Valid inequalities for Lagrangian relaxation in an inventory location problem with stochastic capacity

We developed an efficient heuristic to solve a joint location–distribution–inventory model for a three layered supply chain. A firm must locate distribution centers to supply a commodity to spatially distributed retailers with stochastic demand. The solution approach is based on Lagrangian relaxation, improved with validity constraints derived from the finite set of all possible combinations of mean demand and variance. The optimal solution’s lower bound is found through the optimal solution of the dual problem. The dual gap gives a threshold to the heuristic’s error. The heuristic was applied to 98 instances with an average error threshold of 3%.