Mathematical Methods for Advanced Problems of Inventory Control

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 costs 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. It is well understood that for a stochastic inventory control problem with a fixed cost and a per-unit variable cost, an (s, S) policy is optimal when there is only one supplier. In this work we address the case of multiple suppliers under several different scenarios. We provide a rigorous mathematical proof of the optimality of several inventory control models, which will help managers make better business decisions regarding procurement policies when facing multiple supply sources and/or quantity discounts for big purchases. Broadly, there are two main areas to explore in the realm of inventory control. The first is lost sales and the second is backlog sales. Our study examines both of these crucial areas. Our analysis is concerned with the generalization of the classical (s, S) policy for general demand distributions under a variety of modifications to the classical work of Scarf [36]. In particular, for the lost sales case, 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 the special case of exponential demand density. We also give conditions under which our generalizations of the (s, S) policy reduce to the standard (s, S) policy, even though there are two suppliers involved. Moreover, we provide an explicit solution for the three number policy when the demand distribution is exponential. In the other situation, we are concerned with stochastic inventory control problems with backlog sales during stockout. As was the case for lost sales, we consider both the scenario in which an optimal selection can be made among two suppliers, as well as the scenario in which inventory can be purchased with incremental quantity discounts from a single supplier. We study the problem for arbitrary demand distributions and in infinite horizon. In this case, we first spell out conditions that guarantee the optimization of (s, S) policy for the problem under consideration. If these conditions fail to holds, we also demonstrate that a generalized three parameter policy is optimal in two distinct situations.