Some families of fuzzy numbers closed under the operation of multiplication, and fuzzy numbers of piecewise-polynomial type.

Abstract

In this paper, a fuzzy number on the real line is defined as the one whose membership function is everywhere continuous and has a strict monotonicity on its bounded support. Firstly, it is proved that the family of all fuzzy numbers is closed under the operation of multiplication derived from Zadeh's extension principle. Two kinds of subfamilies of fuzzy numbers closed under the multiplication are presented. For the purpose of describing these two subfamilies, the notion of polynomial type and piecewise-polynomial type fuzzy numbers are introduced. Fundamental formulae for the multiplication among fuzzy numbers of polynomial type or piecewise-polynomial type are given.