By Xiang Li

It presents fuzzy programming method of clear up real-life selection difficulties in fuzzy surroundings. in the framework of credibility concept, it presents a self-contained, finished and updated presentation of fuzzy programming versions, algorithms and functions in portfolio research.

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**Example text**

Fuzzy Sets Syst 1:283– 297 Klir GJ (1999) On fuzzy-set interpretation of possibility theory. Fuzzy Sets Syst 108:263–273 Li X, Liu B (2006a) A sufficient and necessary condition for credibility measures. Int J Uncertain Fuzziness Knowl-Based Syst 14(5):527–535 Li X, Liu B (2006b) The independence of fuzzy variables with applications. Int J Nat Sci Technol 1(1):95–100 Li X, Liu B (2007) Maximum entropy principle for fuzzy variables. Int J Uncertain Fuzziness Knowl-Based Syst 15(2):43–52 Li PK, Liu B (2008a) Entropy of credibility distributions for fuzzy variables.

9 Let ξ = (a, b, c, d) be a symmetric trapezoidal fuzzy variable. 7) that its variance is V [ξ ] = (d − a)2 + (d − a)(c − b) + (c − b)2 /24. 10 Let ξ = EXP(m) be an exponential fuzzy variable. 5 has shown that it has an expected value √ e = ( 6m ln 2)/π. 0031. 11 For a normal fuzzy variable ξ = N (e, σ ), we will prove that it has a variance σ 2 . First, according to the maximality axiom, we have √ Cr (ξ − e)2 ≥ x = Cr{ξ − e ≥ x} for any x ≥ 0. 7) that ∞ V [ξ ] = Cr{ξ − e ≥ √ x} dx 0 ∞ = 2xCr{ξ − e ≥ x} dx 0 ∞ = 2(x − e)ν(x) dx e = σ 2.

5. 3 that Cr{ξ1 ∈ B1 , ξ2 ∈ B2 , . . , ξm ∈ Bm } c = 1 − Cr ξ1 ∈ B1c ∪ ξ2 ∈ B2c ∪ · · · ∪ ξm ∈ Bm 22 1 Credibility Theory = 1 − max Cr ξi ∈ Bic 1≤i≤m = min Cr{ξi ∈ Bi }. 27) holds for any sets B1 , B2 , . . , Bm of real numbers. Then for any (x1 , x2 , . . , xm ) ∈ m , we have ν(x1 , x2 , . . , xm ) = Cr{ξ1 = x1 , ξ2 = x2 , . . , ξm = xm } = min Cr{ξi = xi } 1≤i≤m = min νi (xi ), 1≤i≤m which implies that fuzzy variables ξ1 , ξ2 , . . , ξm are independent. The proof is complete. 18 Fuzzy variables ξ1 , ξ2 , .