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The fixed charge capacitated multicommodity network design problem is a well-known problem, of both practical and theoretical significance. This paper presents an efficient procedure to determine tight upper bounds on the optimal solution of realistically sized problem instances. Feasible solutions are obtained by using a tabu search framework that explores the space of the continuous flow variables by combining pivot moves with column generation, while evaluating the actual mixed integer objective. Computational experiments on a large set of randomly generated test problems show that this procedure outperforms the other available methods and is particularly suited to large problem instances with many commodities.

We present a new solution approach for the multicommodity network flow problem (MCNF) based upon both primal partitioning and decomposition techniques, which simplifies the computations required by the simplex method. The partitioning is performed on an arc-chain incidence matrix of the MCNF, similar within a change of variables to the constraint matrix of the master problem generated in a Dantzig-Wolfe decomposition, to isolate a very sparse, near-triangular working basis of greatly reduced dimension. The majority of the simplex operations performed on the partitioned basis are simply additive and network operations specialized for the nine possible pivot types identified. The columns of the arc-chain incidence matrix are generated by a dual network simplex method for updating shortest paths when link costs change.

We present local-improvement heuristics for a Service Network Design Problem encountered in the motor carrier industry. The scheduled set of vehicle departures determines the right hand side of the capacity constraints of the shipment routing subproblem which is modeled as a multicommodity network flow problem. The heuristics, one for dropping a scheduled service and another for introducing a new service, are based upon subgradients derived from the optimal dual variables of the shipment routing subproblem. The basis of the multicommodity network flow problem is partitioned to facilitate the calculation of the dual variables, reduced costs and subgradients. These are determined in large part by additive and network operations, and only in small part by matrix multiplication. The results of our computational experimentation are presented.