Optimal Policies/Behaviors of a Two-Echelon Serial Inventory System with General Limited Capacities

We study the optimal policies of capacitated two-echelon serial inventory systems. For a system with smaller downstream capacity, we fully characterize the optimal policy using an intuitive backward induction. The key lies in two inductive assumptions: (a) if initially the stock level at the upstream facility is higher than the downstream capacity, then it is meaningless to have more inventories, (b) otherwise, the optimal function is separable in its independent variables. 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 that the upstream/downstream echelon tries to produce/order up to. The numbers of levels and their values depend on the length of remaining horizons and the initial amount of upstream inventories. Our analysis implicates a way to simplify the search of optimal solution.