfied: 1. 2. 3. 4.
u − (α) is abounded left continuous non-decreasing function on (0, 1]; u + (α) is abounded left continuous non-increasing function on (0, 1]; u − (α) and u + (α) are right continuous at α = 0 and left continuous at α = 1. u − (1) ≤ u + (1).
Conversely, if the pair of functions u − (α) and u + (α) satisfy
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