Why Triangular and Trapezoid Membership Functions Are Efficient in Design Applications

In many design problems, it is important to take into account expert knowledge. Expert often describe their knowledge by using imprecise ( fuzzy ) natural-language words like small . To describe this imprecise knowledge in computer-understandable terms, Zadeh came up with special fuzzy methodology -- techniques that have been successful in many applications. This methodology starts with eliciting, from the expert, a membership function corresponding to each imprecise term -- a function that assigns, to each possible value of the corresponding quantity, a degree to which this value satisfies the relevant property (e.g., a degree to which, in the expert\u27s opinion, this value is small). In principle, we can have complex membership functions. However, somewhat surprisingly, in many applications, the simplest membership functions -- of triangular or trapezoid shape -- turned out to be most efficient. There exist some explanations for this surprising empirical phenomenon, but these explanations only work when we use the simplest possible and -operation -- minimum. In this paper, we provide a new, more general explanation which is applicable for all possible and -operations