Optimizing the multi-objective traveling salesman problem with a deep reinforcement learning algorithm using cross fusion attention networks.

The multi-objective traveling salesman problem (MOTSP), a classical type of multi-objective combinatorial optimization problem (MOCOP), is pivotal in numerous real-world applications. However, traditional algorithms often face challenges in efficiently finding satisfactory solutions due to the vast search space and inherent conflicts between objectives. To address this issue, we propose a deep reinforcement learning (DRL) algorithm utilizing a cross fusion attention network (CFAN). The cross fusion attention encoder within the CFAN architecture is designed to capture the relationships between problem instances and weight preferences, thereby constructing unified context features. This enables a single trained CFAN model to solve problems with varying weight preferences. Furthermore, we enhance the model's ability to explore boundary solutions by adjusting the weight distribution. To evaluate the proposed algorithm's effectiveness, we conducted a comparative analysis with classical evolutionary algorithms and advanced DRL approaches across various MOTSP instances. Experimental results demonstrate that CFAN consistently outperforms both categories of algorithms, achieving superior solution quality and generalization capability. In particular, CFAN achieves a 1.43% improvement in the hypervolume (HV) metric over the best-performing DRL algorithm on KroAB instances, a 3.12% improvement on tri-objective problem instances, and a 2.17% improvement on large-scale problem instances. These results highlight the effectiveness of CFAN in handling diverse problem instances.