Ranking interval sets based on inclusion measures and applications to three-way decisions

Three-way decisions provide an approach to obtain a ternary classification of the universe as acceptance region, rejection region and uncertainty region respectively. Interval set theory is a new tool for representing partially known concepts, especially it corresponds to a three-way decision. This paper proposes a framework for comparing two interval sets by inclusion measures. Firstly, we review the basic notations, interpretation and operation of interval sets and classify the orders on interval sets into partial order, preorder and quasi-order. Secondly, we define inclusion measure which indicates the degree to which one interval set is less than another one and construct different inclusion measures to present the quantitative ranking of interval sets. Furthermore, we present similarity measures and distances of interval sets and investigate their relationship with inclusion measures. In addition, we propose the fuzziness measure and ambiguity measure to show the uncertainty embedded in an interval set. Lastly, we study the application of inclusion measures, similarity measures and uncertainty measures of interval sets by a special case of three-way decisions: rough set model and the results show that these measures are efficient to three-way decision processing.