Hop-constrained Minimum Spanning Tree Problems in Nonlinear Cost Network Flows: an Ant Colony Optimization Approach

In this work we address the Hop-Constrained Minimum cost Flow Spanning Tree (HMFST) problem with nonlinear costs. The HMFST problem is an extension of the Hop-Constrained Minimum Spanning Tree problem since it considers flow requirements other than unit flows. We propose a hybrid heuristic, based on Ant Colony Optimization (ACO) and on Local Search (LS), to solve this class of problems given its combinatorial nature and also that the total costs are nonlinearly flow dependent with a fixed-charge component. In order to test the performance of our algorithm we have solved a set of benchmark problems available online and compared the results obtained with the ones reported in the literature for a Multi-Population hybrid biased random key Genetic Algorithm (MPGA). Our algorithm proved to be able to find an optimum solution in more than 75% of the runs, for each problem instance solved, and was also able to improve on many results reported for the MPGA. Furthermore, for every single problem instance we were able to find a feasible solution, which was not the case for the MPGA. Regarding running times, our algorithm improves upon the computational time used by CPLEX and was always lower than that of the MPGA. This work is funded by the ERDF through the Programme COMPETE and by the Portuguese Government through FCT Foundation for Science and Technology, projects PTDC/EGE-GES/099741/2008 and PTDC/EEA-CRO/116014/2009. Marta S. R. Monteiro Faculdade de Economia and LIAAD-INESC Porto L.A., Universidade do Porto, Tel.: +35122-0426240, Rua Dr. Roberto Frias, 4200-464 Porto, Portugal, E-mail: [email protected] Dalila B. M. M. Fontes Faculdade de Economia and LIAAD-INESC Porto L.A., Universidade do Porto, Tel.: +35122-0426240, Rua Dr. Roberto Frias, 4200-464 Porto, Portugal, E-mail: [email protected] Fernando A. C. C. Fontes Faculdade de Engenharia and ISR-Porto, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal, E-mail: [email protected] 2 Marta S.R. Monteiro et al.