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 singlemachine lot-sizing problem with continuous start-up costs. A continuous startup 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) startup 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 state some general properties, study the facetness properties of the model constraints, and present an exponentially-sized family of valid inequalities. We analyze the structure of the extreme points of the convex hull of its feasible region, 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. Mariana Escalante CONICET, Argentina and Departamento de Matemática, FCEIA, Universidad Nacional de Rosario, Argentina Tel.: +54-341-4802649, ext. 225 Fax: +54-341-4802654 E-mail: [email protected] Javier Marenco Instituto de Ciencias, Universidad Nacional de General Sarmiento, Argentina Tel.: +54-11-4469-7528 Fax: +54-11-4469-7501 E-mail: [email protected] Maŕıa del Carmen Varaldo Departamento de Matemática, FCEIA, Universidad Nacional de Rosario, Argentina Tel.: +54-341-4802649, ext. 225 Fax: +54-341-4802654 E-mail: [email protected] 2 Escalante, Marenco and Varaldo