On the square root law for bullwhip: The case of arbitrary lead-times and AR(1) demand

Abstract

We develop a model of a divergent distribution network with one distribution centre and n retailers. Each of the n retailers faces a stochastic first-order autoregressive demand. Furthermore, each retailer uses the Order-Up-To (OUT) policy with minimum mean squared error forecasting to generate replenishment orders that are placed onto the distribution centre (DC). The DC operates a base stock replenishment policy. The base stock policy is an OUT policy with time -invariant forecasts. We assume that there are piece-wise linear and convex inventory holding and backlog costs. We are able to show that the “Square Root Law for Inventories” due to Maister (1977) also holds when the OUT replenishment policy is present. Recall, that Maister’s Square Root Law was derived for the Economic Lot Size model. We assume that piece-wise linear and convex capacity costs are present. These costs are associated with overtime working (above an optimised production capacity) and the cost of un-used, or lost, capacity. These capacity costs have been added to capture the opportunity costs associated with the bullwhip effect. Under such a costing scheme we are also able to show that a “Square Root Law for Bullwhip” also exists when network consolidation occurs. We also consider the impact of different lead-times at each retailer and the distribution centre, in both the decentralised and centralised distribution network. Our analytical results that described the discrete time system behaviour are verified via a spreadsheet simulation model.