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Determining Optimal Points of Membership with Dependent Variables

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

Fuzzy Sets and Systems
Volume 60, Number 1 (November 1993), Pages 19-24.

Abstract

Consider sets X and Y each with associated membership functions. Further suppose that there is a map f:X -> Y. A membership may be induced on Y from the membership on X calculated using the extension principle, and the two memberships on Y may be combined to give an overall membership. The point with the highest membership in X can be determined directly or by back mapping the value with the highest membership in Y. These two methods are equivalent under hypotheses on the membership functions and on f. An example calculation from engineering design is presented.