Combining Two Search Paradigms for Multi-objective Optimization: Two-Phase and Pareto Local Search

  • Jérémie Dubois-Lacoste
  • Manuel López-Ibáñez
  • Thomas Stützle
Part of the Studies in Computational Intelligence book series (SCI, volume 434)

Abstract

In this chapter, we review metaheuristics for solving multi-objective combinatorial optimization problems, when no information about the decision maker’s preferences is available, that is, when problems are tackled in the sense of Pareto optimization. Most of these metaheuristics follow one of the two main paradigms to tackle such problems in a heuristic way. The first paradigm is to rely on Pareto dominance when exploring the search space. The second paradigm is to tackle several single-objective problems to find several solutions that are non-dominated for the original problem; in this case, one may exploit existing, efficient single-objective algorithms, but the performance depends on the definition of the set of scalarized problems. There are also a number of approaches in the literature that combine both paradigms. However, this is usually done in a relatively ad-hoc way. In this chapter, we review two conceptually simple methods representative of each paradigm: Pareto local search and Two-phase local search. The hybridization of these two strategies provides a general framework for engineering stochastic local search algorithms that can be used to improve over the state-of-the-art for several, widely studied problems.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alsheddy, A., Tsang, E.: Guided Pareto local search and its application to the 0/1 multi-objective knapsack problems. In: Caserta, M., Voß, S. (eds.) Proceedings of MIC 2009 The 8th Metaheuristics International Conference. University of Hamburg, Hamburg (2010)Google Scholar
  2. 2.
    Andersen, K., Jörnsten, K., Lind, M.: On bicriterion minimal spanning trees: An approximation. Computers & Operations Research 23(12), 1171–1182 (1996)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Aneja, Y.P., Nair, K.P.K.: Bicriteria transportation problem. Management Science 25(1), 73–78 (1979)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Angel, E.: Approximating the Pareto curve with local search for the bicriteria TSP(1,2) problem. Theoretical Computer Science 310(1-3), 135–146 (2004)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Angus, D., Woodward, C.: Multiple objective ant colony optimization. Swarm Intelligence 3(1), 69–85 (2009)CrossRefGoogle Scholar
  6. 6.
    Applegate, D., Cook, W., Rohe, A.: Chained Lin-Kernighan for large traveling salesman problems. INFORMS Journal on Computing 15(1), 82–92 (2003)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Borges, P.C.: CHESS - changing horizon efficient set search: A simple principle for multiobjective optimization. Journal of Heuristics 6(3), 405–418 (2000)MATHCrossRefGoogle Scholar
  8. 8.
    Coello Coello, C.A., Lamont, G.B., Van Veldhuizen, D.A.: Evolutionary Algorithms for Solving Multi-Objective Problems. Springer, New York (2007)MATHGoogle Scholar
  9. 9.
    Czyzżak, P., Jaszkiewicz, A.: Pareto simulated annealing - a metaheuristic technique for multiple objective combinatorial optimization. Journal of Multi-Criteria Decision Analysis 7(1), 34–47 (1998)CrossRefGoogle Scholar
  10. 10.
    Deb, K.: Multi-Objective Optimization Using Evolutionary Algorithms. Wiley, Chichester (2001)MATHGoogle Scholar
  11. 11.
    Delorme, X., Gandibleux, X., Degoutin, F.: Evolutionary, constructive and hybrid procedures for the bi-objective set packing problem. European Journal of Operational Research 204(2), 206–217 (2010)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Drugan, M.M., Thierens, D.: Path-Guided Mutation for Stochastic Pareto Local Search Algorithms. In: Schaefer, R., Cotta, C., Kołodziej, J., Rudolph, G. (eds.) PPSN XI. LNCS, vol. 6238, pp. 485–495. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  13. 13.
    Du, J., Leung, J.Y.T.: Minimizing total tardiness on one machine is NP-hard. Mathematics of Operations Research 15(3), 483–495 (1990)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Dubois-Lacoste, J.: A study of Pareto and Two-Phase Local Search Algorithms for Biobjective Permutation Flowshop Scheduling. Master’s thesis. IRIDIA, Université Libre de Bruxelles, Belgium (2009)Google Scholar
  15. 15.
    Dubois-Lacoste, J., López-Ibáñez, M., Stützle, T.: Effective Hybrid Stochastic Local Search Algorithms for Biobjective Permutation Flowshop Scheduling. In: Blesa, M.J., Blum, C., Di Gaspero, L., Roli, A., Sampels, M., Schaerf, A. (eds.) HM 2009. LNCS, vol. 5818, pp. 100–114. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Dubois-Lacoste, J., López-Ibáñez, M., Stützle, T.: Adaptive “Anytime” Two-Phase Local Search. In: Blum, C., Battiti, R. (eds.) LION 4. LNCS, vol. 6073, pp. 52–67. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  17. 17.
    Dubois-Lacoste, J., López-Ibáñez, M., Stützle, T.: Automatic configuration of state-of-the-art multi-objective optimizers using the TP+PLS framework. In: Krasnogor, N., et al. (eds.) Proceedings of the Genetic and Evolutionary Computation Conference, GECCO 2011, pp. 2019–2026. ACM press, New York (2011)Google Scholar
  18. 18.
    Dubois-Lacoste, J., López-Ibáñez, M., Stützle, T.: A hybrid TP+PLS algorithm for bi-objective flow-shop scheduling problems. Computers & Operations Research 38(8), 1219–1236 (2011)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Dubois-Lacoste, J., López-Ibáñez, M., Stützle, T.: Improving the anytime behavior of two-phase local search. Annals of Mathematics and Artificial Intelligence 61(2), 125–154 (2011)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Dubois-Lacoste, J., López-Ibáñez, M., Stützle, T.: Pareto Local Search Algorithms for Anytime Bi-Objective Optimization. In: Hao, J.-K., Middendorf, M. (eds.) EvoCOP 2012. LNCS, vol. 7245, pp. 206–217. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  21. 21.
    Ehrgott, M.: Multicriteria optimization. Lecture Notes in Economics and Mathematical Systems, vol. 491. Springer, Berlin (2000)MATHGoogle Scholar
  22. 22.
    Ehrgott, M., Gandibleux, X.: Approximative solution methods for combinatorial multicriteria optimization. TOP 12(1), 1–88 (2004)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Ehrgott, M., Gandibleux, X.: Hybrid metaheuristics for multi-objective combinatorial optimization. In: Blum, C., Blesa, M.J., Roli, A., Sampels, M. (eds.) Hybrid Metaheuristics: An Emergent Approach for Optimization, pp. 221–259. Springer, Berlin (2008)Google Scholar
  24. 24.
    Feo, T.A., Resende, M.G.C.: Greedy randomized adaptive search procedures. Journal of Global Optimization 6, 109–113 (1995)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Gandibleux, X., Mezdaoui, N., Fréville, A.: A Tabu Search Procedure to Solve Multiobjective Combinatorial Optimization Problem. In: Caballero, R., Ruiz, F., Steuer, R. (eds.) Advances in Multiple Objective and Goal Programming. Lecture Notes in Economics and Mathematical Systems, vol. 455, pp. 291–300. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  26. 26.
    Gandibleux, X., Morita, H., Katoh, N.: Use of a Genetic Heritage for Solving the Assignment Problem with Two Objectives. In: Fonseca, C.M., Fleming, P.J., Zitzler, E., Deb, K., Thiele, L. (eds.) EMO 2003. LNCS, vol. 2632, pp. 43–57. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  27. 27.
    García-Martínez, C., Cordón, O., Herrera, F.: A taxonomy and an empirical analysis of multiple objective ant colony optimization algorithms for the bi-criteria TSP. European Journal of Operational Research 180(1), 116–148 (2007)MATHCrossRefGoogle Scholar
  28. 28.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman & Co., San Francisco (1979)MATHGoogle Scholar
  29. 29.
    Garey, M.R., Johnson, D.S., Sethi, R.: The complexity of flowshop and jobshop scheduling. Mathematics of Operations Research 1, 117–129 (1976)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Geiger, M.J.: Decision support for multi-objective flow shop scheduling by the Pareto iterated local search methodology. Computers and Industrial Engineering 61(3), 805–812 (2011)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Glover, F.: Tabu search – Part I. INFORMS Journal on Computing 1(3), 190–206 (1989)MATHCrossRefGoogle Scholar
  32. 32.
    Glover, F.: A Template for Scatter Search and Path Relinking. In: Hao, J.-K., Lutton, E., Ronald, E., Schoenauer, M., Snyers, D. (eds.) AE 1997. LNCS, vol. 1363, pp. 13–51. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  33. 33.
    Grunert da Fonseca, V., Fonseca, C.M., Hall, A.O.: Inferential performance assessment of stochastic optimisers and the attainment function. In: Zitzler, E., Deb, K., Thiele, L., Coello Coello, C.A., Corne, D.W. (eds.) EMO 2001. LNCS, vol. 1993, pp. 213–225. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  34. 34.
    Hamacher, H.W., Ruhe, G.: On spanning tree problems with multiple objectives. Annals of Operations Research 52(4), 209–230 (1994)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Hansen, M.P.: Tabu search for multiobjective optimization: MOTS. In: Climaco, J. (ed.) Proceedings of the 13th International Conference on Multiple Criteria Decision Making (MCDM 1997), pp. 574–586. Springer (1997)Google Scholar
  36. 36.
    Hansen, P., Mladenovic, N.: Variable neighborhood search: Principles and applications. European Journal of Operational Research 130(3), 449–467 (2001)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Ishibuchi, H., Murata, T.: A multi-objective genetic local search algorithm and its application to flowshop scheduling. IEEE Transactions on Systems, Man, and Cybernetics – Part C 28(3), 392–403 (1998)CrossRefGoogle Scholar
  38. 38.
    Jaszkiewicz, A.: Genetic local search for multi-objective combinatorial optimization. European Journal of Operational Research 137(1), 50–71 (2002)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Johnson, D.S.: Optimal two- and three-stage production scheduling with setup times included. Naval Research Logistics Quarterly 1, 61–68 (1954)CrossRefGoogle Scholar
  40. 40.
    Knowles, J.D., Corne, D.: The Pareto archived evolution strategy: A new baseline algorithm for multiobjective optimisation. In: Proceedings of the 1999 Congress on Evolutionary Computation (CEC 1999), pp. 98–105. IEEE Press, Piscataway (1999)Google Scholar
  41. 41.
    Laumanns, M., Thiele, L., Zitzler, E.: Running time analysis of multiobjective evolutionary algorithms on pseudo-boolean functions. IEEE Transactions on Evolutionary Computation 8(2), 170–182 (2004)CrossRefGoogle Scholar
  42. 42.
    Liefooghe, A., Humeau, J., Mesmoudi, S., Jourdan, L., Talbi, E.G.: On dominance-based multiobjective local search: design, implementation and experimental analysis on scheduling and traveling salesman problems. Journal of Heuristics 18(2), 317–352 (2011)CrossRefGoogle Scholar
  43. 43.
    Liefooghe, A., Mesmoudi, S., Humeau, J., Jourdan, L., Talbi, E.G.: A Study on Dominance-Based Local Search Approaches for Multiobjective Combinatorial Optimization. In: Stützle, T., Birattari, M., Hoos, H.H. (eds.) SLS 2009. LNCS, vol. 5752, pp. 120–124. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  44. 44.
    López-Ibáñez, M., Paquete, L., Stützle, T.: Hybrid population-based algorithms for the bi-objective quadratic assignment problem. Journal of Mathematical Modelling and Algorithms 5(1), 111–137 (2006)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    López-Ibáñez, M., Paquete, L., Stützle, T.: Exploratory analysis of stochastic local search algorithms in biobjective optimization. In: Bartz-Beielstein, T., Chiarandini, M., Paquete, L., Preuss, M. (eds.) Experimental Methods for the Analysis of Optimization Algorithms, pp. 209–222. Springer, Berlin (2010)CrossRefGoogle Scholar
  46. 46.
    López-Ibáñez, M., Stützle, T.: The automatic design of multi-objective ant colony optimization algorithms. IEEE Transactions on Evolutionary Computation (2012) (accepted)Google Scholar
  47. 47.
    Lourenço, H.R., Martin, O., Stützle, T.: Iterated local search: Framework and applications. In: Gendreau, M., Potvin, J.Y. (eds.) Handbook of Metaheuristics, 2nd edn. International Series in Operations Research & Management Science, vol. 146, ch. 9, pp. 363–397. Springer, New York (2010)CrossRefGoogle Scholar
  48. 48.
    Lust, T., Teghem, J.: The multiobjective multidimensional knapsack problem: a survey and a new approach. Arxiv preprint arXiv:1007.4063 (2010)Google Scholar
  49. 49.
    Lust, T., Teghem, J.: Two-phase Pareto local search for the biobjective traveling salesman problem. Journal of Heuristics 16(3), 475–510 (2010)MATHCrossRefGoogle Scholar
  50. 50.
    Minella, G., Ruiz, R., Ciavotta, M.: A review and evaluation of multiobjective algorithms for the flowshop scheduling problem. INFORMS Journal on Computing 20(3), 451–471 (2008)MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Paquete, L., Chiarandini, M., Stützle, T.: Pareto local optimum sets in the biobjective traveling salesman problem: An experimental study. In: Gandibleux, X., et al. (eds.) Metaheuristics for Multiobjective Optimisation. Lecture Notes in Economics and Mathematical Systems, vol. 535, pp. 177–200. Springer (2004)Google Scholar
  52. 52.
    Paquete, L., Schiavinotto, T., Stützle, T.: On local optima in multiobjective combinatorial optimization problems. Annals of Operations Research 156, 83–98 (2007)MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Paquete, L., Stützle, T.: A Two-Phase Local Search for the Biobjective Traveling Salesman Problem. In: Fonseca, C.M., Fleming, P.J., Zitzler, E., Deb, K., Thiele, L. (eds.) EMO 2003. LNCS, vol. 2632, pp. 479–493. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  54. 54.
    Paquete, L., Stützle, T.: Design and analysis of stochastic local search for the multiobjective traveling salesman problem. Computers & Operations Research 36(9), 2619–2631 (2009)MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    Parragh, S., Doerner, K.F., Hartl, R.F., Gandibleux, X.: A heuristic two-phase solution approach for the multi-objective dial-a-ride problem. Networks 54(4), 227–242 (2009)MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Ruiz, R., Stützle, T.: A simple and effective iterated greedy algorithm for the permutation flowshop scheduling problem. European Journal of Operational Research 177(3), 2033–2049 (2007)MATHCrossRefGoogle Scholar
  57. 57.
    Schaffer, J.D.: Multiple objective optimization with vector evaluated genetic algorithms. In: Grefenstette, J.J. (ed.) ICGA 1985, pp. 93–100. Lawrence Erlbaum Associates (1985)Google Scholar
  58. 58.
    Serafini, P.: Simulated annealing for multiple objective optimization problems. In: Tzeng, G.H., Yu, P.L. (eds.) Proceedings of the 10th International Conference on Multiple Criteria Decision Making (MCDM 1991), vol. 1, pp. 87–96. Springer (1992)Google Scholar
  59. 59.
    Suppapitnarm, A., Seffen, K., Parks, G., Clarkson, P.: A simulated annealing algorithm for multiobjective optimization. Engineering Optimization 33(1), 59–85 (2000)CrossRefGoogle Scholar
  60. 60.
    Ulungu, E., Teghem, J.: The two phases method: An efficient procedure to solve bi-objective combinatorial optimization problems. Foundations of Computing and Decision Sciences 20(2), 149–165 (1995)MathSciNetMATHGoogle Scholar
  61. 61.
    Ulungu, E., Teghem, J., Fortemps, P., Tuyttens, D.: MOSA method: a tool for solving multiobjective combinatorial optimization problems. Journal of Multi-Criteria Decision Analysis 8(4), 221–236 (1999)MATHCrossRefGoogle Scholar
  62. 62.
    Varadharajan, T.K., Rajendran, C.: A multi-objective simulated-annealing algorithm for scheduling in flowshops to minimize the makespan and total flowtime of jobs. European Journal of Operational Research 167(3), 772–795 (2005)MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    Voudouris, C., Tsang, E.: Guided local search and its application to the travelling salesman problem. European Journal of Operational Research 113(2), 469–499 (1999)MATHCrossRefGoogle Scholar
  64. 64.
    Zilberstein, S.: Using anytime algorithms in intelligent systems. AI Magazine 17(3), 73–83 (1996)Google Scholar
  65. 65.
    Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the strength Pareto evolutionary algorithm for multiobjective optimization. In: Giannakoglou, K., Tsahalis, D., Periaux, J., Papaliliou, K., Fogarty, T. (eds.) Evolutionary Methods for Design, Optimisation and Control, pp. 95–100. CIMNE, Barcelona (2002)Google Scholar
  66. 66.
    Zitzler, E., Thiele, L.: Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study. In: Eiben, A.E., Bäck, T., Schoenauer, M., Schwefel, H.-P. (eds.) PPSN 1998. LNCS, vol. 1498, pp. 292–301. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  67. 67.
    Zitzler, E., Thiele, L.: Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto evolutionary algorithm. IEEE Transactions on Evolutionary Computation 3(4), 257–271 (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Jérémie Dubois-Lacoste
    • 1
  • Manuel López-Ibáñez
    • 1
  • Thomas Stützle
    • 1
  1. 1.IRIDIA, CoDE, Université Libre de Bruxelles (ULB)BrusselsBelgium

Personalised recommendations