Controller design and reduction of bullwhip for a model supply chain system using z-transform analysis

In this work, a discrete time series model of a supply chain system is derived using material balances and information flow. Transfer functions for each unit in the supply chain are obtained by z-transform. The entire chain can be modeled by combining these transfer functions into a close loop transfer function for the network. The model proves to be very useful in revealing the dynamics characteristic of the system. The system can be viewed as a linear discrete system with lead time and operating constraints. The stability of the system can be analyzed using the characteristic equation. Controllers are designed using frequency analysis. The bullwhip effect, i.e. magnification of amplitudes of demand perturbations from the tail to upstream levels of the supply chain, is a very important phenomenon for supply chain systems. We proved that intuitive operation of a supply chain system with demand forecasting will cause bullwhip. Moreover, lead time alone would not cause bullwhip. It does so only when accompanied by demand forecasting. Furthermore, we show that by implementing a proportional intergral or a cascade inventory position control and properly synthesizing the controller parameters, we can effectively suppress the bullwhip effect. Moreover, the cascade control structure is superior in meeting customer demand due to its better tracking of long term trends of customer demand.