Dantzig–Wolfe Decomposition for the Facility Location and Production Planning Problem

Abstract There are several mathematical models proposed for the facility location and production planning problem in the literature. However, some of these models disregard what products each customer has ordered and neglect critical production-related constraints and setup decisions while some others do not well define the connection cost between customers and facilities. In this study, we propose two mathematical models to overcome the disadvantages aforementioned, along with their reformulations by item decomposition to improve lower bounds. We demonstrate that the pricing subproblems of the item decomposition are related to uncapacitated lot-sizing problems with the Wagner–Whitin property. This property is employed to enhance the performance of column generation for the item decomposition. Our computational results show that this item decomposition method can improve lower bounds over other classical lower bounding techniques, such as linear programming relaxation and model reformulation. Additionally, we implement the proposed item decomposition method to other benchmark problems in the literature and observe that our proposed method can improve the benchmark solutions with a statistical significance.