Fuzzy numbers revisited: operations on extensional fuzzy numbers
Krzysztof Siminski
TL;DR
The paper tackles the problem of performing arithmetic and comparisons on fuzzy numbers without the computational burden and shape distortion caused by Zadeh's extension. It introduces extensional fuzzy numbers (EFN) defined by $x_S(y)=S(x,y)$ using fuzzy $\star$-similarity relations, enabling closed-form operations and flexible, potentially unbounded supports. It defines addition, multiplication, inverses, and a suite of comparison operators $=_S, <_S, \leq_S, >_S, \geq_S$ for EFN, and demonstrates their use in practical tasks. Through sorting and fuzzy-graph shortest-path applications, including a Floyd–Warshall-based approach and multiple illustrative examples, the work shows EFN can efficiently handle uncertainty without discretisation, with a publicly available C++ implementation on GitHub.
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.