Linear programming based decomposition methods for inventory distribution systems

We consider an inventory distribution system consisting of one warehouse and multiple retailers. The retailers face random demand and are supplied by the warehouse. The warehouse replenishes its stock from an external supplier. The objective is to minimize the total expected replenishment, holding and backlogging cost over a finite planning horizon. The problem can be formulated as a dynamic program, but this dynamic program is difficult to solve due to its high dimensional state variable. It has been observed in the earlier literature that if the warehouse is allowed to ship negative quantities to the retailers, then the problem decomposes by the locations. One way to exploit this observation is to relax the constraints that ensure the nonnegativity of the shipments to the retailers by associating Lagrange multipliers with them, which naturally raises the question of how to choose a good set of Lagrange multipliers. In this paper, we propose efficient methods that choose a good set of Lagrange multipliers by solving linear programming approximations to the inventory distribution problem. Computational experiments indicate that the inventory replenishment policies obtained by our approach can outperform several standard benchmarks by significant margins.