# Approximation algorithms for the joint replenishment problem with deadlines

## Abstract

The Joint Replenishment Problem (\({\hbox {JRP}}\)) is a fundamental optimization problem in supply-chain management, concerned with optimizing the flow of goods from a supplier to retailers. Over time, in response to demands at the retailers, the supplier ships orders, via a warehouse, to the retailers. The objective is to schedule these orders to minimize the sum of ordering costs and retailers’ waiting costs. We study the approximability of \({\hbox {JRP-D}}\), the version of \({\hbox {JRP}}\) with deadlines, where instead of waiting costs the retailers impose strict deadlines. We study the integrality gap of the standard linear-program (LP) relaxation, giving a lower bound of \(1.207\), a stronger, computer-assisted lower bound of \(1.245\), as well as an upper bound and approximation ratio of \(1.574\). The best previous upper bound and approximation ratio was \(1.667\); no lower bound was previously published. For the special case when all demand periods are of equal length, we give an upper bound of \(1.5\), a lower bound of \(1.2\), and show APX-hardness.

### Keywords

Joint replenishment problem NP-completeness APX-hardness Approximation algorithms## Notes

### Acknowledgments

We would like to thank Łukasz Jeż, Dorian Nogneng, Jiří Sgall, and Grzegorz Stachowiak for stimulating discussions and useful comments. We are also grateful to anonymous reviewers of earlier versions of this manuscript for pointing out several mistakes and suggestions for improving the presentation. A preliminary version of this work appeared in the Proceedings of the 40th International Colloquium on Automata, Languages and Programming (ICALP’13). Research supported by NSF Grants CCF-1217314, CCF-1117954, OISE-1157129; EPSRC grants EP/J021814/1 and EP/D063191/1; FP7 Marie Curie Career Integration Grant; Royal Society Wolfson Research Merit Award; and Polish National Science Centre Grant DEC-2013/09/B/ST6/01538.

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