The Optimal Inventory Policy for Batch Ordering

We consider a single product dynamic inventory problem in which the demands in each period are independent and identically distributed random variables. There is a constant lead time, a discount factor 0 ≦ α ≦ 1, a unit ordering cost c and an expected holding and penalty cost function L(·) for which −c(l − α)y + L(y) is unimodal, and total backlogging of unfilled demand. In addition each order for stock must be in some nonnegative integral multiple of Q, a fixed positive constant. It is shown that the (k, Q) policy is optimal for the finite and infinite period models. With the (k, Q) policy if the initial inventory on hand and on order in a period is less than k, an order is placed for the smallest multiple of Q that will bring the inventory on hand and on order to at least k; otherwise, no order is placed. The optimal value of k is easy to compute and is the same for the finite and infinite period models. The results are generalized to the case where the demand distributions and cost functions vary over time. The (k, Q) policy is not optimal in general for the case where there is also a fixed charge for placing an order.