Computational Algorithm and Tools of Fuzzy Arithmetic Based on the Principle of Maximum Entropy

Abstract

The standard fuzzy arithmetic introduced by Zadeh is not always suitable for solving applied problems in various fields of science, technology and business. Even in the case of a small sequence of arithmetic operations (for example, when calculating forecast estimates of business profits), the support of the resulting fuzzy number becomes very large. It is difficult to make a decision about investing if the maximum and minimum of the expected profit differ several times. Therefore, the authors developed an algorithm which forms the result of an arithmetic operation as a subset of a fuzzy binary relation and which implements the principle of maximum entropy. This algorithm allows performing arithmetic operations on fuzzy numerical sets with an arbitrary form of the membership function. Compared to standard fuzzy arithmetic, the proposed algorithm provides a much smaller increase in the support of the resulting fuzzy number. We implemented the algorithm in an add-in for Microsoft Excel, which readers can download and use for applied calculations for free.

Algorithms for assessing the qualitative and quantitative risks of lending to small and medium-sized businesses based on fuzzy calculus

“…We propose to use discretized fuzzy numbers with an arbitrary form of the membership function and algorithm of fuzzy arithmetic based on the maximum entropy principle. As we showed in a special study (Sveshnikov & Bocharnikov, 2022), these methods are the most suitable for solving applied problems.…”

“…For aggregating we propose to use the Sugeno fuzzy integral (Sugeno, 1972) (s)∫ from the fuzzy membership function h:X→[0,1], defined on the discrete set , along the fuzzy measure g(•) is represented as follows:…”