Skip to main content

Advertisement

Springer Nature Link
Log in

Many-objective evolutionary algorithm based on dynamic mating and strengthened fitness selection mechanism

  • Published:
The Journal of Supercomputing Aims and scope Submit manuscript

Abstract

Many-objective evolutionary algorithms have demonstrated their superiority in dealing with many-objective optimization problems. However, their performance in handling many objective optimization problems can be significantly affected due to the sensitivity on the curvature of Pareto front. This paper proposes the dynamic mating selection strategy and strengthened fitness selection mechanism to solve many objective optimization problems (MaOEA-DMSF). In MaOEA-DMSF, a dynamic mating selection strategy is proposed to select appropriate mating population, which can generate high-quality offspring with a higher probability. In addition, a strengthened fitness selection mechanism is proposed to improve convergence without deterioration in population diversity. To verify the effectiveness of the proposed MaOEA-DMSF, a series of experiments are carried out against eight state-of-the-art many-objective optimization algorithms on three widely used benchmark test suites. Experimental results demonstrate that the proposed MaOEA-DMSF has higher competitiveness compared with peer competitors.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Data availability

No datasets were generated or analysed during the current study.

References

  1. He Z, Yen GG (2016) Many-objective evolutionary algorithms based on coordinated selection strategy. IEEE Trans Evol Comput 21(2):220–233. https://doi.org/10.1109/TEVC.2016.2598687

    Article  MATH  Google Scholar 

  2. Deng Q, Kang Q, Zhou M, Wang X, Albeshri A (2024) Activation function-assisted objective space mapping to enhance evolutionary algorithms for large-scale many-objective optimization. IEEE Trans Syst Man Cybern Syst. https://doi.org/10.1109/TSMC.2024.3454051

    Article  MATH  Google Scholar 

  3. Cui Z, Qu C, Zhang Z, Jin Y, Cai J, Zhang W, Chen J (2024) An adaptive interval many-objective evolutionary algorithm with information entropy dominance. Swarm Evol Comput 91:101749. https://doi.org/10.1016/j.swevo.2024.101749

    Article  MATH  Google Scholar 

  4. Yang Y, Zhang C, Liu Y, Ning J, Guo Y (2024) Deep reinforcement learning assisted novelty search in Voronoi regions for constrained multi-objective optimization. Swarm Evol Comput 91:101732. https://doi.org/10.1016/j.swevo.2024.101732

    Article  MATH  Google Scholar 

  5. Deb K, do Val Lopes CL, Martins FVC, Wanner EF (2023) Identifying pareto fronts reliably using a multistage reference-vector-based framework. IEEE Trans Evol Comput 28(1):252–266. https://doi.org/10.1109/TEVC.2023.3246922

    Article  Google Scholar 

  6. Suresh A, Deb K (2024) Machine learning based prediction of new Pareto-optimal solutions from pseudo-weights. IEEE Trans Evol Comput 28(5):1351–1365. https://doi.org/10.1109/TEVC.2023.3319494

    Article  MATH  Google Scholar 

  7. Liu Y, Zhu N, Li M (2020) Solving many-objective optimization problems by a Pareto-based evolutionary algorithm with preprocessing and a penalty mechanism. IEEE Trans Cybern 51(11):5585–5594. https://doi.org/10.1109/TCYB.2020.2988896

    Article  MATH  Google Scholar 

  8. Trivedi A, Srinivasan D, Sanyal K, Ghosh A (2016) A survey of multiobjective evolutionary algorithms based on decomposition. IEEE Trans Evol Comput 21(3):440–462. https://doi.org/10.1109/TEVC.2016.2608507

    Article  MATH  Google Scholar 

  9. Li K, Deb K, Zhang Q, Kwong S (2014) An evolutionary many-objective optimization algorithm based on dominance and decomposition. IEEE Trans Evol Comput 19(5):694–716. https://doi.org/10.1109/TEVC.2014.2373386

    Article  MATH  Google Scholar 

  10. Li L, Yen GG, Sahoo A, Chang L, Gu T (2021) On the estimation of pareto front and dimensional similarity in many-objective evolutionary algorithm. Inf Sci 563:375–400. https://doi.org/10.1016/j.ins.2021.03.008

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu Y, Ishibuchi H, Masuyama N, Nojima Y (2019) Adapting reference vectors and scalarizing functions by growing neural gas to handle irregular Pareto fronts. IEEE Trans Evol Comput 24(3):439–453. https://doi.org/10.1109/TEVC.2019.2926151

    Article  MATH  Google Scholar 

  12. Xiang Y, Zhou Y, Yang X, Huang H (2019) A many-objective evolutionary algorithm with Pareto-adaptive reference points. IEEE Transa Evol Comput 24(1):99–113. https://doi.org/10.1109/tevc.2019.2909636

    Article  MATH  Google Scholar 

  13. Ishibuchi H, Setoguchi Y, Masuda H, Nojima Y (2016) Performance of decomposition-based many-objective algorithms strongly depends on Pareto front shapes. IEEE Trans Evol Comput 21(2):169–190. https://doi.org/10.1109/TEVC.2016.2587749

    Article  MATH  Google Scholar 

  14. Jain H, Deb K (2013) An evolutionary many-objective optimization algorithm using reference-point based nondominated sorting approach, part II: handling constraints and extending to an adaptive approach. IEEE Trans Evol Comput 18(4):602–622. https://doi.org/10.1109/TEVC.2013.2281535

    Article  MATH  Google Scholar 

  15. He Z, Yen GG, Zhang J (2013) Fuzzy-based Pareto optimality for many-objective evolutionary algorithms. IEEE Trans Evol Comput 18(2):269–285. https://doi.org/10.1109/TEVC.2013.2258025

    Article  MATH  Google Scholar 

  16. Deb K, Mohan M, Mishra S (2005) Evaluating the ϵ-domination based multi-objective evolutionary algorithm for a quick computation of Pareto-optimal solutions. Evol Comput 13(4):501–525. https://doi.org/10.1162/106365605774666895

    Article  MATH  Google Scholar 

  17. Yang S, Li M, Liu X, Zheng J (2013) A grid-based evolutionary algorithm for many-objective optimization. IEEE Trans Evol Comput 17(5):721–736. https://doi.org/10.1109/TEVC.2012.2227145

    Article  MATH  Google Scholar 

  18. Xiang Y, Zhou Y, Li M, Chen Z (2017) A vector angle-based evolutionary algorithm for unconstrained many-objective optimization. IEEE Trans Evol Comput 21(1):131–152. https://doi.org/10.1109/TEVC.2016.2587808

    Article  MATH  Google Scholar 

  19. Zhou S, Dai Y, Chen Z (2024) Dominance relation selection and angle-based distribution evaluation for many-objective evolutionary algorithm. Swarm Evol Comput 86:101515. https://doi.org/10.1016/j.swevo.2024.101515

    Article  MATH  Google Scholar 

  20. Palakonda V, Kang JM, Jung H (2022) An adaptive neighborhood based evolutionary algorithm with pivot-solution based selection for multi-and many-objective optimization. Inf Sci 607:126–152. https://doi.org/10.1016/j.ins.2022.05.119

    Article  MATH  Google Scholar 

  21. Zhou J, Zou J, Yang S, Zheng J, Gong D, Pei T (2021) Niche-based and angle-based selection strategies for many-objective evolutionary optimization. Inf Sci 571:133–153. https://doi.org/10.1016/j.ins.2021.04.050

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang Q, Li H (2007) MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evol Comput 11(6):712–731. https://doi.org/10.1109/TEVC.2007.892759

    Article  MATH  Google Scholar 

  23. Han X, Chao T, Yang M, Li M (2024) A steady-state weight adaptation method for decomposition-based evolutionary multi-objective optimisation. Swarm Evol Comput 89:101641. https://doi.org/10.1016/j.swevo.2024.101641

    Article  MATH  Google Scholar 

  24. Li G, Wang GG, Xiao RB (2022) A novel adaptive weight algorithm based on decomposition and two-part update strategy for many-objective optimization. Inf Sci 615:323–347. https://doi.org/10.1016/j.ins.2022.09.057

    Article  MATH  Google Scholar 

  25. Yuan Y, Xu H, Wang B, Zhang B, Yao X (2015) Balancing convergence and diversity in decomposition-based many-objective optimizers. IEEE Trans Evol Comput 20(2):180–198. https://doi.org/10.1109/TEVC.2015.2443001

    Article  MATH  Google Scholar 

  26. Wang Z, Zhang Q, Zhou A, Gong M, Jiao L (2015) Adaptive replacement strategies for MOEA/D. IEEE Trans Cybern 46(2):474–486. https://doi.org/10.1109/TCYB.2015.2403849

    Article  MATH  Google Scholar 

  27. Wu Y, Wei J, Ying W, Lan Y, Cui Z, Wang Z (2022) A collaborative decomposition-based evolutionary algorithm integrating normal and penalty-based boundary intersection methods for many-objective optimization. Inf Sci 616:505–525. https://doi.org/10.1016/j.ins.2022.10.136

    Article  MATH  Google Scholar 

  28. Pang LM, Ishibuchi H, Shang K (2022) Use of two penalty values in multiobjective evolutionary algorithm based on decomposition. IEEE Trans Cybern 53(11):7174–7186. https://doi.org/10.1109/TCYB.2022.3182167

    Article  MATH  Google Scholar 

  29. Zhao Q, Guo Y, Yao X, Gong D (2022) Decomposition-based multiobjective optimization algorithms with adaptively adjusting weight vectors and neighborhoods. IEEE Trans Evol Comput 27(5):1485–1497. https://doi.org/10.1109/TEVC.2022.3201890

    Article  MATH  Google Scholar 

  30. He X, Zhou Y, Chen Z, Zhang Q (2018) Evolutionary many-objective optimization based on dynamical decomposition. IEEE Trans Evol Comput 23(3):361–375. https://doi.org/10.1109/TEVC.2018.2865590

    Article  MATH  Google Scholar 

  31. Liu C, Zhao Q, Yan B, Elsayed S, Ray T, Sarker R (2018) Adaptive sorting-based evolutionary algorithm for many-objective optimization. IEEE Trans Evol Comput 23(2):247–257. https://doi.org/10.1109/TEVC.2018.2848254

    Article  MATH  Google Scholar 

  32. Bosman PA, Thierens D (2003) The balance between proximity and diversity in multiobjective evolutionary algorithms. IEEE Trans Evol Comput 7(2):174–188. https://doi.org/10.1109/TEVC.2003.810761

    Article  MATH  Google Scholar 

  33. While L, Hingston P, Barone L, Huband S (2006) A faster algorithm for calculating hypervolume. IEEE Trans Evol Comput 10(1):29–38. https://doi.org/10.1109/TEVC.2010.2077298

    Article  MATH  Google Scholar 

  34. Zitzler E and Künzli S (2004) Indicator-based selection in multiobjective search. In: International conference on parallel problem solving from nature, pp 832-842. Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-30217-9_84

  35. Zapotecas-Martínez S, López-Jaimes A, García-Nájera A (2019) LIBEA: a lebesgue indicator-based evolutionary algorithm for multi-objective optimization. Swarm Evol Comput 44:404–419. https://doi.org/10.1016/j.swevo.2018.05.004

    Article  MATH  Google Scholar 

  36. García JLL, Monroy R, Hernández VAS, Coello CAC (2021) COARSE-EMOA: an indicator-based evolutionary algorithm for solving equality constrained multi-objective optimization problems. Swarm Evol Comput 67:100983. https://doi.org/10.1016/j.swevo.2021.100983

    Article  MATH  Google Scholar 

  37. Falcón-Cardona JG, Ishibuchi H, Coello CAC, Emmerich M (2021) On the effect of the cooperation of indicator-based multiobjective evolutionary algorithms. IEEE Trans Evol Comput 25(4):681–695. https://doi.org/10.1109/TEVC.2021.3061545

    Article  MATH  Google Scholar 

  38. Pour PA, Bandaru S, Afsar B, Emmerich M, Miettinen K (2023) A performance indicator for interactive evolutionary multiobjective optimization methods. IEEE Trans Evol Comput 28(3):778–787. https://doi.org/10.1109/TEVC.2023.3272953

    Article  MATH  Google Scholar 

  39. Liang Z, Luo T, Hu K, Ma X, Zhu Z (2020) An indicator-based many-objective evolutionary algorithm with boundary protection. IEEE Trans Cybern 51(9):4553–4566. https://doi.org/10.1109/TCYB.2019.2960302

    Article  MATH  Google Scholar 

  40. Liu Z, Wang H, Xi Y (2022) Performance indicator-based adaptive model selection for offline data-driven multiobjective evolutionary optimization. IEEE Trans Cybern 53(10):6263–6276. https://doi.org/10.1109/TCYB.2022.3170344

    Article  MATH  Google Scholar 

  41. Shang K, Ishibuchi H (2020) A new hypervolume-based evolutionary algorithm for many-objective optimization. IEEE Trans Evol Comput 24(5):839–852. https://doi.org/10.1109/TEVC.2020.2964705

    Article  MATH  Google Scholar 

  42. Palakonda V, Mallipeddi R (2020) An evolutionary algorithm for multi and many-objective optimization with adaptive mating and environmental selection. IEEE Access 8:82781–82796. https://doi.org/10.1109/ACCESS.2020.2991752

    Article  MATH  Google Scholar 

  43. Palakonda V, Kang JM (2023) Pre-DEMO: preference-inspired differential evolution for multi/many-objective optimization. IEEE Trans Syst Man Cybern Syst 53(12):7618–7630. https://doi.org/10.1109/TSMC.2023.3298690

    Article  MATH  Google Scholar 

  44. Shen J, Wang P, Wang X (2020) A controlled strengthened dominance relation for evolutionary many-objective optimization. IEEE Trans Cybern 52(5):3645–3657. https://doi.org/10.1109/TCYB.2020.3015998

    Article  MATH  Google Scholar 

  45. Palakonda V, Kang JM, Jung H (2024) Clustering-aided grid-based one-to-one selection-driven evolutionary algorithm for multi/many-objective optimization. IEEE Access 12:120612–120623. https://doi.org/10.1109/ACCESS.2024.3398415

    Article  MATH  Google Scholar 

  46. Sun Y, Xue B, Zhang M, Yen GG (2018) A new two-stage evolutionary algorithm for many-objective optimization. IEEE Trans Evol Comput 23(5):748–761. https://doi.org/10.1109/TEVC.2018.2882166

    Article  MATH  Google Scholar 

  47. Steuer RE (1986) Multiple criteria optimization: theory, computation and application. Wiley, New York, NY, USA

    MATH  Google Scholar 

  48. Li M, Yang S, Liu X (2015) Pareto or non-Pareto: Bi-criterion evolution in multiobjective optimization. IEEE Trans Evol Comput 20(5):645–665. https://doi.org/10.1109/TEVC.2015.2504730

    Article  MATH  Google Scholar 

  49. Li M, Yang S, Liu X (2015) Bi-goal evolution for many-objective optimization problems. Artif Intell 228:45–65. https://doi.org/10.1016/j.artint.2015.06.007

    Article  MathSciNet  MATH  Google Scholar 

  50. Zitzler E, Thiele L (1999) Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans Evol Comput 3(4):257–271. https://doi.org/10.1109/4235.797969

    Article  MATH  Google Scholar 

  51. Zitzler E, Laumanns M, Thiele L (2001) SPEA2: improving the strength Pareto evolutionary algorithm. TIK Report. https://doi.org/10.3929/ethz-a-004284029

    Article  MATH  Google Scholar 

  52. Li K, Fialho A, Kwong S, Zhang Q (2013) Adaptive operator selection with bandits for a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evol Comput 18(1):114–130. https://doi.org/10.1109/TEVC.2013.2239648

    Article  MATH  Google Scholar 

  53. Wang R, Zhang Q, Zhang T (2016) Decomposition-based algorithms using Pareto adaptive scalarizing methods. IEEE Trans Evol Comput 20(6):821–837. https://doi.org/10.1109/TEVC.2016.2521175

    Article  MATH  Google Scholar 

  54. Yu K, Liang J, Qu B, Luo Y, Yue C (2021) Dynamic selection preference-assisted constrained multiobjective differential evolution. IEEE Trans Syst Man Cybern Syst 52(5):2954–2965. https://doi.org/10.1109/TSMC.2021.3061698

    Article  MATH  Google Scholar 

  55. Liu Y, Gong D, Sun X, Zhang Y (2017) Many-objective evolutionary optimization based on reference points. Appl Soft Comput 50:344–355. https://doi.org/10.1016/j.asoc.2016.11.009

    Article  MATH  Google Scholar 

  56. He Z, Yen GG (2015) Many-objective evolutionary algorithm: objective space reduction and diversity improvement. IEEE Trans Evol Comput 20(1):145–160. https://doi.org/10.1109/TEVC.2015.2433266

    Article  MATH  Google Scholar 

  57. Sun Y, Yen GG, Yi Z (2018) IGD indicator-based evolutionary algorithm for many-objective optimization problems. IEEE Trans Evol Comput 23(2):173–187. https://doi.org/10.1109/TEVC.2018.2791283

    Article  MATH  Google Scholar 

  58. Li W, Chen Y, Dong Y, Huang Y (2024) A solution potential-based adaptation reference vector evolutionary algorithm for many-objective optimization. Swarm Evol Comput 84(101451):1–15. https://doi.org/10.1016/j.swevo.2023.101451

    Article  MATH  Google Scholar 

  59. Deb K, Thiele L, Laumanns M, Zitzler E (2002) Scalable multi-objective optimization test problems. In: Proceedings of the 2002 congress on evolutionary computation. CEC'02, 1:825–830. https://doi.org/10.1109/CEC.2002.1007032

  60. Huband S, Hingston P, Barone L, While L (2006) A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans Evol Comput 10(5):477–506. https://doi.org/10.1109/TEVC.2005.861417

    Article  MATH  Google Scholar 

  61. Cheng R, Li M, Tian Y, Zhang X, Yao X (2017) Benchmark functions for the CEC'2017 competition on evolutionary many-objective optimization. https://api.semanticscholar.org/CorpusID:13842279

  62. Zhu Q, Lin Q, Du Z, Liang Z, Wang W, Zhu Z, Chen J, Huang P, Ming Z (2016) A novel adaptive hybrid crossover operator for multiobjective evolutionary algorithm. Inf Sci 345:177–198. https://doi.org/10.1016/j.ins.2016.01.046

    Article  MATH  Google Scholar 

  63. Mallipeddi R, Suganthan PN, Pan QK, Tasgetiren MF (2011) Differential evolution algorithm with ensemble of parameters and mutation strategies. Appl Soft Comput 11(2):1679–1696. https://doi.org/10.1016/j.asoc.2010.04.024

    Article  MATH  Google Scholar 

  64. Deb K, Goyal M (1996) A combined genetic adaptive search (GeneAS) for engineering design. Comput Sci Inf 26:30–45. https://doi.org/10.1007/978-3-662-03423-1_27

    Article  MATH  Google Scholar 

  65. Wang Y, Cai Z, Zhang Q (2011) Diferential evolution with composite trial vector generation strategies and control parameters. IEEE Trans Evol Comput 15:55–66. https://doi.org/10.1109/TEVC.2010.2087271

    Article  MATH  Google Scholar 

  66. Zitzler E, Thiele L, Laumanns M, Fonseca CM, Da Fonseca VG (2003) Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evol Comput 7:117–132. https://doi.org/10.1109/TEVC.2003.810758

    Article  MATH  Google Scholar 

  67. While L, Hingston P, Barone L, Huband S (2006) A faster algorithm for calculating hypervolume. IEEE Trans Evol Comput 10:29–38. https://doi.org/10.1109/TEVC.2005.851275

    Article  MATH  Google Scholar 

  68. Tian Y, Cheng R, Zhang X, Xi Y (2017) PlatEMO: a MATLAB platform for evolutionary multi-objective optimization [educational forum]. IEEE Comput Intell Mag 12(4):73–87. https://doi.org/10.1109/MCI.2017.2742868

    Article  MATH  Google Scholar 

  69. Tan KC, Chew YH, Lee LH (2006) A hybrid multiobjective evolutionary algorithm for solving vehicle routing problem with time windows. Comput Optim Appl 34:115–151. https://doi.org/10.1007/s10589-005-3070-3

    Article  MathSciNet  MATH  Google Scholar 

  70. Solomon MM (1987) Algorithms for the vehicle routing and scheduling problems with time window constraints. Oper Res 35:254–265. https://doi.org/10.1287/opre.35.2.254

    Article  MathSciNet  MATH  Google Scholar 

  71. Palakonda V, Kang J, Jung H (2024) Benchmarking real-world many-objective problems: a problem suite with baseline results. IEEE Acess 12:49275–49290. https://doi.org/10.1109/ACCESS.2024.3383916

    Article  Google Scholar 

  72. Jain H, Deb K (2014) An evolutionary many-objective optimization algorithm using reference-point based nondominated sorting approach, part II: handling constraints and extending to an adaptive approach. IEEE Trans Evol Comput 18(4):602–622. https://doi.org/10.1109/TEVC.2013.2281534

    Article  MATH  Google Scholar 

  73. Tanabe R, Ishibuchi H (2020) An easy-to-use real-world multi-objective optimization problem suite. Appl Soft Comput 89:106078. https://doi.org/10.1016/j.asoc.2020.106078

    Article  MATH  Google Scholar 

  74. Goel T, Vaidyanathan R, Haftka RT, Shyy W, Queipo NV, Tucker K (2007) Response surface approximation of Pareto optimal front in multi-objective optimization. Comput Methods Appl Mech Eng 196(4–6):879–893. https://doi.org/10.1016/j.cma.2006.07.010

    Article  MATH  Google Scholar 

  75. Vaidyanathan R, Tucker PK, Papila N, Shyy W (2004) Computational-fluid-dynamics-based design optimization for single-element rocket injector. J Propuls Power 20(4):705–717

    Article  MATH  Google Scholar 

  76. Zapotecas-Martínez S, García-Nájera A, Menchaca-Méndez A (2023) Engineering applications of multi-objective evolutionary algorithms: a test suite of box-constrained real-world problems. Eng Appl Artif Intell 123:106192. https://doi.org/10.1016/j.engappai.2023.106192

    Article  MATH  Google Scholar 

  77. Tanino T, Tanaka T, Inuiguchi M, Miettinen K (2003) Graphical illustration of Pareto optimal solutions. Multi-objective programming and goal programming: theory and applications. Springer, Berlin Heidelberg, pp 197–202

    Chapter  MATH  Google Scholar 

  78. Thiele L, Miettinen K, Korhonen PJ, Molina J (2009) A preference-based evolutionary algorithm for multi-objective optimization. Evol Comput 17(3):411–436. https://doi.org/10.1007/978-3-642-19893-9_15

    Article  MATH  Google Scholar 

  79. Chen YS (2017) Performance enhancement of multiband antennas through a two-stage optimization technique. Int J RF Microwave Comput Aided Eng 27(2):e21064. https://doi.org/10.1002/mmce.21064

    Article  Google Scholar 

  80. Tafesse Azene Y (2011) Work roll system optimisation using thermal analysis and genetic algroithm. http://dspace.lib.cranfield.ac.uk/handle/1826/7117

  81. Musselman K, Talavage J (1980) A tradeoff cut approach to multiple objective optimization. Oper Res 28(6):1424–1435. https://doi.org/10.1287/opre.28.6.1424

    Article  MathSciNet  MATH  Google Scholar 

  82. Deb K, Jain H (2013) An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: solving problems with box constraints. IEEE Trans Evol Comput 18(4):577–601. https://doi.org/10.1109/TEVC.2013.2281535

    Article  MATH  Google Scholar 

Download references

Funding

This research is partly supported by the National Natural Science Foundation of China under Project Code (62176146).

Author information

Authors and Affiliations

  1. School of Computer Science and Engineering, Xi’an University of Technology, Xi’an, 710048, China

    Wei Li & Wenhao Tang

  2. Shaanxi Key Laboratory for Network Computing and Security Technology, Xi’an, 710048, China

    Lei Wang

Authors
  1. Wei Li
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Wenhao Tang
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Lei Wang
    View author publications

    You can also search for this author inPubMed Google Scholar

Contributions

Wei Li and Lei Wang wrote the main manuscript text. Wenhao Tang prepared all figures and tables. All authors reviewed the manuscript.

Corresponding author

Correspondence to Wei Li.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Ethical approval

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix

The mathematical models for the real-world many-objective problems considered in this paper are as follows:

1 Car side impact design problem

$$ \begin{aligned} Wc = & 1.98 + 4.9 \times x1 + 6.67 \times x2 + 6.98 \times x3 \\ & + 4.01 \times x4 + 1.78 \times x5 + 10 - 5 \times x6 + 2.73 \times x7 \\ \end{aligned} $$
$$Pfp=4.72-0.5\times x4-0.19\times x2x3$$
$$Vil=0.5\times (VMBP(x)+VFD(x))$$
$${g}_{1}(x)=1-1.16+0.3717\times {x}_{2}{x}_{4}+0.0092928\times {x}_{3}\ge 0$$
$$ \begin{aligned} g_{2} \left( x \right) = & 0.32 - 0.261 + 0.0159 \times x_{1} x_{2} + 0.06486 \times x_{1} \\ & + 0.019 \times x_{2} x_{7} - 0.0144 \times x_{3} x_{5} - 0.0154464 \times x_{6} \ge 0 \\ \end{aligned} $$
$$ \begin{aligned} g_{3} \left( x \right) = & 0.32 - 0.214 - 0.0082 \times x_{5} + 0.04520 \times x_{1} + 0.0135168 \times x_{1} \\ & - 0.03099 \times x_{2} x + 6 + 0.018 \times x_{2} x_{7} - 0.007176 \times x_{3} \\ & - 0.0232 \times x_{3} + 0.00364 \times x_{5} x_{6} + 0.018 \times x_{2}^{2} \ge 0 \\ \end{aligned} $$
$$ \begin{aligned} g_{4} \left( x \right) = & 0.32 - 0 - 74 + 0.61 \times x_{2} + 0.031296 \times x_{3} \\ & + 0.031872 \times x7 - 0.227 \times x_{2}^{2} \ge 0 \\ \end{aligned} $$
$${g}_{5}(x)=32-28.98-3.818\times {x}_{3}+4.2\times {x}_{1}{x}_{2}-1.27296\times {x}_{6}+2.68065\times {x}_{7}\ge 0$$
$$ \begin{aligned} g_{6} \left( x \right) = & 32 - 33.86 - 2.95 \times x_{3} + 5.057 \times x_{1} x_{2} \\ & + 3.795 \times x_{2} + 3.4431 \times x_{7} - 1.45728 \ge 0 \\ \end{aligned} $$
$${g}_{7}(x)=32-46.36+9.9\times {x}_{2}+4.4505\times {x}_{1}\ge 0$$
$${g}_{8}(x)=4-{f}_{2}(x)\ge 0$$
$${g}_{9}(x)=9.9-VMBP(x)\ge 0$$
$$g10(x)=15:7-VFD(x)\ge 0$$
$$VMBP(x)=10.58-0.674\times x1x2-0.67275\times x2$$
$$ S_{c} = \mathop \sum \limits_{i = 1}^{10} max\left\{ {g_{i} \left( x \right),0} \right\} $$
$${f}_{1}(x)={W}_{c}( Minimize )$$
$${f}_{2}(x)={P}_{fp}( Minimize )$$
$${f}_{3}(x)={V}_{il}( Minimize )$$
$${f}_{4}(x)={S}_{c}( Minimize )$$

where \(x_{1} \in \left[ {0.5,1.5} \right],\;x_{2} \in \left[ {0.45,1.35} \right],\;x_{3} \in \left[ {0.5,1.5} \right],\;x_{4} \in \left[ {0.5,1.5} \right],\)

$$ x_{5} \in \left[ {0.875,2.625} \right],\;x_{6} \in \left[ {0.4,1.2} \right], \;x_{7} \in \left[ {0.4,1.2} \right] $$

2 Liquid-rocket single-element injector design

$$ \begin{aligned} D_{c} c = & 0.692 + 0.477 \times x_{1} - 0.687 \times x_{2} - 0.080 \times x_{3} \\ & - 0.0650 \times x_{4} - 0.167 \times x_{1}^{2} - 0.0129 \times x_{2} x_{1} + 0.0796 \\ & \times x_{2}^{2} - 0.0634 \times x_{3} x_{1} - 0.0257 \times x_{3} x_{2} + 0.0877 \times x_{3}^{2} \\ & - 0.0521 \times x_{4} x_{1} + 0.00156 \times x_{4} x_{2} + 0.00198 \times x_{4} x_{3} \\ & + 0.0184 \times x_{4}^{2} \\ \end{aligned} $$
$$ \begin{aligned} TS_{max} = & 0.758 + 0.358 \times x_{1} - 0.807 \times x_{2} + 0.0925 \times x_{3} \\ & - 0.0468 \times x_{4} - 0.172 \times x_{1}^{2} + 0.0106 \times x_{2} x_{1} + 0.0697 \times x_{2}^{2} \\ & - 0.146 \times x_{3} x_{1} - 0.0416 \times x_{3} x_{2} + 0.102 \times x_{3}^{2} - 0.0694 \times x_{4} x_{1} \\ & - 0.00503 \times x_{4} x_{2} + 0.0151 \times x_{4} x_{3} + 0.0173 \times x_{4}^{2} \\ \end{aligned} $$
$$ \begin{aligned} TP_{max} = & 0.370 - 0.205 \times x_{1} + 0.0307 \times x_{2} + 0.108 \times x_{3} + 1.019 \times x_{4} \\ & - 0.135 \times x_{1}^{2} + 0.0141 \times x_{2} x_{1} + 0.0998 \times x_{2}^{2} + 0.208 \times x_{3} x_{1} \\ & - 0.0301 \times x_{3} x_{2} - 0.226 \times x_{3}^{2} + 0.353 \times x_{4} x_{1} - 0.0497 \times x_{4} x_{3} \\ & - 0.423 \times x_{4}^{2} + 0.202 \times x_{2} x_{1}^{2} - 0.281 \times x_{3} x_{1}^{2} - 0.342 \times 1x_{2}^{2} \\ & - 0.245 \times x_{3} x_{2}^{2} + 0.281 \times x_{2} *x_{3}^{2} - 0.184 \times x_{2} x_{4}^{2} + 0.281 \times x3x_{2} x_{1} \\ \end{aligned} $$
$$ \begin{aligned} T_{is} = & 0.153 - 0.322 \times x_{1} + 0.396 \times x_{2} + 0.424 \times x_{3} + 0.0226 \times x_{4} \\ & + 0.175 \times x_{1}^{2} + 0.0185 \times x_{2} x_{1} - 0.0701 \times x_{2}^{2} - 0.251 \times x_{3} x_{1} \\ & + 0.179 \times x3x_{2} + 0.0150 \times x_{3}^{2} + 0.0134 \times x_{4} x_{1} + 0.0296 \times x_{4} x_{2} \\ & + 0.0752 \times x_{4} x_{3} + 0.0192 \times x_{2}^{2} \\ \end{aligned} $$
$${f}_{1}(x)={D}_{cc}( Minimize )$$
$${f}_{2}(x)=T{S}_{max}( Minimize )$$
$${f}_{3}(x)=T{P}_{max}( Minimize )$$
$${f}_{4}(x)={T}_{is}( Minimize )$$

where \({x}_{1}\in [\text{0,20}],{x}_{2}\in [\text{0,40}],{x}_{3}\in [-\text{40,0}],\) and \({x}_{4}\in [\text{0.01,0.02}]\).

3 Location of a pollution monitoring system

$$f({x}_{1},{x}_{2})=-{u}_{1}({x}_{1},{x}_{2}){u}_{2}({x}_{1},{x}_{2}){u}_{3}({x}_{1},{x}_{2})$$
$${u}_{1}({x}_{1},{x}_{2})3\times {(1-{x}_{1})}^{2}{e}^{-{x}_{1}^{2}-{({x}_{2}+1)}^{2}}$$
$${u}_{2}({x}_{1},{x}_{2})-10\times (\frac{{x}_{1}}{4-{x}_{1}^{3}-{x}_{2}^{5}}){e}^{-{x}_{1}^{2}{x}_{2}^{2}}$$
$${u}_{3}({x}_{1},{x}_{2})\frac{1}{3}{e}^{-{({x}_{1}+1)}^{2}-{x}_{2}^{2}}$$
$${f}_{1}({x}_{1},{x}_{2})f({x}_{1},{x}_{2})$$
$${f}_{2}({x}_{1},{x}_{2})f({x}_{1}-1.2,{x}_{2}-1.5)$$
$${f}_{3}({x}_{1},{x}_{2})f({x}_{1}+0.3,{x}_{2}-3.0)$$
$${f}_{4}({x}_{1},{x}_{2})f({x}_{1}-1.0,{x}_{2}+0.5)$$
$${f}_{5}({x}_{1},{x}_{2})f({x}_{1}-0.5,{x}_{2}-1.7)$$

where \({x}_{1}\in [-\text{4.9,3.2}],{x}_{2}\in [-\text{3.5,6}]\)

4 Ultra-wideband antenna design

$$ \begin{aligned} VSWR_{p} = & 502.94 - 27.18 \times \left( {\frac{{w_{1} - 20.0}}{0.5}} \right) + 43.08 \times \left( {\frac{{l_{1} - 20.0}}{2.5}} \right) \\ & + 47.75 \times \left( {a_{1} - 6.0} \right) + 32.25 \times \left( {\frac{{b_{1} - 5.5}}{0.5}} \right) + 31.67 \times \left( {a_{2} - 11.0} \right) \\ & - 36.19 \times \left( {\frac{{w_{1} - 20.0}}{0.5}} \right)\left( {\frac{w2 - 2.5}{{0.5}}} \right) - 39.44 \times \left( {\frac{w1 - 20.0}{{0.5}}} \right)\left( {a1 - 6.0} \right) \\ & + 57.45 \times \left( {a1 - 6.0} \right)\left( {\frac{b1 - 5.5}{{0.5}}} \right) \\ \end{aligned} $$
$$ \begin{aligned} VSWR_{WL} = & 130.53 + 45.97 \times \left( {\frac{{l_{1} - 20.0}}{2.5}} \right) - 52.93 \times \left( {\frac{{w_{1} - 20.0}}{0.5}} \right) \\ & - 78.93 \times \left( {a_{1} - 6.0} \right) + 79.22 \times \left( {a_{2} - 11.0} \right) + 47.23 \times \left( {\frac{{w_{1} - 20.0}}{0.5}} \right)\left( {a_{1} - 6.0} \right) \\ & - 40.61 \times \left( {\frac{{w_{1} - 20.0}}{0.5}} \right)\left( {a_{2} - 11.0} \right) - 50.62 \times \left( {a_{1} - 6.0} \right)\left( {a_{2} - 11.0} \right) \\ \end{aligned} $$
$$ \begin{aligned} VSWR_{Wi} = & 203.16 - 42.75 \times \left( {\frac{w1 - 20.0}{{0.5}}} \right) + 56.67 \times \left( {a1 - 6.0} \right) \\ & + 19.88\left( {\frac{b1 - 5.5}{{0.5}}} \right) - 12.89 \times \left( {a2 - 11.0} \right) \\ & - 35.09 \times \left( {a1 - 6.0} \right)\left( {\frac{b1 - 5.5}{{0.5}}} \right) \\ & - 22.91 \times \left( {\frac{b1 - 5.5}{{0.5}}} \right)\left( {a2 - 11.0} \right) \\ \end{aligned} $$
$$ \begin{aligned} Ga_{p} = & 0.76 - 0.06 \times \left( {\frac{l1 - 20.0}{{2.5}}} \right) + 0.03 \times \left( {\frac{12 - 2.5}{{0.5}}} \right) \\ & + 0.02 \times (a2 - 0.76 - 0.06 \times \left( {\frac{l1 - 20.0}{{2.5}}} \right) \\ & + 0.03 \times \left( {\frac{12 - 2.5}{{0.5}}} \right) + 0.02 \times \left( {a2 - 11.0} \right) - 0.02 \times \left( {\frac{62 - 6.5}{{0.5}}} \right) \\ & - 0.03 \times \left( {\frac{d2 - 12.0}{{0.5}}} \right) + 0.03 \times \left( {\frac{l1 - 20.0}{{2.5}}} \right)\left( {\frac{w1 - 20.0}{{0.5}}} \right) \\ & - 0.02 \times \left( {\frac{l1 - 20.0}{{2.5}}} \right)\left( {\frac{(2 - 2.5}{{0.5}}} \right) + 0.02 \times \left( {\frac{l1 - 20.0}{{2.5}}} \right)\left( {\frac{b2 - 6.5}{{0.5}}} \right) \\ \end{aligned} $$
$$ \begin{aligned} Fd_{EH} = & 1.08 - 0.12 \times \left( {\frac{l1 - 20.0}{{2.5}}} \right) - 0.26 \times \left( {\frac{w1 - 20.0}{{0.5}}} \right) \\ & - 0.05 \times \left( {a2 - 11.0} \right) - 0.12 \times \left( {\frac{b2.6.5}{{0.5}}} \right) \\ & + 0.08 \times \left( {a1 - 6.0} \right)\left( {\frac{62 - 6.5}{{0.5}}} \right) + 0.07 \times \left( {a2 - 6.0} \right)\left( {\frac{b2 - 5.5}{{0.5}}} \right) \\ \end{aligned} $$
$${f}_{1}(x)=VSW{R}_{p}( Minimize )$$
$${f}_{2}(x)=-VSW{R}_{WL} (Maximize)$$
$${f}_{3}(x)=-VSW{R}_{Wi} (Maximize)$$
$${f}_{4}(x)=-G{a}_{p} (Maximize)$$
$${f}_{5}(x)=F{d}_{EH} (Minimize)$$

where \({a}_{1}\in [\text{5,7}],{a}_{2}\in [\text{10,12}],{b}_{1}\in [\text{5,6}],{b}_{2}\in [\text{6,7}],{d}_{1}\in [\text{3,4}],{d}_{2}\in [\text{11.5,12.5}],\)

\({l}_{1}\in \left[\text{17.5,22.5}\right], {l}_{2}\in [\text{2,3}],{w}_{1}\in [\text{17.5,22.5}],\) and \({w}_{2}[\text{2,3}].\)

Among these decision variables \({a}_{1},{b}_{1},{l}_{1},{l}_{2},{w}_{1},{w}_{2}\) are the primary parameters. The remaining four parameters are determined as follows:

$$\begin{array}{cc}{a}_{2}& ={l}_{1}\times {l}_{2}\times {w}_{1}\times {w}_{2}\\ {b}_{2}& ={l}_{1}\times {l}_{2}\times {w}_{1}\times {a}_{1}\\ {d}_{1}& ={l}_{1}\times {b}_{1}\times {w}_{2}\times {a}_{2}\\ {d}_{2}& ={w}_{1}\times {a}_{1}\times {b}_{1}\times {w}_{2}\end{array}$$

5 Single-pass Work roll cooling design problem

$$ \begin{aligned} \Delta T_{s} = & - 370.8659 + 3.92666667 \times x_{1} + 0.046222222 \times x_{1}^{2} + 3.02845679 \times x_{2} \\ & - 0.00130938 \times x_{2}^{2} + 0.0000001 \times x_{3} - 0.025 \times x_{3}^{2} \\ & - 7.0603175 \times x_{4} + 0.085079365 \times x_{4}^{2} + 26.9230797 \times x_{5} \\ & + 3.44362939 \times x_{5}^{2} + 3.99055556 \times x_{6} - 0.041944444 \times x_{6}^{2} \\ & - 0.801944 \times x_{7} + 0.004701389 \times x_{7}^{2} \\ \end{aligned} $$
$$ \begin{aligned} Rt_{s} = & 140.63997 - 10.082111 \times x_{1} + 0.064200000 \times x_{1}^{2} - 1.2915963 \times x_{2} \\ & + 0.002292222 \times x_{2}^{2} + 0.0014444 \times x_{3} - 0.000003333 \times x_{3}^{2} \\ & + 12.1149569 \times x_{4} - 0.21604354 \times x_{4}^{2} - 387.34458 \times x_{5} \\ & + 145.751168 \times x_{5}^{2} - 2.6677500 \times x_{6} + 0.0386875 \times x_{6}^{2} \\ & - 1.4585556 \times x_{7} + 0.01198229 \times x_{7}^{2} \\ \end{aligned} $$
$$ \begin{aligned} \Delta T_{9d} = & - 819.98558 + 9.97380000 \times x_{1} - 0.31336444 \times x_{1}^{2} + 1.43803309 \times x_{2} \\ & - 0.59030e^{ - 3} \times x_{2}^{2} - 0.13506667 \times x_{3} + 0.095348889 \times x_{3}^{2} \\ & - 5.1424209 \times x_{4} + 0.062201542 \times x_{4}^{2} + 52.8477445 \times x_{5} \\ & - 23.073031 \times x_{5}^{2} + 0.050088889 \times x_{6} - 0.00849611 \times x_{6}^{2} \\ & - 0.22464444 \times x_{7} + 0.001426597 \times x_{7}^{2} \\ \end{aligned} $$
$$ \begin{aligned} Rt_{9d} = & 913.51841 - 18.435556 \times x_{1} + 0.5084 \times x_{1}^{2} - 5.0893x_{2} \\ & + 0.002138444 \times x_{2}^{2} x_{2}^{2} - 3.9332778 \times x_{3} - 0.13521667 \times x_{3}^{2} \\ & + 14.6337279 \times x_{4} - 0.18028299 \times x_{4}^{2} - 248.51971 \times x_{5} \\ & + 101.248057 \times x_{5}^{2} + 0.1710833 \times x_{6} + 0.0171375 \times x_{6}^{2} \\ & + 0.108888 \times x_{7} - 0.17812e^{ - 3} \times x_{7}^{2} \\ \end{aligned} $$
$$ \begin{aligned} \Delta T_{15d} = & - 533.55904 + 9.74111 \times x_{1} - 0.35888889 \times x_{1}^{2} + 1.00235802 \times x_{2} \\ & - 0.40321e^{ - 3} \times x_{2}^{2} - 0.51055556 \times x_{3} + 0.0949444 \times x_{3}^{2} - 3.9777324 \times x_{4} \\ & + 0.04591383 \times x_{4}^{2} + 51.7361959 \times x_{5} - 28.708807 \times x_{5}^{2} - 1.0575000 \times x_{6} \\ & + 0.001652778 \times x_{6}^{2} - 0.29222 \times x_{7} + 0.0022048 \times x_{7}^{2} \\ \end{aligned} $$
$$ \begin{aligned} Rt_{15d} = & 316.50677 - 19.781444 \times x_{1} + 0.553933333 \times x_{1}^{2} - 4.0415037 \times x_{2} \\ & + 0.001683704 \times x_{2}^{2} - 5.6866667 \times x_{3} - 0.07520000 \times x_{3}^{2} \\ & + 11.4528209 \times x_{4} - 0.13340408 \times x_{4}^{2} - 199.10688 \times x_{5} \\ & + 97.5171611 \times x_{5}^{2} + 0.867388889 \times x_{6} + 0.009866667 \times x_{6}^{2} \\ & + 0.738388889 \times x_{7} - 0.00434792 \times x_{7}^{2} \\ \end{aligned} $$
$${f}_{1}(x)=\Delta {T}_{s}( Minimize )$$
$${f}_{2}(x)=R{t}_{s}( Minimize )$$
$${f}_{3}(x)=\Delta {T}_{9d}( Minimize )$$
$${f}_{4}(x)=R{t}_{9d}( Minimize )$$
$${f}_{5}(x)=\Delta {T}_{15d}( Minimize )$$
$${f}_{6}(x)=R{t}_{15d}( Minimize )$$

where the decision variables \({x}_{1}\in \left[\text{5,15}\right],{x}_{2}\in [\text{950,1250}],{ x}_{3}\in [\text{15,50}],{x}_{4}\in [\text{10,30}],{x}_{5}\in [\text{0.14,1.256}],{ x}_{6}\in [\text{40,80}],{x}_{7}\in [\text{20,100}]\).

6 Water resource planning

$${C}_{dn}=106780.37\times ({x}_{2}+{x}_{3})+61704.67$$
$${C}_{fc}=3000\times {x}_{1}$$
$${C}_{tf}=\frac{(305700\times 289\times {x}_{2})}{(0.06*2289{)}^{0.65}}$$
$${C}_{afd}=2250\times 2289\times {e}^{-39.75\times {x}_{2}+9.9\times {x}_{3}+2.74)}$$
$${C}_{af}=25\times (\frac{1.39}{{x}_{1}{x}_{2}}+4940\times {x}_{3}-80)$$
$${g}_{1}(x)=\frac{0.00139}{{x}_{1}{x}_{2}}+4.94\times {x}_{3}-0.08\le 1.0$$
$${g}_{2}(x)=\frac{0.000306}{{x}_{2}{x}_{2}}+1.082\times {x}_{3}-0.0986\le 1.0$$
$${g}_{3}(x)=\frac{12.307}{{x}_{1}{x}_{2}}+49408.24\times {x}_{3}+4051.02\le 5000$$
$${g}_{4}(x)=\frac{2.{x}_{1}}{{x}_{1}{x}_{2}}+8046.33\times {x}_{3}-696.71\le 16000$$
$${g}_{5}(x)=\frac{2.138}{{x}_{1}{x}_{2}}+7883.39\times {x}_{3}-705.04\le 10000$$
$${g}_{6}(x)=\frac{0.417}{{x}_{1}{x}_{2}}+1721.26\times {x}_{3}-136.54\le 2000$$
$${g}_{7}(x)=\frac{0.164}{{x}_{1}{x}_{2}}+631.13\times {x}_{3}-54.48\le 550$$
$$ S_{c} = \sum\nolimits_{i = 1}^{10} {max} \left\{ {g_{i} \left( x \right),0} \right\} $$
$${f}_{1}(x)={C}_{dn}( Minimize )$$
$${f}_{2}(x)={C}_{fc}( Minimize )$$
$${f}_{3}(x)={C}_{tf}( Minimize )$$
$${f}_{4}(x)={C}_{afd}( Minimize )$$
$${f}_{5}(x)={C}_{af}( Minimize )$$
$${f}_{6}(x)={S}_{c}( Minimize )$$

where \({x}_{1}\in [\text{0.01,0.45}],{x}_{2}\in [\text{0.01,0.10}],and {x}_{3}\in [\text{0.01,0.10}]\)

7 Car cab design problem

$$ \begin{aligned} W_{c} = & 1.98 + 4.9 \times x_{1} + 6.67 \times x_{2} + 6.98 \times x_{3} + 4.01 \times x_{4} \\ & + 1.78 \times x_{5} + 0.00001 \times x_{6} + 2.73 \times x_{7} \\ \end{aligned} $$
$$ g_{1} \left( x \right) = 1 - \left( {1.16 - 0.3717 \times x_{2} x_{4} - 0.00931 \times x_{2} x_{10} - 0.484 \times x_{3} x_{9} + 0.01343 \times x_{6} } \right) \ge 0 $$
$$ \begin{aligned} g_{2} \left( x \right) = & 0.32 - \left( {0.261 - 0.0159 \times x_{1} x_{2} } \right. - 0.188 \times x_{1} x_{8} \\ & - 0.019 \times x_{2} x_{7} + 0.0144 \times x_{3} x_{5} + 0.8757 \times x_{5} x_{10} \\ & + 0.08045 \times x_{6} x_{9} + 0.00139 \times x_{8} x_{11} \left. { + 0.00001575 \times x_{10} x_{11} } \right) \ge 0 \\ \end{aligned} $$
$$ \begin{aligned} g_{3} \left( x \right) = & 0.32 - \left( {0.261 - 0.0159 \times x_{1} x_{2} } \right. - 0.188 \times x_{1} x_{8} \\ & - 0.019 \times x_{2} x_{7} + 0.0144 \times x_{3} x_{5} + 0.8757 \times x_{5} x_{10} \\ & + 0.08045 \times x_{6} x_{9} + 0.00139 \times x_{8} x_{11} \left. { + 0.00001575 \times x_{10} x_{11} } \right) \ge 0 \\ \end{aligned} $$
$$ \begin{aligned} g_{4} \left( x \right) = & 0.32 - \left( {0.74 - 0.61 \times x_{2} } \right. - 0.163 \times x_{3} x_{8} \\ & + 0.001232 \times x_{3} x_{10} - 0.166 \times x_{7} x_{9} \left. { + 0.227 \times x_{2} x_{2} } \right) \ge 0 \\ \end{aligned} $$
$$ g_{5} \left( x \right) = 32 - \left( {\frac{URD \times MRD \times LRD}{3}} \right) \ge 0 $$
$$ \begin{aligned} URD = & 28.98 + 3.818 \times x_{3} - 4.2 \times x_{1} x_{2} + 0.0207 \times x_{5} x_{10} \\ & + 6.63 \times x_{6} x_{9} - 7.77 \times x_{7} x_{8} + 0.32 \times x_{9} x_{10} \\ \end{aligned} $$
$$ \begin{aligned} MRD = & 33.86 + 2.95 \times x_{3} + 0.1792 \times x_{10} - 5.057 \times x_{1} x_{2} \\ & - 11 \times x_{2} x_{8} - 0.0215 \times x_{5} x_{10} - 9.98 \times x_{7} x_{8} + 22 \times x_{8} x_{9} \\ \end{aligned} $$
$$LRD=46.36-9.9\times {x}_{2}-12.9\times {x}_{1}{x}_{8}+0.1107\times {x}_{3}{x}_{10}$$
$$ \begin{aligned} g_{6} \left( x \right) = & 32 - \left( {4.72 - 0.5 \times x} \right. - 4 - 0.19 \times x_{2} x_{3} - 0.0122 \times x_{4} x_{10} \\ & + 0.009325 \times x_{6} x_{10} \left. { + 0.000191 \times x_{11} x_{11} } \right) \ge 0 \\ \end{aligned} $$
$$ \begin{aligned} g_{7} \left( x \right) = & 4 - \left( {10.58 - 0.674 \times x_{1} x_{2} } \right. - 1.95 \times x_{2} x_{8} \\ & + 0.02054 \times x_{3} x_{10} - 0.0198 \times x_{4} x_{10} \left. { + 0.028 \times x_{6} x_{10} } \right) \ge 0 \\ \end{aligned} $$
$$ \begin{aligned} g_{8} \left( x \right) = & 9.9 - \left( {16.45 - 0.489 \times x_{3} x_{7} } \right. - 0.84 \times x_{5} x_{6} + 0.043 \times x_{9} x_{10} \\ & - 0.0556 \times x_{9} x_{11} \left. { - 0.000786 \times x_{11} x_{11} } \right) \ge 0 \\ \end{aligned} $$
$$ F_{2} = maxg_{1} \left( x \right),0 $$
$$ F_{3} = maxg_{2} \left( x \right),0 $$
$$ F_{4} = maxg_{3} \left( x \right),0 $$
$$ F_{5} = maxg_{4} \left( x \right),0 $$
$$ F_{6} = maxg_{5} \left( x \right),0 $$
$$ F_{7} = maxg_{6} \left( x \right),0 $$
$$ F_{8} = maxg_{7} \left( x \right),0 $$
$$ F_{9} = maxg_{8} \left( x \right),0 $$
$$ f_{1} \left( x \right) = W_{c} \left( { Minimize } \right) $$
$$ f_{2} \left( x \right) = F_{2} \left( {Minimize } \right) $$
$$ f_{3} \left( x \right) = F_{3} \left( {Minimize } \right) $$
$$ f_{4} \left( x \right) = F_{4} \left( {Minimize } \right) $$
$$ f_{5} \left( x \right) = F_{5} \left( {Minimize } \right) $$
$$ f_{6} \left( x \right) = F_{6} \left( {Minimize } \right) $$
$$ f_{7} \left( x \right) = F_{7} \left( {Minimize } \right) $$
$$ f_{8} \left( x \right) = F_{8} \left( {Minimize } \right) $$
$$ f_{9} \left( x \right) = F_{9} \left( {Minimize } \right) $$

where \( x_{1} \in \left[ {0.5,1.5} \right],x_{2} \in \left[ {0.45,1.35} \right],x_{3} \in \left[ {0.5,1.5} \right],x_{4} \in \left[ {0.5,1.5} \right],x_{5} \in \left[ {0.875,2.625} \right],\)

$$ x_{6} \in \left[ {0.4,1.2} \right], x_{7} \in \left[ {0.4,1.2} \right] $$

Multi-pass Work roll cooling design problem

$$ \begin{aligned} \Delta T_{sP1} = & - 3078.7483 + 3.92666667 \times x_{1} + 0.046222222 \times x_{1}^{2} + 2.39944444 \times x_{2} \\ & + 4.9 \times x_{3} - 0.02511111 \times x_{3}^{2} - 7.0603175 \times x_{4} + 0.085079365 \times x_{4}^{2} \\ & + 433.703704 \times x_{5} + 3.99055556 \times x_{6} - 0.04194444 \times x_{6}^{2} \\ & - 0.80194444 \times x_{7} + 0.004701389 \times x_{7}^{2} \\ \end{aligned} $$
$$ \begin{aligned} \Delta T_{SP2} = & - 74017.666 + 3.92666667 \times x_{8} + 0.046222222 \times x_{8}^{2} + 124.916389 \times x_{9} \\ & - 0.05286111 \times x_{9}^{2} + 4.9 \times x_{10} - 0.02511111 \times x_{10}^{2} - 9.3007256 \times x_{11} \\ & + 0.112235828 \times x_{11}^{2} + 2560.67901 \times x_{12} - 4322.5309 \times x_{12}^{2} \\ & + 1.07972222 \times x_{13} - 0.02115278 \times x_{13}^{2} - 0.90302431 \times x_{14} \\ & + 0.004701389 \times x_{14}^{2} \\ \end{aligned} $$
$$ \begin{aligned} Rt_{sP2} = & - 69381.097 - 9.7710000 \times x_{8} + 0.419755556 \times x_{8}^{2} + 120.479931 \times x_{9} \\ & - 0.05048750 \times x_{9}^{2} + 1.91477778 \times x_{10} - 0.18294444 \times x_{10}^{2} \\ & + 37.1468753 \times x_{11} - 0.46127891 \times x_{11}^{2} - 28760.889 \times x_{12} \\ & + 53404.1667 \times x_{12}^{2} + 22.5628611 \times x_{13} - 0.14073750 \times x_{13}^{2} \\ & - 1.9127026 \times x_{14} + 0.013371181 \times x_{14}^{2} \\ \end{aligned} $$
$$ \begin{aligned} \Delta T_{sP3} = & - 90392.107 + 3.92666667 \times x_{15} + 0.046222222 \times x_{15}^{2} + 163.144722 \times x_{16} \\ & - 0.07365278 \times x_{16}^{2} + 4.9 \times x_{17} - 0.02511111 \times x_{17}^{2} \\ & - 7.0603175 \times x_{18} + 0.085079365 \times x_{18}^{2} + 68.7345679 \times x_{19} \\ & + 297.839506 \times x_{19}^{2} + 3.99055556 \times x_{20} - 0.04194444 \times x_{20}^{2} \\ & - 0.95991111x_{21} + 0.004701389 \times x_{21}^{2} \\ \end{aligned} $$
$$ \begin{aligned} Rt_{sP3} = & 167054.217 - 9.771 \times x_{15} + 0.419755556 \times x_{15}^{2} - 297.75003 \times x_{16} \\ & + 0.134493056 \times x_{16}^{2} + 1.91477778 \times x_{17} - 0.18294444 \times x_{17}^{2} \\ & + 17.2142766 \times x_{18} - 0.21967166 \times x_{18}^{2} - 11074.571 \times x_{19} \\ & + 12297.3765 \times x_{19}^{2} - 3.3344167 \times x_{20} + 0.044243056 \times x_{20}^{2} \\ & - 2.0744939 \times x_{21} + 0.013371181 \times x_{21}^{2} \\ \end{aligned} $$
$$ \begin{aligned} \Delta T_{sP4} = & - 78565.254 + 3.92666667 \times x_{22} + 0.046222222 \times x_{22}^{2} \\ & + 152.096806 \times x_{23} - 0.07365278 \times x_{23}^{2} + 4.9 \times x_{24} - 0.02511111 \times x_{24}^{2} \\ & - 7.0603175 \times x_{25} + 0.085079365 \times x_{25}^{2} - 80.185185 \times x_{26} \\ & + 297.839506 \times x_{26}^{2} + 3.99055556 \times x_{27} - 0.04194444 \times x_{27}^{2} \\ & - 0.99705208 \times x_{28} + 0.004701389 \times x_{28}^{2} \\ \end{aligned} $$
$$ \begin{aligned} Rt_{sP4} = & 149025.120 - 9.771 \times x_{22} + 0.419755556 \times x_{22}^{2} - 277.57607 \times x_{23} \\ & + 0.134493056 \times x_{23}^{2} + 1.91477778 \times x_{24} - 0.18294444 \times x_{24}^{2} \\ & + 17.2142766 \times x_{25} - 0.21967166 \times x_{25}^{2} - 17223.259 \times x_{26} \\ & + 12297.3765 \times x_{26}^{2} - 3.3344167 \times x_{27} + 0.044243056 \times x_{27}^{2} \\ & - 2.1801262 \times x_{28} + 0.013371181 \times x_{28}^{2} \\ \end{aligned} $$
$$ \begin{aligned} \Delta T_{sP5} = & - 868.02142 + 3.92666667 \times x_{29} + 0.046222222 \times x_{29}^{2} \\ & + 0.926388889 \times x_{30} + 4.9 \times x_{31} - 0.02511111 \times x_{31}^{2} \\ & - 7.0603175 \times x_{32} + 0.085079365 \times x_{32}^{2} - 324.41358 \times x_{33} \\ & + 297.839506 \times x_{33}^{2} + 3.99055556 \times x_{34} - 0.0419333 \times x_{34}^{2} \\ & - 1.0194307 \times x_{35} + 0.004701389 \times x_{35}^{2} \\ \end{aligned} $$
$$ \begin{aligned} Rt_{sP5} = & 16371.3330 - 9.771 \times x_{29} + 0.419755556 \times x_{29}^{2} - 1.4530278 \times x_{30} \\ & + 1.91477778 \times x_{31} - 0.18294444 \times x_{31}^{2} + 17.2142766 \times x_{32} \\ & - 0.21967166 \times x_{32}^{2} - 27307.108 \times x_{33} + 12297.3765 \times x_{33}^{2} - 3.3344167 \times x_{34} \\ & + 0.044243056 \times x_{34}^{2} - 2.2437730 \times x_{35} + 0.013371181 \times x_{35}^{2} \\ \end{aligned} $$
$$ f_{1} \left( x \right) = \Delta T_{sP1} \left( {Minimize} \right) $$
$$ f_{2} \left( x \right) = Rt_{sP1} \left( {Minimize} \right) $$
$$ f_{3} \left( x \right) = \Delta T_{sP2} \left( {Minimize} \right) $$
$$ f_{4} \left( x \right) = Rt_{sP2} \left( {Minimize} \right) $$
$$ f_{5} \left( x \right) = \Delta T_{sP3} \left( {Minimize} \right) $$
$$ f_{6} \left( x \right) = Rt_{sP3} \left( {Minimize} \right) $$
$$ f_{7} \left( x \right) = \Delta T_{sP4} \left( {Minimize} \right) $$
$$ f_{8} \left( x \right) = Rt_{sP4} \left( {Minimize} \right) $$
$$ f_{9} \left( x \right) = \Delta T_{sP5} \left( {Minimize} \right) $$
$$ f_{1} 0\left( x \right) = Rt_{sP5} \left( {Minimize} \right) $$

where the decision variables \(x_{1} \in \left[ {5,15} \right],x_{2} \in \left[ {1230,1250} \right],x_{3} \in \left[ {10,30} \right],x_{4} \in \left[ {15,50} \right],\)

$$ x_{5} \in \left[ {0.14,0.2} \right],x_{6} \in \left[ {40,80} \right],x_{7} \in \left[ {65,75} \right],x_{8} \in \left[ {5,15} \right],x_{9} \in \left[ {1155,1195} \right],x_{10} \in \left[ {10,30} \right], $$
$$ x_{11} \in \left[ {15,50} \right],x_{12} \in \left[ {0.17,0.29} \right],x_{13} \in \left[ {40,80} \right],x_{14} \in \left[ {75.75,85.75} \right],x_{15} \in \left[ {5,15} \right], $$
$$ x_{16} \in \left[ {1080,1120} \right],x_{17} \in \left[ {10,30} \right],x_{18} \in \left[ {15,50} \right],x_{19} \in \left[ {0.32,0.44} \right],x_{20} \in \left[ {40,80} \right], $$
$$ x_{21} \in \left[ {82.25,92.25} \right],x_{22} \in \left[ {5,15} \right],x_{23} \in \left[ {1005,1045} \right],x_{24} \in \left[ {10,30} \right],x_{25} \in \left[ {15,50} \right],x_{26} \in \left[ {0.57,0.69} \right], $$
$$ x_{27} \in \left[ {40,80} \right],x_{28} \in \left[ {86.2,96.2} \right],x_{29} \in \left[ {5,15} \right],x_{30} \in \left[ {950,970} \right],x_{31} \in \left[ {10,30} \right],x_{32} \in \left[ {15,50} \right], $$

\(x_{33} \in \left[ {0.98,1.1} \right],x_{34} \in \left[ {40,80} \right], and x_{35} \in \left[ {88.58,98.58} \right]\).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, W., Tang, W. & Wang, L. Many-objective evolutionary algorithm based on dynamic mating and strengthened fitness selection mechanism. J Supercomput 81, 440 (2025). https://doi.org/10.1007/s11227-024-06821-3

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11227-024-06821-3

Keywords