A fuzzy ranking method for fuzzy numbers

Ranking fuzzy numbers is one of very important research topics in fuzzy set theory because it is a base of decision-making in applications. However, fuzzy numbers may not be easily ordered into one sequence according to their magnitudes because they represent uncertain values. When two fuzzy numbers overlap with each other, a fuzzy number may not be considered absolutely larger than the other. That is, even when a fuzzy number may be considered larger than the other, it may also be considered smaller than the other. It means that for a given set of fuzzy numbers, several ranking sequences possibly exist. However, most of the existing ranking methods produce only one ranking sequence. They ignore other possible sequences due to the overlap between fuzzy numbers. We propose a ranking method which generates possible ranking sequences of given fuzzy numbers. Our method takes a viewpoint from users, and uses it for evaluation of fuzzy numbers. Fuzzy numbers will be ranked based on the evaluations and a fuzzy set of sequences of fuzzy numbers will be produced as a ranking results. Numeric examples and comparisons with other methods are also presented.