A simple proof for optimality of (s, S) policies in infinite-horizon inventory systems

The classical proofs for the existence of a stationary ( s, S ) inventory policy that minimizes the total discounted or average cost over an infinite horizon are lengthy because they depend heavily on the optimality results for corresponding finite-horizon models. This note presents a simpler alternative. Since optimal stationary ( s, S ) policies are relatively simple to characterize, it is easy to construct a solution to the optimality equation which is satisfied by an ( s, S ) policy or an equivalent variant thereof. For the discounted model, the proof characterizes an ( s, S ) policy that is optimal for all initial inventory positions. This policy can be generated by a simple existing algorithm. For the average-cost model, the optimality proof is completed with some additional arguments, which are simple but novel, to overcome the normal difficulties encountered in models with unbounded one-step expected costs.