Approximation Algorithms for the Capacitated Multi-Item Lot-Sizing Problem via Flow-Cover Inequalities Retsef Levi Sloan School of Management, MIT, Cambridge, MA, 02139, USA email: retsef@mit.edu Andrea Lodi University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy. email: andrea.lodi@unibo.it Maxim Sviridenko IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598. email: sviri@us.ibm.com We study the classical capacitated multi-item lot-sizing problem with hard capacities. There are N items, each of which has specified sequence of demands over a finite planning horizon of T discrete periods; the demands are known in advance but can vary from period to period. All demands must be satisfied on time. Each order incurs a time-dependent fixed 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 find 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 flow-cover inequalities; the proposed LP can be solved to optimality in polynomial time via an efficient 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 first constant approximation algorithm for the problem. We also describe an interesting on-the-fly variant of the algorithm that does not require solving the LP a-priori with all the flow-cover inequalities. As a by-product we obtain the first theoretical proof regarding the strength of flow-cover inequalities in capacitated inventory models. Key words: Inventory management; Approximation ; Flow-cover inequalities ; Algorithms MSC2000 Subject Classification: Primary: 90B05; Secondary: 68W25, OR/MS subject classification: Primary: inventory/production , approximations/heuristics; Secondary: produc- tion/scheduling, approximations/heuristics 1. Introduction The issue of capacity constraints arises in many practical and theoretical inventory management problems as well as in problems in other application domains, such as facility location problems. In most practical inventory systems there exist capacity constraints that limit the quantities that one can order, ship or produce. Unfortunately, it is often the case that models with capacity constraints are computationally far more challenging than their counterpart models with no capacity constraints. In particular, in many problems with capacity constraints computing optimal policies and sometimes even feasible policies is a very challenging task. In recent years there has been an immense amount of work to develop integer programming methods for solving hard, large-scale deterministic inventory management problems. (We refer the reader to the recent book of Pochet and Wolsey [20].) A major part of this work has been focused on constructing strong formulations for the corresponding inventory models. In fact, it is essential to have an integer pro- gramming formulation with a strong linear programming relaxation. Stronger formulations are achieved by identifying valid inequalities that are satisfied by all feasible integral solutions and cut off fractional solutions. Another key aspect within an integer programming framework is the ability to construct good feasible integer solutions to the corresponding model. This has been known to have a huge impact on decreasing the computational effort involved. In models with capacity constraints, finding good feasible solutions can be very challenging. In this paper, we study the classical capacitated multi-item lot-sizing problem, which is an extension of the single-item economic lot-sizing problem [20]. Next, we propose a novel facility location type linear program (LP), and show how to round its optimal solution to a feasible integral solution with cost that is guaranteed to be at most twice the optimal cost. This is called a 2-approximation algorithm, that is, 1