# Credibilistic Programming: An Introduction to Models and Applications (Uncertainty and Operations Research)

Language: English

Pages: 144

ISBN: 364236375X

Format: PDF / Kindle (mobi) / ePub

It provides fuzzy programming approach to solve real-life decision problems in fuzzy environment. Within the framework of credibility theory, it provides a self-contained, comprehensive and up-to-date presentation of fuzzy programming models, algorithms and applications in portfolio analysis.

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α-optimistic value for a triangular fuzzy variable is ξsup (α) = 2αb + (1 − 2α)c, if α ≤ 0.5 (2α − 1)a + (2 − 2α)b, if α > 0.5 which is shown by Fig. 4.4. Example 4.4 Suppose that ξ = (a, b, c, d) is a trapezoidal fuzzy variable. It follows from the credibility inversion theorem that ⎧ 1, if r ≤ a ⎪ ⎪ ⎪ ⎪ ⎪ (2b − a − r)/2(b − a), if a < r ≤ b ⎨ if b < r ≤ c Cr{ξ ≥ r} = 0.5, ⎪ ⎪ ⎪ (d − r)/2(d − c), if c < r ≤ d ⎪ ⎪ ⎩ 0, if r > d. 4.1 Optimistic Value 77 Fig. 4.4 Optimistic value of a

when s = t; (b) for any 0 ≤ s ≤ 1 and 0 ≤ t ≤ 1, we have T (s, t) = T (1 − s, 1 − t). Remark 6.1 It is easy to prove that the cross-entropy is permutationally symmetric, that is, the value does not change if the outcomes are labeled differently. 6.1 Cross-Entropy 121 Remark 6.2 The concept of cross-entropy measures the divergence of a fuzzy variable from a priori one instead of the distance. Hence, the symmetry property is not necessary. That is, we may have D[ξ ; η] = D[η; ξ ]. For example,

reduced by another event with measure zero. Theorem 1.6 (Liu 2004) Let {Ai } be a sequence of events with lim Cr{Ai } = 0. i→∞ Then for any event B, we have lim Cr{B ∪ Ai } = lim Cr{B\Ai } = Cr{B}. i→∞ i→∞ (1.4) Proof For any events Ai and B, it follows from the monotonicity axiom and the subadditivity theorem that Cr{B} ≤ Cr{B ∪ Ai } ≤ Cr{B} + Cr{Ai }, and Cr{B\Ai } ≤ Cr{B} ≤ Cr{B\Ai } + Cr{Ai }. 6 1 Credibility Theory Letting i → ∞. According to the squeeze theorem, we have lim Cr{B

entropy. However, there may be another type of information, for example, a prior credibility function, which may be based on intuition or experience with the problem. If both the a prior credibility function and the moment constraints are given, which credibility function should we choose? The following minimum crossentropy principle tells us that: out of all credibility functions satisfying given moment constraints, choose the one that is closest to the given a priori credibility function. There

ν, and fuzzy variable η has a credibility function μ. According to Definition 6.2, the crossentropy can be simulated as the numerical integration of function T (ν, μ) = ν ln(ν/μ) + (1 − ν) ln (1 − ν)/(1 − μ) . 126 6 Cross-Entropy Minimization Model Fig. 6.2 The cross-entropy simulation with variable parameter N Randomly generate vectors y i and calculate the objective values fi = f (x, y i ) for all i = 1, 2, . . . , N . Furthermore, calculate the credibilities νi and μi for all i = 1, 2, .