Models and Algorithms for Optimizing Thresholds in Fuzzy Representation-Based Three-Way Decision

Many models and methods have been developed to determine numerical thresholds in three-way decision. However, there are challenges such as local optimum-based thresholds, initial point-related optimal solutions and a limited model generalization. To address these challenges, we design a penalty mechanism-based approximate model and solving algorithm. First, a standard optimization model and its approximate model are established by means of a penalty mechanism. Moreover, three properties of the approximate model are explored, and the relationships between both types of models are analyzed. Second, a penalty mechanism-based particle swarm optimization (PMPSO) algorithm is designed to solve general frameworks with our established models, and comparative experiments are conducted to verify the effectiveness and advantages of the algorithm. Third, we generalize the established model by fuzzy loss representations, and take linguistic intuitionistic fuzzy numbers (LIFNs) as a representation to establish general thresholddetermined models based on single and multiple ranking measure functions of LIFNs. Then, we prove the existence and uniqueness of the optimal solution and develop a three-way decision method with the PMPSO algorithm. Finally, an illustrative example and more comparative analyses are considered to demonstrate the effectiveness of our method.