Consequences of order crossover under order-up-to inventory policies

Order crossover occurs whenever replenishment orders do not arrive in the sequence in which they were placed. This paper argues that order crossover is becoming more prevalent and analyzes the dangers of ignoring it. We present an exact iterative algorithm for computing the distribution of the number of orders outstanding, and formulae for the inventory shortfall distribution (the quantity of inventory in replenishment at the start of a period) and the more common lead-time demand distribution, which are different when order crossover is possible. The lead-time demand distribution can have much higher variability than the shortfall distribution. We show that basing inventory policies on the lead-time demand distribution – rather than the shortfall distribution – can lead to significantly higher inventory cost, even if the probability of order crossover is small. We give an alternative proof to that of Zalkind, which shows that the variance of shortfall is less than the variance of the standard lead-time demand.