Solving Multi - Item Capacitated Lot - Sizing Problems with Setup Times by Branch - and - Cut 1

We study the multi–item capacitated lot–sizing problem with setup times (MCL). This model appears often in practice, either in its pure form or with additional constraints, but it is very difficult to solve to optimality. In MCL demand for multiple items must be met over a time horizon, items compete for a shared capacity, and each setup uses up some of this capacity. By analyzing the polyhedral structure of simplified models obtained from a single time period of MCL, we obtain strong valid inequalities for MCL. These inequalities are the first reported that consider demand for multiple items and the joint capacity restriction simultaneously. We report successful computational results with these inequalities in a branch–and–cut algorithm; these results suggest that our contributions significantly advance the state of the art in solving MCL.