The multi-item capacitated lot-sizing problem with safety stocks and demand shortage costs

We address a multi-item capacitated lot-sizing problem with setup times, safety stock and demand shortages. Demand cannot be backlogged, but can be totally or partially lost. Safety stock is an objective to reach rather than an industrial constraint to respect. The problem is np-hard. We propose a Lagrangian relaxation of the resource capacity constraints. We develop a dynamic programming algorithm to solve the induced sub-problems. An upper bound is also proposed using a Lagrangian heuristic with several smoothing algorithms. Some experimental results showing the effectiveness of the approach are reported.