Note: Optimality Conditions for an (s, S) Policy with Proportional and Lump-Sum Penalty Costs
指出Aneja和Noori关于(s, S)策略最优性的证明有误，通过将问题等价转化为典型缺货问题，给出了一个简单的凸性条件来保证该策略的最优性。
We consider the optimality of the (s, S) policy for a periodic-review stochastic inventory problem with two types of shortage costs. The problem may arise in a rush-order application at a bank branch where the emergency provision costs during a foreign currency stockout are represented by proportional and lump-sum penalties. Aneja and Noori (1987) analyzed this problem and presented a set of conditions for the convexity of a particular function and made a claim about the K-convexity of another function to prove the optimality of the (s, S) policy. We show that because the function that is claimed to be K-convex is actually concave over a subset of its domain, Aneja and Noori's arguments cannot be used to prove the optimality of the (s, S) policy. However, we argue that Aneja and Noori's problem is equivalent to the typical lost-sales problem, and using this equivalence, we .nd a simple convexity condition that assures the optimality of the (s, S) policy.