Tournament selection is a widely used method for parent selection in multi- and many-objective evolutionary algorithms (MOEAs). The tournament size determines the strength of selection pressure based on each algorithm’s internal criterion; larger sizes increase the likelihood of selecting highly ranked solutions, while smaller sizes promote diversity in parent selection. Despite its importance, many MOEAs adopt a fixed tournament size of two without justification, potentially limiting performance across different phases of the search. This study investigates the effects of both static and dynamic tournament sizing strategies on optimization behavior. In the static setting, several fixed values are tested; in the dynamic setting, two scheduling strategies are employed: one that gradually increases the tournament size (ITS) and one that decreases it (DTS) as the search progresses. Experimental results on DTLZ and WFG benchmark problems with three, five, and nine objectives reveal that static tournament sizes of more than two tend to show higher optimization performance than the typical size of two, although the effect varies by base algorithm. ITS generally outperforms static sizing, especially for AGE-MOEA and AGE-MOEA-II, where its benefits become more pronounced as the number of objectives increases. These findings highlight the importance of tournament sizing as a key design factor in enhancing the performance of MOEAs.

1. [1] K. Miettinen, “Nonlinear multiobjective optimization,” Springer, 1998.
2. [2] C. A. Coello Coello, G. B. Lamont, and D. A. Van Veldhuizen, “Evolutionary algorithms for solving multi-objective problems,” Springer, 2007.
3. [3] K. Deb, “Multi-objective optimization using evolutionary algorithms,” John Wiley & Sons, 2001.
4. [4] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II,” IEEE Trans. on Evolutionary Computation, Vol.6, No.2, pp. 182-197, 2002. https://doi.org/10.1109/4235.996017
5. [5] A. Brindle, “Genetic algorithms for function optimization,” Ph.D. thesis, University of Alberta, 1980. https://doi.org/10.7939/R3FB4WS2W
6. [6] S. L. Yadav and A. Sohal, “Comparative study of different selection techniques in genetic algorithm,” Int. J. of Engineering, Science and Mathematics, Vol.6, No.3, pp. 174-180, 2017.
7. [7] Y. Tian, R. Cheng, X. Zhang, Y. Su, and Y. Jin, “A strengthened dominance relation considering convergence and diversity for evolutionary many-objective optimization,” IEEE Trans. on Evolutionary Computation, Vol.23, No.2, pp. 331-345, 2019. https://doi.org/10.1109/TEVC.2018.2866854
8. [8] A. Panichella, “An adaptive evolutionary algorithm based on noneuclidean geometry for many-objective optimization,” Proc. of the Genetic and Evolutionary Computation Conf., pp. 595-603, 2019. https://doi.org/10.1145/3321707.3321839
9. [9] A. Panichella, “An improved Pareto front modeling algorithm for large-scale many-objective optimization,” Proc. of the Genetic and Evolutionary Computation Conf., pp. 565-573, 2022. https://doi.org/10.1145/3512290.3528732
10. [10] N. Beume, B. Naujoks, and M. Emmerich, “SMS-EMOA: Multiobjective selection based on dominated hypervolume,” European J. of Operational Research, Vol.181, No.3, pp. 1653-1669, 2007. https://doi.org/10.1016/j.ejor.2006.08.008
11. [11] J. Bader and E. Zitzler, “HypE: An algorithm for fast hypervolume-based many-objective optimization,” Evolutionary Computation, Vol.19, No.1, pp. 45-76, 2011. https://doi.org/10.1162/EVCO_a_00009
12. [12] Y. Sun, G. G. Yen, and Z. Yi, “IGD indicator-based evolutionary algorithm for many-objective optimization problems,” IEEE Trans. on Evolutionary Computation, Vol.23, No.2, pp. 173-187, 2019. https://doi.org/10.1109/TEVC.2018.2791283
13. [13] H. Seada and K. Deb, “A unified evolutionary optimization procedure for single, multiple, and many objectives,” IEEE Trans. on Evolutionary Computation, Vol.20, No.3, pp. 358-369, 2016. https://doi.org/10.1109/TEVC.2015.2459718
14. [14] M. Elarbi, S. Bechikh, A. Gupta, L. Ben Said, and Y.-S. Ong, “A new decomposition-based NSGA-II for many-objective optimization,” IEEE Trans. on Systems, Man, and Cybernetics: Systems, Vol.48, No.7, pp. 1191-1210, 2018. https://doi.org/10.1109/TSMC.2017.2654301
15. [15] C. Bian and C. Qian, “Better running time of the nondominated sorting genetic algorithm II (NSGA-II) by using stochastic tournament selection,” Proc. Parallel Problem Solving From Nature, pp. 428-441, 2022. https://doi.org/10.1007/978-3-031-14721-0_30
16. [16] K. Deb, L. Thiele, M. Laumanns, and E. Zitzler, “Scalable multiobjective optimization test problems,” Proc. the Congress on Evolutionary Computation, Vol.1, pp. 825-830, 2002. https://doi.org/10.1109/CEC.2002.1007032
17. [17] S. Huband, P. Hingston, L. Barone, and L. While, “A review of multiobjective test problems and a scalable test problem toolkit,” IEEE Trans. on Evolutionary Computation, Vol.10, No.5, pp. 477-506, 2006. https://doi.org/10.1109/TEVC.2005.861417
18. [18] Y. Nakanishi, M. Doi, and H. Sato, “The impact of tournament size on the performance of evolutionary multi-objective and many-objective algorithms,” Joint 13th Int. Conf. on Soft Computing and Intelligent Systems and 25th Int. Symp. on Advanced Intelligent Systems, 2024. https://doi.org/10.1109/SCISISIS61014.2024.10760202
19. [19] H. Sato and H. Ishibuchi, “Evolutionary many-objective optimization: Difficulties, approaches, and discussions,” IEEJ Trans. on Electrical and Electronic Engineering, Vol.18, No.7, pp. 1048-1058, 2023. https://doi.org/10.1002/tee.23796
20. [20] K. Ikeda, H. Kita, and S. Kobayashi, “Failure of Pareto-based MOEAs: Does non-dominated really mean near to optimal?” Proc. of the Congress on Evolutionary Computation, Vol.2, pp. 957-962, 2001. https://doi.org/10.1109/CEC.2001.934293
21. [21] Y. Yuan, H. Xu, B. Wang, and X. Yao, “A new dominance relation-based evolutionary algorithm for many-objective optimization,” IEEE Trans. on Evolutionary Computation, Vol.20, No.1, pp. 16-37, 2016. https://doi.org/10.1109/TEVC.2015.2420112
22. [22] Q. Zhang and H. Li, “MOEA/D: A multiobjective evolutionary algorithm based on decomposition,” IEEE Trans. on Evolutionary Computation, Vol.11, No.6, pp. 712-731, 2007. https://doi.org/10.1109/TEVC.2007.892759
23. [23] L. While, P. Hingston, L. Barone, and S. Huband, “A faster algorithm for calculating hypervolume,” IEEE Trans. on Evolutionary Computation, Vol.10, No.1, pp. 29-38, 2006. https://doi.org/10.1109/TEVC.2005.851275
24. [24] I. Das and J. E. Dennis, “Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems,” SIAM J. on Optimization, Vol.8, No.3, pp. 631-657, 1998. https://doi.org/10.1137/S1052623496307510
25. [25] K. Deb and H. Jain, “An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: Solving problems with box constraints,” IEEE Trans. on Evolutionary Computation, Vol.18, No.4, pp. 577-601, 2014. https://doi.org/10.1109/TEVC.2013.2281535
26. [26] A. Hussain and S. Cheema, “A new selection operator for genetic algorithms that balances between premature convergence and population diversity,” Croatian Operational Research Review, Vol.11, pp. 107-119, 2020. https://doi.org/10.17535/crorr.2020.0009
27. [27] Y. Lavinas, C. Aranha, T. Sakurai, and M. Ladeira, “Experimental analysis of the tournament size on genetic algorithms,” IEEE Int. Conf. on Systems, Man, and Cybernetics, pp. 3647-3653, 2018. https://doi.org/10.1109/SMC.2018.00617
28. [28] M. Nicolau, “Application of a simple binary genetic algorithm to a noiseless testbed benchmark,” Proc. of the 11th Annual Conf. Companion on Genetic and Evolutionary Computation Conf., pp. 2473-2478, 2009. https://doi.org/10.1145/1570256.1570346
29. [29] G. R. Harik, F. G. Lobo, and D. E. Goldberg, “The compact genetic algorithm,” IEEE Trans. on Evolutionary Computation, Vol.3, No.4, pp. 287-297, 1999. https://doi.org/10.1109/4235.797971
30. [30] N. Hansen, A. Auger, S. Finck, and R. Ros, “Real-parameter black-box optimization benchmarking 2010: Experimental setup,” INRIA, Research Report No.RR-7215, 2010. https://inria.hal.science/inria-00462481
31. [31] N. Hansen, S. Finck, R. Ros, and A. Auger, “Real-parameter black-box optimization benchmarking 2009: Noiseless functions definitions,” INRIA, Research Report No.RR-6829, 2009. https://inria.hal.science/inria-00362633v2
32. [32] E. Zitzler, K. Deb, and L. Thiele, “Comparison of multiobjective evolutionary algorithms: Empirical results,” Evol. Comput., Vol.8, No.2, pp. 173-195, 2000. https://doi.org/10.1162/106365600568202
33. [33] J. Blank and K. Deb, “Pymoo: Multi-objective optimization in python,” IEEE Access, Vol.8, pp. 89497-89509, 2020. https://doi.org/10.1109/ACCESS.2020.2990567
34. [34] H. Ishibuchi, H. Masuda, Y. Tanigaki, and Y. Nojima, “Modified distance calculation in generational distance and inverted generational distance,” Proc. Evolutionary Multi-Criterion Optimization, Vol.9019, pp. 110-125, 2015. https://doi.org/10.1007/978-3-319-15892-1_8
35. [35] J. R. Schott, “Fault tolerant design using single and multicriteria genetic algorithm optimization,” Master’s thesis, Massachusetts Institute of Technology, 1995. http://hdl.handle.net/1721.1/11582
36. [36] J. Demšar, “Statistical comparisons of classifiers over multiple data sets,” J. of Machine Learning Research, Vol.7, No.1, pp. 1-30, 2006.
37. [37] M. Li and X. Yao, “An empirical investigation of the optimality and monotonicity properties of multiobjective archiving methods,” Proc. 10th Int. Conf., Evolutionary Multi-Criterion Optimization, Vol.11411, pp. 254-266, 2019. https://doi.org/10.1007/978-3-030-12598-1_2