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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.

The network design problem is a multicommodity minimal cost network flow problem with fixed costs on the arcs, i.e., a structured linear mixed-integer programming problem. It has various applications, such as construction of new links in transportation networks, topological design of computer communication networks, and planning of empty freight car transportation on railways. We present a Lagrangean heuristic within a branch-and-bound framework as a method for finding the exact optimal solution of the uncapacitated network design problem with single origins and destinations for each commodity (the simplest problem in this class, but still NP-hard). The Lagrangean heuristic uses a Lagrangean relaxation as subproblem, solving the Lagrange dual with subgradient optimization, combined with a primal heuristic (the Benders subproblem) yielding primal feasible solutions. Computational tests on problems of various sizes (up to 1000 arcs, 70 nodes and 138 commodities or 40 nodes and 600 commodities) and of several different structures lead to the conclusion that the method is quite powerful, outperforming for example a state-of-the-art mixed-integer code, both with respect to problem size and solution time.

The multicommodity network-flow model concerns routing of a number of commodities through a capacitated network at minimal cost. In the basic model, it is assumed that for each commodity, the flow can be routed on any path connecting its origin and its destination. In telecommunication applications, where a commodity represents a communication pair, there are often additional time-delay or reliability requirements on paths that are used for routing. These requirements may vary by communication pair, represented by different priority classes. In this paper, we extend the basic multicommodity network-flow model to include such side constraints on paths. The extended problem is NP-hard with the constrained shortest-path problem as a special case. To solve the extended model, we use a column-generation approach, in which the solution is built up successively by path generation. The side constraints are efficiently handled in the path-generation subproblem. We further discuss various enhancements of this approach. Computational results show that the column-generation approach provides an efficient way for solving the extended model, even for fairly large networks.

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