Credibilistic Programming: An Introduction to Models and by Xiang Li

By Xiang Li

It offers fuzzy programming method of resolve real-life selection difficulties in fuzzy surroundings. in the framework of credibility conception, it offers a self-contained, accomplished and updated presentation of fuzzy programming versions, algorithms and functions in portfolio research.

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

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Proof If fuzzy variables ξ1 , ξ2 , . . 17 and the duality axiom that Cr (ξ1 ∈ B1 ) ∪ (ξ2 ∈ B2 ) ∪ · · · ∪ (ξm ∈ Bm ) c = 1 − Cr ξ1 ∈ B1c , ξ2 ∈ B2c , . . , ξm ∈ Bm = 1 − min Cr ξi ∈ Bic 1≤i≤m = max Cr{ξi ∈ Bi }. 28) holds. For any sets B1 , B2 , . . , Bm of real numbers, it follows from the duality axiom that Cr{ξ1 ∈ B1 , ξ2 ∈ B2 , . . , ξm ∈ Bm } c = 1 − Cr ξ1 ∈ B1c ∪ ξ2 ∈ B2c ∪ · · · ∪ ξm ∈ Bm = 1 − max Cr ξi ∈ Bic 1≤i≤m = min Cr{ξi ∈ Bi }. 17, fuzzy variables ξ1 , ξ2 , . . , ξm are independent.

2 Credibility functions for ξ1 , ξ2 and ξ3 difficult to compare the returns arising from the second stock and the first stock. As a result, if we take portfolios x = (1, 0, 0) and y = (0, 1, 0), it is difficult to decide which one is better. In fact, it follows form the independence that fuzzy vector (ξ1 , ξ2 ) has a joint credibility function ν = ν1 ∧ ν2 . 5. t. x1 + x2 + x3 = 1 ⎪ ⎩ x1 , x2 , x3 ≥ 0. It is easy to calculate that the optimal portfolio is x ∗ = (0, 1, 0). t. x1 + x2 + x3 = 1 ⎪ ⎩ x1 , x2 , x3 ≥ 0.

If not, we regenerate a point randomly until a feasible one is obtained. We repeat this procedure pop-size times, and generate the first population. The initialization process is summarized as follows. 1 (Initialization Process) Step 1. Step 2. Step 3. Step 4. Step 5. Set i = 1. Randomly generate a chromosome from the predetermined region. If it is feasible, set i = i + 1. Otherwise, go to step 2. If i ≤ pop-size, go to step 2. Return the initialized population vi , i = 1, 2, . . , pop-size. 3 Evaluation Function Evaluation function assigns each chromosome a probability of reproduction so that its likelihood of being selected is proportional to its fitness relative to the other chromosomes in the population.

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