A method for ranking fuzzily fuzzy numbers

Ranking fuzzy numbers is one of the very important research topics in fuzzy set theory because it is a base of decision-making in all application areas. However, fuzzy numbers cannot be easily arranged in order of magnitude because they represent uncertain and vague values. When two fuzzy numbers overlap with each other, a fuzzy number may not be considered absolutely larger than the other. That is, even though a fuzzy number may be considered larger than the other, it may also be considered smaller than the other. It means that, even when we consider only two fuzzy numbers, two sequences may be the ranking result at the same time. 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 fuzzy numbers, and represents them with fuzzy sets. Some numeric examples are also presented to show how our method ranks fuzzy numbers