Fuzzy numbers revisited: operations on extensional fuzzy numbers

TL;DR

This paper introduces a new approach to operations on extensional fuzzy numbers, addressing computational complexity and feature preservation issues in fuzzy arithmetic, with practical implementation and examples.

Contribution

It defines operations and relational operators for extensional fuzzy numbers, offering an alternative to traditional fuzzy set methods and improving computational efficiency.

Findings

Operations on extensional fuzzy numbers are effectively defined.

The approach reduces computational complexity compared to Zadeh's extension rule.

Practical C++ implementation demonstrates applicability.

Abstract

Fuzzy numbers are commonly represented with fuzzy sets. Their objective is to better represent imprecise data. However, operations on fuzzy numbers are not as straightforward as maths on crisp numbers. Commonly, the Zadeh's extension rule is applied to elaborate a result. This can produce two problems: (1) high computational complexity and (2) for some fuzzy sets and some operations the results is not a fuzzy set with the same features (eg. multiplication of two triangular fuzzy sets does not produce a triangular fuzzy set). One more problem is the fuzzy spread -- fuzziness of the result increases with the number of operations. These facts can severely limit the application field of fuzzy numbers. In this paper we would like to revisite this problem with a different kind of fuzzy numbers -- extensional fuzzy numbers. The paper defines operations on extensional fuzzy numbers and relational operators (=, >, >=, <, <=) for them. The proposed approach is illustrated with several applicational examples. The C++ implementation is available from a public GitHub repository.