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Fuzzy Measurement of Sustainability by Yannis A. Phillis

By Yannis A. Phillis

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It then remains to consider cases in which xi ≥ 0 and y ≥ xi. The membership function of f (A ) is written μf (A )(y) = max min[e − x1 , e −2 ( y − x1 ) ] . x1 y ≥ x1 ≥ 0 Now fix y and consider the exponential terms in the right side of the above equation. Both are continuous functions of x1. As x1 increases from 0 to y the first term, e−x1, is continuous and decreasing in x1, taking values in the interval [1, e−y]. The second term, e−2(y − x1), is increasing in x1 and equals e−2y for x1 = 0 and 1 for x1 = y.

Won et al. (2002) proved that xn+1 is an increasing function of the input variables under the following conditions: Condition MC1: The membership functions assigned to the inputs are piecewise differentiable, in the sense that they are continuous on the corresponding domains and differentiable at all but a countable number of points. Moreover, for any pair of linguistic values A and B belonging to the same fuzzy partition, if A < B then 1 dμ A ( x ) 1 dμ B ( x ) ≤ , μ A ( x) dx μ B ( x) dx for all x where μA(x) and μB(x) are differentiable and nonzero.

7b. Functions of several variables can be viewed as functions of one composite variable or vector. In set theory, several sets can be combined using the Cartesian product. The Cartesian product X1 × … × Xn of n crisp sets X1, …, Xn is a set whose elements are the ordered pairs (x1, …, xn) for all x1 ∈ X1, …, xn ∈ Xn. Symbolically, X1 × … × Xn = {(x1, …, xn) | x1 ∈ X1, …, xn ∈ Xn }. , X1 = … = Xn = X, the Cartesian product is denoted Xn. Consider now the fuzzy sets A1 and A2 in X1 and X2 respectively.

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