Dynamic supply chain design: square root law for Bullwhip

We present a methodology to dynamically design supply chain networks taking into account both inventory and capacity (bullwhip) related costs. We assume i.i.d. demand and the “order-up-to” policy is used to place replenishment orders with a lead-time of one period. Our methodology is complete, analytical and exact. We illustrate its application via a “toy” numerical example that seems to suggest that a “square root law for bullwhip costs” exists. We provide a sketch of a proof that shows this is indeed the case.