An optimization-based formulation for three-way decisions.

Three-way decisions were originally derived from decision-theoretic rough sets (DTRSs). A fundamental and important problem on DTRSs operating in multiple environments is how to determine the threshold parameter pairs based on loss functions. In this paper, we establish a theoretic and systematic framework for the optimization of the threshold parameters in DTRSs. A general optimization-based approach is developed to determine the threshold pair for DTRSs with various semantics. First, with the aid of the Karush–Kuhn–Tucker (KKT) condition, we propose two optimization models and prove that these models are mathematically equivalent to previous DTRS models. Second, the proposed models are extended to new models using intervals, triangular fuzzy numbers, and intuitionistic fuzzy numbers of loss functions. We discuss the relations among these loss functions based on nonlinear ranking approaches to fuzzy numbers. Third, three types of extended optimization models for three fuzzy environments are constructed, and the uniqueness of their optimal solutions is analyzed and proven. In addition, we utilize optimization techniques to search for their optimal solutions and to determine the threshold pairs. Finally, an example and related comparisons are presented to show the effectiveness and superiority of our approach.