On the bullwhip and inventory variance produced by an ordering policy

The Bullwhip Problem in supply chains is first outlined. A discrete control theory model of a generic model of a replenishment rule is presented. From this model, an analytical expression for bullwhip is derived that is directly equivalent to the common statistical measure often used in simulation, statistical and empirical studies to quantify the bullwhip effect. This analytical expression clearly shows that bullwhip can be reduced by taking a fraction of the error in the inventory position and pipeline position, rather then account for all of the errors every time an ordering decision is made as is common in many scheduling systems. Furthermore, increasing the average age of the forecast reduces bullwhip, as does reducing the production lead-time. Next an analytical expression for the variance of the inventory position is derived and used together with the bullwhip expression to determine suitable ordering system designs that minimise both bullwhip and inventory variance for a range of weightings between the two variances. The relationship between our Bullwhip metric and the control theory metric Noise Bandwidth is highlighted. This contribution then derives and exploits an analytical expression for the Integral of Time * Absolute Error (ITAE) criterion often used to quantify inventory responsiveness in response to a deterministic demand. The results form a Decision Support System that can be used to design balanced supply chain ordering decisions and clearly shows how the Bullwhip Effect can be reduced, at the expense of increased inventory recovery times. Thus, this paper presents the general solution to the bullwhip problem on a sound mathematical basis.