A mirror-image lead time inventory model

The production-operations literature is rife with the inventory model where the demand rate, D, is iid, but where the lead time, L, is deterministic. Here we present a mirror-image lead time model where the reverse is true; that is, where D is deterministic and L is iid. This lead time model is interesting: first, it is subject to order crossover; second, because of order crossover, the optimal order cycle and order quantity could be smaller; and third, the optimal cost incurs inventory savings due to the reduction in the variance of the lead times in crossover situations. We perform this analysis in the context of three inventory reorder point, order quantity models that differ only in the characterisation of the shortage cost. These are Model 1 that uses the shortage cost per stockout occasion; Model 2 with the shortage cost per unit short; and Model 3, where the shortage cost is per unit short per unit time. Also, we develop response surfaces (for the normal approximation) for the optimal cost, C*, the optimal order quantity, Q*, and the optimal safety stock factor, , in terms of the problem parameters (ordering cost per order, holding cost per unit per unit time, the particular shortage cost, and the standard deviation of the parent lead time). However, we find that the normal approximation is good in Model 1 and adequate for Model 3. For Model 2, the normal approximation understates the cost, because of the large penalties associated with the long right tail of the distribution of the demand during the effective lead time when the parent lead time is exponentially distributed. Nevertheless, the normal approximation could serve to provide a bound for the total cost and could be more convenient, albeit biased slightly upward in the case of Model 3 and significantly downward in the case of Model 2 than a straightforward simulation of the problem.