The fixed charge transportation problem: a strong formulation based on Lagrangian decomposition and column generation

A new and strong convexified formulation of the fixed charge transportation problem is provided. This formulation is obtained by integrating the concepts of Lagrangian decomposition and column generation. The decomposition is made by splitting the shipping variables into supply and demand side copies, while the columns correspond to extreme flow patterns for single sources or sinks. It is shown that the combination of Lagrangian decomposition and column generation provides a formulation whose strength dominates those of three other convexified formulations of the problem. Numerical results illustrate that our integrated approach has the ability to provide strong lower bounds. The Lagrangian decomposition yields a dual problem with an unbounded set of optimal solutions. We propose a regularized column generation scheme which prioritizes an optimal dual solution with a small-norm. We further demonstrate numerically that information gained from the strong formulation can also be used for constructing a small-sized core problem which yields high-quality upper bounds.