Approximating Alpha-cuts With the Vertex Method

Kevin N. Otto, Andrew D. Lewis and Erik K. Antonsson

Fuzzy Sets and Systems
Volume 55, Number 1 (April 1993), Pages 43-50.


If f:Rn->R is continuous and monotonic in each variable, and if µi is a fuzzy number on the ith coordinate, then the membership on R induced by f and by the membership on Rn given by µ(x)=min(µ1(x1),...,µn(xn)) can be evaluated by determining the membership at the endpoints of the level cuts of each µi. Here more general conditions are given for both the function f and the manner in which the fuzzy numbers µi are combined so that this simple method for computing induced membership may be used. In particular, a geometric condition is given so that the alpha-cuts computed when the fuzzy numbers are combined using min is an upper bound for the actual induced membership.