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Approximating $$\alpha$$-cuts with the vertex method. (English) Zbl 0931.26010
Summary: If $$f\colon\mathbb{R}^n\to\mathbb{R}$$ is continuous and monotonic in each variable, and if $$\mu_i$$ is a fuzzy number on the $$i$$th coordinate, then the membership on $$\mathbb{R}$$ induced by $$f$$ and by the membership on $$\mathbb{R}^n$$ given by $$\mu(x)=\min(\mu_1(x^1),\cdots,\mu_n(x^n))$$ can be evaluated by determining the membership at the endpoints of the level cuts of each $$\mu_i$$. Here more general conditions are given for both the function $$f$$ and the manner in which the fuzzy numbers $$\{\mu_i\}$$ are combined, so that this simple method for computing induced membership may be used. In particular, a geometric condition is given under which the $$\alpha$$-cuts computed when the fuzzy numbers are combined using min is an upper bound for the actual induced membership.

##### MSC:
 2.6e+51 Fuzzy real analysis 3e+72 Theory of fuzzy sets, etc.
##### Keywords:
membership; level cuts; fuzzy numbers
Full Text:
##### References:
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