Capacitated Lot Sizing Problems
The previous chapter considered the economic lot sizing problem when no limit exists on the production quantity in any time period. The absence of production capacities permitted demonstrating the optimality of a solution satisfying the so-called zero-inventory ordering (ZIO) property, which eliminates the possibility of production occurring in a period in which inventory was held from the prior period. This, in turn, enabled fully characterizing a solution based solely on the production periods, i.e., the periods in which production is positive. Equally importantly, this property permitted characterizing an optimal solution as a sequence of independent regeneration intervals (RIs), where an RI is dened by a pair of successive production periods (s; u) with u > s such that is1 = 0, iu1 = 0, and it 6= 0 for t = s; s+1; : : : ; u 2, where it is the inventory level at the end of period t. This denition of an RI combined with the ZIO property implies that within an RI, in the absence of backordering, production only occurs in the rst period of the RI; for example, for the RI dened by the period pair (s; u), the production quantity in period s must equal the sum of the demands in periods s through u 1. Otherwise, the ZIO property would be violated.