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Annals of Operations Research

, Volume 147, Issue 1, pp 23–41 | Cite as

Two-phase method and Lagrangian relaxation to solve the Bi-Objective Set Covering Problem

  • Christian Prins
  • Caroline ProdhonEmail author
  • Roberto Wolfler Calvo
Article

Abstract

This paper deals with the Bi-Objective Set Covering Problem, which is a generalization of the well-known Set Covering Problem. The proposed approach is a two-phase heuristic method which has the particularity to be a constructive method using the primal-dual Lagrangian relaxation to solve single objective Set Covering problems. The results show that this algorithm finds several potentially supported and unsupported solutions. A comparison with an exact method (up to a medium size), shows that many Pareto-optimal solutions are retrieved and that the other solutions are well spread and close to the optimal ones. Moreover, the method developed compares favorably with the Pareto Memetic Algorithm proposed by Jaszkiewicz.

Keywords

Multi-objective combinatorial optimization Lagrangean relaxation Set covering problem 

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References

  1. Balas, E., and A. Ho. (1980). “Set Covering Using Cutting Planes, Heuristics and Subgradient Optimisation: A Computational Study.” Mathematical Programming Study, 12, 37–60.Google Scholar
  2. Beasley, J.E. (1987). “An Algorithm for Set Covering Problem.” European Journal of Operational Research 31, 85–93.CrossRefGoogle Scholar
  3. Caprara, A., M. Fischetti, and P. Toth. (1999). “A Heuristic Method for the Set Covering Problems.” OR 47, 730–743.Google Scholar
  4. Ceria, S., P. Nobili, and A. Sassano. (1998). “A Lagrangian-Based Heuristic for Large-Scale Set Covering Problems.” Mathematical Programming, 81, 215–228.Google Scholar
  5. Chvátal, V. (1979). “A Greedy Heuristic for the Set Covering Problem.” Mathematics of Operations Research 4(3), 233–235.Google Scholar
  6. Cordone, R., F. Ferrandi, D. Sciuto, and R. Wolfler Calvo. (2001). “An Efficient Heuristic Approach to Solve the Unate Covering Problem.” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 20(12), 1377–1388.CrossRefGoogle Scholar
  7. Daskin, M.S. and E.H. Stren. (1981). “A Hierarchical Objective Set Covering Model for Emergency Medical Service Vehicle Deployment.” Transportation Science, 15(2), 137–152.CrossRefGoogle Scholar
  8. Ehrgott, M. and X. Gandibleux. (2000). “A Survey and Annoted Bibliography of Multiobjective Combinatorial Optimization.” OR Spektrum, 22, 425–460.Google Scholar
  9. Gandibleux, X. Personal communication.Google Scholar
  10. Gandibleux, X., D. Vancoppenolle, and D. Tuyttens. (1998). “A First Making Use of GRASP for Solving MOCO Problems.” In Proceeding of the 14th International Conference on Multiple Criteria Decision-Making pp. 8–12, Charlottesville, USA.Google Scholar
  11. Jaszkiewicz, A. (2001). “A Comparative Study of Multiple-Objective Metaheuristics on the Bi-Objective Set Covering Problem and the Pareto Memetic Algorithm.” Working Paper RA-003/01, Institute of Computing Science, Poznań University of Technology ul.Piotrowo 3a,60-965 Poznań, Poland.Google Scholar
  12. Jaszkiewicz, A. (2003). “Do Multiple Objective Metaheuristics Deliver on their Promises? A Computational Experiment on the Set Covering Problem.” IEEE Transactions on Evolutionary Computation, 7(2), 133–143.CrossRefGoogle Scholar
  13. Liu, Y.H. (1993). “A Heuristic Algorithm for the Multi-Criteria Set-Covering Problems.” Mathematic Letter 6(3), 21–23.CrossRefGoogle Scholar
  14. MCDM society. (2001). http://www.terry.uga.edu/mcdm.
  15. Nemhauser, G.L. and L.A. Wolsey. (1988). Integer and Combinatorial Optimization. John Wiley and Sons, New York.Google Scholar
  16. Steuer, R.E., L.R. Gardiner, and J. Gray. (1996). “A Bibliographic Survey of the Activities and International Nature of Multiple Criteria Decision Making.” Journal of Multi-Criteria Decision Analysis, 5, 195–217.CrossRefGoogle Scholar
  17. Tuyttens, D., J. Teghem, PH. Fortemps, and K. Van Nieuwenhuyze. (2000). “Performance of the Mosa Method for the Bicriteria Assignment Problem.” Journal of Heuristics 6, 295–310.CrossRefGoogle Scholar
  18. Visee, M., J. Teghem, M. Pirlot, and E.L. Ulungu. (1998). “Two-Phases Method and Branch and Bound Procedures to Solve the Bi-Objective Knapsack Problem.” Journal of Global Optimization, 12, 139–155.CrossRefGoogle Scholar
  19. Zitzler, E. (1999). Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications. PhD thesis, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland.Google Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Christian Prins
    • 1
  • Caroline Prodhon
    • 1
    • 2
    Email author
  • Roberto Wolfler Calvo
    • 1
  1. 1.Institute Charles DelaunayUniversity of Technology of TroyesTroyes CedexFrance
  2. 2.ISTIT, University of Technology of TroyesTroyes CedexFrance

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