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Download e-book for kindle: Credibilistic Programming: An Introduction to Models and by Xiang Li

By Xiang Li

ISBN-10: 364236375X

ISBN-13: 9783642363757

ISBN-10: 3642363768

ISBN-13: 9783642363764

It offers fuzzy programming method of resolve real-life choice difficulties in fuzzy setting. in the framework of credibility idea, it presents a self-contained, finished and updated presentation of fuzzy programming versions, algorithms and purposes in portfolio analysis.

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

Example text

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. 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.

For any α1 x1 + α2 x2 = x, we have x1 < a1 or x2 < a2 . Then it follows from the Zadeh extension principle that ν(x) = 0. Case 2. a ≤ x < b. It follows from the Zadeh extension theorem that ν(x) = sup a1 ≤y1 ≤b1 ,a2 ≤y2 ≤b2 ,α1 y1 +α2 y2 =x ν1 (y1 ) ∧ ν2 (y2 ) . Since ν1 is increasing on the interval [a1 , b1 ] and ν2 is increasing on the interval [a2 , b2 ], the vector (y1 , y2 ) which maximizes the binary function ν1 ∧ ν2 should satisfy the following equations α1 y1 + α2 y2 = x, (y1 − a1 )/(b1 − a1 ) = (y2 − a2 )/(b2 − a2 ).

Conversely, we assume Cr{ξ = e} = 1. Then we immediately have Cr{(ξ − e)2 ≥ r} = 0 for any r > 0. Thus ∞ V [ξ ] = Cr (ξ − e)2 ≥ r dr = 0. 0 The theorem is proved. Let ξ be a fuzzy variable that takes values in [a, b], but whose credibility function is otherwise arbitrary. If its expected value is given, above theorem tells us that its minimum variance is zero. On the other hand, what is the possible maximum variance? The following maximum variance theorem will answer this question. 4 (Li et al.

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


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