Convex-cone-based comparisons of and difference evaluations for fuzzy sets

Abstract

This paper focuses on how to compare two fuzzy sets and, from the viewpoint of set optimization, proposes eight types of fuzzy-set relations based on a convex cone as new comparison criteria of fuzzy sets. Then, difference evaluation functions for fuzzy sets are introduced. Under suitable assumptions of certain compactness and stability of fuzzy sets, we show that these functions correspond well to the fuzzy-set relations. In addition, through transforming these functions stepwise, we deal with numerical calculation methods of them in particular cases. Consequently, we can judge whether each fuzzy-set relation holds or not for given two fuzzy sets with the aid of computers.

Keywords:

• Fuzzy set
• set relation
• scalarization
• calculation method

Acknowledgements

The authors are grateful to the guest editors of the special issue for the encouragement to submit the paper to this journal and to the anonymous referee for a lot of valuable comments. The authors also would like to dedicate the paper to Professor Johannes Jahn on the occasion of his 65th birthday.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by JSPS KAKENHI (Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science) [grant number 26400194].