Characterization of the Bullwhip Effect in Linear, Time-Invariant Supply Chains: Some Formulae and Tests

The authors analyze the bullwhip effect in multistage, decentralized supply chains operated with linear and time-invariant inventory management policies; the focus is on robustness. The supply chain is modeled as a single-input, single-output control system driven by arbitrary customer demands. The authors derive robust analytical conditions to predict the presence of the bullwhip effect and bound its magnitude, based only on the way inventories are managed. These results hold independently of the customer demand. The authors also characterize the stream of orders placed at any stage of the chain when the customer demand process is known and ergodic and give an exact formula for the variance of the orders placed. This formula generalizes existing results by broadening the class of inventory replenishment policies and customer demand processes to which it applies. The authors also show that the bullwhip effect can be mitigated by introducing commitments for future orders into the ordering policies.