An inventory model for items with demands satisfied from stock or by special deliveries

In this paper, a mathematical model is developed for the analysis of optimal replenishment policies for items which experience lumpy demands. In order to avoid disrupting the inventory system, a maximum issue quantity of w units is specified such that customer demands with sizes exceeding w units will be filtered out of the inventory system and treated as special orders to be satisfied by placing special replenishment orders to a higher echelon. Special deliveries are then arranged to ship the goods to the customers. Customer orders with sizes less than or equal to w units will be met from stock. The control discipline is the (s, S) inventory policy with continuous review and the nature of the customer demands is approximated by a discrete compound Poisson distribution. In order to reduce the annual replenishment cost, it is also specified that if the available inventory is below the order-up-to level S at the time when a special replenishment order is placed, such a replenishment order will also raise the available inventory level to S. The optimal values of the control parameters, s and S, are determined. The theoretical results obtained are illustrated with a numerical example.