A Comparison of Constraint Handling Techniques on NSGA-II

Abstract

Almost all real-world and engineering problems involve multi-objective optimization of some sort that is often constrained. To solve these constrained multi-objective optimization problems, constrained multi-objective optimization evolutionary algorithms (CMOEAs) are enlisted. These CMOEAs require specific constraint handling techniques. This study aims to address and test the most successful constraint handling techniques, seven different penalty constraint techniques, as applied to the Non-dominated Sorting Genetic Algorithm II (NSGA-II). In this paper, NSGA-II is chosen because of its high popularity amongst evolutionary algorithms. Inverted Generational Distance and Hypervolume are the main metrics that are discussed to compare the constraint handling techniques. NSGA-II is applied on 13 constrained multi-objective problems known as CF1-CF10, C1-DTLZ1, C2-DTLZ2, and C3-DTLZ4. The result of IGD and HV values are compared and the feasibility proportions of each combination on each problem are shown. The results of simulation present interesting findings that have been presented at the end of paper as discussion and conclusion.

This is a preview of subscription content, access via your institution.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

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

References

  1. 1.

    Ali MM, Khompatraporn C, Zabinsky ZB (2005) A numerical evaluation of several stochastic algorithms on selected continuous global optimization test problems. J Global Optim 31:635–672

    MathSciNet  Article  Google Scholar 

  2. 2.

    Asafuddoula M, Ray T, Sarker R, Alam K (2012) An adaptive constraint handling approach embedded MOEA/D. 2012 IEEE congress on evolutionary computation, 2012. IEEE, pp 1–8

  3. 3.

    Back T (1996) Evolutionary algorithms in theory and practice: evolution strategies, evolutionary programming, genetic algorithms. Oxford University Press, Oxford

  4. 4.

    Carvalho RD, Saldanha RR, Gomes B, Lisboa AC, Martins A (2012) A multi-objective evolutionary algorithm based on decomposition for optimal design of Yagi-Uda antennas. IEEE Trans Magn 48:803–806

    Article  Google Scholar 

  5. 5.

    Coello CAC (2002) Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art. Comput Methods Appl Mech Eng 191:1245–1287

    MathSciNet  Article  Google Scholar 

  6. 6.

    Coello CAC, Lamont GB, Van Veldhuizen DA (2007) Evolutionary algorithms for solving multi-objective problems. Springer

  7. 7.

    Da Ronco CC, Ponza R, Benini EJAOCMIE (2014) Aerodynamic shape optimization in aeronautics: a fast and effective multi-objective approach. Arch Comput Methods Eng 21:189–271

    MathSciNet  Article  Google Scholar 

  8. 8.

    Deb K (2001) Multi objective optimization using evolutionary algorithms. Wiley

  9. 9.

    Deb K, Datta R (2010) A fast and accurate solution of constrained optimization problems using a hybrid bi-objective and penalty function approach. In: IEEE congress on evolutionary computation IEEE, pp 1–8

  10. 10.

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

    Article  Google Scholar 

  11. 11.

    Deb K, Thiele L, Laumanns M, Zitzler E (2002b) Scalable multi-objective optimization test problems. In: Proceedings of the 2002 congress on evolutionary computation. CEC’02 (Cat. No. 02TH8600), IEEE, pp 825–830

  12. 12.

    Deb K (2000) An efficient constraint handling method for genetic algorithm. Comput Methods Appl Mech Eng 186:311–338

    Article  Google Scholar 

  13. 13.

    Dentcheva D, Wolfhagen E (2016) Two-stage optimization problems with multivariate stochastic order constraints. Math Oper Res 41:1–22

    MathSciNet  Article  Google Scholar 

  14. 14.

    Erlebach T, Kellerer H, Pferschy U (2002) Approximating multiobjective knapsack problems. Manage Sci 48:1603–1612

    Article  Google Scholar 

  15. 15.

    Fan Z, Fang Y, Li W, Lu J, Cai X, Wei C (2017) A comparative study of constrained multi-objective evolutionary algorithms on constrained multi-objective optimization problems. In: 2017 IEEE congress on evolutionary computation (CEC), 2017. IEEE, pp 209–216

  16. 16.

    Fonseca CM, Fleming PJ (1995) Multiobjective optimization and multiple constraint handling with evolutionary algorithms 1: a Unified formulation

  17. 17.

    Gandomi AH, Yang X-S, Talatahari S, Alavi AH (2013) Metaheuristic algorithms in modeling and optimization, pp 1–24

  18. 18.

    Gandomi AH, Yang X-S (2012) Evolutionary boundary constraint handling scheme. Neural Comput Appl 21:1449–1462

    Article  Google Scholar 

  19. 19.

    Herrmann JW, Lee CY, Hinchman J (1995) Global job shop scheduling with a genetic algorithm. Prod Oper Manag 4:30–45

    Article  Google Scholar 

  20. 20.

    Hiroyasu T, Miki M, Watanabe SJPOI (1999) Divided range genetic algorithms in multiobjective optimization problems. Proc IWES 99:57–65

    Google Scholar 

  21. 21.

    Hoffmeister F, Sprave J (1996) Problem-independent handling of constraints by use of metric penalty functions

  22. 22.

    Ji B, Yuan X, Yuan YJITOC (2017) Modified NSGA-II for solving continuous berth allocation problem: Using multiobjective constraint-handling strategy. IEEE Trans Cybern 47:2885–2895

    Article  Google Scholar 

  23. 23.

    Joines JA, Houck CR (1994) On the use of non-stationary penalty functions to solve nonlinear constrained optimization problems with GA’s. In: Proceedings of the First IEEE conference on evolutionary computation. IEEE world congress on computational intelligence. IEEE, pp 579–584

  24. 24.

    Jozefowiez N, Laporte G, Semet F (2012) A generic branch-and-cut algorithm for multiobjective optimization problems: application to the multilabel traveling salesman problem. INFORMS J Comput 24:554–564

    MathSciNet  Article  Google Scholar 

  25. 25.

    Köksalan M, Phelps S (2007) An evolutionary metaheuristic for approximating preference-nondominated solutions. INFORMS J Comput 19:291–301

    MathSciNet  Article  Google Scholar 

  26. 26.

    Leguizamón G, Coello CAC (2008) Boundary search for constrained numerical optimization problems with an algorithm inspired by the ant colony metaphor. IEEE Trans Evol Comput 13:350–368

    Article  Google Scholar 

  27. 27.

    Li M, Zheng J (2009) Spread assessment for evolutionary multi-objective optimization. In: International conference on evolutionary multi-criterion optimization. Springer, pp 216–230

  28. 28.

    Loganathan G, Sherali HD (1987) A convergent interactive cutting-plane algorithm for multiobjective optimization. Oper Res 35:365–377

    MathSciNet  Article  Google Scholar 

  29. 29.

    Mallipeddi R, Suganthan PN (2010) Ensemble of constraint handling techniques. IEEE Trans Evol Comput 14:561–579

    Article  Google Scholar 

  30. 30.

    Masin M, Bukchin Y (2008) Diversity maximization approach for multiobjective optimization. Oper Res 56:411–424

    MathSciNet  Article  Google Scholar 

  31. 31.

    Mete HO, Zabinsky ZB (2014) Multiobjective interacting particle algorithm for global optimization. INFORMS J Comput 26:500–513

    MathSciNet  Article  Google Scholar 

  32. 32.

    Mezura-Montes E, Coello CAC (2011) Constraint-handling in nature-inspired numerical optimization: past, present and future. Swarm Evolut Comput 1:173–194

    Article  Google Scholar 

  33. 33.

    Michalewicz Z, Schoenauer M (1996) Evolutionary algorithms for constrained parameter optimization problems. Evol Comput 4:1–32

    Article  Google Scholar 

  34. 34.

    Morales AK, Quezada CV (1998) A universal eclectic genetic algorithm for constrained optimization. In: Proceedings of the 6th European congress on intelligent techniques and soft computing, pp 518–522

  35. 35.

    Müller J (2017) Socemo: surrogate optimization of computationally expensive multiobjective problems. INFORMS J Comput 29:581–596

    MathSciNet  Article  Google Scholar 

  36. 36.

    Phelps S, Köksalan M (2003) An interactive evolutionary metaheuristic for multiobjective combinatorial optimization. Manage Sci 49:1726–1738

    Article  Google Scholar 

  37. 37.

    Rathnayake UJJOI, Sciences O (2016) Review of binary tournament constraint handling technique in NSGA II for optimal control of combined sewer systems. J Inf Optim Sci 37:37–49

    Google Scholar 

  38. 38.

    Rathnayake US, Tanyimboh TJ (2012) Optimal control of combined sewer systems using SWMM 50. WIT Trans Built Environ 122:87–96

    Article  Google Scholar 

  39. 39.

    Rauner MS, Gutjahr WJ, Heidenberger K, Wagner J, Pasia J (2010) Dynamic policy modeling for chronic diseases: metaheuristic-based identification of pareto-optimal screening strategies. Oper Res 58:1269–1286

    MathSciNet  Article  Google Scholar 

  40. 40.

    Ray T, Tai K, Seow CJEO (2001) An evolutionary algorithm for multiobjective optimization. Eng Optim 33:399–424

    Article  Google Scholar 

  41. 41.

    Richardson JT, Palmer MR, Liepins GE, Hilliard MR (1989) Some guidelines for genetic algorithms with penalty functions. In: Proceedings of the 3rd international conference on genetic algorithms, pp 191–197

  42. 42.

    Riquelme N, Von Lücken C, Baran B (2015) Performance metrics in multi-objective optimization. 2015 Latin American computing conference (CLEI). IEEE, pp 1–11

  43. 43.

    Runarsson TP, Yao X (2000) Stochastic ranking for constrained evolutionary optimization. IEEE Trans Evol Comput 4:284–294

    Article  Google Scholar 

  44. 44.

    Runarsson TP, Yao XJ (2005) Search biases in constrained evolutionary optimization. IEEE Trans Syst Man Cybern 35:233–243

    Article  Google Scholar 

  45. 45.

    Sajedi S, Huang Q, Gandomi AH, Kiani B (2017) Reliability-based multiobjective design optimization of reinforced concrete bridges considering corrosion effect. ASCE-ASME J Risk Uncert Eng Syst A Civ Eng 3:04016015

    Google Scholar 

  46. 46.

    Schott JR (1995) Fault tolerant design using single and multicriteria genetic algorithm optimization. Air Force Inst of Tech Wright-Patterson AFB OH

  47. 47.

    Snyman F, Helbig M (2017) Solving constrained multi-objective optimization problems with evolutionary algorithms. In: International conference on swarm intelligence. Springer, pp 57–66

  48. 48.

    Sourd F, Spanjaard O (2008) A multiobjective branch-and-bound framework: application to the biobjective spanning tree problem. INFORMS J Comput 20:472–484

    MathSciNet  Article  Google Scholar 

  49. 49.

    Srinivas N, Deb KJEC (1994) Muiltiobjective optimization using nondominated sorting in genetic algorithms. Evol Comput 2:221–248

    Article  Google Scholar 

  50. 50.

    Stidsen T, Andersen KA, Dammann B (2014) A branch and bound algorithm for a class of biobjective mixed integer programs. Manage Sci 60:1009–1032

    Article  Google Scholar 

  51. 51.

    Takahama T, Sakai S, Iwane N (2005) Constrained optimization by the ε constrained hybrid algorithm of particle swarm optimization and genetic algorithm. In: Australasian joint conference on artificial intelligence. Springer, pp 389–400

  52. 52.

    Tanaka M, Watanabe H, Furukawa Y, Tanino T (1995) GA-based decision support system for multicriteria optimization. In: IEEE International conference on systems man and cybernetics. Institute of electrical engineers INC (IEEE), pp 1556–1561

  53. 53.

    Tang Z, Hu X, Périaux J (2019) Multi-level hybridized optimization methods coupling local search deterministic and global search evolutionary algorithms, pp 1–37

  54. 54.

    Tian Y, Cheng R, Zhang X, Jin Y (2017) PlatEMO: a MATLAB platform for evolutionary multi-objective optimization [educational forum]. IEEE Comput Intell Mag 12:73–87

    Article  Google Scholar 

  55. 55.

    Van Veldhuizen DA (1999) Multiobjective evolutionary algorithms: classifications, analyses, and new innovations. Air Force Inst of Tech Wright-Pattersonafb Oh School of Engineering

  56. 56.

    Wang H (2012) Zigzag search for continuous multiobjective optimization. INFORMS J Comput 25:654–665

    MathSciNet  Article  Google Scholar 

  57. 57.

    Woldesenbet YG, Yen GG, Tessema BG (2009) Constraint handling in multiobjective evolutionary optimization. IEEE Cong Evol Comput 13:514–525

    Article  Google Scholar 

  58. 58.

    Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1:67–82

    Article  Google Scholar 

  59. 59.

    Xiao J, Xu J, Shao Z, Jiang C, Pan L (2007) A genetic algorithm for solving multi-constrained function optimization problems based on KS function. In: 2007 IEEE Congress on evolutionary computation. IEEE, pp 4497–4501

  60. 60.

    Yang X-S, Cui Z, Xiao R, Gandomi AH, Karamanoglu M (2013) Swarm intelligence and bio-inspired computation: theory and applications, Newnes

  61. 61.

    Zabinsky ZB (2010) Random search algorithms. Wiley Encyclopedia of Operations Research and Management Science

  62. 62.

    Zabinsky ZB (2013) Stochastic adaptive search for global optimization, Springer Science & Business Media

  63. 63.

    Zames G, Ajlouni N, Ajlouni N, Ajlouni N, Holland J, Hills W, Goldberg D (1981) Genetic algorithms in search, optimization and machine learning. Inf Technol J 3:301–302

    Google Scholar 

  64. 64.

    Zhang G, Su Z, Li M, Yue F, Jiang J, Yao XJITOR (2017) Constraint handling in NSGA-II for solving optimal testing resource allocation problems. 66:1193–1212

    Google Scholar 

  65. 65.

    Zitzler E, Thiele L (1998) An evolutionary algorithm for multiobjective optimization: the strength pareto approach, vol 43

Download references

Funding

The authors confirm that there is no source of funding for this study.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Amir H. Gandomi.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (DOCX 34 kb)

Appendix

Appendix

CF1:

$$ {\text{Min}}\quad f_{1\,\,} (x) = x_{1} + \frac{2}{{\left| {J_{1} } \right|}}\sum\limits_{{j \in J_{1} }} {\left( {x_{j} - x_{1}^{{0.5\left( {1.0 + \frac{3(j - 2)}{n - 2}} \right)}} } \right)} $$
$$ {\text{Min}}\quad f_{2\,\,} (x) = 1 - x_{1} + \frac{2}{{\left| {J_{2} } \right|}}\sum\limits_{{j \in J_{2} }} {\left( {x_{j} - x_{1}^{{0.5(1.0 + \frac{3(j - 2)}{n - 2})}} } \right)^{2} } $$
$$ {\text{s}}.{\text{t}}.\quad f_{1} + f_{2} - a\left| {\sin \left[ {N\pi (f_{1} - f_{2} + 1)} \right]} \right| - 1 \ge 0 $$

where \( J_{1} = \left\{ {j\left| {j\,{\text{is}}\,{\text{odd}}\,{\text{and}}\,2 \le j \le n} \right.} \right\} \) and \( J_{1} = \left\{ {j\left| {j\,{\text{is}}\,{\text{even}}\,{\text{and}}\quad 2 \le j \le n} \right.} \right\} \). N is an integer and \( a \ge \frac{1}{2N} \).

CF2:

$$ {\text{Min}}\quad f_{1\,\,} (x) = x_{1} + \frac{2}{{\left| {J_{1} } \right|}}\sum\limits_{{j \in J_{1} }} {\left( {x_{j} - \sin \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)} \right)^{2} } $$
$$ {\text{Min}}\quad f_{2\,\,} (x) = 1 - \sqrt {x_{1} } + \frac{2}{{\left| {J_{2} } \right|}}\sum\limits_{{j \in J_{2} }} {\left( {x_{j} - \cos \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)} \right)^{2} } $$

S.t.

$$ \frac{t}{{1 + e^{4\left| t \right|} }} \ge 0 $$

where \( J_{1} = \left\{ {j\left| {j\,{\text{is}}\,{\text{odd}}\,{\text{and}}\quad 2 \le j \le n} \right.} \right\} \) and \( J_{2} = \left\{ {j\left| {j\,{\text{is}}\,{\text{even}}\,{\text{and}}\quad 2 \le j \le n} \right.} \right\} \).

CF3:

$$ {\text{Min}}\quad f_{1} = x_{1} + \frac{2}{{\left| {J_{1} } \right|}}\left( {4\sum\limits_{{j \in J_{1} }} {y_{j}^{2} } - 2\mathop \prod \limits_{{j \in J_{1} }} \cos \left( {\frac{{20y_{j} \pi }}{\sqrt j }} \right) + 2} \right) $$
$$ {\text{Min}}\quad f_{2} = 1 - x_{1}^{2} + \frac{2}{{\left| {J_{2} } \right|}}\left( {4\sum\limits_{j \in J2} {y_{j}^{2} } - 2\mathop \prod \limits_{{j \in J_{2} }} \cos \left( {\frac{{20y_{j} \pi }}{\sqrt j }} \right) + 2} \right) $$

S.t.

$$ f_{2} + f_{1}^{2} - a\sin \left[ {N\pi (f_{1}^{2} - f_{2} + 1)} \right] - 1 \ge 0 $$

where \( y_{j} = x_{j} - \sin \left( {6\pi x_{1} + \frac{j\pi }{n}} \right),\quad j = 2, \ldots ,n \), and \( J_{1} ,J_{2} \) are same as mentioned above equations.

CF4:

$$ {\text{Min}}\quad f_{1} = x_{1} + \sum\limits_{{j \in J_{1} }} {h_{j} (y_{j} )} $$
$$ {\text{Min}}\quad f_{2} = 1 - x_{1} + \sum\limits_{{j \in J_{2} }} {h_{j} (y_{j} )} $$

S.t.

$$ \frac{t}{{1 + e^{4\left| t \right|} }} \ge 0 $$

where \( y_{j} = x_{j} - \sin \left( {6\pi x_{1} + \frac{j\pi }{n}} \right),\quad j = 2, \ldots ,n \), \( h_{2} (t) = \left\{ {\begin{array}{*{20}l} {\left| t \right|} \hfill & {{\text{if}}\;t < \frac{3}{2}\left( {1 - \frac{\sqrt 2 }{2}} \right)} \hfill \\ {0.125 + (t - 1)^{2} } \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right. \), and \( t = x_{2} - \sin \left( {6\pi x_{1} + \frac{2\pi }{n}} \right) - 0.5x_{1} + 0.25 \)\( J_{1} ,J_{2} \) are same as mentioned above equations.

CF5:

$$ {\text{Min}}\quad f_{1} = x_{1} + \sum\limits_{{j \in J_{1} }} {h_{j} (y_{j} )} $$
$$ {\text{Min}}\quad f_{2} = 1 - x_{1} + \sum\limits_{{j \in J_{2} }} {h_{j} (y_{j} )} $$

S.t.

$$ x_{2} - 0.8x_{1} \sin (6\pi x_{1} + \frac{2\pi }{n}) - 0.5x_{1} + 0.25 \ge 0 $$

where \( y_{j} = \left\{ {\begin{array}{*{20}l} {x_{j} - 0.8x_{1} \cos \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)\quad {\text{if}}\;j \in J_{1} } \hfill \\ {x_{j} - 0.8x_{1} \sin \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)\quad {\text{if}}\;j \in J_{2} } \hfill \\ \end{array} } \right. \) and \( h_{2} (t) = \left\{ {\begin{array}{*{20}l} {\left| t \right|} \hfill & {{\text{if}}\;t < \frac{3}{2}\left( {1 - \frac{\sqrt 2 }{2}} \right)} \hfill \\ {0.125 + (t - 1)^{2} } \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right. \)\( h_{j} (t) = 2t^{2} - \cos (4\pi t) + 1 \) for j = 3,4,…,n

CF6:

$$ {\text{Min}}\quad f_{1} = x_{1} + \sum\limits_{{j \in J_{1} }} {y_{j}^{2} } $$
$$ {\text{Min}}\quad f_{2} = (1 - x_{1} )^{2} + \sum\limits_{{j \in J_{2} }} {y_{j} }^{2} $$

S.t:

$$ x_{2} - 0.8x_{1} \sin \left( {6\pi x_{1} + \frac{2\pi }{n}} \right) - sign(0.5(1 - x_{1} ) - (1 - x_{1} )^{2} )\sqrt {\left| {0.5(1 - x_{1} ) - (1 - x_{1} )^{2} } \right|} \ge 0 $$
$$ x_{4} - 0.8x_{1} \sin \left( {6\pi x_{1} + \frac{4\pi }{n}} \right) - sign(0.25(1 - x_{1} ) - 0.5(1 - x_{1} )^{2} )\sqrt {\left| {0.25(1 - x_{1} ) - 0.5(1 - x_{1} )^{{}} } \right|} \ge 0 $$
$$ y_{j} = \left\{ {\begin{array}{*{20}l} {x_{j} - 0.8x_{1} \cos \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)\quad {\text{if}}\;j \in J_{1} } \hfill \\ {x_{j} - 0.8x_{1} \sin \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)\quad {\text{if}}\;j \in J_{2} } \hfill \\ \end{array} } \right. $$

CF7:

$$ {\text{Min}}\quad f_{1} = x_{1} + \sum\limits_{{j \in J_{1} }} {h_{j} (y_{j} )} $$
$$ {\text{Min}}\quad f_{2} = 1 - x_{1} + \sum\limits_{{j \in J_{2} }} {h_{j} (y_{j} )} $$
$$ y_{j} = \left\{ {\begin{array}{*{20}l} {x_{j} - 0.8x_{1} \cos \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)\quad {\text{if}}\;j \in J_{1} } \hfill \\ {x_{j} - 0.8x_{1} \sin \left( {6\pi x_{1} + \frac{j\pi }{n}} \right)\quad {\text{if}}\;j \in J_{2} } \hfill \\ \end{array} } \right. $$

S.t.

$$ x_{2} - \sin (6\pi x_{1} + \frac{2\pi }{n}) - sign(0.5(1 - x_{1} ) - (1 - x_{1} )^{2} )\sqrt {\left| {0.5(1 - x_{1} ) - (1 - x_{1} )^{2} } \right|} \ge 0 $$
$$ x_{4} - \sin (6\pi x_{1} + \frac{4\pi }{n}) - sign(0.25\sqrt {(1 - x_{1} )} - 0.5(1 - x_{1} )^{{}} )\sqrt {\left| {0.5(1 - x_{1} ) - (1 - x_{1} )^{2} } \right|} \ge 0 $$

where \( h_{2} (t) = h_{4} (t) = t^{2} \) and \( h_{j} (t) = 2t^{2} - \cos (4\pi t) + 1 \) for j = 3,5,6,…,n.

CF8:

$$ \begin{aligned} Min\,f_{1} = \cos (0.5x_{1} \pi )\cos (0.5x_{2} \pi ) + \frac{2}{{\left| {J_{1} } \right|}}\sum\limits_{{j \in J_{1} }} {\left( {x_{j} - 2x_{2} \sin \left( {2\pi x_{1} + \frac{j\pi }{n}} \right)} \right)^{2} } \hfill \\ Min\,f_{2} = \cos (0.5x_{1} \pi )\sin (0.5x_{2} \pi ) + \frac{2}{{\left| {J_{2} } \right|}}\sum\limits_{{j \in J_{2} }} {\left( {x_{j} - 2x_{2} \sin \left( {2\pi x_{1} + \frac{j\pi }{n}} \right)} \right)^{2} } \hfill \\ Min\,f_{3} = \sin (0.5x_{1} \pi ) + \frac{2}{{\left| {J_{3} } \right|}}\sum\limits_{{j \in J_{3} }} {\left( {x_{j} - 2x_{2} \sin \left( {2\pi x_{1} + \frac{j\pi }{n}} \right)} \right)^{2} } \hfill \\ \end{aligned} $$

S.t.

$$ \frac{{f_{1}^{2} + f_{2}^{2} }}{{1 - f_{3}^{2} }} - a\left| {\sin \left[ {N\pi \left( {\frac{{f_{1}^{2} + f_{2}^{2} }}{{1 - f_{3}^{2} }} + 1} \right)} \right]} \right| - 1 \ge 0 $$

where

$$ \begin{aligned} J_{1} = \left\{ {\left. j \right|3 \le j \le n,and\,j - 1\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ J_{2} = \left\{ {\left. j \right|3 \le j \le n,and\,j - 2\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ J_{3} = \left\{ {\left. j \right|3 \le j \le n,and\,j\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ \end{aligned} $$

CF9:

$$ \begin{aligned} Min\,f_{1} = \cos (0.5x_{1} \pi )\cos (0.5x_{2} \pi ) + \frac{2}{{\left| {J_{1} } \right|}}\sum\limits_{{j \in J_{1} }} {\left( {x_{j} - 2x_{2} \sin \left( {2\pi x_{1} + \frac{j\pi }{n}} \right)} \right)^{2} } \hfill \\ Min\,f_{2} = \cos (0.5x_{1} \pi )\sin (0.5x_{2} \pi ) + \frac{2}{{\left| {J_{2} } \right|}}\sum\limits_{{j \in J_{2} }} {\left( {x_{j} - 2x_{2} \sin \left( {2\pi x_{1} + \frac{j\pi }{n}} \right)} \right)^{2} } \hfill \\ Min\,f_{3} = \sin (0.5x_{1} \pi ) + \frac{2}{{\left| {J_{3} } \right|}}\sum\limits_{{j \in J_{3} }} {\left( {x_{j} - 2x_{2} \sin \left( {2\pi x_{1} + \frac{j\pi }{n}} \right)} \right)^{2} } \hfill \\ \end{aligned} $$

S.t.

$$ \frac{{f_{1}^{2} + f_{2}^{2} }}{{1 - f_{3}^{2} }} - a\sin \left[ {N\pi \left( {\frac{{f_{1}^{2} + f_{2}^{2} }}{{1 - f_{3}^{2} }} + 1} \right)} \right] - 1 \ge 0 $$

where

$$ \begin{aligned} J_{1} = \left\{ {\left. j \right|3 \le j \le n,and\,j - 1\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ J_{2} = \left\{ {\left. j \right|3 \le j \le n,and\,j - 2\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ J_{3} = \left\{ {\left. j \right|3 \le j \le n,and\,j\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ \end{aligned} $$

CF10:

$$ \begin{aligned} Min\,f_{1} = \cos (0.5x_{1} \pi )\cos (0.5x_{2} \pi ) + \frac{2}{{\left| {J_{1} } \right|}}\sum\limits_{{j \in J_{1} }} {\left[ {4y_{j}^{2} - \cos (8\pi y_{j} ) + 1} \right]} \hfill \\ Min\,f_{2} = \cos (0.5x_{1} \pi )\sin (0.5x_{2} \pi ) + \frac{2}{{\left| {J_{2} } \right|}}\sum\limits_{{j \in J_{2} }} {\left[ {4y_{j}^{2} - \cos (8\pi y_{j} ) + 1} \right]} \hfill \\ Min\,f_{3} = \sin (0.5x_{1} \pi ) + \frac{2}{{\left| {J_{3} } \right|}}\sum\limits_{{j \in J_{3} }} {\left[ {4y_{j}^{2} - \cos (8\pi y_{j} ) + 1} \right]} \hfill \\ \end{aligned} $$

S.t.

$$ \frac{{f_{1}^{2} + f_{2}^{2} }}{{1 - f_{3}^{2} }} - a\sin \left[ {N\pi (\frac{{f_{1}^{2} + f_{2}^{2} }}{{1 - f_{3}^{2} }} + 1)} \right] - 1 \ge 0 $$

where

$$ \begin{aligned} J_{1} = \left\{ {\left. j \right|3 \le j \le n,and\,j - 1\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ J_{2} = \left\{ {\left. j \right|3 \le j \le n,and\,j - 2\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ J_{3} = \left\{ {\left. j \right|3 \le j \le n,and\,j\,is\,a\,multiplication\,of\,3} \right\} \hfill \\ \end{aligned} $$

and \( y_{j} = x_{j} - 2x_{2} \sin \left( {2\pi x_{1} + \frac{j\pi }{n}} \right) \)

C1_DTLZ1:

Regarding to [11]:DTLZ1 test problem is as follows:

$$ \begin{aligned} Min\,f_{1} (x) = \frac{1}{2}x_{1} x_{2} x_{M - 1} (1 + g(x_{M} )) \hfill \\ Min\,f_{2} (x) = \frac{1}{2}x_{1} x_{2} (1 - x_{M - 1} )(1 + g(x_{M} )) \hfill \\ \end{aligned} $$
$$ \begin{aligned} Min\,f_{M - 1} (x) = \frac{1}{2}x_{1} (1 - x_{2} )(1 + g(x_{M} )) \hfill \\ Min\,f_{M} (x) = \frac{1}{2}(1 - x_{1} )(1 + g(x_{M} )) \hfill \\ 0 \le x_{i} \le 1 \hfill \\ \end{aligned} $$

where \( g(x_{M} ) = 100(\left| {x_{M} } \right| + \sum\limits_{{x_{i} \in x_{M} }} {(x_{i} - 0.5)^{2} - \cos (20\pi (x_{i} - 0.5)))} \). Objective functions of C1-DTLZ1 is same as DTLZ1, only one constraint is added as follows:

$$ c(x) = 1 - \frac{{f_{M} (x)}}{0.6} - \sum\nolimits_{i = 1}^{M - 1} {\frac{{f_{i} (x)}}{0.5}} \ge 0 $$

C2_DTLZ2:

$$ \begin{aligned} Min\,f_{1} (x) = (1 + g(x_{M} ))\cos (x_{1} \pi /2) \ldots \cos (x_{M - 1} \pi /2) \hfill \\ Min\,f_{2} (x) = (1 + g(x_{M} ))\cos (x_{1} \pi /2) \ldots \sin (x_{M - 1} \pi /2) \hfill \\ \end{aligned} $$
$$ Min\,f_{M(x)} = (1 + g(x_{M} ))\sin (x_{1} \pi /2) $$
$$ 0 \le x_{i} \le 1,\;{\text{for}}\;i = 1, \ldots ,n $$
$$ g(x_{M} ) = \sum\limits_{{x_{i} \in X_{M} }} {(x_{i} - 0.5)^{2} } $$

Objective functions of C2-DTLZ2 is same as DTLZ2, only one constraint is added as follows:

$$ c(x) = \hbox{max} \left\{ {\max_{i = 1}^{M} } \right.[(f_{i} (x) - 1)^{2} + \sum\limits_{j = 1,j \ne i}^{M} {f_{j}^{2} } - r^{2} \,\left. {} \right],[\sum\limits_{i = 1}^{M} {(f_{i}^{{}} } (x) - 1/\sqrt M )^{2} - r^{2} ]\left. {} \right\} $$

where r = 0.4

C3_DTLZ4:

In [11], DTLZ2 has been modified by mapping a meta-variable:\( x_{i} \to x_{i}^{\alpha } \) where \( \alpha = 100 \).

$$ c(x) = \frac{{f_{j}^{2} }}{4} + \sum\limits_{i = 1,i \ne j}^{M} {f_{i}^{2} (x) - 1 \ge 0,\quad \forall j = 1, \ldots ,M} $$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hobbie, J.G., Gandomi, A.H. & Rahimi, I. A Comparison of Constraint Handling Techniques on NSGA-II. Arch Computat Methods Eng (2021). https://doi.org/10.1007/s11831-020-09525-y

Download citation