Solution of the Convexly Combined Form of Sup·min and Inf·max Composite Fuzzy Relations Equation

Abstract

This paper deals with the problem to find the solutions of the convexly combined form of Sup·min and Inf·max composite fuzzy relations equation, λ·(x_??_R)+λ·(x_??_R)=y, and provides the method to solve it. First, the convexly combined form of Sup·min and Inf·max composite fuzzy relations equation is formulated by applying a notion of fuzzy convex combination. Furthermore, a mixed fuzzy input-output relational system of pessimistic and optimistic compositions is discussed, which is represented in the convexly combined form of Sup·min and Inf·max composite fuzzy relations equation. Second, we show the necessary and sufficient condition on which the set of solutions is not empty. Third, we provide the method to find the solution which satisfies an optimum condition. Finally, a numerical example is shown.