# Scheduling parallel batching machines in a sequence

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## Abstract

Motivated by the application of scheduling a sequence of locks along a waterway, we consider a scheduling problem where multiple parallel batching machines are arranged in a sequence and process jobs that travel along this sequence. We investigate the computational complexity of this problem. More specifically, we show that minimizing the sum of completion times is strongly NP-hard, even for two identical machines and when all jobs travel in the same direction. A second NP-hardness result is obtained for a different special case where jobs all travel at an identical speed. Additionally, we introduce a class of so-called synchronized schedules and investigate special cases where the existence of an optimum solution which is synchronized can be guaranteed. Finally, we reinforce the claim that bidirectional travel contributes fundamentally to the computational complexity of this problem by describing a polynomial time procedure for a setting with identical machines and where all jobs travel in the same direction at equal speed.

## Keywords

Machine scheduling Complexity Parallel batching machine Machine sequence## Notes

### Acknowledgements

This research has been partially funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office. This work was carried out when both authors were affiliated with KU Leuven. We thank the associate editor and the referees for their careful reading of the manuscript.

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