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 threshold-determined 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.

Presently, intelligent games have emerged as a substantial research area. Nonetheless, the slow convergence of intelligent wargame training and the low success rates of agents against specific rules present challenges. In this article, we propose a game confrontation algorithm combining the multiple attribute decision making (MADM) approach from management science and reinforcement learning (RL) technology. This integration enables us to combine the strengths of both approaches and addresses the above issues effectively. This study conducts experiments using the algorithm that integrates MADM and RL techniques to gather confrontation data from the red and blue sides within the winning-first wargame platform. The data is then analyzed using the weight calculation method of intuitionistic fuzzy numbers to determine each intelligent opponent agent's threat level from the perspective of MADM. The threat level calculated by MADM is used to construct the reward function for the red side. The simulation results demonstrate that the algorithm combining MADM and RL proposed in this study outperforms classical RL algorithms regarding intelligence. This approach effectively addresses issues, such as the convergence difficulty, caused by random initialization and the sparse rewards for agent neural networks in wargame environments with large maps. Combining the MADM method from management with the RL algorithm in control can lead to cross-disciplinary innovation in academic fields, which provides innovative research values for intelligent wargame design and RL algorithm improvements.

Due to the effectiveness and advantages of interval-valued intuitionistic fuzzy sets (IVIFSs) in evaluating uncertainty and risk, we introduce IVIFSs into loss functions of decision-theoretic rough sets (DTRSs) and propose an optimization-based approach to interval-valued intuitionistic fuzzy three-way decisions. First, based on the classical DTRSs and two previous optimization models, we construct a new concise linear programming model for simultaneously determining the threshold pair. Our model is mathematically equivalent to the DTRSs and the previous models under the Karush-Kuhn-Tucker (KKT) condition. Second, we extend the constructed model via the IVIFSs of loss functions and we discuss the relations between these loss functions based on a similarity measure function-based ranking method and a multiple score function-based ranking method for IVIFSs. Third, we develop our extended models via two ranking methods and we prove the existence and uniqueness of the optimal solution of the model. The optimization-based method, along with its algorithm for three-way decisions, is designed in an interval-valued intuitionistic fuzzy environment. Compared to the latest existing methods, our method has three advantages (see Advantages 1-3). Finally, an illustrative example is considered, and the advantages of our approach are demonstrated by this example.

A novel dynamic intuitionistic fuzzy TOPSIS method is proposed to evaluate the agents when the expert evaluation values are Intuitionistic Fuzzy Numbers (IFNs). Similar to Positive Ideal Solution (PIS) in traditional TOPSIS, Positive Ideal Agent (PIA) is an ideal agent which achieves the best values in all the expert evaluations. Negative Ideal Agent (NIA) is the worst in every expert's evaluation. Then agent similarity to PIA and that to NIA are calculated. The relative ideal closeness of each agent is defined as a combination of the corresponding two similarities, which indicates the agent's relative capabilities. To deal with varied numbers of agents and experts, the update mechanisms are presented. A numerical example demonstrates the validity of our approach.