By Didier Dubois and Henri Prade (Eds.)
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Extra resources for Fuzzy Sets and Systems Theory and Applications
We must evaluate the degree of possibility for x ʦ ޒ, fuzzily restricted to belong to ˜ ()ޒ, to be greater than y ʦ ޒfuzzily restricted to belong to N M ʦᏼ ˜ ʦ ᏼ()ޒ. The degree of possibility of M у N is defined as v (M у N) = sup x, y : x у y min(µM (x), µN (y)) (27) This formula is an extension of the inequality x у y according to the extension principle. It is a degree of possibility in the sense that when a pair (x, y) exists such that x у y and µM (x) = µN (y) = 1, then v (M у N) = 1 , Since M and N are convex fuzzy numbers, it can be seen on Fig.
N}. Then xˆ is a strict relative maximum point of the mathematical programming problem: Maximize min µi(xi) i = l, n subject to f(x1, . , xn) = f( xˆ 1 , . . , xˆ n ) = f( xˆ ). Note that this theorem gives only sufficient conditions for relative maximum points. Moreover, it is a local version of Lemma I with different hypotheses. b. Properties of * If * is commutative, so is * . If * is associative, so is * . ) Distributivity of * over ʜ, ˜ (])ޒ3, M ∀(M, N, P) ∈ [ ᏼ * (N ʜ P) = (M * N) ʜ (M * P) (obvious).
They can be extended to evaluate a whole fuzzy partition in order to give a rating of the total amount of ambiguity that arises when deciding to which ∑ m µ Ai ( x ) = 1 . For such a of A1, . . , Am an element x belongs. We have i=1 fuzzy partition, the measure of fuzziness is (Capocelli and De Luca, 1973) d(A1, . . , Am) = ∑ ∑ υ (µ X m i=1 j=1 Aj ( xi )), where is any continuous and strictly concave function in [0, 1]. When (A1, . . , A m ) is an ordinary partition of X, d(A 1, . . , A m) = 0.