A reinforcement learning-enhanced discrete zebra optimization algorithm for solving the traveling salesman problem

In this study, a novel hybrid algorithm named the Reinforced Zebra Optimization Algorithm (RZOA) is proposed to solve the Traveling Salesman Problem (TSP). Initially, a discrete version of the Zebra Optimization Algorithm, referred to as DZOA, was developed, in which the continuous relationships of the original ZOA were transformed into a set of discrete operators to effectively update the routes. Subsequently, to enhance the intelligence of the operator selection process and to maintain a dynamic balance between exploration and exploitation, a mechanism based on Deep Reinforcement Learning was designed. Within the framework of the proposed RZOA, each zebra agent adaptively and self-learnedly makes decisions using a Deep Q-Network (DQN). Each agent observes both its own state and the overall population state, employs the DQN to select the optimal operator, and improves its decision-making policy dynamically through an experience replay memory. The state–action–reward structure was carefully designed to ensure an intelligent and balanced learning behavior between local and global search processes. The results of numerical experiments conducted on 42 standard benchmark datasets from the TSPLIB repository demonstrate that the proposed RZOA exhibits significant superiority over the compared algorithms in terms of solution quality, convergence speed, and performance stability. In particular, the proposed method achieves near-optimal performance with an average Percentage Deviation of the Best solution (PDB) below 1% and an average Percentage Deviation of the Average solution (PDA) typically below 0.5% for small and medium-scale instances, while maintaining PDA values within 4–5% for large-scale problems. Moreover, RZOA outperforms competing algorithms in the majority of benchmark cases, demonstrating superior robustness and consistency. Furthermore, the Friedman and Wilcoxon statistical analyses confirm this superiority at a 95% confidence level. Overall, by integrating the global search capability of the Zebra Optimization Algorithm with the adaptive decision-making power of the DQN, the proposed RZOA provides a novel, intelligent, and efficient approach for solving complex combinatorial optimization problems.