Improved base-stock approximations for independent stochastic lead tims with order crossover

Order crossovers occur when replenishment orders arrive in a sequence that is different than the one in which they were placed. Order crossovers require that optimal reorder levels be set with regard to the inventory shortfall distribution rather than the lead-time demand distribution. Assuming periodic review and independent lead times, this paper suggests simple approximations of the shortfall distribution by showing that the variance of the number of orders outstanding is bounded above by the standard deviation of lead time divided by √3. Using this bound in a normal approximation improves significantly upon the common practice of basing policies on the lead-time demand distribution. A negative binomial approximation of the shortfall, based on its exact variance, offers even greater improvement, at the cost of some additional informational and computational requirements.