How to explain the efficiency of triangular and trapezoid membership functions in applications to design

It is well known that expert knowledge is very important for solving design problems. However, expert knowledge is not easy to describe in precise terms, since experts often use imprecise (“fuzzy”) words from natural language such as “small” or “large”. In order to describe such knowledge in precise terms - which would be understandable to a computer - Lotfi Zadeh came up with a special methodology that he called fuzzy. This methodology had many successful applications, in particular, applications to design. The first stage of the general fuzzy methodology is eliciting, from the expert, a membership function corresponding to each imprecise term, i.e., a function that assigns, to each possible value of the corresponding quantity, a degree to which this value satisfies this property (e.g., a degree to which, in the expert's opinion, this given value is small). If we follow the expert's opinion very closely, we often come up with very complex membership functions. However, surprisingly, in many applications, the simplest membership functions - of triangular or trapezoid shape - turned out to be more efficient than the supposedly more adequate complex ones. This is counterintuitive: the closer we follow the expert’s opinion, the worse our result. Some explanations for this seemingly counterintuitive phenomenon have been proposed earlier. However, these explanations only work when we use the simplest possible “and”-operation - minimum, while this phenomenon has been observed for other “and”-operations as well. In this paper, we provide a new, more general explanation for the above phenomenon, an explanation that works for all possible “and”-operations.