A switch system of three-stage supply chain with time delay

As the order-up-to policy in the supply chain, each stage of the supply chain has different shipment quantity to the next stage according to the actual inventory level, and the supply chain turns out to be a switch system. In this paper, the switch system of the three-stage supply chain contains a distributor, a wholesaler, the retailers and the external demand. The distributor has a maximum fixed inventory storage, and the delay time (e.g. transportation time) is considered. Then the state equation of the switch system of the three-stage supply chained is established, and all the switching rules are listed. As the variation of the switching rule, the number of switching matrixes is up to 64, it is difficult to determine the stable state of the switch system , then a numerical example is given to illustrate the stable property of the switch system.