Variance amplification: The equivalence of discrete and continuous time analyses
Abstract
The “Bullwhip Effect” is the well-known phenomenon where the variance of the order rates is amplified as the orders proceed through the supply chain. This effect is a severe problem in many industries. We solve the differential equations for an inventory balancing and ordering policy to obtain the continuous time domain evolution of the free response of the system. Using the free response, we calculate the bullwhip effect and inventory variation produced by the continuous time representation of the replenishment policy. We then compare our continuous time results to known results for the discrete time representation of the system. Studying the same system in the discrete time domain shows the solution to be managerially equivalent to the continuous case, since the answers are similar to each other. Whilst the discrete and continuous analyses give slightly different expressions for bullwhip and inventory variance, they have the same structure. Therefore, in practical situations, either domain can be used to study a supply chain.