A Lower Bound of the Choquet Integral Integrated Within Martins’ Algorithm

  • Hugo Fouchal
  • Xavier Gandibleux
  • Fabien Lehuédé
Conference paper
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 648)


The problem investigated in this work concerns the integration of a decision-maker preference model within an exact algorithm in multiobjective combinatorial optimization. Rather than computing the complete set of efficient solutions and choosing a solution afterwards, our aim is to efficiently compute one solution satisfying the decision maker preferences elicited a priori. The preference model is based on the Choquet integral. The reference optimization problem is the multiobjective shortest path problem, where Martins’ algorithm is used. A lower bound of the Choquet integral is proposed that aims to prune useless partial paths at the labeling stage of the algorithm. Various procedures exploiting the proposed bound are presented and evaluated on a collection of benchmarks. Numerical experiments show significant improvements compared to the exhaustive enumeration of solutions.


Choquet integral Multiobjective optimization Shortest paths 



The authors would like to thank the regional council of Pays de la Loire (France), MILES project, for their support of this research.


  1. J. Branke, K. Deb, K. Miettinen, and R. Slowinski. Multiobjective Optimization: Interactive and Evolutionary Approaches, volume 5252 of Lecture Notes in Computer Science. Springer, 2008. 470 p.Google Scholar
  2. M. Ehrgott and X. Gandibleux. A survey and annoted bibliography of multiobjective combinatorial optimization. OR Spektrum, 22:425–460, 2000.Google Scholar
  3. G. Evans. An overview of techniques for solving multiobjective mathematical problems. Management Science, 30(11):1268–1282, 1984.CrossRefGoogle Scholar
  4. J. Figueira, S. Greco, and M. Ehrgott. Multiple Criteria Decision Analysis: State of the Art Surveys. Springer Verlag, Boston, Dordrecht, London, 2005. 1045 p.Google Scholar
  5. L. Galand and P. Perny. Search for Choquet-optimal paths under uncertainty. In 23rd conference on Uncertainty in Artificial Intelligence, pages 125–132, Vancouver, 7 2007. AAAI Press.Google Scholar
  6. L. Galand, P. Perny, and O. Spanjaard. Choquet-based optimisation in multiobjective shortest path and spanning tree problems. European Journal of Operational Research, 204(2):303–315, 2010.CrossRefGoogle Scholar
  7. L. Galand and O. Spanjaard. OWA-based search in state space graphs with multiple cost functions. In 20th International Florida Artificial Intelligence Research Society Conference, pages 86–91. AAAI Press, 2007.Google Scholar
  8. X. Gandibleux, F. Beugnies, and S. Randriamasy. Martins’ algorithm revisited for multi-objective shortest path problems with a maxmin cost function. 4OR: A Quarterly Journal of Operations Research, 4(1):47–59, 2006.Google Scholar
  9. M. Grabisch. Fuzzy integral in multicriteria decision making. Fuzzy Sets and Systems, 69:279–298, 1995.CrossRefGoogle Scholar
  10. M. Grabisch, I. Kojadinovic, and P. Meyer. A review of capacity identification methods for Choquet integral based multi-attribute utility theory – applications of the kappalab R package. European Journal of Operational Research, 186(2):766–785, 2008.CrossRefGoogle Scholar
  11. M. Grabisch and C. Labreuche. A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. Annals of Operations Research, 175:247–286, 2010.CrossRefGoogle Scholar
  12. P. Hansen. Bicriterion path problems. In G. Fandel and T. Gal, editors, Multiple Criteria Decision Making Theory and Application, volume 177 of Lecture Notes in Economics and Mathematical Systems, pages 109–127. Springer Verlag, 1979.Google Scholar
  13. P. Kouvelis and G. Yu. Robust Discrete Optimization and Its Applications, volume 14 of Nonconvex Optimization and its Applications. Kluwer Academic Publishers, Dordrecht, 1997. 356 p.Google Scholar
  14. C. Labreuche and F. Lehuédé. MYRIAD: a tool suite for MCDA. In Proceedings of EUSFLAT’05, Barcelona, September 7-9, 2005.Google Scholar
  15. F. Lehuédé, M. Grabisch, C. Labreuche, and P. Savéant. Integration and propagation of a multi-criteria decision making model in constraint programming. Journal of Heuristics, 12(4-5):329–346, September 2006.Google Scholar
  16. E.Q.V. Martins. On a multicriteria shortest path problem. European Journal of Operational Research, 16(2):236–245, 1984.CrossRefGoogle Scholar
  17. P. Perny and O. Spanjaard. A preference-based approach to spanning trees and shortest paths problems. European Journal of Operational Research, 162(3):584–601, 2005.CrossRefGoogle Scholar
  18. A. Przybylski, X. Gandibleux, and M. Ehrgott. Two phase algorithms for the bi-objective assignment problem. European Journal of Operational Research, 185(2):509–533, 2008.CrossRefGoogle Scholar
  19. S. Randriamasy, X. Gandibleux, J. Figueira, and P. Thomin. Device and a method for determining routing paths in a communication network in the presence of selection attributes. Patent 11/25/04. #20040233850. Washington, DC, USA.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Hugo Fouchal
    • 1
    • 2
  • Xavier Gandibleux
    • 1
  • Fabien Lehuédé
    • 2
  1. 1.LINAUniversité de NantesNantes Cedex-03France
  2. 2.IRCCyNÉcole des Mines de NantesNantesFrance

Personalised recommendations