Linear-programming extended formulations for the single-item lot-sizing problem with backlogging and constant capacity

Recently, several authors [ 8, 10] have argued for the use of extended formulations to tighten production planning models. In this work we present two linear-programming extended formulations of the constant-capacity lot-sizing problem with backlogging. The first one applies to the problem with a general cost function and has O(n(3)) variables and constraints. This improves on the more straightforward O(n(4)) Florian and Klein [ 2] type formulation. The second one applies when the costs satisfy the Wagner-Whitin property but it has O(n(2)) variables and O(n(3)) constraints. As a by-product, we positively answer an open question of Miller and Wolsey [ 4] about the tightness of an extended formulation for the continuous mixing set.