Fuzzy relation equations on a finite set

Evolutionary algorithm determining defuzzyfication operators

“…Please note that, in general, f (1) is not necessarily less than g (1). Thus, we can reach improper intervals, already discussed in the framework of the extended interval arithmetic by Kaucher in [21].…”

“…where [f (1),g (1)] or [g(1),f (1)] may be empty, depending on the sign of f (1) − g (1) (if it is less than zero, the other is empty).…”

“…for any continuous, linear functionalφ on C( [0,1]) there is a function of bounded variation, h, such that:φ…”

“…Defuzzyfication is the main operation of fuzzy controllers and fuzzy inference systems [1,2,25] where fuzzy inference rules appear. If the consequent parts of fuzzy rules are fuzzy, a defuzzyfication process is required, in the course of which real numbers are attached to membership functions.…”

“…Fuzzy numbers [1][2][3] are very special fuzzy sets defined on the universe of all real numbers, R. They are of great importance in fuzzy systems. Triangular and trapezoidal (or triangularly and trapezoidally shaped) fuzzy numbers are usually used in applications or the so-called (L,R) numbers with two shape functions, L and R, proposed by Dubois and Prade [4] in 1978 as a restricted class of membership functions.…”