Equivalence class in the set of fuzzy numbers and its application in decision-making problems

Since the discovery of fuzzy sets, the arithmetic operations of fuzzy numbers (Zadeh [7, 8]) which may be viewed as a generalization of interval arithmetic (Moore [5]) have emerged as an important area of research within the theory of fuzzy sets (Mizumoto and Tanaka [4], Dubois and Prade [2]). The arithmetic operations of fuzzy numbers have been performed either by extension principle [7, 8] or by using alpha cuts as discussed by Dubois and Prade [2]. If two triangular (linear) fuzzy numbers are added or subtracted by applying extension principle, then the result is again a triangular fuzzy number. However, when we take other operations like multiplication or division, then the result is not a linear triangular fuzzy number. Thus these operations are not closed in the sense that operations of two same types (triangular/trapezoidal, etc.) of fuzzy numbers may not necessarily result in fuzzy number of that type (triangular/trapezoidal, etc.). Hence it is cumbersome to find the membership function of the arithmetical operations of large number of fuzzy numbers (may be of same type) by these principles. In the present paper, an attempt is made to overcome such type of difficulties. In Section 3, a relation between two fuzzy numbers is defined and it has been proved that this relation divides the whole set of fuzzy numbers into equivalence classes. In Section 4, arithmetic operations in a particular class are defined. These operations are different from the arithmetic operations of fuzzy numbers, developed by extension principle or alpha cut. It has been proved and also verified through examples that these arithmetic operations are closed in their respective