M. O'Heigeartaigh J.K. Lenstra A.H.G. Rinnooy Kan Combinatorial Optimization: Annotated Bibliographies 1985 Wiley Amsterdam

Abstract

Advancement in fuzzy arithmetic is essential to advancing the applicability of fuzzy set theory (FST) to approximate reasoning problems arose in engineering, medicine, managerial science and many other domains. In the recent FST literature, many of the noteworthy progresses in fuzzy arithmetic have been built upon increasingly sophisticated mathematical concepts and structures, leading to highly non-trivial results. In this article, we take a different approach, of offering a simple, concrete framework that could serve as an entry point for the classically-trained engineers (those who might not be well-acquainted with deep theoretical FST advances) to understand and use fuzzy arithmetic in their applications. Specifically, the contributions of this work are: (i) We extend a foundational model previously proposed in Ngan (2018) to arrive at a simple fuzzy arithmetic framework, in which each key element within the framework possesses simple, concrete meaning to these classically-trained engineers; (ii) We describe how to use these key elements as building blocks to create various concrete fuzzy arithmetic operators that are adaptive to various assumptions. This means that the classically-trained engineers can now easily customize and use fuzzy arithmetic robustly in their application domains; (iii) We demonstrate that the proposed framework can overcome the possibility of generating pathological results that were seen in other fuzzy arithmetic approaches in the FST literature; (iv) We demonstrate the utility of the proposed framework in solving multiple-criteria-decision-making (MCDM) problems concretely and in an easy-to-interpret manner (which are highly important considerations for the users of MCDM). Last but not least, we offer a new avenue for extending this fuzzy arithmetic framework to higher-order fuzzy settings, such as the type-2 fuzzy and other type-2-like fuzzy settings. Thus, our framework can potentially provide a unified, robust approach for developing fuzzy arithmetic in a wide range of FST settings.