Primal-Dual Algorithms for Deterministic Inventory Problems

Abstract

We consider several classical models in deterministic inventory theory: the single-item lot-sizing problem, the joint replenishment problem, and the multistage assembly problem. These inventory models have been studied extensively, and play a fundamental role in broader planning issues, such as the management of supply chains. For each of these problems, we wish to balance the cost of maintaining surplus inventory for future demand against the cost of replenishing inventory more frequently. For example, in the joint replenishment problem, demand for several commodities is specified over a discrete finite planning horizon, the cost of maintaining inventory is linear in the number of units held, but the cost incurred for ordering a commodity is independent of the size of the order; furthermore, there is an additional fixed cost incurred each time a nonempty subset of commodities is ordered. The goal is to find a policy that satisfies all demands on time and minimizes the overall holding and ordering cost. We shall give a novel primal-dual framework for designing algorithms for these models that significantly improve known results in several ways: the performance guarantees for the quality of the solutions improve on or match previously known results; the performance guarantees hold under much more general assumptions about the structure of the costs, and the algorithms and their analysis are significantly simpler than previous known results. Finally, our primal-dual framework departs from the structure of previously studied primal-dual approximation algorithms in significant ways, and we believe that our approach may find applications in other settings. More specifically, we provide 2-approximation algorithms for the joint replenishment problem and for the assembly problem, and solve the single-item lot-sizing problem to optimality. The results for the joint replenishment and the lot-sizing problems also hold for their generalizations with back orders allowed. As a byproduct of our work, we prove known and new upper bounds on the integrality gap of some linear-programming (LP) relaxations of the abovementioned problems.

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