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Fuzzy expert systems attempt to model the cognitive processes of human experts. They currently accomplish this by capturing knowledge in the form of linguistic propositions. Real-world problems dictate the need to include mathematical knowledge as well. Pattern matching is a critical part of the inference procedure in expert systems. Matches are made between data clauses, premise clauses, and conclusion clauses, forming an inference chain. Preprocessing the clauses may generate intervals of real numbers which are compared in the fuzzy matching algorithm. These same intervals may be used in arithmetic expressions. The purpose of this article is to devise a method for incorporating arithmetic expressions into inference process of Fuzzy Expert Systems. Interval arithmetic is used to evaluate these expressions. Logical relations between intervals are analyzed using probability theory.0

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In this study we introduce two new concepts: (1) a new approach to construct clusters and (2) a methodology to compute similarities between numerical vectors based on clusters. The new approach to construct clusters is based on a variable‐distance threshold. There are several important domains where using commonly utilized fixed distance threshold clustering method might create clusters contradicting human expert reasoning. The domains where variable‐distance threshold clustering is more suitable are discussed and explained. In addition, we introduce a new concept for computing similarities between two numerical vectors, based on a membership in corresponding clusters. Such a concept constitutes an appropriate tool under greater degree of uncertainty where model structure is vague, and data are unreliable. First, we describe the procedure for construction of variable‐distance threshold clusters. Then we provide several models for computing the similarity between the two numerical vectors based on clusters. Several examples are included to illustrate the practical application of the models.