Inventory Management with Stochastic Lead Time
Mathematics of Operations Research , vol.
This article analyzes a continuous time back-ordered inventory system with stochastic demand and stochastic
delivery lags for placed orders. This problem in general has an innite dimensional state space and is hence
intractable. We first obtain the set of minimal conditions for reducing such a system's state space to one-
dimension and show how this reduction is done. Next, by modeling demand as a diffusion process, we
reformulate the inventory control problem as an impulse control problem. We simplify the impulse control
problem to a Quasi-Variation Inequality (QVI). Based on the QVI formulation, we obtain the optimality of
the (s, S) policy and the limiting distribution of the inventory level. We also obtain the long run average
cost of such an inventory system. Finally, we provide a method to solve the QVI formulation. Using a set
of computational experiments, we show that signicant losses are incurred in approximating a stochastic
lead time system with a fixed lead time system, thereby highlighting the need for such stochastic lead time
models. We also provide insights into the dependence of this value loss on various problem parameters.