Some properties of discrete lot sizing and scheduling problem with setup times using integral demand

Discrete lot sizing and scheduling problem (DLSP) is the problem of determining a minimal cost production schedule such that dynamic demand is fulfilled without backlogging in a single stage manufacturing process. The problem requires solution of the sequencing and sizing of production lots for a number of different items over a discrete and finite planning horizon. Especially, when we consider setup times in the model, the problem is known to be NP-hard. We developed a formulation different from the conventional DLSP, in that setup costs are charged only when production begins with a new production lot of different items, while the general DLSP charges the setup costs whenever a new production lot begins. We assumed that the demand could be met by the integral quantity with DLSP instead of binary quantity. To support our arguments several propositions are introduced