The single-item lot-sizing polytope with continuous start-up costs and uniform production capacity

In this work we consider the uniform capacitated single-item single-machine lot-sizing problem with continuous start-up costs. A continuous start-up cost is generated in a period whenever there is a nonzero production in the period and the production capacity in the previous period is not saturated. This concept of start-up does not correspond to the standard (discrete) start-up considered in previous models, thus motivating a polyhedral study of this problem. In this work we explore a natural integer programming formulation for this problem. We consider the polytope obtained as convex hull of the feasible points in this problem. We state some general properties, study whether the model constraints define facets, and present an exponentially-sized family of valid inequalities for it. We analyze the structure of the extreme points of this convex hull, their adjacency and bounds for the polytope diameter. Finally, we study the particular case when the demands are high enough in order to require production in all the periods. We provide a complete description of the convex hull of feasible solutions in this case and show that all the inequalities in this description are separable in polynomial time, thus proving its polynomial time solvability. Copyright Springer Science+Business Media New York 2015