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The submodular joint replenishment problem

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Abstract

The joint replenishment problem is a fundamental model in supply chain management theory that has applications in inventory management, logistics, and maintenance scheduling. In this problem, there are multiple item types, each having a given time-dependent sequence of demands that need to be satisfied. In order to satisfy demand, orders of the item types must be placed in advance of the due dates for each demand. Every time an order of item types is placed, there is an associated joint setup cost depending on the subset of item types ordered. This ordering cost can be due to machine, transportation, or labor costs, for example. In addition, there is a cost to holding inventory for demand that has yet to be served. The overall goal is to minimize the total ordering costs plus inventory holding costs. In this paper, the cost of an order, also known as a joint setup cost, is a monotonically increasing, submodular function over the item types. For this general problem, we show that a greedy approach provides an approximation guarantee that is logarithmic in the number of demands. Then we consider three special cases of submodular functions which we call the laminar, tree, and cardinality cases, each of which can model real world scenarios that previously have not been captured. For each of these cases, we provide a constant factor approximation algorithm. Specifically, we show that the laminar case can be solved optimally in polynomial time via a dynamic programming approach. For the tree and cardinality cases, we provide two different linear programming based approximation algorithms that provide guarantees of three and five, respectively.

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Acknowledgments

We thank the refereeing team for their numerous insightful comments that significantly improved the quality and exposition of the paper. The research of the first author was supported by NSF grants CCF-0832782, CCF-1017688 and NSERC grant PGS-358528. The research of the second author was partially conducted while he was a student at the Operations Research Center at Massachusetts Institute of Technology and was supported by a National Defense Science and Engineering Graduate Fellowship from the Air Force Office of Scientific Research. The research of the third author was partially supported by NSF grant CMMI-0846554 (CAREER Award), an AFOSR award FA9550-11-1-0150, an SMA grant and the Buschbaum Research Fund of MIT. The research of the fourth author was supported by NSF grants CCF-0832782 and CCF-1017688.

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Correspondence to Adam N. Elmachtoub.

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Cheung, M., Elmachtoub, A.N., Levi, R. et al. The submodular joint replenishment problem. Math. Program. 158, 207–233 (2016). https://doi.org/10.1007/s10107-015-0920-3

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