By Xiang Li

It offers fuzzy programming method of clear up real-life choice difficulties in fuzzy surroundings. in the framework of credibility thought, it offers a self-contained, finished and updated presentation of fuzzy programming versions, algorithms and purposes in portfolio research.

**Read Online or Download Credibilistic Programming: An Introduction to Models and Applications (Uncertainty and Operations Research) PDF**

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IEEE Trans Syst Man Cybern, Part A 38(2):299– 308 Delgado M, Cuellar MP, Pegalajar MC (2008) Multiobjective hybrid optimization and training of recurrent neural networks. 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.

It is solved that y2 = x(c2 − b2 ) + α1 ((c1 − b1 )c2 − c1 (c2 − b2 )) . α1 (c1 − b1 ) + α2 (c2 − b2 ) 26 1 Credibility Theory Taking it into the credibility function ν2 , we get ν(x) = (c − x)/2(c − b). Case 4. 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 ).

5 Let us reconsider the portfolio selection problem. Suppose that there are three stocks and the returns are independent triangular fuzzy variables ξ1 = (0, 3, 6), ξ2 = (2, 3, 4), and ξ3 = (−1, 0, 1). See Fig. 2. It is clear that the second stock has a better return than the third stock since it follows from the credibility inversion theorem that Cr{ξ2 ≥ ξ3 } = 1. However, it is 38 2 Credibilistic Programming Fig. 2 Credibility functions for ξ1 , ξ2 and ξ3 difficult to compare the returns arising from the second stock and the first stock.