Optimal policies of a two-echelon serial inventory system with general limited capacities

We study optimal policies of capacitated two-echelon serial inventory systems under periodic review. For a system with smaller downstream capacity, we fully characterise the optimal policy as a further modified echelon base stock policy using an intuitive backward induction. The key lies in the magnitude relation between the initial upstream stock level and the downstream capacity. For a system with smaller upstream capacity, we demonstrate that the optimal policy is of a more complex structure where there can be at most four/five target levels up to which the upstream/downstream echelon tries to produce/order. The numbers of levels and their values depend on the length of remaining horizons and the amount of initial upstream inventories. We also specify these potential target levels and then suggest a way to simplify the search of optimal solutions.