It presents fuzzy programming method of clear up real-life choice difficulties in fuzzy atmosphere. in the framework of credibility conception, it presents a self-contained, accomplished and up to date presentation of fuzzy programming versions, algorithms and purposes in portfolio research.

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

5)(d − c + b − a). Now, we examine the cross-entropy of fuzzy variable τ from ξ . First, now we have d D[τ ; ξ ] = zero. five ln zero. 5/ν(x) + zero. five ln zero. five/ 1 − ν(x) dx a d = d −0. five ln ν(x) − zero. five ln 1 − ν(x) dx − a ln 2 dx a d = (a − d) ln 2 − zero. five a ln ν(x) + ln 1 − ν(x) dx. 6. 1 Cross-Entropy 123 in addition, it follows from the concept of integration by way of elements that b ln ν(x) + ln 1 − ν(x) dx = −2(b − a), a c b d c ln ν(x) + ln 1 − ν(x) dx = ln zero. five + ln zero. five dx = 2(c − b) ln zero. five, b ln ν(x) + ln 1 − ν(x) dx = −2(d − c), c which means that D[τ ; ξ ] = (d − c + b − a)(1 − ln 2). Theorem 6. 1 For any fuzzy variables ξ and η, we've got D[ξ ; η] ≥ zero and the equality holds if and provided that ξ and η are identically allotted. evidence permit ν and μ be the credibility services of fuzzy variables ξ and η, respectively. It follows from the nonnegativity of functionality T (s, t) that D[ξ ; η] = ∞ −∞ T ν(x), μ(x) dx ≥ zero. moreover, the equality holds if and provided that T (ν(x), μ(x)) = zero, i. e. , ν(x) = μ(x) for all x ∈ , which suggests that ξ and η are identically disbursed. If ξ and η are discrete fuzzy variables, the theory should be proved similarly. The evidence is whole. Theorem 6. 2 permit ξ and η be non-stop fuzzy variables. For any actual numbers a and b, we have now D[aξ + b; aη + b] = |a|D[ξ ; η]. facts the belief is trivial while a = zero. In what follows, we think = zero. permit ν and μ be the credibility features of fuzzy variables ξ and η, respectively. It follows from Definition 6. 2 that D[aξ + b; aη + b] = = ∞ −∞ ∞ −∞ T ν (x − b)/a , μ (x − b)/a dx |a|T ν(t), μ(t) dt = |a|D[ξ ; η]. The facts is whole. 124 6 Cross-Entropy Minimization version Theorem 6. three allow τ = (a, b) be an equipossible fuzzy variable. Then for any non-stop fuzzy variable ξ taking values in [a, b], we've got D[ξ, τ ] = H [τ ] − H [ξ ]. facts permit ν be the credibility functionality of ξ . It follows from the definition of crossentropy that b D[ξ, τ ] = ν(x) ln 2ν(x) + 1 − ν(x) ln 2 − 2ν(x) dx a b = ln 2 + ν(x) ln ν(x) + 1 − ν(x) ln 1 − ν(x) dx a = (b − a) ln 2 − H [ξ ] = H [τ ] − H [ξ ]. The facts is entire. Theorem 6. four permit τ be an equipossible fuzzy variable taking values in {x1 , x2 , . . . , xn }. For any fuzzy variable ξ taking values in {x1 , x2 , . . . , xn }, we have now D[ξ, τ ] = H [τ ] − H [ξ ]. facts it can be proved equally with Theorem 6. three. 6. 2 minimal Cross-Entropy precept in lots of actual difficulties, the credibility functionality of a fuzzy variable is unavailable other than a few partial details, for instance, second constraints, that may be in response to observations. therefore, the utmost entropy precept tells us that out of the entire credibility services gratifying given constraints, pick out the person who has greatest entropy. although, there's one other form of info, for instance, a previous credibility functionality, that may be according to instinct or event with the matter. If either the a previous credibility functionality and the instant constraints are given, which credibility functionality may still we elect?