# Single-machine serial-batch delivery scheduling with two competing agents and due date assignment

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## Abstract

We consider a set of single-machine batch delivery scheduling problems involving two competing agents under two due date assignment models. Belonging to one of the two agents, each job is processed and delivered in a batch to its agent, where the jobs in each batch come from the same agent. The jobs in a batch are processed sequentially and the processing time of a batch is equal to the sum of the processing times of the jobs in it. A setup time is required at the start of each batch. The dispatch date of a job equals the delivery date of the batch it is in, i.e., the completion time of the last job in the batch. There is no capacity limit on each delivery batch, and the cost per batch delivery is fixed and independent of the number of jobs in the batch. The due date of each job is a decision variable, which is to be assigned by the decision maker using one of two due date models, namely the common and unrestricted due date models. Given the due date assignment model, the overall objective is to minimize one agent’s scheduling criterion, while keeping the other agent’s criterion value from exceeding a threshold given in advance. Two kinds of scheduling criteria are involved: (i) the total cost comprising the earliness, tardiness, job holding, due date assignment, and batch delivery costs; and (ii) the total cost comprising the earliness, weighted number of tardy jobs, job holding, due date assignment, and batch delivery costs. For each of the problems considered, we show that it is \(\mathcal {NP}\)-hard in the ordinary sense and admits a fully polynomial-time approximation scheme.

## Keywords

Scheduling Due date assignment Batch delivery Two agents Dynamic programming## Notes

### Acknowledgements

We thank the Editor, an Associate Editor, and two anonymous referees for their helpful comments on earlier versions of our paper. This paper was supported in part by the National Natural Science Foundation of China under Grant Numbers 11561036, 71501024, 71532007, and 71520107002; and in part by Project funded by China Postdoctoral Science Foundation under Grant Number 2017M612099. Cheng was supported in part by The Hong Kong Polytechnic University under the Fung Yiu King-Wing Hang Bank Endowed Professorship in Business Administration.

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