Credibilistic Programming: An Introduction to Models and by Xiang Li

By Xiang Li

It presents fuzzy programming method of clear up real-life choice difficulties in fuzzy surroundings. in the framework of credibility concept, it presents a self-contained, finished and up to date presentation of fuzzy programming versions, algorithms and purposes in portfolio analysis.

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IEEE Trans Syst Man Cybern, Part B 38(2):381–403 Ehrgott M (2000) Multicriteria optimization. Springer, Berlin Ewald G, Kurek W, Brdys MA (2008) Grid implementation of a parallel multiobjective genetic algorithm for optimized allocation of chlorination stations in drinking water distribution systems: Chojnice case study. IEEE Trans Syst Man Cybern, Part C 38(4):497–509 Farina M, Amato P (2004) A fuzzy definition of “optimality” for many-criteria optimization problems. IEEE Trans Syst Man Cybern, Part A 34(2):315–326 Goldberg DE (1989) Genetic algorithms & engineering optimization.

X ≥ c. For any α1 x1 + α2 x2 = x, we have x1 ≥ c1 or x2 ≥ c2 . It follows from the Zadeh extension theorem that ν(x) = 0. 13 Similarly, suppose that trapezoidal fuzzy variables ξ1 = (a1 , b1 , c1 , d1 ) and ξ2 = (a2 , b2 , c2 , d2 ) are independent. Then for any nonnegative real numbers α1 and α2 , we have α1 ξ1 + α2 ξ2 = (α1 a1 + α2 a2 , α1 b1 + α2 b2 , α1 c1 + α2 c2 , α1 d1 + α2 d2 ). 24 Suppose that exponential fuzzy variables ξ1 = EXP(m1 ) and ξ2 = EXP(m2 ) are independent. Then for any α1 > 0 and α2 > 0, we have α1 ξ1 + α2 ξ2 = E(α1 m1 + α2 m2 ).

Xm ) be a solution vector in the solution space satisfying x1 + x2 + · · · + xm = 1 xi ≥ 0, i = 1, 2, . . , m. 16) We may encode the solution by a chromosome v = (v1 , v2 , . . , vm ) satisfying vi ≥ 0, i = 1, 2, . . , m. 17) Then the encoding and decoding processes are determined by the equations x1 = v1 , v1 + v2 + · · · + vm i = 1, 2, . . , m. 2 Initialization Define an integer pop-size as the size of population, which generally depends on the nature of the problem. Randomly generate pop-size chromosomes for the initialized population.

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