On the optimization of initial order quantities

A lot-sizing problem arises from the determination of initial batch quantities in the case of a finite production rate when demand cannot be satisfied until each batch is made available, assuming no backlogging to be allowed. In this paper the optimal procedure for the general problem of initial order quantities is discussed. An approach in terms of a present value principle taking all payments into consideration is applied in order to derive the optimal solution. By applying the present value measure, the opportunity cost for the initial inventory build-up is incorporated automatically. This is considered to be a more accurate approach than using traditional average cost measures. It is shown that the production runs are divided into two phases, the initial transient phase followed by the stationary phase. With the production rate being greater than the demand rate, a positive idle time between two consecutive production runs is needed in the stationary phase but not within the initial transient phase. To discontinue the sales period is never optimal. Optimal batch quantities increase during the transient phase and then remain constant. To solve the problem, an approach is developed in a framework of the Lagrangian function and Kuhn-Tucker conditions.