Supply chain aperiodicity, bullwhip and stability analysis with Jury's inners

Abstract

Jury (1974, Inners and the Stability of Dynamic Systems. New York: John Wiley) presents a novel method for the analysis of dynamical systems based on matrices of the coefficients of the systems transfer function and its ‘inners’. Here, we exploit his procedure for an analysis of a supply chain replenishment or ordering decision known as the order-up-to policy. We study the discrete-time case and generalize the classical order-up-to policy by the addition of two independent proportional controllers in the policy's feedback loops. The addition of the proportional controllers is well known to allow the order-up-to policy to eliminate the bullwhip problem and we quantify this herein using Jury's inners approach. However, care has to be taken with the use of independent controllers as they can introduce stability problems. This is because the roots of the characteristic equation become complex, and they may even move out of the unit circle in the z-plane. We identify the conditions of stability using Jury's inners approach. We also investigate further the root distribution in the characteristic equation to identify the conditions under which the order-up-to policy is aperiodic. An aperiodic system has only a limited number of maxima and minima in its dynamic response. Thus, aperiodicity is an important characteristic of a supply chain replenishment policy as it will not induce rogue seasonality.