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

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 l₂ 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^∞ 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