H<sup>∞</sup> Control of a State Matrix Model of Closed-loop Supply Chain and its Bullwhip Effect

Abstract

A state matrix model of a closed-loop supply chain based on remanufacturing is built, which includes two types of inventories: the actual product inventory in a manufacturer and the virtual inventory used by a customer. A new method to quantify the bullwhip effect is presented and described by a l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm of the discrete-time transfer function of the closed-loop supply chain which the output is the variance of the expected cost of the manufacturer and the input is the variance of the customer's demand in downstream. It is the minimum of the up-bounded of the bullwhip effect. To reduce the bullwhip effect, applying linear matrix inequality approach, an H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> control strategy is designed in the worse case that the end customer's demand is the most fluctuation. The essence of the method is that the optimum strategy of the supply chain managements in the worse case. To prove it, a number example is given