Integer and combinatorial optimization
Integer and combinatorial optimization
The convex hull of two core capacitated network design problems
Mathematical Programming: Series A and B
Lot-sizing with constant batches: formulation and valid inequalities
Mathematics of Operations Research
Capacitated facility location: valid inequalities and facets
Mathematics of Operations Research
Capacitated facility location: separation algorithms and computational experience
Mathematical Programming: Series A and B - Special issue on computational integer programming
Strengthening integrality gaps for capacitated network design and covering problems
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Fully Polynomial Approximation Schemes for Single-Item Capacitated Economic Lot-Sizing Problems
Mathematics of Operations Research
Shipping Multiple Items by Capacitated Vehicles: An Optimal Dynamic Programming Approach
Transportation Science
Production Planning by Mixed Integer Programming (Springer Series in Operations Research and Financial Engineering)
Primal-Dual Algorithms for Deterministic Inventory Problems
Mathematics of Operations Research
Progressive Interval Heuristics for Multi-Item Capacitated Lot-Sizing Problems
Operations Research
Algorithms for capacitated rectangle stabbing and lot sizing with joint set-up costs
ACM Transactions on Algorithms (TALG)
Approximation and Online Algorithms
Journal of Discrete Algorithms
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We study the classical multi-item capacitated lot-sizing problemwith hard capacities. There are Nitems, each of which has specified sequence of demands over a finite planning horizon of discrete Tperiods; 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 costregardless 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, being the first constant approximation algorithm for the problem. We also describe an interesting on-the-flyvariant of the algorithm that does not require to solve 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. We believe that some of the novel algorithmic ideas proposed in this paper have a promising potential in constructing strong LP relaxations and LP-based approximation algorithms for other inventory models, and for the capacitated facility location problem.