Approximate Dynamic Programming Methods for an Inventory Allocation Problem under Uncertainty

In this paper, we propose two approximate dynamic programming methods to optimize the distribution operations of a company manufacturing a certain product at multiple production plants and shipping it to different customer locations for sale. We begin by formulating the problem as a dynamic program. Our first approximate dynamic programming method uses a linear approximation of the value function, and computes the parameters of this approximation by using the linear programming representation of the dynamic program. Our second method relaxes the constraints that link the decisions for different production plants. Consequently, the dynamic program decomposes by the production plants. Computational experiments show that the proposed methods are computationally attractive, and in particular, the second method performs significantly better than standard benchmarks. Supply chain systems with multiple production plants provide protection against demand uncertainty and opportunities for production smoothing by allowing the demand at a particular customer location to be satisfied through different production plants. However, managing these types of supply chains requires careful planning. When planning the distribution of products to the customer locations, one has to consider many factors, such as the current inventory levels, forecasts of future production quantities and forecasts of customer demands. The decisions for different production plants interact and a decision that maximizes the immediate benefit is not necessarily the “best” decision. In this paper, we consider the distribution operations of a company manufacturing a certain product at multiple production plants and shipping it to different customer locations for sale. A certain amount of production occurs at the production plants at the beginning of each time period. Before observing the realization of the demand at the customer locations, the company decides how much product should be shipped to the customer locations and how much product should be stored at the production plants. Once a certain amount of product is shipped to a particular customer location, revenue is earned on the sales and shortage cost is incurred on the unsatisfied demand. The left over product is disposed at a salvage value. ∗School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, †[email protected].