Download Credibilistic Programming: An Introduction to Models and by Xiang Li PDF

By Xiang Li

It presents fuzzy programming method of remedy real-life selection difficulties in fuzzy atmosphere. in the framework of credibility concept, it presents a self-contained, accomplished and updated presentation of fuzzy programming types, algorithms and functions in portfolio analysis.

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

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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. That is, the chromosomes with higher fitness will have more chance to produce offspring.

2, which implies that E[ξ2 + η2 ] < E[ξ2 ] + E[η2 ]. 3 For a triangular fuzzy variable ξ = (a, b, c), we will prove that its expected value is E[ξ ] = (a + 2b + c)/4. 5) Especially, if ξ is a symmetric fuzzy variable with b − a = c − b, we have E[ξ ] = b. First, we assume a ≥ 0. For any r ≥ 0, according to the credibility inversion theorem, we have ⎧ 1, if 0 ≤ r ≤ a ⎪ ⎪ ⎨ (2b − a − r)/2(b − a), if a < r ≤ b Cr{ξ ≥ r} = (c − r)/2(c − b), if b < r ≤ c ⎪ ⎪ ⎩ 0, if c < r < +∞. Then it follows from the definition of expected value that c E[ξ ] = Cr{ξ ≥ r} dr = (a + 2b + c)/4.

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. In this case, the optimal portfolio is x ∗ = (1, 0, 0). 3 Genetic Algorithm For a general credibilistic programming model, if the credibilistic mappings have analytical expressions and the objective and constraint functions have good mathematical properties, such as differentiability and convexity, we can design efficient solution algorithms by using the Kuhn-Tucker conditions.

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