Economic Order Quantities and Multiple Set-Up Costs

We consider the problem of scheduling the production of a single product at each instant during a time horizon of length T (≦ ∞) so as to minimize the average cost per unit time; backlogging of demand and disposal of stock are not allowed. Two types of costs are incurred; the holding cost per unit time is assumed to be proportional to the inventory level while the ordering cost is that associated with the multiple set-up cost function (see (3) and Figure 1). By studying a special case (K = 0 in (3)) in §2, the analysis of the problem practically becomes that of the standard economic lot size model so that we can rigorously derive some not too widely known results about the lot size model while exploring our model. In particular, we obtain planning horizon theorems and characterize the form of an optimal production schedule for T < ∞ and T = ∞. In §3 the special assumption is removed and similar, but slightly weaker, theorems are established. For example, an optimal production schedule may have several orders of two different sizes in contrast to the special case where every order is the same size. In addition, simple algorithms for finding optimal policies for both T < ∞ and T = ∞ are given.