Optimal Policies for Inventory Systems with Piecewise-Linear Concave Ordering Costs

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We study infinite-horizon stochastic inventory problems with general demand distributions and piecewise linear concave ordering costs. Such costs arise in the important cases of quantity discounts or multiple suppliers. We consider the case of concave cost involving two linear segments. This corresponds to the case of one supplier with a fixed cost, a variable cost up to a given order quantity, and a quantity discount beyond that or, equivalently, the case of two suppliers, one with a low fixed cost along with a high variable cost and the other with a high fixed cost along with a low variable cost. We show that certain three and four parameter generalizations of the classical (s,S) policy are optimal. Our contributions consist of generalizing the demand, solving a functional Bellman equation for the value function that arises in the infinite-horizon framework, and providing an explicit solution in a special case of the exponential demand. We also give conditions under which our generalizations of the (s,S) policy reduce to the standard (s,S) policy. Finally and importantly, our method is also new in the sense that we construct explicitly the value function and we do not therefore need to utilize the notion of K-convexity used in the literature of inventory problems with fixed costs.