A Duality-Based Relaxation and Decomposition Approach for Inventory Distribution Systems

We propose a new method for making the inventory replenishment decisions in distribution systems. In particular, we consider distribution systems consisting of multiple retailers that face random demand and a warehouse that supplies the retailers. The method that we propose is based on formulating the distribution problem as a dynamic program and relaxing the constraints that ensure the nonnegativity of the shipments to the retailers by associating Lagrange multipliers with them. We show that our method provides lower bounds on the value functions and a good set of values for the Lagrange multipliers can be obtained by maximizing a concave function in a relatively straightforward manner. Computational experiments indicate that our method can provide significant improvements over the traditional approaches for making the inventory replenishment decisions, in terms of both the tightness of the lower bounds on the value functions and the performance of the policies. This paper proposes a new method for making the inventory replenishment decisions in a distribution system consisting of multiple retailers and a warehouse. The retailers face random demand and the demand that cannot be met is backlogged. The warehouse supplies the retailers, and replenishes itself from an external supplier. We are interested in finding a policy to supply the retailers and to replenish the warehouse so as to minimize the total expected cost over a finite planning horizon. Two standard approaches for making the inventory replenishment decisions in distribution systems are due to Clark and Scarf (1960) and Federgruen and Zipkin (1984b). In particular, the seminal paper by Clark and Scarf (1960) introduces the balance assumption, which amounts to assuming that it is never desirable to redistribute the total amount of retailer inventory among the retailers even if it is allowed to do so. Under the balance assumption, it is possible to show that the optimal inventory replenishment policy can be found by focusing on one installation at a time. Federgruen and Zipkin (1984b) propose relaxing the requirement that the shipments to the retailers are nonnegative. Under the assumption that negative shipments to the retailers are allowed, it is also possible show that the optimal inventory replenishment policy can be found by focusing on one installation at a time. Since this assumption is a relaxation on the original problem, their relaxation strategy obtains lower bounds on the value functions. There has been much computational work showing that the inventory replenishment policies obtained under the balance assumption of Clark and Scarf (1960) and the relaxation strategy of Federgruen and Zipkin (1984b) perform quite well. Nevertheless, the comprehensive computational work by Dogru (2005), among others, indicates that there are still many practically important settings where the balance assumption and the relaxation strategy remain inadequate. We shortly list some of these settings. This paper proposes a viable alternative when such inadequacies arise. The method that we propose in this paper is based on formulating the distribution problem as a dynamic program and relaxing the constraints that ensure the nonnegativity of the shipments to the retailers. Although a similar idea is used by the relaxation strategy of Federgruen and Zipkin (1984b), the novel aspect of our method is that it explicitly associates Lagrange multipliers with the relaxed constraints, whereas Federgruen and Zipkin (1984b) simply drop the constraints without using any penalty terms. It turns out that the presence of the Lagrange multipliers significantly improves the lower bounds on the value functions and the performance of the policies. Furthermore, we show that a good set of values for the Lagrange multipliers can be obtained by maximizing a concave function in a relatively straightforward manner. Computational experiments indicate that although our method does not always provide better performance than the inventory replenishment policies obtained under the balance assumption of Clark and Scarf (1960), it can perform well when the balance assumption remains inadequate. Furthermore, the relaxation strategy of Federgruen and Zipkin (1984b) can be obtained simply by setting all of the Lagrange multipliers in our method to zero, whereas our method tightens the lower bounds on the value functions by adjusting the Lagrange multipliers over a set that includes zero. As a result, our method naturally improves the lower bounds on the value functions that are computed under the relaxation strategy of Federgruen and Zipkin (1984b). There is extensive literature on distribution systems. The balance assumption introduced by Clark and Scarf (1960) plays a crucial role in this area and it recurrently appears in many settings. Eppen and Schrage (1981) consider a distribution system consisting of identical retailers that face normally