Set-valued Preference Relation and Its Properties

Classical preference relation depicts the fact that alternative a is better than alternative $b_{,}$ and fuzzy preference relation describes the degree to which a is better than b. Furthermore, in order to describe in what aspects a is better than $b_{,}$ the concepts of set-valued binary relation, set-valued preference relation, set-valued weak or strict preference relation are proposed in this paper. Then the structure characterization of set-valued preference relation is presented. Next, a general method of inducing a set-valued preference relation from an ordered information system is developed. Particularly, for a three-valued ordered information system, the structural and quantitative properties of the induced set-valued preference relation matrix are discussed under different partial order relations. Finally, a numerical example is given to illustrate the application of related concepts and properties in data analysis, which shows the rationality and effectiveness of the proposed algorithm.