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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:
26E50 Fuzzy real analysis
03E72 Theory of fuzzy sets, etc.
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