An Integrated Mathematical Programming and Reinforcement Learning Algorithm for the Flexible Job Shop Scheduling with Variable Lot-sizing

The Flexible Job Shop Scheduling Problem with Variable Lot-Sizing (FJSP-VLS) extends the Flexible Job shop Scheduling Problem (FJSP) by permitting variable lot-sizing for jobs of the same type across different operations. This approach provides enhanced flexibility compared to the conventional method of consistent lot-sizing. However, existing algorithms face efficiency bottlenecks when scaling to real-world production scenarios. To address this challenge, the authors propose an Integrated Mathematical Programming and Reinforcement Learning (IMPRL) algorithm that synergistically combines a dual-attention neural network with Proximal Policy Optimization (PPO) for adaptive scheduling, coupled with a Mixed Integer Linear Programming (MILP) model for joint lot-sizing and machine optimization. Extensive experiments on 10 benchmark-derived instance classes demonstrate IMPRL’s superiority: it reduces makespan by 6.86% (up to 11.87% for 30$\times$ 10 instances) compared to TOP PDR, achieves 9.81% improvement in generalization tests, and maintains solution quality while being an order-of-magnitude faster than MILP and GA-MH${}_{\text{ER}}$ approaches. The algorithm’s hierarchical architecture effectively resolves inconsistencies in sublot completion times, while the case study fully demonstrates its practicality in large-scale FJSP-VLS implementations. The key managerial insights derived from the research findings are also highlighted, along with an acknowledgment of the algorithm’s limitations.