Characterizing order processes of ( s , S ) and ( r , nQ ) policies

In this paper we provide a novel approach to characterize the order process of an (s, S) and an (r, nQ) inventory policy. These inventory policies are commonly used when a fixed order cost is present. We provide two distinct perspectives. First, we analyze the time between subsequent replenishments, as well as the size of the replenishment orders, and characterize the distribution of both. Second, we look at the number of units ordered over an interval t, and compare it with the demand observed over the same period. Both perspectives are complementary and provide useful information for the upstream supplier observing this order process. We are in particular interested in the impact of the batching parameter (Q or S − s) on the order variance amplification ratio. It is commonly accepted in the supply chain literature that batching of orders creates a bullwhip effect. Our analysis shows that the answer is more subtle. For instance, the squared coefficient of variation (scv) of time between subsequent orders is smaller than or equal to the scv of time between subsequent demand arrivals for both policies. The order quantities can exhibit either variance amplification or dampening, compared to the demand quantities, depending on the distribution of the demand sizes and the value of the batching parameter. The bullwhip ratio, defined as the variance of the number of units ordered compared to the variance of the number of units demanded, strongly depends on the time window in which these numbers are observed. Our methodology is based on the properties of the batch-Markovian arrival process.