Two- and three-index formulations of the minimum cost multicommodity k-splittable flow problem

The Multicommodity Flow Problem (MCFP) considers the efficient routing of commodities from their origins to their destinations, subject to capacity restrictions and edge costs. This paper studies the NP-hard Minimum Cost Multicommodity k-splittable Flow Problem in which each commodity may use at most k paths between its origin and its destination. The problem has applications in transportation problems where a number of commodities must be routed, using a limited number of transportation units at each destination. Based on a three-index formulation from (Truffot et al., 2005) we present a new two-index formulation for the problem, and solve both formulations through branch-and-price. The three-index algorithm by Truffot et al. is improved by introducing a heuristic method to reach a feasible solution by eliminating some symmetry. A novel branching strategy for the two-index formulation is presented, forbidding subpaths in the branching children. The proposed heuristic for the three-index algorithm improves the performance, but the three-index algorithm is still outperformed by the two-index algorithm, both with respect to running time and to the number of solved test instances.