Effective network formulations for lot sizing with backlogging in two-level serial supply chains

This study considers the serial lot sizing problem with backlogging in two-level supply chains to determine when and how much to order at a warehouse and ship to a retailer over a T-period planning horizon so that the external known demand occurring at the retailer is satisfied and the total cost at all levels is minimized. In particular, the uncapacitated two-level serial lot sizing problem with backlogging and the two-level serial lot sizing problem with cargo capacity and backlogging are formulated using effective shortest-path network representations, which define the convex hull of their feasible solutions. These representations lead to efficient algorithms with O(T-3) time for the uncapacitated problem and O(T-6) time for the capacitated problem. Furthermore, a tight reformulation with O(T-3) variables and O(T-2) constraints (resp.O(T-6) variables and O(T-5) constraints) is proposed for the uncapacitated (resp.capacitated) problem.