# zbMATH — the first resource for mathematics

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:
 [1] Buckley, J.; Qu, Y., On using α-cuts to evaluate fuzzy equations, Fuzzy sets and systems, 38, 309-312, (1990) · Zbl 0723.04006 [2] Dong, W.; Shah, H., Vertex method for computing functions on fuzzy variables, Fuzzy sets and systems, 24, 65-78, (1987) · Zbl 0634.94025 [3] Dong, W.; Wong, F.S., The vertex method and its use in earthquake engineering, (), 173-192 [4] Dong, W.M.; Wong, F.S., Fuzzy weighted averages and implementation of the extension principle, Fuzzy sets and systems, 21, 183-199, (1987) · Zbl 0611.65100 [5] Dubois, D.; Prade, H., Fuzzy real algebra: some results, Fuzzy sets and systems, 2, 327-348, (1979) · Zbl 0412.03035 [6] Dubois, D.; Prade, H., Fuzzy sets and systems: theory and applications, (1980), Academic Press New York · Zbl 0444.94049 [7] Dubois, D.; Prade, H., Fuzzy numbers: an overview, (), 3-39 [8] Dubois, D.; Prade, H., Possibility theory: an approach to the computerized processing of information, (1988), Plenum Press New York [9] Negoiţǎ, C.V.; Ralescu, D.A., Applications of fuzzy sets to systems analysis, (1975), Halsted Press New York · Zbl 0326.94002 [10] Nguyen, H.T., A note on the extension principle for fuzzy sets, J. math. anal. appl., 64, 369-380, (1978) · Zbl 0377.04004 [11] Novák, Vilém, Fuzzy sets and their applications, (1989), Adam Hilger Philadelphia, PA, (Published in English by IOP Publishing Ltd.) · Zbl 0683.94018 [12] Wood, K.L.; Antonsson, E.K., Computations with imprecise parameters in engineering design: background and theory, ASME J. mechanisms, transmissions, and automation in design, 111, 616-625, (1989) [13] Wood, K.L.; Antonsson, E.K., Modeling imprecision and uncertainty in preliminary engineering design, Mechanism and machine theory, 25, 305-324, (1990) [14] Wood, K.L.; Otto, K.N.; Antonsson, E.K., Engineering design calculations with fuzzy parameters, (), 52, 434-445, (1992), Yokohama, Japan
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.