Approximation Algorithms for the Capacitated Multi-Item Lot-Sizing Problem via Flow-Cover Inequalities

We study the classical capacitated multi-item lot-sizing problem with hard capacities. There are N items, each of which has speci ed sequence of demands over a nite planning horizon of T discrete periods; the demands are known in advance but can vary from period to period. All demands must be satis ed on time. Each order incurs a time-dependent xed ordering cost regardless of the combination of items or the number of units ordered, but the total number of units ordered cannot exceed a given capacity C. On the other hand, carrying inventory from period to period incurs holding costs. The goal is to nd a feasible solution with minimum overall ordering and holding costs. We show that the problem is strongly NP-hard, and then propose a novel facility location type LP relaxation that is based on an exponentially large subset of the well-known ow-cover inequalities; the proposed LP can be solved to optimality in polynomial time via an ef cient separation procedure for this subset of inequalities. Moreover, the optimal solution of the LP can be rounded to a feasible integer solution with cost that is at most twice the optimal cost; this provides a 2-approximation algorithm which is the rst constant approximation algorithm for the problem. We also describe an interesting on-they variant of the algorithm that does not require solving the LP a-priori with all the ow-cover inequalities. As a by-product we obtain the rst theoretical proof regarding the strength of ow-cover inequalities in capacitated inventory models. ∗[email protected]. Sloan School of Management, MIT, Cambridge, MA, 02139. †[email protected]. D.E.I.S., University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy. Supported in part by the EU projects ADONET (contract n. MRTN-CT-2003-504438) and ARRIVAL (contract n. FP6-021235-2). ‡[email protected]. IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598.