It offers fuzzy programming method of remedy real-life selection difficulties in fuzzy atmosphere. in the framework of credibility idea, it presents a self-contained, finished and up to date presentation of fuzzy programming types, algorithms and purposes in portfolio research.

self reliant. for instance, take a credibility area (Θ, A, Cr) to be {θ1 , θ2 , θ3 } with Cr{θ1 } = 0.7, Cr{θ2 } = 0.3 and Cr{θ3 } = 0.2. outline fuzzy variables ⎧ ⎧ ⎨ 1, if θ = θ1 ⎨ zero, if θ = θ1 η1 (θ ) = zero, if θ = θ2 ξ1 (θ ) = zero, if θ = θ2 ⎩ ⎩ 2, if θ = θ3 , three, if θ = θ3 . one could end up that E[ξ1 ] = 0.9, E[η1 ] = 0.8, and E[ξ1 + η1 ] = 1.9, which means that E[ξ1 + η1 ] > E[ξ1 ] + E[η1 ]. nevertheless, if we outline ⎧ ⎨ zero, if θ = θ1 ξ2 (θ ) = 1, if θ = θ2 ⎩ 2, if θ = θ3 , ⎧ ⎨ 0,.

Ξ and η Empty set The set of genuine numbers greatest operator minimal operator common quantifier Existential quantifier ix Chapter 1 Credibility thought the concept that of fuzzy set used to be initialized via Zadeh (1965) through club functionality in 1965. on the way to degree the opportunity of a fuzzy occasion happens, Zadeh proposed the suggestions of threat degree (Zadeh 1978) and necessity degree (Zadeh 1979). it really is proved that either threat degree and necessity degree fulfill the houses of.

Theorems, entropy optimization version and its crisp equivalents, fuzzy simulation, and purposes in portfolio choice challenge. 5.1 Entropy This part introduces the definitions of entropy for discrete fuzzy variable and non-stop fuzzy variable, respectively. Definition 5.1 (Li and Liu 2008a) consider that ξ is a discrete fuzzy variable taking values in {x1 , x2 , . . .}. Then its entropy is outlined by way of ∞ H [ξ ] = S Cr{ξ = xi } (5.1) i=1 the place S(t) = −t ln t − (1 − t) ln(1 − t). it's.

[aξ + b] = |a|H [ξ ]. (5.4) evidence consider that fuzzy variables ξ and aξ + b have credibility capabilities ν and μ, respectively. If a = zero, then (5.4) is trivial. in a different way, for any x ∈ , one could turn out that μ(x) = Cr{aξ + b = x} = Cr ξ = (x − b)/a = ν (x − b)/a . It follows from the definition of entropy that H [aξ + b] = = +∞ −∞ +∞ −∞ S ν(x − b)/a dx |a|S ν(x) dx = |a|H [ξ ]. the theory is proved. 5.2 greatest Entropy precept enable ξ be a fuzzy variable with a few given details, for.

|=x ν1 (y1 ) ∧ ν2 (y2 ) sup y1 ≥0,y2 ≥0,y1 −y2 =x ν1 (y1 ) ∧ ν2 (y2 ) = ν1 (x) ∧ ν2 (0) = ν1 (x) for any x ≥ zero. in keeping with Definition 7.1, we've got ∞ d(ξ, η) = zero √ 6 ln 2 ν1 (x) dx = m1 . π equally, if m1 < m2 , we will turn out that √ 6 ln 2 d(ξ, η) = m2 . π more often than not, the gap among exponential fuzzy variables is √ d(ξ, η) = 6 ln 2 (m1 ∨ m2 ). π Theorem 7.1 (Li and Liu 2008b) For any fuzzy variables ξ, η and τ , we now have (a) (b) (c) (d) (Nonnegativity) d(ξ, η) ≥ 0;.