A Production-Inventory Control Model with a Mixture of Back-Orders and Lost-Sales

We consider a production-inventory control model of finite capacity, in which backordering up to a certain level is allowed. We assume that there exist two possible production rates. The control is based on two critical stock-levels and prescribes to change the production rate used only when one of these levels is reached. A fixed cost is associated with every switch-over. The rate at which customers arrive and the distribution of the demand of an arriving customer depend on the production rate used at that moment. A formula for the long-run average expected costs per unit time is obtained as a function of the chosen critical levels. From this formula we derive expressions for various interesting operating characteristics of this system, amongst which are the joint stationary distribution of the processes describing the production rate used and the inventory of the system and the average number of switch-overs and lost-sales per unit time.