Non-singleton fuzzification made simpler

Abstract

Non-singleton fuzzification is used in rule-based fuzzy systems when the measurements that activate them are imperfect or uncertain or when their inputs are words. It models such measurements or words as fuzzy numbers or more general fuzzy sets so that, regardless of the cause of a measurement’s or word’s imperfections or uncertainties, they are treated within the framework of fuzzy sets and systems. Since 2011, there has been a resurgence of interest in both type-1 and interval type-2 non-singleton fuzzy systems. This paper removes a computational bottleneck associated with computing the firing level or firing interval for such fuzzy systems, by providing closed-form formulas for them when the involved fuzzy sets are trapezoidal or triangular, which are widely used fuzzy sets. This is done for both the minimum and product t-norms. It is also demonstrated that a non-singleton fuzzy system that uses the product t-norm has the potential to outperform a non-singleton fuzzy system that uses the minimum t-norm. The results in this paper greatly simplify non-singleton fuzzy systems, which should make them much more popular.