A Lagrangian Heuristic Based Branch-and-Bound Approach for the Capacitated Network Design Problem

The capacitated network design problem is a multicommodity minimal cost network flow problem with fixed charges on the arcs and is well known to be NP-hard. The problem type is very common in the context of transportation networks, telecommunication networks, etc. In this paper we propose an efficient method for this problem, based on a Lagrangian heuristic within a branch-and-bound framework. The Lagrangian heuristic uses a Lagrangian relaxation to obtain easily solved subproblems and solves the Lagrangian dual by subgradient optimization. It also includes techniques for finding primal feasible solutions. The Lagrangian heuristic is then embedded into a branch-and-bound scheme that yields further primal improvements. Special penalty tests and cutting criteria are developed. The branch-and-bound scheme can either be an exact method that guarantees the optimal solution of the problem or be a quicker heuristic. The method has been tested on networks of various structures and sizes. Computational comparisons between this method and a state-of-the-art mixed-integer code are presented. The method is found to be capable of generating good feasible solutions to large-scale problems within reasonable time and data storage limits.