Inventory Resonances in Multi-echelon Supply Chains

We consider multi-echelon supply chains, typically consisting of retailer, distributor, manufacturer, etc. The inventory of each actor is subject to its own delay in delivery time, and each chooses its own inventory replenishment strategy. We model this interaction with a system of linear differential delay equations. When put in matrix form the system is triangular and we derive an exact solution for the homogeneous system. The solution is characterized by a set of Lambert modes, obtained from the solution to a transcendental equation. For a complete set of solutions the modes must be independent. In case the modes fail to be independent we find the additional solutions needed for a complete solution. These additional solutions exhibit a new version of the bullwhip effect for which even if the long-term behaviour of the solutions is stable there is a transient surge in inventory. We discuss the practical import of this theoretical prediction.