An Approximate Analytic Solution of the Inventory Balance Delay Differential Equation

Abstract

In this paper we present an analytic investigation of the continuous time representation of the inventory balance equation supplemented by an order-up-to replenishment policy. The adopted model parameters describe the startup of a distribution facility facing a constant demand. The exact solution is approximated by a complex exponential function containing integration constants that are dependent on the principal mode of the LambertWfunction. We give a detailed review of the solution strategy emphasising some underexposed components, and provide a fair discussion of its limitations. In particular we give a detailed study on the matching of the exact and the approximate analytic solution. We also derive and analyse damped non-oscillatory solutions that have been neglected in the literature. Although the model is prone to the well known permanent inventory deficit, it serves as a solid foundation for future improvements