On the Polyhedral Structure of Two-Level Lot-Sizing Problems with Supplier Selection

In this paper, we study a two-level lot-sizing problem with supplier selection (LSS). This NP-hard problem arises in different production planning and supply chain management applications. We first present a dynamic programming algorithm for LSS that is polynomial when the number of plants is fixed. We use this algorithm to describe the convex hull of solutions to the problem in an extended space of variables. We then investigate a smaller “multi-commodity” extended formulation, show that it has fractional vertices, and present a family of valid inequalities to strengthen it. Next, we explore the polyhedral structure of the formulation of the problem with traditional variables. We derive several families of strong valid inequalities for this formulation, and give conditions under which they are facet-defining. We prove that these new families of facet-defining inequalities are sufficient to describe the convex hull of the problem with two plants and two periods. Finally, we show numerically that incorporating these valid inequalities within a cut-and-branch framework leads to significant improvement in computation.