Advertisement

Journal of Heuristics

, Volume 25, Issue 1, pp 107–139 | Cite as

Use of a goal-constraint-based approach for finding the region of interest in multi-objective problems

  • Ricardo LandaEmail author
  • Giomara Lárraga
  • Gregorio Toscano
Article
  • 104 Downloads

Abstract

This paper presents a hybrid approach that combines an evolutionary algorithm with a classical multi-objective optimization technique to incorporate the preferences of the decision maker into the search process. The preferences are given as a vector of goals, which represent the desirable values for each objective. The proposed approach enhances the goal-constraint technique in such a way that, instead of use the provided \(\varepsilon \) values to compute the upper bounds of the restated problem, it uses only the information of the vector of goals to generate the constraints. The bounds of the region of interest are obtained using an efficient constrained evolutionary optimization algorithm. Then, an interpolation method is placed in charge of populating such a region. It is worth noting that although goal-constraint is able to obtain the bounds of problems regardless of their number objectives, the interpolation method adopted in this paper is restricted to bi-objective problems. The proposed approach was validated using problems from the ZDT, DTLZ, and WFG benchmarks. In addition, it was compared with two well-known algorithms that use the g-dominance approach to incorporate the preferences of the decision maker. The results corroborate that the incorporation of a priori preferences into the proposed approach is useful to direct the search efforts towards the decision’s maker region of interest.

References

  1. Allmendinger, R., Li, X., Branke, J.: Reference point-based particle swarm optimization using a steady-state approach. In: Li, X., Kirley, M., Zhang, M., Green, D., Ciesielski, V., Abbass, H., Michalewicz, Z., Hendtlass, T., Deb, K., Tan, K.C., Branke, J., Shi, Y. (eds.) Simulated Evolution and Learning, 7th International Conference, SEAL 2008, pp. 200–209, Lecture Notes in Computer Science, vol. 5361. Springer, Melbourne (2008)Google Scholar
  2. Alves, M.J., Costa, J.P.: An exact method for computing the Nadir values in multiple objective linear programming. Eur. J. Oper. Res. 198(2), 637–646 (2009)MathSciNetzbMATHGoogle Scholar
  3. Becerra, R.L.: Algoritmos culturales aplicados a optimización con restricciones y optimización multiobjetivo. Master’s thesis, Electrical Engineering, CINVESTAV-IPN (2002)Google Scholar
  4. Becerra, R.L., Coello, C.A.C.: Solving hard multiobjective optimization problems using \(\varepsilon \)-constraint with cultured differential evolution. In: Runarsson, T.P., Beyer, H.-G., Burke, E., Merelo-Guervós, J.J., Whitley, L.D., Yao, X. (eds.) Parallel Problem Solving from Nature—PPSN IX, 9th International Conference, pp. 543–552. Lecture Notes in Computer Science, vol. 4193 Springer, Reykjavik, September (2006)Google Scholar
  5. Becerra, R.L., Coello, C.A.C., Hernández-Díaz, A.G., Caballero, R., Molina, J.: Alternative techniques to solve hard multi-objective optimization problems. In: Thierens, D. (ed.) 2007 Genetic and Evolutionary Computation Conference (GECCO’2007), vol. 1, pp. 757–764, ACM Press, London (2007)Google Scholar
  6. Benayoun, R., de Montgolfier, J., Tergny, J., Laritchev, O.: Linear programming with multiple objective functions: step method (stem). Math. Program. 1(1), 366–375 (1971)MathSciNetzbMATHGoogle Scholar
  7. Bechikh, S.: Incorporating decision maker’s preference information in evolutionary multi-objective optimization. Ph.D. thesis, High Institute of Management of Tunis, University of Tunis, Tunisia, (2013)Google Scholar
  8. Branke, J., Deb, K., Dierolf, H., Osswald, M.: Finding knees in multi-objective optimization. In: Yao, X., Burke, E.K., Lozano, J.A., Smith, J., Merelo-Guervós, J.J., Bullinaria, J.A., Rowe, J.E., Tiňo, P., Kabán, A., Schwefel, H.-P. (eds.) Parallel Problem Solving from Nature—PPSN VIII, pp. 722–731. Springer, Berlin (2004)Google Scholar
  9. Cagnina, L.C., Esquivel, S.C.: Solving hard multiobjective problems with a hybridized method. J. Comput. Sci. Technol. 10(3), 843–866 (2010)Google Scholar
  10. Coello, C.A.C., Pulido, G.T., Lechuga, M.S.: Handling multiple objectives with particle swarm optimization. IEEE Trans. Evol. Comput. 8(3), 256–279 (2004)Google Scholar
  11. Coello, C.A.C., Lamont, G.B., Van Veldhuizen, D.A.: Evolutionary Algorithms for Solving Multi-objective Problems, 2nd edn. Springer, New York (2007). ISBN 978-0-387-33254-3zbMATHGoogle Scholar
  12. Cohon, J.L., Marks, D.H.: A review and evaluation of multiobjective programing techniques. Water Resour. Res. 11(2), 208–220 (1975)Google Scholar
  13. Corne, D.W., Knowles, J.D., Oates, M.J.: The pareto envelope-based selection algorithm for multiobjective optimization. In: Schoenauer, M., Deb, K., Rudolph, G., Yao, X., Lutton, E., Merelo, J.J., Schwefel, H.-P. (eds.) Proceedings of the Parallel Problem Solving from Nature VI Conference, pp. 839–848, Lecture Notes in Computer Science No. 1917 . Springer, Paris (2000)Google Scholar
  14. Cvetković, D., Parmee, I.C.: Use of preferences for GA-based multi-objective optimisation. In: Proceedings of the 1st Annual Conference on Genetic and Evolutionary Computation, vol. 2, GECCO’99, pp. 1504–1509. Morgan Kaufmann Publishers Inc, San Francisco (1999)Google Scholar
  15. Cvetković, D., Parmee, I.C.: Preferences and their application in evolutionary multiobjective optimisation. IEEE Trans. Evol. Comput. 6(1), 42–57 (2002)Google Scholar
  16. Deb, K.: Solving goal programming problems using multi-objective genetic algorithms. In: Proceedings of the 1999 Congress on Evolutionary Computation—CEC99 (Cat. No. 99TH8406), vol. 1, p. 84 (1999)Google Scholar
  17. Deb, K.: Multi-objective Optimization Using Evolutionary Algorithms. Wiley, Chichester (2001). ISBN 0-471-87339-XzbMATHGoogle Scholar
  18. Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)Google Scholar
  19. Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable multi-objective optimization test problems. In: Congress on Evolutionary Computation (CEC’2002), vol. 1, pp. 825–830. IEEE Service Center , Piscataway, NJ (2002)Google Scholar
  20. Deb, K., Sundar, J.: Reference point based multi-objective optimization using evolutionary algorithms. In: Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation, GECCO ’06, pp. 635–642. ACM , New York, NY (2006)Google Scholar
  21. Deb, K., Kumar, A.: Interactive evolutionary multi-objective optimization and decision-making using reference direction method. In: Proceedings of the 9th Annual Conference on Genetic and Evolutionary Computation, GECCO ’07, pp. 781–788. ACM, New York, NY (2007)Google Scholar
  22. Deb, K., Kumar, A.: Light beam search based multi-objective optimization using evolutionary algorithms. In: 2007 IEEE Congress on Evolutionary Computation, pp. 2125–2132 (2007)Google Scholar
  23. Deb, Kalyan, Miettinen, Kaisa.: A review of nadir point estimation procedures using evolutionary approaches: a tale of dimensionality reduction. Technical report, 01 (2009)Google Scholar
  24. Dessouky, M.I., Ghiassi, M., Davis, W.J.: Estimates of the minimum nondominated criterion values in multiple-criteria decision-making. Eng. Costs Prod. Econ. 10(2), 95–104 (1986)Google Scholar
  25. Díaz-Manríquez, A., Pulido, G.T., Becerra, R.L.: A long-term memory approach for dynamic multiobjective evolutionary algorithms. In: ECTA and FCTA 2011—Proceedings of the International Conference on Evolutionary Computation Theory and Applications and the Proceedings of the International Conference on Fuzzy Computation Theory and Applications (Parts of the International Joint Conference on Computational Intelligence IJCCI 2011), Paris, France, 24–26 October, pp. 333–337 (2011)Google Scholar
  26. Ehrgott, M., Tenfelde-Podehl, D.: Computation of ideal and nadir values and implications for their use in MCDM methods. Eur. J. Oper. Res. 151(1), 119–139 (2003)MathSciNetzbMATHGoogle Scholar
  27. Fernández, E., Leyva, J.C.: A method based on multiobjective optimization for deriving a ranking from a fuzzy preference relation. Eur. J. Oper. Res. 154(1), 110–124 (2004)MathSciNetzbMATHGoogle Scholar
  28. Fliege, J., Drummond, L.M.G., Svaiter, B.F.: Newton’s method for multiobjective optimization. SIAM J. Optim. 20(2), 602–626 (2009)MathSciNetzbMATHGoogle Scholar
  29. Fonseca, C.M., Fleming, P.J.: Genetic algorithms for multiobjective optimization: formulation discussion and generalization. In: Proceedings of the 5th International Conference on Genetic Algorithms, pp. 416–423. Morgan Kaufmann Publishers Inc, San Francisco, CA (1993)Google Scholar
  30. Greenwood, G.W., Hu, X., D’Ambrosio, J.G.: Fitness functions for multiple objective optimization problems: combining preferences with pareto rankings. In: Belew, R.K., Vose, M.D. (eds.) FOGA, pp. 437–455. Morgan Kaufmann, Burlington (1996)Google Scholar
  31. Harada, K., Sakuma, J., Kobayashi, S.: Local search for multiobjective function optimization: pareto descent method. In: Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation, GECCO ’06, pp. 659–666. ACM, New York, NY (2006)Google Scholar
  32. Hillermeier, C.: Nonlinear Multiobjective Optimization: A Generalized Homotopy Approach. International Series on Numerical Mathematics, vol. 25. Birkhäuser, Basel (2001)zbMATHGoogle Scholar
  33. Holland, J.H.: Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control and Artificial Intelligence. MIT Press, Cambridge, MA (1992)Google Scholar
  34. Horn, J., Nafpliotis, N., Goldberg, D.E.: A niched pareto genetic algorithm for multiobjective optimization. In: Proceedings of the First IEEE Conference on Evolutionary Computation, IEEE World Congress on Computational Intelligence, vol. 1, pp. 82–87. IEEE Service Center, Piscataway, NJ (1994)Google Scholar
  35. Huband, S., Hingston, P., Barone, L., While, L.: A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans. Evol. Comput. 10(5), 477–506 (2006)zbMATHGoogle Scholar
  36. Jin, Y., Okabe, T., Sendho, B.: Adapting weighted aggregation for multiobjective evolution strategies. In: Zitzler, E., Thiele, L., Deb, K., Coello, C.A., Corne, D. (eds.) Evolutionary Multi-criterion Optimization, pp. 96–110. Springer, Berlin (2001)Google Scholar
  37. Jin, Y., Sendhoff, B.: Incorporation of fuzzy preferences into evolutionary multiobjective optimization. In: Proceedings of the 4th Annual Conference on Genetic and Evolutionary Computation, GECCO’02, pp. 683–683. Morgan Kaufmann Publishers Inc, San Francisco, CA (2002)Google Scholar
  38. Kennedy, J., Eberhart, R.: Particle swarm optimization. In: IEEE International Conference on Neural Networks, vol. 4, pp. 1942–1948 (1995)Google Scholar
  39. Korhonen, P., Salo, S., Steuer, R.E.: A heuristic for estimating Nadir criterion values in multiple objective linear programming. Oper. Res. 45(5), 751–757 (1997)zbMATHGoogle Scholar
  40. Landa, R., Coello, C.A.C., Toscano-Pulido, G.: Goal-constraint: incorporating preferences through an evolutionary \(\epsilon \)-constraint based method. In: 2013 IEEE Congress on Evolutionary Computation (CEC’2013), pp. 741–747, Cancún, México, 20–23, IEEE Press, ISBN 978-1-4799-0454-9 (2013)Google Scholar
  41. Lara, A., Sanchez, G., Coello, C.A.C., Schutze, O.: HCS: a new local search strategy for memetic multiobjective evolutionary algorithms. IEEE Trans. Evol. Comput. 14(1), 112–132 (2010)Google Scholar
  42. Lárraga Maldonado, G.: Incorporating preferences through an evolutionary \(\varepsilon \)-constraint based method. Master’s thesis, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional (2014)Google Scholar
  43. Martín, A., Schütze, O.: Pareto tracer: a predictor–corrector method for multi-objective optimization problems. Eng. Optim. 50(3), 516–536 (2018)MathSciNetGoogle Scholar
  44. Martin, B., Goldsztejn, A., Granvilliers, L., Jermann, C.: On continuation methods for non-linear bi-objective optimization: towards a certified interval-based approach. J. Glob. Optim. 64(1), 3–16 (2016)MathSciNetzbMATHGoogle Scholar
  45. Metev, B., Vassilev, V.: A method for nadir point estimation in MOLP problems. Cybern. Inf. Technol. 3, 1 (2003)MathSciNetGoogle Scholar
  46. Miettinen, K.: Nonlinear Multiobjective Optimization. International Series in Operations Research and Management Science. Kluwer Academics Publishers, Boston, MA (1998)Google Scholar
  47. Molina, J., Santana, L.V., Hernández-Díaz, A.G., Coello, C.A.C., Caballero, R.: g-dominance: reference point based dominance for multiobjective metaheuristics. Eur. J. Oper. Res. 197(2), 685–692 (2009)zbMATHGoogle Scholar
  48. Pareto, V.: Cours d’Economie Politique. Droz, Geneve (1896)Google Scholar
  49. Ranjithan, S.R., Chetan, S.K., Dakshima, H.K.: Constraint method-based evolutionary algorithm (CMEA) for multiobjective optimization. In: Zitzler, E., Deb, K., Thiele, L., Coello, C.A.C., Corne, D. (eds.) First International Conference on Evolutionary Multi-criterion Optimization, pp. 299–313. Lecture Notes in Computer Science No. 1993, Springer (2001)Google Scholar
  50. Rao, S.M.: Tchebycheff method-based evolutionary algorithm for multiobjective optimization. Ph.D. thesis, North Carolina State University (2003)Google Scholar
  51. Rekiek, B., de Lit, P., Delchambre, A.: Hybrid assembly line design and user’s preferences. Int. J. Prod. Res. 40(5), 1095–1111 (2002)zbMATHGoogle Scholar
  52. Rudolph, G., Schütze, O., Grimme, C., Domínguez-Medina, C., Trautmann, H.: Optimal averaged hausdorff archives for bi-objective problems: theoretical and numerical results. Comput. Optim. Appl. 64(2), 589–618 (2016)MathSciNetzbMATHGoogle Scholar
  53. Ruiz, A.B., Saborido, R., Luque, M.: A preference-based evolutionary algorithm for multiobjective optimization: the weighting achievement scalarizing function genetic algorithm. J. Glob. Optim. 62(1), 101–129 (2015)MathSciNetzbMATHGoogle Scholar
  54. Said, L.B., Bechikh, S., Ghedira, K.: The r-dominance: a new dominance relation for interactive evolutionary multicriteria decision making. IEEE Trans. Evol. Comput. 14(5), 801–818 (2010)Google Scholar
  55. Santana-Quintero, L.V., Ramírez-Santiago, N., Coello, C.A.C., Luque, J.M., Hernández-Díaz, A.G.: A new proposal for multiobjective optimization using particle swarm optimization and rough sets theory. In: Runarsson, T.P., Beyer, H.-G., Burke, E., Merelo-Guervós, J.J., Whitley, L.D., Yao, X. (eds.) Parallel Problem Solving from Nature—PPSN IX, Lecture Notes in Computer Science, vol. 4193, pp. 483–492. Springer, Berlin (2006)Google Scholar
  56. Schaffer, J.D.: Multiple objective optimization with vector evaluated genetic algorithms. In: Proceedings of the 1st International Conference on Genetic Algorithms, pp. 93–100. L. Erlbaum Associates Inc, Hillsdale, NJ (1985)Google Scholar
  57. Schütze, O., Dell’Aere, A., Dellnitz, M.: On continuation methods for the numerical treatment of multi-objective optimization problems. In: Branke, J., Deb, K., Miettinen, K., Steuer, R.E. (eds.) Practical Approaches to Multi-objective Optimization, Number 04461 in Dagstuhl Seminar Proceedings, Dagstuhl, Germany. Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany (2005)Google Scholar
  58. Schütze, O., Coello, C.A.C., Mostaghim, S., Talbi, E.-G., Dellnitz, M.: Hybridizing evolutionary strategies with continuation methods for solving multi-objective problems. Eng. Optim. 40(5), 383–402 (2008)MathSciNetGoogle Scholar
  59. Schütze, O., Esquivel, X., Lara, A., Coello, C.A.C.: Using the averaged Hausdorff distance as a performance measure in evolutionary multiobjective optimization. IEEE Trans. Evol. Comput. 16(4), 504–522 (2012)Google Scholar
  60. Schütze, O., Hernández, V.A.S., Trautmann, H., Rudolph, G.: The hypervolume based directed search method for multi-objective optimization problems. J. Heuristics 22(3), 273–300 (2016)Google Scholar
  61. Srigiriraju, K.C.: Noninferior surface tracing evolutionary algorithm (NSTEA) for multi objective optimization. Master’s thesis, North Carolina State University, Raleigh, NC (2000)Google Scholar
  62. Srinivas, N., Deb, K.: Muiltiobjective optimization using nondominated sorting in genetic algorithms. Evol. Comput. 2(3), 221–248 (1994)Google Scholar
  63. Thiele, L., Miettinen, K., Korhonen, P.J., Molina, J.: A preference-based evolutionary algorithm for multi-objective optimization. Evol. Comput. 17(3), 411–436 (2009)Google Scholar
  64. Toscano, G., Landa, R., Lárraga, G., Leguizamón, G.: On the use of stochastic ranking for parent selection in differential evolution for constrained optimization. Soft Comput. 21, 1–17 (2016)Google Scholar
  65. Van Veldhuizen, D.A.: Multiobjective evolutionary algorithms: classifications, analyses, and new innovations. Ph.D. thesis, Department of Electrical and Computer Engineering. Graduate School of Engineering. Air Force Institute of Technology, Wright-Patterson AFB, Ohio (1999)Google Scholar
  66. Wang, H.: Zigzag search for continuous multiobjective optimization. INFORMS J. Comput. 25(4), 654–665 (2013)MathSciNetGoogle Scholar
  67. Zhang, Q., Li, H.: MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans. Evol. Comput. 11(6), 712–731 (2007)Google Scholar
  68. Zhang, X., Tian, Y., Jin, Y.: A knee point-driven evolutionary algorithm for many-objective optimization. IEEE Trans. Evol. Comput. 19(6), 761–776 (2015)Google Scholar
  69. Zitzler, E., Deb, K., Thiele, L.: Comparison of multiobjective evolutionary algorithms: empirical results. Evol. Comput. 8(2), 173–195 (2000)Google Scholar
  70. Zitzler, E., Thiele, L.: Multiobjective evolutionary algorithms: a comparative case study and the strength pareto approach. IEEE Trans. Evol. Comput. 3(4), 257–271 (1999)Google Scholar
  71. Zitzler, E., Künzli, S.: Indicator-based selection in multiobjective search. In: Yao, X., Burke, E.K., Lozano, J.A., Smith, J., Merelo-Guervós, J.J., Bullinaria, J.A., Rowe, J.E., Tiňo, P., Kabán, A., Schwefel, H.-P. (eds.) Parallel Problem Solving from Nature—PPSN VIII, pp. 832–842. Springer, Berlin (2004)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Centro de Investigacion y de Estudios Avanzados del Instituto Politecnico NacionalCd. VictoriaMexico

Personalised recommendations