Discrete-Time Inventory Problems with Lead-Time and Order-Time Constraint

We consider a discrete-time inventory problem with infinite horizon, with a fixed lead time of several periods. When there is no restriction on the times of order, in particular when it is possible to make an order while some previous order is not yet delivered, then it is possible to reduce the problem to a standard one with no lead time (more precisely a lead time of one period) by considering the inventory position instead of the real inventory. This is a well-known result, which requires however the possibility of backlog. We consider here the same problem, this time imposing a constraint on the order times. It is not possible to make an order while some previous deliveries are pending. With this constraint, the problem cannot be reduced to a standard one for the inventory position.We solve the problem and show cases when the optimal policy is of (s, S) type.