By Xiang Li

It presents fuzzy programming method of resolve real-life choice difficulties in fuzzy surroundings. in the framework of credibility idea, it offers a self-contained, complete and up to date presentation of fuzzy programming types, algorithms and functions in portfolio research.

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

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**Extra resources for Credibilistic Programming: An Introduction to Models and Applications (Uncertainty and Operations Research)**

**Example text**

Fm , fuzzy variables f1 (ξ1 ), f2 (ξ2 ), . . , fm (ξm ) are also independent. Proof For any (x1 , x2 , . . 17, we have Cr f1 (ξ1 ) = x1 , f2 (ξ2 ) = x2 , . . , fm (ξm ) = xm = Cr ξ1 ∈ f1−1 (x1 ), ξ2 ∈ f2−1 (x2 ), . . , ξm ∈ fm−1 (xm ) = min Cr ξi ∈ fi−1 (xi ) 1≤i≤m = min Cr fi (ξi ) = xi . 1≤i≤m Thus f1 (ξ1 ), f2 (ξ2 ), . . , fm (ξm ) are independent. The proof is complete. 20 If fuzzy variables ξ and η are independent, then it follows from above theorem that fuzzy variables ξ 2 and (η + 1)3 are also independent.

Therefore, we need to define a credibilistic mapping from the collection of fuzzy variables to the set of real numbers, such that we can rank fuzzy variables according to the natural ordership of real numbers. 9), if the credibilistic mappings U, U1 , U2 , . . t. Ui gi (x, ξ ) ≤ 0, i = 1, 2, . . , n. 11) is a crisp nonlinear programming model since the objective function and constraints are both well defined. In what follows, we will call it a credibilistic programming model. 2 Credibilistic Programming 37 credibilistic mappings including the expected value operator, optimistic value, pessimistic value, entropy, cross-entropy, and distance.

5, 0, if x ∈ {−1, 1} otherwise. 15 Fuzzy variables ξ1 , ξ2 , . . , ξm are said to be identically distributed if and only if each pair of them are identically distributed. 27 Suppose that ξi = (a, b, c), i = 1, 2, . . , m are independent and identically distributed triangular fuzzy variables. For any positive numbers x1 , x2 , . . , xm with x1 + x2 + · · · + xm = 1, we have ξ1 x1 + ξ2 x2 + · · · + ξm xm = (a, b, c). That is, the weighted sum is identically distributed with each element. The conclusion still holds if the triangular fuzzy variables are changed to be equipossible fuzzy variables, trapezoidal fuzzy variables, normal fuzzy variables, or exponential fuzzy variables.