An (s, r, S) diffusion inventory model with exponential leadtimes and order cancellations

We consider an inventory model of diffusion type for a single item, based on a so-called (s, r, S) policy, which is a refinement of the classical (s, S) policy. The content level process behaves like a reflected Brownian motion with negative drift between jumps, at which replenishments are supplied which take the current content up to some prespecified level S. The process W starts at W(0) = S but is not bounded from above; the inventory is supposed to have infinite capacity. Whenever the content level drops to level s an order is placed to take the inventory back to level S, which the supplier will carry out after some random leadtime. However, if during a leadtime W reaches again a certain prespecified level r (due to its intrinsic fluctuations), the order is cancelled and a penalty is paid. To assess the performance of this inventory system, one needs to compute several expected total discounted cost functionals of W: set-up cost (composed of the cost of actual replenishments and those of cancellations), variable delivery cost, holding cost, penalties on lost demands. All these functionals are derived in closed form as functions of the system primitives and in particular of the decision variables S, r and s. We also give some examples of numerical optimizations based on these results.