# A MULTI-POPULATION GENETIC ALGORITHM FOR TREE-SHAPED NETWORK DESIGN PROBLEMS

### Dalila B. M. M. Fontes, José Fernando Gonçalves

#### Abstract

In this work we propose a multi-population genetic algorithm for tree-shaped network design problems using random keys. Recent literature on finding optimal spanning trees suggests the use of genetic algorithms. Furthermore, random keys encoding has been proved efficient at dealing with problems where the relative order of tasks is important. Here we propose to use random keys for encoding trees. The topology of these trees is restricted, since no path from the root vertex to any other vertex may have more than a pre-defined number of arcs. In addition, the problems under consideration also exhibit the characteristic of flows. Therefore, we want to find a minimum cost tree satisfying all demand vertices and the pre-defined number of arcs. The contributions of this paper are twofold: on one hand we address a new problem, which is an extension of the well known NP-hard hop-constrained MST problem since we also consider determining arc flows such that vertices requirements are met at minimum cost and the cost functions considered include a fixed cost component and a nonlinear flow routing component; on the other hand, we propose a new genetic algorithm to efficiently find solutions to this problem.

#### References

- Ahuja, R. and Orlin, J. (2001). Multi-exchange neighborhood structures for the capacitated minimum spanning tree problem. Mathematical Programming, 91:71-97.
- Bean, J. (1994). Genetics and random keys for sequencing and optimization. ORSA Journal on Computing, 6:154-160.
- Cormen, T. H., Leiserson, C. E., Rivest, R. L., and Stein, C. (2001). Introduction to algorithms. MIT press Cambridge, MA, 2nd edition.
- Dahl, G., Gouveia, L., and Requejo, C. (2006). On formulations and methods for the hop-constrained minimum spanning tree problem. In Pardalos, P. M. and Resende, M., editors, Handbooks of Telecommunications, pages 493-515. Springer.
- Deering, S. E., D., D. E., and Farinacci (1994). An architecture for wide-area multicast routing. Proceedings of SIGCOMM.
- Fontes, D. B. M. M. (2009). Optimal hop-constrained trees for nonlinear cost flow networks. INFOR, to appear.
- Fontes, D. B. M. M. and Gonc¸alves, J. F. (2007). Heuristic solutions for general concave minimum cost network flow problems. Networks, 50:67-76.
- Fontes, D. B. M. M., Hadjiconstantinou, E., and Christofides, N. (2003). Upper bounds for single source uncapacitated minimum concave-cost network flow problems. Networks, 41:221-228.
- Gen, M., Cheng, R., and Oren, S. (2001). Network design techniques using adapted genetic algorithms. Advances in Engineering Software, 32:731-744.
- Gen, M., Kumar, A., and Kim, R. (2005). Recent network design techniques using evolutionary algorithms. International Journal of Production and Economics, 98:251-261.
- Gonc¸alves, J. (2007). A hybrid genetic algorithmheuristic for a two-dimensional orthogonal packing problem. European Journal of Operational Research, 183:1212-1229.
- Gonc¸alves, J. and Almeida, J. (2002). A hybrid genetic algorithm for assembly line balancing. Journal of Heuristics, 8:629-642.
- Gonc¸alves, J., Mendes, J., and Resende, M. (2005). A hybrid genetic algorithm for the job shop scheduling problem. European Journal of Operational Research, 167:77-95.
- Gonc¸alves, J. and Resende, M. (2004). An evolutionary algorithm for manufacturing cell formation. Computers and Industrial Engineering, 47:247-273.
- Gouveia, L., Paias, A., and Sharma, D. (2008). Modeling and solving the rooted distance-constrained minimum spanning tree problem. Computers & Operations Research, 35:600-613.
- Gouveia, L. and Requejo, C. (2001). A new lagrangean relaxation approach for the hop-constrained minimum spanning tree problem. European Journal of Operational Research, 132:539-552.
- Han, L., Wang, Y., and Guo, F. (MAY 2005). A new genetic algorithm for the degree-constrained minimum spanning tree problem. IEEE International Workshop on VlSI Design and Video Technology, pages 125-128.
- Lacerda, E. and Medeiros, M. (2006). A genetic algorithm for the capacitated minimum spanning tree problem. IEEE Congress on Evolutionary Computation, 1-6:725-729.
- LeBlanc, L. and Reddoch, R. (1990). Reliable link topology/capacity design and routing in backbone telecommunication networks. First ORSA telecommunications SIG conference.
- Raidl, G. and Julstrom, B. (2003). Edge sets: An effective evolutionary coding of spanning trees. IEEE Transactions on Evolutionary Computation, 7:225-239.
- Thompson, E., Paulden, T., and Smith, D. (2007). The dandelion code: A new coding of spanning trees for genetic algorithms. IEEE Transactions on Evolutionary Computation, 11:91-100.
- Woolston, K. and Albin, S. (1988). Design of centralized networks with reliability and availability constraints. Computers & Operations Research, 15:207-217.
- Zeng, Y. and Wang, Y. (2003). A new genetic algorithm with local search method for degree-constrained minimum spanning tree problems. Proceedings of ICCIMA, pages 218-222.

#### Paper Citation

#### in Harvard Style

B. M. M. Fontes D. and Fernando Gonçalves J. (2009). **A MULTI-POPULATION GENETIC ALGORITHM FOR TREE-SHAPED NETWORK DESIGN PROBLEMS** . In *Proceedings of the International Joint Conference on Computational Intelligence - Volume 1: ICEC, (IJCCI 2009)* ISBN 978-989-674-014-6, pages 177-182. DOI: 10.5220/0002281101770182

#### in Bibtex Style

@conference{icec09,

author={Dalila B. M. M. Fontes and José Fernando Gonçalves},

title={A MULTI-POPULATION GENETIC ALGORITHM FOR TREE-SHAPED NETWORK DESIGN PROBLEMS},

booktitle={Proceedings of the International Joint Conference on Computational Intelligence - Volume 1: ICEC, (IJCCI 2009)},

year={2009},

pages={177-182},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0002281101770182},

isbn={978-989-674-014-6},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Joint Conference on Computational Intelligence - Volume 1: ICEC, (IJCCI 2009)

TI - A MULTI-POPULATION GENETIC ALGORITHM FOR TREE-SHAPED NETWORK DESIGN PROBLEMS

SN - 978-989-674-014-6

AU - B. M. M. Fontes D.

AU - Fernando Gonçalves J.

PY - 2009

SP - 177

EP - 182

DO - 10.5220/0002281101770182