Explicit solution of inventory problems with delivery lags

We consider a continuous-time inventory system with fixed delivery lag, subject to a demand modelled as a diffusion process with drift. Excess demand is backlogged. We prove that the optimal ordering policy is a function of the sum of the stack on hand and the stock ordered but not yet delivered. Moreover, we state a relation linking the value function when orders are pending with the value function when no order is pending. As a consequence, the (a priori) infinite dimensional Quasi-Variational inequality satisfied by the value function reduces to a finite dimensional one. The one-product inventory problem is then solved explicitly in the case of linear holding and shortage costs with fixed and proportional ordering cost. The optimal policy is determined; it is an (s, S) policy applied to the sum of stock on hand and orders pending, which means that when this sum decays below a critical level s, an order to level S is placed.