Single item lot-sizing problems with backlogging on a single machine at a finite production rate

In this paper we consider a single item lot-sizing problem with backlogging on a single machine at a finite production rate. The objective is to minimize the total cost of setup, stockholding and backlogging to satisfy a sequence of discrete demands. Both varying demands over a finite planning horizon and fixed demands at regular intervals over an infinite planning horizon are considered. We have characterized the structure of an optimal procedure schedule for both cases. As a consequence of this characterization, a dynamic programming algorithm is proposed for the computation of an optimal production schedule for the varying demands case and a simpler one for the fixed demands case.