By Xiang Li

ISBN-10: 364236375X

ISBN-13: 9783642363757

ISBN-10: 3642363768

ISBN-13: 9783642363764

It presents fuzzy programming method of clear up real-life selection difficulties in fuzzy atmosphere. in the framework of credibility thought, it presents a self-contained, complete and up to date presentation of fuzzy programming versions, algorithms and functions in portfolio analysis.

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**Extra info for Credibilistic Programming: An Introduction to Models and Applications**

**Sample text**

If a = 0, then the equation holds trivially. If a > 0, we have ∞ E[aξ ] = Cr{aξ ≥ r} dr − 0 ∞ = 0 −∞ 0 Cr{ξ ≥ r/a} dr − −∞ 0 ∞ =a Cr{aξ ≤ r} dr Cr{ξ ≤ r/a} dr 0 Cr{ξ ≥ s} ds − a −∞ 0 Cr{ξ ≤ s} ds = aE[ξ ]. 1 that ∞ E[aξ ] = Cr{aξ ≥ r} dr − 0 ∞ = Cr{ξ ≤ r/a} dr − 0 = −a 0 −∞ = aE[ξ ]. 1 Expected Value 49 Step 2: We prove that E[ξ + b] = E[ξ ] + b for any real number b. If b ≥ 0, according to the duality axiom of credibility measure, we have ∞ E[ξ + b] = 0 Cr{ξ + b ≥ r} dr − −∞ 0 ∞ = 0 Cr{ξ ≥ r − b} dr − −∞ 0 ∞ = −b 0 = Cr{ξ ≥ s} ds + −b −b Cr{ξ ≥ s} ds − Cr{ξ ≤ r − b} dr Cr{ξ ≤ s} ds −∞ 0 Cr{ξ + b ≤ r} dr Cr{ξ ≤ s} ds + E[ξ ] −b = E[ξ ] + b.

1 (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 ).

Initialize a real number α ∈ (0, 1). Step 2. Calculate the objective values fi for all chromosomes. 3 Genetic Algorithm 41 Step 3. Reorder these chromosomes according to their objective values. Step 4. Set i = 1. Step 5. Calculate the evaluation value for the ith chromosome Eval(vi ) = α(1 − α)i−1 . Step 6. If i < pop-size, set i = i + 1, and goto step 5. 4 Selection Process During each successive generation, a proportion of the existing population is selected to breed a new generation. The selection process is based on spinning the roulette wheel pop-size times, and selecting a single chromosome at each time.

### Credibilistic Programming: An Introduction to Models and Applications by Xiang Li

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