Probabilistic method of fuzzy number comparison

Abstract

The subject of fuzzy numbers comparison is not very prevalent in modern research. A number of older publications propose a number of well-known methods that broadly fall into one of three classes. First is index-based comparison, which maps fuzzy numbers into a real value and ranks them accordingly, such as Adamo, Yager, Chang, Liu and Wang, etc. Sec-ondly, there are methods that propose fuzzy number ranking based on distance (e.g. Hamming distance) from certain reference sets, such as Kerre and Jain. Third category is rare and pro-poses a specific pair-wise ranking approach in particular circumstances. New methods are being proposed, but this is not a frequent occurrence. Thus, a new method of fuzzy numbers comparison is proposed that takes into account a confidence probability of a comparison. In the paper a generalized method for computing or-dering relation between fuzzy numbers regardless of the specifics of membership function is provided. An example of its usage is considered for triangular fuzzy numbers that is one of the most common ways of expressing uncertainty. The results are also compared to some of the existing methods. Formal properties of the relation, based on the proposed comparison meth-od are discussed and proven. The operational semantics of a logical operator or a function that implements the new method in software is considered for the “less than” operator and described with a state diagram. Other relations, such as “greater than” and “equals” are also discussed. Research materials provide some insights into certain properties of the proposed meth-od and particular hurdles when implementing it in software systems, such as using smooth analytically defined membership function and caching certain intermediate computation re-sults.