In this work we address the multi-item capacitated lot-sizing problem with lost sales. Demand cannot be backlogged, but can be totally or partially lost. This problem is NP-hard. We propose different mathematical formulations. To nd a near optimal solution, different heuristic approaches are developed. They are mainly based on a Lagrangian relaxation of the resource capacity constraints. Computational results will be presented in order to compare our approaches to monolithic resolutions of the proposed models using a commercial solver.