A hybrid whale optimization algorithm for distributed no-wait flow-shop scheduling problem with batch delivery

Enterprises have increasingly focused on integrated production and transportation problems, recognizing their potential to enhance cohesion across different decision-making levels. The whale optimization algorithm, with its advantages such as minimal parameter control, has garnered attention. In this study, a hybrid whale optimization algorithm (HWOA) is designed to settle the distributed no-wait flow-shop scheduling problem with batch delivery (DNWFSP-BD). Two objectives are considered concurrently, namely, the minimization of the makespan and total energy consumption. In the proposed algorithm, four vectors are proposed to represent a solution, encompassing job scheduling, factory assignment, batch delivery and speed levels. Subsequently, to generate high-quality candidate solutions, a heuristic leveraging the Largest Processing Time (LPT) rule and the NEH heuristic is introduced. Moreover, a novel path-relinking strategy is proposed for a more meticulous search of the optimal solution neighborhood. Furthermore, an insert-reversed block operator and variable neighborhood descent (VND) are introduced to prevent candidate solutions from converging to local optima. Finally, through comprehensive comparisons with efficient algorithms, the superior performance of the HWOA algorithm in solving the DNWFSP-BD is conclusively demonstrated.