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

It presents fuzzy programming method of clear up real-life choice difficulties in fuzzy setting. in the framework of credibility idea, it presents a self-contained, complete and up to date presentation of fuzzy programming versions, algorithms and functions in portfolio research.

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N include fuzzy parameters. suppose that x is a call vector, and ξ is a fuzzy vector, then the overall fuzzy programming version may be written as max f (x, ξ ) s. t. gi (x, ξ ) ≤ zero, i = 1, 2, . . . , n. (2. nine) instance 2. four during this instance, we examine the portfolio choice challenge. The time period portfolio refers to any choice of monetary resources corresponding to shares, bonds, and funds. Portfolio can be held via person traders or controlled via monetary execs, banks and different monetary associations.

S 1−s ⎠ + t 2 (1 − t)2 (6. four) − Then the next houses approximately T (s, t) could be simply proved (see Fig. 6. 1): (a) it's a strictly convex functionality with recognize to (s, t) and attains its minimal worth 0 whilst s = t; (b) for any zero ≤ s ≤ 1 and nil ≤ t ≤ 1, now we have T (s, t) = T (1 − s, 1 − t). comment 6. 1 you'll be able to turn out that the cross-entropy is permutationally symmetric, that's, the worth doesn't switch if the results are categorised another way. 6. 1 Cross-Entropy 121 comment 6. 2 the idea that of cross-entropy measures the divergence of a fuzzy variable from a priori one rather than the gap.

Five = lim Cr{Ai }. i→∞ (c) based on the duality axiom, we've lim Cr Aci < zero. five. i→∞ Then it follows from end (a) that Cr Ac = lim Cr Aci . i→∞ back, it follows from the duality axiom that Cr{A} = lim Cr{Ai }. i→∞ 1. 1 Credibility degree 7 (d) in line with the duality axiom, we have now Cr{Ac } < zero. five. Then it follows from end (b) that Cr{A} = 1 − Cr Ac = 1 − lim Cr Aci = lim Cr{Ai }. i→∞ i→∞ The evidence is entire. instance 1. four often conversing, the credibility degree is neither reduce semicontinuous nor top semicontinuous.

Instance three. 21 think that there are 4 shares with autonomous general fuzzy returns. desk three. three indicates the anticipated values and traditional variances for those fuzzy amounts. If the investor want to maximize the funding go back with possibility point 1. forty four, then we get the subsequent mean-variance version ⎧ max 1. 1x1 + 1. 3x2 + 1. 5x3 + x4 ⎪ ⎪ ⎪ ⎨ s. t. x + 1. 1x + 1. 3x + 1. 4x ≤ 1. 2 1 2 three four ⎪ x + x + x + x = 1 1 2 three four ⎪ ⎪ ⎩ x1 , x2 , x3 , x4 ≥ zero. We use the Matlab functionality Linprog to resolve the linear programming version.

Five. For this situation, in accordance with the duality axiom, we've got c Cr ξ1 ∈ B1c ∪ ξ2 ∈ B2c ∪ · · · ∪ ξm ∈ Bm ≤ zero. five. in addition, it follows from Theorem 1. three that Cr{ξ1 ∈ B1 , ξ2 ∈ B2 , . . . , ξm ∈ Bm } c = 1 − Cr ξ1 ∈ B1c ∪ ξ2 ∈ B2c ∪ · · · ∪ ξm ∈ Bm 22 1 Credibility conception = 1 − max Cr ξi ∈ Bic 1≤i≤m = min Cr{ξi ∈ Bi }. 1≤i≤m Conversely, think that (1. 27) holds for any units B1 , B2 , . . . , Bm of actual numbers. Then for any (x1 , x2 , . . . , xm ) ∈ m , we now have ν(x1 , x2 , . . . , xm ) = Cr{ξ1 = x1 , ξ2 = x2 , .

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