# 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.

## References

- Alimonti, P., & Kann, V. (2000). Some APX-completeness results for cubic graphs.
*Theoretical Computer Science*,*237*(1–2), 123–134.CrossRefGoogle Scholar - Arkin, E., Joneja, D., & Roundy, R. (1989). Computational complexity of uncapacitated multi-echelon production planning problems.
*Operations Research Letters*,*8*(2), 61–66.CrossRefGoogle Scholar - Becchetti, L., Marchetti-Spaccamela, A., Vitaletti, A., Korteweg, P., Skutella, M., & Stougie, L. (2009). Latency-constrained aggregation in sensor networks.
*ACM Transactions on Algorithms*,*6*(1), 13:1–13:20.CrossRefGoogle Scholar - Bienkowski, M., Byrka, J., Chrobak, M., Jeż, Ł., & Sgall, J. (2014). Better approximation bounds for the joint replenishment problem. In:
*Proceedings of the of the 25th ACM-SIAM Symposium on Discrete Algorithms (SODA)*, pp. 42–54.Google Scholar - Brito, C., Koutsoupias, E., & Vaya, S. (2012). Competitive analysis of organization networks or multicast acknowledgement: How much to wait?
*Algorithmica*,*64*(4), 584–605.CrossRefGoogle Scholar - Buchbinder, N., Kimbrel, T., Levi, R., Makarychev, K., & Sviridenko, M. (2008). Online make-to-order joint replenishment model: Primal dual competitive algorithms. In:
*Proceedings of the 19th ACM-SIAM Symposium on Discrete Algorithms (SODA)*, pp. 952–961.Google Scholar - Khanna, S., Naor, J., & Raz, D. (2002). Control message aggregation in group communication protocols. In:
*Proceedings of the 29th International Colloquium on Automata, Languages and Programming (ICALP)*, pp. 135–146.Google Scholar - Levi, R., Roundy, R., & Shmoys, D.B. (2005). A constant approximation algorithm for the one-warehouse multi-retailer problem. In:
*Proceedings of the Sixteenth Annual ACM-SIAM symposium on Discrete Algorithms (SODA)*, pp. 365–374.Google Scholar - Levi, R., Roundy, R., & Shmoys, D. B. (2006). Primal-dual algorithms for deterministic inventory problems.
*Mathematics of Operations Research*,*31*(2), 267–284.CrossRefGoogle Scholar - Levi, R., Roundy, R., Shmoys, D. B., & Sviridenko, M. (2008). A constant approximation algorithm for the one-warehouse multiretailer problem.
*Management Science*,*54*(4), 763–776.CrossRefGoogle Scholar - Levi, R., & Sviridenko, M. (2006). Improved approximation algorithm for the one-warehouse multi-retailer problem. In:
*Proceedings of the 9th International Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX)*, pp. 188–199.Google Scholar - Nonner, T., & Souza, A. (2009). Approximating the joint replenishment problem with deadlines.
*Discrete Mathematics, Algorithms and Applications*,*1*(2), 153–174.CrossRefGoogle Scholar