Energy-efficient distributed flexible job shop scheduling with variable processing speeds: a Q-learning hyper-heuristic algorithm

This paper addresses the energy-efficient distributed flexible job shop scheduling problem (EEDFJSP) with the dual objectives of minimizing makespan and total energy consumption (TEC). In this problem, jobs are assigned to multiple factories, each containing several machines that can operate at variable speed levels, leading to a trade-off between productivity and energy efficiency. To solve this complex multi-objective optimization problem, a Q-learning-based hyper-heuristic algorithm (QL-HH) is proposed. The algorithm incorporates a hybrid initialization strategy combining four heuristic rules (NR2, longest remaining workload, minimum completion time, and speed-level heuristics) to generate a high-quality and diverse initial population. A Q-learning framework with an improved dynamic ε-greedy mechanism is developed to adaptively select eight low-level heuristic (LLH) operators according to the current state of each individual, which is characterized by normalized objective values and crowding distance. Additionally, critical path-based speed adjustment operators are designed to reduce energy consumption without increasing makespan. Extensive experiments are conducted on 23 benchmark instances, including high-flexibility and low-flexibility datasets derived from classical FJSP instances. The results demonstrate that QL-HH consistently outperforms four state-of-the-art algorithms (QMOEA/D-AWA, MOEA/D, NSGA-Ⅱ, and KBEA) in terms of hypervolume (HV) and inverted generational distance (IGD), with statistical significance confirmed by Wilcoxon and Friedman tests.