Set-theoretic models of three-way decision

Abstract

The theory of three-way decision is about a philosophy of thinking in threes, a methodology of working with threes, and a mechanism of processing in threes. We approach a whole through three parts, in terms of three units, or from three perspectives. A trisecting–acting–outcome (TAO) model of three-way decision involves trisecting a whole into three parts and acting on the three parts, in order to produce an optimal outcome. In this paper, we further explore the TAO model in a set-theoretic setting and make three new contributions. The first contribution is an examination of three-way decision with nonstandard sets for representing concepts under the two kinds of objective/ontic and subjective/epistemic uncertainty. The second contribution is an introduction of an evaluation-based framework of three-way decision. We present a classification of trisections and investigate the notion of an evaluation space. The third contribution is, within the proposed framework, a systematical study of three-way decision with rough sets, interval sets, fuzzy sets, shadowed sets, rough fuzzy sets, interval fuzzy sets (or equivalently, vague sets, interval-valued fuzzy sets, intuitionistic fuzzy sets), and soft sets.

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Acknowledgements

This work was supported in part by a Discovery Grant from NSERC, Canada. The author thanks Professors Witold Pedrycz and Shyi-Ming Chen for their encouragements during the preparation of the paper. The author is grateful to reviewers for their constructive and critical comments.

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Yao, Y. Set-theoretic models of three-way decision. Granul. Comput. 6, 133–148 (2021). https://doi.org/10.1007/s41066-020-00211-9

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Keywords

  • Three-way decision
  • Interval set
  • Rough set
  • Fuzzy set
  • Shadowed set
  • Rough fuzzy set
  • Interval fuzzy set
  • Soft set
  • Intuitionistic fuzzy set
  • Vague set
  • Interval-valued fuzzy set