A polyhedral study of the single-item lot-sizing problem with continuous start-up costs

In this work we consider the 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. We introduce a natural integer programming formulation for this problem, we study some general properties and facet-inducing inequalities of the associated polytope, and we state relationships with known lotsizing polytopes.