A robust and stable analytical solution to the production and inventory control problem via a z-transform approach


A discrete linear control theory model of a generic model of a replenishment rule is presented. The replenishment rule, which we term a Deziel Eilon-Automatic Pipeline, Inventory and Order Based Production Control System (DE-APIOBPCS), is guaranteed to be stable and has some special properties relating to how it reacts to variations in the distribution of production /distribution lead-times and variable yield rates. These properties are highlighted via z-transform analysis and managerial implementations are highlighted. From our z-transform model, an analytical expression for Bullwhip is derived that is directly equivalent to the common statistical measure, the Coefficient of Variation (COV) 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 (or Work In Progress or `orders placed but not yet received') position, rather then account for all of the error every time an ordering decision is placed as is common in many scheduling decisions. Furthermore, increasing the average age of the forecast reduces Bullwhip, as does reducing the production lead-time. We then derive an analytic expression for inventory variance using the same procedure to identify the closed-form Bullwhip expression. 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, not always at the cost of increasing inventory holdings, by increasing the average age of the exponential forecast and the time to correct inventory and WIP deviations from target. Thus we present a solution to the production and inventory control problem via a z-transform approach.

Pre-prints of the 12th International Working Seminar of Production Economics, 18th-22nd February, Igls, Austria, Vol. 1, 37-37