Framework for Many-Objective Test Problems with Both Simple and Complicated Pareto-Set Shapes

  • Dhish Kumar Saxena
  • Qingfu Zhang
  • João A. Duro
  • Ashutosh Tiwari
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6576)


Test problems have played a fundamental role in understanding the strengths and weaknesses of the existing Evolutionary Multi-objective Optimization (EMO) algorithms. A range of test problems exist which have enabled the research community to understand how the performance of EMO algorithms is affected by the geometrical shape of the Pareto front (PF), i.e., PF being convex, concave or mixed. However, the shapes of the Pareto Set (PS) of most of these test problems are rather simple (linear or quadratic), even though the real-world engineering problems are expected to have complicated PS shapes. The state-of-the-art in many-objective optimization problems (those involving four or more objectives) is rather worse. There is a dearth of test problems (even those with simple PS shapes) and the algorithms that can handle such problems. This paper proposes a framework for continuous many-objective test problems with arbitrarily prescribed PS shapes. The behavior of two popular EMO algorithms namely NSGAII and MOEA/D has also been studied for a sample of the proposed test problems. It is hoped that this paper will promote an integrated investigation of EMO algorithms for their scalability with objectives and their ability to handle complicated PS shapes with varying nature of the PF.


Evolutionary Many-objective Optimization Pareto-set shapes 


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  1. 1.
    Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer, Boston (1999)zbMATHGoogle Scholar
  2. 2.
    Kalyanmoy, D.: Multi-Objective Optimization Using Evolutionary Algorithms. John Wiley & Sons, Inc., New York (2001)zbMATHGoogle Scholar
  3. 3.
    Deb, K.: Multi-objective genetic algorithms: Problem difficulties and construction of test problems. Evolutionary Computation 7, 205–230 (1999)CrossRefGoogle Scholar
  4. 4.
    Zitzler, E., Deb, K., Thiele, L.: Comparison of multiobjective evolutionary algorithms: Empirical results. Evolutionary Computation 8, 173–195 (2000)CrossRefGoogle Scholar
  5. 5.
    Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable Test Problems for Evolutionary Multi-Objective Optimization. In: Abraham, A., Jain, R., Goldberg, R. (eds.) Evolutionary Multiobjective Optimization: Theoretical Advances and Applications, pp. 105–145. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Kasprzak, E., Lewis, K.: An approach to facilitate decision trade-offs in pareto solution sets. Journal of Engineering Valuation and Cost Analysis 3, 173–187 (2000)Google Scholar
  7. 7.
    Hillermeier, C.: Nonlinear Multiobjective Optimization: A Generalized Homotopy Approach. Birkhäuser-Verlag, Basel (2000) ISBN 978-3764364984zbMATHGoogle Scholar
  8. 8.
    Li, H., Zhang, Q.: Multiobjective Optimization Problems With Complicated Pareto Sets, MOEA/D and NSGA-II. IEEE Transactions on Evolutionary Computation 13, 284–302 (2009)CrossRefGoogle Scholar
  9. 9.
    Okabe, T., Jin, Y., Olhofer, M., Sendhoff, B.: On test functions for evolutionary 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.) PPSN 2004. LNCS, vol. 3242, pp. 792–802. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. 10.
    Huband, S., Hingston, P., Barone, L., While, L.: A review of multiobjective test problems and a scalable test problem toolkit. IEEE Transactions on Evolutionary Computation 10, 477–506 (2006)CrossRefzbMATHGoogle Scholar
  11. 11.
    Deb, K., Sinha, A., Kukkonen, S.: Multi-objective test problems, linkages, and evolutionary methodologies. In: Genetic and Evolutionary Computation Conference (GECCO), pp. 1141–1148 (2006)Google Scholar
  12. 12.
    Li, H., Zhang, Q.: A multiobjective differential evolution based on decomposition for multiobjective optimization with variable linkages. In: Runarsson, T.P., Beyer, H.-G., Burke, E.K., Merelo-Guervós, J.J., Whitley, L.D., Yao, X. (eds.) PPSN 2006. LNCS, vol. 4193, pp. 583–592. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Zhang, Q., Zhou, A., Jin, Y.: Rm-meda: A regularity model-based multiobjective estimation of distribution algorithms. IEEE Transactions on Evolutionary Computation 21, 41–63 (2008)CrossRefGoogle Scholar
  14. 14.
    Ishibuchi, H., Hitotsuyanagi, Y., Tsukamoto, N., Nojima, Y.: Many-objective test problems to visually examine the behavior of multiobjective evolution in a decision space. In: Schaefer, R., Cotta, C., Kołodziej, J., Rudolph, G. (eds.) PPSN XI. LNCS, vol. 6239, pp. 91–100. Springer, Heidelberg (2010)Google Scholar
  15. 15.
    Deb, K., Agrawal, S., Pratap, A., Meyarivan, T.: A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6, 182–197 (2002)CrossRefGoogle Scholar
  16. 16.
    Zhang, Q., Li, H.: MOEA/D: A multi-objective evolutionary algorithm based on decomposition. IEEE Transactions on Evolutionary Computation 11, 712–731 (2007)CrossRefGoogle Scholar
  17. 17.
    Hughes, E.: Evolutionary many-objective optimisation: many once or one many? In: IEEE Congress on Evolutionary Computation, vol. 1, pp. 222–227. IEEE, Los Alamitos (2005)Google Scholar
  18. 18.
    Knowles, J., Corne, D.W.: Quantifying the effects of objective space dimension in evolutionary multiobjective optimization. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds.) EMO 2007. LNCS, vol. 4403, pp. 757–771. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  19. 19.
    Purshouse, R.C., Fleming, P.J.: On the evolutionary optimization of many conflicting objectives. IEEE Transactions on Evolutionary Computation 11, 770–784 (2007)CrossRefGoogle Scholar
  20. 20.
    Purshouse, R.C., Fleming, P.J.: Evolutionary many-objective optimization: An exploratory analysis. In: IEEE Congress on Evolutionary Computation, pp. 2066–2073 (2003)Google Scholar
  21. 21.
    Ishibuchi, H., Tsukamoto, N., Nojima, Y.: Evolutionary many-objective optimization: A short review. In: IEEE Congress on Evolutionary Computation, pp. 2424–2431 (2008)Google Scholar
  22. 22.
    Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Dordrecht (1999) ISBN 0-7923-8278-1zbMATHGoogle Scholar
  23. 23.
    Das, I., Dennis, J.E.: Normal-boundary intersection: A new method for generating the pareto surface in nonlinear multicriteria optimization problems. SIAM J. on Optimization 8, 631–657 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Veldhuizen, D.A.V., Lamont, G.B.: Multiobjective evolutionary algorithm research: A history and analysis. Technical Report TR-98-03, 1998, Dept. Elec. Comput. Eng., Graduate School of Eng., Air Force Inst. Technol., Wright-Patterson, AFB, OH (1998)Google Scholar
  25. 25.
    Bandyopadhyay, S., Saha, S., Maulik, U., Deb, K.: A simulated annealing-based multiobjective optimization algorithm: Amosa. IEEE Transactions on Evolutionary Computation 12, 269–283 (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Dhish Kumar Saxena
    • 1
  • Qingfu Zhang
    • 2
  • João A. Duro
    • 1
  • Ashutosh Tiwari
    • 1
  1. 1.Manufacturing DepartmentCranfield UniversityBedforshireUK
  2. 2.Department of Computing and Electronic SystemsUniversity of EssexColchesterUK

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