Approximate solutions of capacitated fixed-charge minimum cost network flow problems

This article proposes two approximate methods to solve capacitated, single-commodity and fixed-charge network flow problems. First, relaxed problems obtained by the Lagrange relaxation method are shown to form a minimum spanning tree problem, yielding a lower bound to the original problem. The subgradient optimization technique can be then applied to improve this bound by updating the Lagrange multipliers. The second part of the article presents a new, simple, and efficient heuristic to find a feasible and approximate solution that is also an upper bound to the original problem.