Scheduling customer orders on unrelated parallel machines to minimise total weighted completion time

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This paper addresses the scheduling problem for customer orders on a set of unrelated parallel machines. Each order consists of multiple product types with various workloads that can be assigned to and processed by the machines. The objective is to minimise the total weighted completion time of all customer orders. Several optimality properties are developed, and an easily computable lower bound is established. Besides, for two important special cases, the corresponding optimal schedules are further provided. Inspired by these results, three heuristic algorithms are proposed, and their worst case performances are proved to be bounded. The effectiveness of the lower bound and the proposed algorithms are demonstrated through numerical experiments. This study brings new perspectives to the management of differentiated customers in complicated production environment.