Skip to main content
SpringerLink
Log in

A comparative study of the evolutionary many-objective algorithms

  • Review
  • Published:
Progress in Artificial Intelligence Aims and scope Submit manuscript

Abstract

The many-objective optimization problem (MaOP) is widespread in real life. It contains multiple conflicting objectives to be optimized. Many evolutionary many-objective (EMaO) algorithms are proposed and developed to solve it. The EMaO algorithms have received extensive attentions and in-depth studies. At the beginning of this paper, the challenges of designing EMaO algorithms are first summarized. Based on the optimization strategies, the existing EMaO algorithms are classified. Characteristics of each class of algorithms are interpreted and compared in detail. Their applicability for different types of MaOPs is discussed. Next, the numerical experiment was implemented to test the performance of typical EMaO algorithms. Their performance is analyzed from the perspectives of solution quality, convergence speed and the approximation of the Pareto front. Performance of different algorithms on different kind of test cases is analyzed, respectively. At last, the researching statuses of existing algorithms are summarized. The future researching directions of the EMaO algorithm are prospected.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Ishibuchi, H., Setoguchi, Y., Masuda, H., et al.: Performance of decomposition-based many-objective algorithms strongly depends on Pareto front shapes. IEEE Trans. Evol. Comput. 21(2), 169–190 (2017)

    Article  Google Scholar 

  2. Srinivas, S.N., Deb, K.: Muiltiobjective optimization using nondominated sorting in genetic algorithms. Evolut. Comput. 2(3), 221–248 (1994)

    Article  Google Scholar 

  3. Horn, J., Nafpliotis, N., Goldberg, D.E.: A niched Pareto genetic algorithm for multiobjective optimization. In: IEEE World Congress on Computational Intelligence, Proceedings of the First IEEE Conference on Evolutionary Computation, 1994. IEEE Xplore, vol. 1, pp. 82–87

  4. 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)

    Article  Google Scholar 

  5. Deb, K., Pratap, A., Agarwal, S., et al.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)

    Article  Google Scholar 

  6. Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: improving the strength Pareto evolutionary algorithm. In: TIK-Report 103, Computer Engineering and Networks Laboratory (TIK), Department of Electrical Engineering, ETH, Zurich (2001)

  7. Zhang, Q., Li, H.: MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans. Evol. Comput. 11(6), 712–731 (2008)

    Article  Google Scholar 

  8. Deb, K., Jain, H.: An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, Part I: solving problems with box constraints. IEEE Trans. Evolut. Comput. 18(4), 577–601 (2014)

    Article  Google Scholar 

  9. He, Z., Yen, G.G.: Visualization and performance metric in many-objective optimization. IEEE Trans. Evol. Comput. 20(3), 386–402 (2016)

    Article  Google Scholar 

  10. Ibrahim, A., Rahnamayan, S., Martin, M.V., et al.: 3D-RadVis: visualization of Pareto front in many-objective optimization. In: Evolutionary Computation. IEEE, 2016

  11. Freitas, A.R.R., Silva, R.C.P.: On the visualization of trade-offs and reducibility in many-objective optimization. In: Companion Publication of the Conference on Genetic and Evolutionary Computation. ACM, (2014)

  12. Tanigaki, Y., Narukawa, K., Nojima, Y., et al.: Preference-based NSGA-II for many-objective knapsack problems. In: International Symposium on Soft Computing and Intelligent Systems, pp. 637–642. IEEE, (2004)

  13. Mohammadi, A., Omidvar, M.N., Li, X., et al.: Integrating user preferences and decomposition methods for many-objective optimization. In: Evolutionary Computation, pp. 421–428. IEEE, (2014)

  14. Walker, D.J., Everson, R.M., Fieldsend, J.E.: Visualisation and ordering of many-objective populations. In: IEEE congress on evolutionary computation (CEC), pp. 1–8. IEEE, (2010)

  15. Ibrahim, A., Rahnamayan, S., Martin, M.V., et al.: 3D-RadVis: visualization of Pareto front in many-objective optimization. In: IEEE Congress on Evolutionary Computation, pp. 736–745. IEEE, (2016)

  16. He, Z., Yen, G.G.: An improved visualization approach in many-objective optimization. In: IEEE Congress on Evolutionary Computation, pp. 1618–1625. IEEE, (2016)

  17. Silva, R., Salimi, A., Li, M., et al.: Visualization and analysis of tradeoffs in many-objective optimization: a case study on the interior permanent magnet motor design. IEEE Trans. Magn. 52(3), 1–4 (2016)

    Article  Google Scholar 

  18. Jiang, S., Yang, S.: A strength pareto evolutionary algorithm based on reference direction for multiobjective and many-objective optimization. IEEE Trans. Evolut. Comput. 21(3), 329–346 (2017)

    Article  MathSciNet  Google Scholar 

  19. Cheng, R., Jin, Y., Olhofer, M., et al.: A reference vector guided evolutionary algorithm for many-objective optimization. In: IEEE Transactions on Evolutionary Computation, (2016)

  20. Asafuddoula, M., Ray, T., Sarker, R.: A decomposition-based evolutionary algorithm for many objective optimization. IEEE Trans. Evol. Comput. 19(3), 445–460 (2014)

    Article  MATH  Google Scholar 

  21. Liu, H.L., Gu, F., Zhang, Q.: Decomposition of a multiobjective optimization problem into a number of simple multiobjective subproblems. IEEE Trans. Evol. Comput. 18(3), 450–455 (2014)

    Article  Google Scholar 

  22. Britto, A., Mostaghim, S., Pozo, A.: Archive based multi-swarm algorithm for many-objective problems. In: 2014 Brazilian Conference on Intelligent Systems (BRACIS), pp. 79–84. IEEE, (2014)

  23. Cai, X., Li, Y., Fan, Z., et al.: An external archive guided multiobjective evolutionary algorithm based on decomposition for combinatorial optimization. IEEE Trans. Evol. Comput. 19(4), 508–523 (2015)

    Article  Google Scholar 

  24. Yuan, Y., Xu, H., Wang, B., et al.: Balancing convergence and diversity in decomposition-based many-objective optimizers. IEEE Trans. Evolut. Comput. 20(2), 180–198 (2016)

    Article  Google Scholar 

  25. Zhang, X., Tian, Y., Jin, Y.: A knee point-driven evolutionary algorithm for many-objective optimization. IEEE Trans. Evolut. Comput. 19(6), 761–776 (2014)

    Article  Google Scholar 

  26. Zou, J., Fu, L., Zheng, J., et al.: A many-objective evolutionary algorithm based on rotated grid. Appl. Soft Comput. 67, 596–609 (2018)

    Article  Google Scholar 

  27. Yang, S., Li, M., Liu, X., et al.: A grid-based evolutionary algorithm for many-objective optimization. IEEE Trans. Evolut. Comput. 17(5), 721–736 (2013)

    Article  Google Scholar 

  28. Qi, Y., Ma, X., Liu, F., et al.: Moea/d with adaptive weight adjustment. Evolut. Comput. 22(2), 231–264 (2014)

    Article  Google Scholar 

  29. Liu, H.L., Chen, L., Zhang, Q., et al.: An evolutionary many-objective optimisation algorithm with adaptive region decomposition. In: 2016 IEEE Congress on Evolutionary Computation (CEC), pp. 4763–4769. IEEE, (2016)

  30. Cai, X., Yang, Z., Fan, Z., et al.: Decomposition-based-sorting and angle-based-selection for evolutionary multiobjective and many-objective optimization. IEEE Trans. Cybern. 47(9), 2824–2837 (2017)

    Article  Google Scholar 

  31. He, Z., Yen, G.G.: Diversity improvement in decomposition-based multi-objective evolutionary algorithm for many-objective optimization problems. In: 2014 IEEE International Conference on Systems, Man and Cybernetics (SMC), pp. 2409–2414. IEEE, (2014)

  32. Zhu, C., Cai, X., Fan, Z., et al.: A two-phase many-objective evolutionary algorithm with penalty based adjustment for reference lines. In: IEEE Congress on Evolutionary Computation, pp. 2161–2168. IEEE, (2016)

  33. Chugh, T., Jin, Y., Miettinen, K., et al.: A surrogate-assisted reference vector guided evolutionary algorithm for computationally expensive many-objective optimization. IEEE Trans. Evolut. Comput. 22(1), 129–142 (2018)

    Article  Google Scholar 

  34. Zhou, C., Dai, G., Zhang, C., et al.: Entropy based evolutionary algorithm with adaptive reference points for many-objective optimization problems. Inf. Sci. 465, 232–247 (2018)

    Article  MathSciNet  Google Scholar 

  35. Wu, M., Li, K., Kwong, S., et al.: Matching-based selection with incomplete lists for decomposition multi-objective optimization. IEEE Trans. Evolut. Comput. 21, 554–568 (2017)

    Article  Google Scholar 

  36. He, Z., Yen, G.G.: Many-objective evolutionary algorithms based on coordinated selection strategy. IEEE Trans. Evolut. Comput. 21(2), 220–233 (2017)

    Article  Google Scholar 

  37. Li, K., Deb, K., Zhang, Q., et al.: An evolutionary many-objective optimization algorithm based on dominance and decomposition. IEEE Trans. Evolut. Comput. 19(5), 694–716 (2015)

    Article  Google Scholar 

  38. Ray, T., Asafuddoula, M., Isaacs, A.A.: Steady state decomposition based quantum genetic algorithm for many objective optimization. In: Evolutionary Computation, pp. 2817–2824. IEEE, (2013)

  39. Laumanns, M., Thiele, L., Deb, K., et al.: Combining convergence and diversity in evolutionary multiobjective optimization. Evolut. Comput. 10(3), 263–282 (2002)

    Article  Google Scholar 

  40. Sato, H., Aguirre, H.E., Tanaka, K.: Self-controlling dominance area of solutions in evolutionary many-objective optimization. Lect. Notes Comput. Sci. 6457(2), 455–465 (2010)

    Article  Google Scholar 

  41. Li, M., Zheng, J., Shen, R., et al.: A grid-based fitness strategy for evolutionary many-objective optimization. In: Proceedings of Genetic and Evolutionary Computation Conference, GECCO 2010, Portland, Oregon, USA, pp. 463–470. (2010)

  42. Pierro, F.D., Khu, S.T., Savic, D.A.: An investigation on preference order ranking scheme for multiobjective evolutionary optimization. IEEE Trans. Evolut. Comput. 11(1), 17–45 (2007)

    Article  Google Scholar 

  43. Farina, M., Amato, P.: A fuzzy definition of optimality for many-criteria optimization problems. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum 34(3), 315–326 (2004)

    Article  Google Scholar 

  44. He, Z., Yen, G.G., Zhang, J.: Fuzzy-based pareto optimality for many-objective evolutionary algorithms. IEEE Trans. Evolut. Comput. 18(2), 269–285 (2014)

    Article  Google Scholar 

  45. Li, M., Zheng, J., Li, K., et al.: Enhancing diversity for average ranking method in evolutionary many-objective optimization. In: Parallel Problem Solving from Nature, PPSN XI, pp. 647–656. Springer, Berlin (2010)

  46. Zou, X., Chen, Y., Liu, M., et al.: A new evolutionary algorithm for solving many-objective optimization problems. IEEE Trans. Syst. Man Cybern. Part B Cybern. Publ. IEEE Syst. Man Cybern. Soc. 38(5), 1402–1412 (2008)

    Article  Google Scholar 

  47. Kukkonen, S., Lampinen, J.: Ranking-dominance and many-objective optimization. In: IEEE Congress on Evolutionary Computation, CEC 2007, pp. 3983–3990. IEEE, (2007)

  48. Yuan, Y., Xu, H., Wang, B., et al.: A new dominance relation-based evolutionary algorithm for many-objective optimization. IEEE Trans. Evolut. Comput. 20(1), 16–37 (2016)

    Article  Google Scholar 

  49. Gong, D.W., Sun, J., Miao, Z.: A set-based genetic algorithm for interval many-objective optimization problems. IEEE Trans. Evolut. Comput. 99, 1 (2016)

    Google Scholar 

  50. Deb, K., Pratap, A., Agarwal, S., et al.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evolut. Comput. 6(2), 182–197 (2002)

    Article  Google Scholar 

  51. Deb, K., Jain, H.: Handling many-objective problems using an improved NSGA-II procedure. In: 2012 IEEE Congress on Evolutionary Computation (CEC), pp. 1–8. IEEE, (2012)

  52. Das, I., Dennis, J.E.: Normal-boundary intersection: a new method for generating the pareto surface in nonlinear multicriteria optimization problems. SIAM J. Opt. 8(3), 631–657 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  53. Seada, H., Deb, K.: U-NSGA-III: a unified evolutionary optimization procedure for single, multiple, and many objectives: proof-of-principle results. In: International Conference on Evolutionary Multi-Criterion Optimization, pp. 34–49. Springer, Cham, (2015)

  54. Li, B., Li, J., Tang, K., et al.: An improved two archive algorithm for many-objective optimization. IEEE Trans. Evolut. Comput. 19(4), 524–541 (2015)

    Article  Google Scholar 

  55. Zhao, H., Xiao, J.: A new many-objective evolutionary algorithm based on self-adaptive differential evolution. In: Ninth International Conference on Natural Computation, pp. 601–605. IEEE, (2013)

  56. Cheng, J., Yen, G.G., Zhang, G.: A many-objective evolutionary algorithm with enhanced mating and environmental selections. IEEE Trans. Evolut. Comput. 19(4), 592–605 (2015)

    Article  Google Scholar 

  57. Li, M., Yang, S., Liu, X.: Shift-based density estimation for pareto-based algorithms in many-objective optimization. IEEE Trans. Evolut. Comput. 18(3), 348–365 (2014)

    Article  Google Scholar 

  58. Elarbi, M., Bechikh, S., Gupta, A., et al.: A new decomposition-based NSGA-II for many-objective optimization. IEEE Trans. Syst. Man. Cybern. Syst. 48(7), 1191–1210 (2018)

    Article  Google Scholar 

  59. Tanigaki, Y., Narukawa, K., Nojima, Y., et al.: Preference-based NSGA-II for many-objective knapsack problems. In: 15th International Symposium on Soft Computing and Intelligent Systems (SCIS), 2014 Joint 7th International Conference on and Advanced Intelligent Systems (ISIS), pp. 637–642. IEEE, (2014)

  60. Bandyopadhyay, S., Mukherjee, A.: An Algorithm for many-objective optimization with reduced objective computations: a study in differential evolution. IEEE Trans. Evolut. Comput. 19(3), 400–413 (2015)

    Article  Google Scholar 

  61. Murata, T., Taki, A.: Examination of the performance of objective reduction using correlation-based weighted-sum for many objective knapsack problems. In: International Conference on Hybrid Intelligent Systems, pp. 175–180. IEEE, (2010)

  62. Guo, X., Wang, X., Wang, M., et al.: A new objective reduction algorithm for many-objective problems: employing mutual information and clustering algorithm. In: Eighth International Conference on Computational Intelligence and Security, pp. 11–16. IEEE, (2012)

  63. Saxena, D.K., Duro, J.A., Tiwari, A., et al.: Objective reduction in many-objective optimization: linear and nonlinear algorithms. IEEE Trans. Evolut. Comput. 17(1), 77–99 (2013)

    Article  Google Scholar 

  64. Freitas, A.R.R., Fleming, P.J., Guimaraes, F.G.: A non-parametric harmony-based objective reduction method for many-objective optimization. In: 2013 IEEE International Conference on Systems, Man, and Cybernetics (SMC), pp. 651–656, IEEE, (2013)

  65. Pal, M., Saha, S., Bandyopadhyay, S.: Clustering based online automatic objective reduction to aid many-objective optimization. In: IEEE Congress on Evolutionary Computation, pp. 1131–1138. IEEE, (2016)

  66. Li, Y., Liu, H.L., Gu, F. An objective reduction algorithm based on hyperplane approximation for many-objective optimization problems. In: 2016 IEEE Congress on Evolutionary Computation (CEC), pp. 2470–2476. IEEE, (2016)

  67. Yuan, Y., Ong, Y.S., Gupta, A., et al.: Objective reduction in many-objective optimization: evolutionary multiobjective approaches and comprehensive analysis. IEEE Trans. Evolut. Comput. 22(2), 189–210 (2018)

    Article  Google Scholar 

  68. He, Z., Yen, G.G.: Many-objective evolutionary algorithm: objective space reduction and diversity improvement. IEEE Trans. Evolut. Comput. 20(1), 145–160 (2016)

    Article  Google Scholar 

  69. Dai, G., Zhou, C., Wang, M., et al.: Indicator and reference points co-guided evolutionary algorithm for many-objective optimization problems. Knowl.-Based Syst. 140, 50–63 (2018)

    Article  Google Scholar 

  70. Bader, J., Zitzler, E.: HypE: an algorithm for fast hypervolume-based many-objective optimization. Evolut. Comput. 19(1), 45–76 (2011)

    Article  Google Scholar 

  71. Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: multiobjective selection based on dominated hypervolume. Eur. J. Oper. Res. 181(3), 1653–1669 (2007)

    Article  MATH  Google Scholar 

  72. Yuan, Y., Xu, H., Wang, B.: Evolutionary many-objective optimization using ensemble fitness ranking. In: Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation, pp. 669–676. ACM, (2014)

  73. Xiao, J., Wang, K.: Ranking-based elitist differential evolution for many-objective optimization. In: 2013 5th International Conference on Intelligent Human–Machine Systems and Cybernetics (IHMSC), vol. 1, pp. 310–313. IEEE, (2013)

  74. Li, B., Tang, K., Li, J., et al.: Stochastic ranking algorithm for many-objective optimization based on multiple indicators. IEEE Trans. Evolut. Comput. 20(6), 924–938 (2016)

    Article  Google Scholar 

  75. Li, F., Liu, J., Tan, S., et al.: R2-mopso: a multi-objective particle swarm optimizer based on r2-indicator and decomposition. In: 2015 IEEE Congress on Evolutionary Computation (CEC), pp. 3148–3155. IEEE, (2015)

  76. Gómez, R.H., Coello, C.A.C.: MOMBI: a new metaheuristic for many-objective optimization based on the R2 indicator. In: 2013 IEEE Congress on Evolutionary Computation (CEC), pp. 2488–2495. IEEE, (2013)

  77. Díaz-Manríquez, A., Toscano-Pulido, G., Coello, C.A.C., et al.: A ranking method based on the R2 indicator for many-objective optimization. In: 2013 IEEE Congress on Evolutionary Computation (CEC), pp. 1523–1530. IEEE, (2013)

  78. Deb, K., Thiele, L., Laumanns, M., et al.: Scalable test problems for evolutionary multiobjective optimization. In: Evolutionary Multiobjective Optimization, pp. 105–145, (2006)

  79. Huband, S., Hingston, P., Barone, L., et al.: A review of multiobjective test problems and a scalable test problem toolkit. Evolut. Comput. IEEE Trans. 10(5), 477–506 (2006)

    Article  Google Scholar 

  80. Zitzler, E., Künzli, S.: Indicator-based selection in multiobjective search. In: International Conference on Parallel Problem Solving from Nature, pp. 832–842. Springer, Berlin, (2004)

  81. Deb, K., Mohan, M., Mishra, S.: A fast multi-objective evolutionary algorithm for finding well-spread pareto-optimal solutions. KanGAL Rep. 2003002, 1–18 (2003)

    Google Scholar 

  82. Ishibuchi, H., Masuda, H., Tanigaki, Y., et al.: Difficulties in specifying reference points to calculate the inverted generational distance for many-objective optimization problems. In: IEEE, pp. 170–177 (2015)

  83. Knowles, J., Corne, D.: On metrics for comparing nondominated sets. In: Proceedings of the 2002 Congress on Evolutionary Computation, CEC ‘02, pp. 711–716. IEEE, (2002)

  84. Ishibuchi, H., Masuda, H., Nojima, Y.: A study on performance evaluation ability of a modified inverted generational distance indicator. In: Conference on Genetic and Evolutionary Computation, pp. 695–702. IEEE, (2015)

  85. Williamson, D.F., Parker, R.A., Kendrick, J.S.: The box plot: a simple visual method to interpret data. Ann. Intern. Med. 110(11), 916–921 (1989)

    Article  Google Scholar 

  86. Richtárik, P., Takáč, M.: Parallel coordinate descent methods for big data optimization. Math. Program. 156(1–2), 433–484 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  87. Jones, J.J.: Earnings management during import relief investigations. J. Account. Res. 29(2), 193–228 (1991)

    Article  MathSciNet  Google Scholar 

  88. Demšar, J.: Statistical comparisons of classifiers over multiple data sets. J. Mach. Learn. Res. 7, 1–30 (2006)

    MathSciNet  MATH  Google Scholar 

  89. Knowles, J., Thiele, L., Zitzler, E.: A tutorial on the performance assessment of stochastic multiobjective optimizers. Tik Rep. 214, 327–332 (2006)

    Google Scholar 

  90. García, S., Molina, D., Lozano, M., et al.: A study on the use of non-parametric tests for analyzing the evolutionary algorithms’ behaviour: a case study on the CEC’2005 special session on real parameter optimization. J. Heuristics 15(6), 617 (2009)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation Program of China (61572116, 61572117, 61502089) and the Special Fund for Fundamental Research of Central Universities of Northeastern University (N150408001, N161606003).

Author information

Authors and Affiliations

  1. School of Computer Science and Engineering, Northeastern University, Shenyang, 110819, People’s Republic of China

    Haitong Zhao, Changsheng Zhang & Bin Zhang

  2. School of Information Science and Engineering, Shenyang Ligong University, Shenyang, People’s Republic of China

    Jiaxu Ning

  3. Department of Computer Science, IOWA State University, Ames, IA, 50010, USA

    Peng Sun

  4. Sam’s Club Technology Wal-mart Inc., Bentonville, AR, 72712, USA

    Yunfei Feng

Authors
  1. Haitong Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Changsheng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jiaxu Ning
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Peng Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Yunfei Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changsheng Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to juris- 967 dictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, H., Zhang, C., Ning, J. et al. A comparative study of the evolutionary many-objective algorithms. Prog Artif Intell 8, 15–43 (2019). https://doi.org/10.1007/s13748-019-00174-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13748-019-00174-2

Keywords

Search

Navigation