An extended graphical method with reinforcement learning for distributed job shop scheduling problems with transfers

Modern manufacturing often requires collaboration across multiple factories due to the dispersion of specialized equipment and differences in processing capacity. In such scenarios, jobs must be processed across different sites, and cross-factory transfers significantly impact scheduling performance. This paper studies the distributed job shop scheduling problem with transfers and proposes an Extended Graphical Method with Reinforcement Learning (EGMRL) to effectively address it. The main innovations of EGMRL are fourfold. First, a transfer-zone is introduced into the graphical method, enabling explicit modeling of both intra-factory and inter-factory transportation times. Second, a layered path search algorithm is developed to accelerate path exploration, thereby improving computational efficiency while maintaining accuracy. Third, a Q-learning–based adaptive strategy dynamically guides job deletion and reinsertion according to the inter-factory state, enhancing adaptability across different problem scales. Finally, a tabu search module is integrated as a local improvement strategy to refine factory-level schedules and prevent premature convergence. Comprehensive experiments on 240 extended benchmark instances and a real-world engineering case study demonstrate that EGMRL consistently outperforms four competitive algorithms in terms of solution quality and stability. Furthermore, the results suggest that the extended graphical method provides promising new solution approaches for tackling scheduling problems with other practical constraints, such as worker–machine collaboration and sequence-dependent setup times. • Proposes an extended graphical method by introducing a transfer zone to model both intra- and inter-factory transportation. • Develops a layered path search algorithm to accelerate path exploration and reduce computational effort. • Integrates a Q-learning–based adaptive strategy to guide job deletion and reinsertion dynamically. • Proposes an Extended Graphical Method with reinforcement learning for DJSPT.