Bullwhip Behaviour as a Function of the Lead-Time for the Order-Up-To Policy Under ARMA Demand

We consider how lead-times create the bullwhip effect in an inventory replenishment system. The inventory system is a combination of a specific demand process, a forecasting method, and a replenishment policy. Previous studies have evaluated the bullwhip effect when some of the system parameters such the forecasting model, demand process, or the lead-time are changed. This type of analyses has proved practically valuable, indicating which improvement measures can be taken. However, the specific causes of a bullwhip behavior are often difficult to grasp. It is often assumed that longer lead times lead to more bullwhip; herein we show this is not always so. We study the order-up-to (OUT) replenishment policy, with general auto-regressive moving average (ARMA(p,q)) demand processes, conditionally expected forecasting, and general lead-times. Using the eigenvalues of the demand process we study the effect of eigenvalue ordering on the bullwhip metric. The positivity of the demand impulse response determines whether the bullwhip produced is increasing in the lead-time. We illustrate our results by studying the ARMA(2,2) demand process.