March  2021, 17(2): 1001-1023. doi: 10.3934/jimo.2020009

Non-dominated sorting methods for multi-objective optimization: Review and numerical comparison

1. 

School of science, Southwest University of Science and Technology, Mianyang 621010, China

2. 

School of Management, Guangzhou University, Guangzhou 510006, China

* Corresponding author: Changzhi Wu, C.Wu@exchange.curtin.edu.au

Received  October 2018 Revised  September 2019 Published  January 2020

In multi-objective evolutionary algorithms (MOEAs), non-domina-ted sorting is one of the critical steps to locate efficient solutions. A large percentage of computational cost of MOEAs is on non-dominated sorting for it involves numerous comparisons. By now, there are more than ten different non-dominated sorting algorithms, but their numerical performance comparing with each other is not clear yet. It is necessary to investigate the advantage and disadvantage of these algorithms and consequently give suggestions to specific users and algorithm designers. Therefore, a comprehensively numerical study of non-dominated sorting algorithms is presented in this paper. Firstly, we design a population generator. This generator can generate populations with specific features, such as population size, number of Pareto fronts and number of points in each Pareto front. Then non-dominated sorting algorithms were tested using populations generated in certain structures, and results were compared with respect to number of comparisons and time consumption. Furthermore, In order to compare the performance of sorting algorithms in MOEAs, we embed them into a specific MOEA, dynamic sorting genetic algorithm (DSGA), and use these variations of DSGA to solve some multi-objective benchmarks. Results show that dominance degree sorting outperforms the other methods, fast non-dominance sorting performs the worst and the other sorting algorithms performs equally.

Citation: Qiang Long, Xue Wu, Changzhi Wu. Non-dominated sorting methods for multi-objective optimization: Review and numerical comparison. Journal of Industrial & Management Optimization, 2021, 17 (2) : 1001-1023. doi: 10.3934/jimo.2020009
References:
[1]

A. Cheng and L. Cheng-Chew, Optimizing System-On-Chip verifications with multi-objective genetic evolutionary algorithms, Journal of Industrial and Management Optimization, 10 (2014), 383-396.  doi: 10.3934/jimo.2014.10.383.  Google Scholar

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H. B. FangQ. WangY. C. Tu and M. F. Horstemeyer, An efficient non-dominated sorting method for evolutionary algorithms, Evolutionary Computation, 16 (2008), 355-384.   Google Scholar

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F. A. Fortin, S. Grenier and M. Parizeau, Generalizing the Improved Run-time Complexity Algorithm for Non-Dominated Sorting, Proceedings of the 15th annual conference on Genetic and evolutionary computation, 2013. Google Scholar

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J. Himanshu and K. Deb, An evolutionary many-objective optimization algorithm using reference-point based nondominated sorting approach, part II: Handling constraints and extending to an adaptive approach, IEEE Transations Evolutionary Computation, 18 (2014), 602-622.   Google Scholar

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M. T. Jensen, Reducing the run-time complexity of multiobjective EAs: The NSGA-II and other algorithms, IEEE Transactions on Evolutionary Computation, 7 (2003), 503-515.   Google Scholar

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M. Kent and E. Keedwell, Deductive sort and climbing sort: New methods for non-dominated sorting, Evolutionary Computation, 20 (2012), 1-26.   Google Scholar

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T. C. Koopmans and others, Activity Analysis of Production and Allocation, , Wiley New York, 1951. Google Scholar

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[14]

Q. Long, W. N. Xu and K. Q. Zhao, Dynamic sorting genetic algorithm for multi-objective optimization, Swarm and Evolutionary Computation, (2017). Google Scholar

[15]

B. Maxim and A. Shalyto, A provably asymptotically fast version of the generalized Jensen algorithm for non-dominated sorting, International Conference on Parallel Problem Solving from Nature, Springer, Cham, 2014. Google Scholar

[16]

S. Nidamarthi and K. Deb, Muiltiobjective optimization using nondominated sorting in genetic algorithms, Evolutionary Computation, 2 (1994), 221-248.   Google Scholar

[17]

G. Patrik and A. Syberfeldt, A new algorithm using the non-dominated tree to improve non-dominated sorting, Evolutionary Computation, 26 (2018), 89-116.   Google Scholar

[18]

K. Samuel and J. Gillis, Mathematical methods and theory in games, programming, and economics, Physics Today, 13 (1960), 54. Google Scholar

[19]

C. ShiZ. Y. YanZ. Z. Shi and L. Zhang, A fast multi-objective evolutionary algorithm based on a tree structure, Applied Soft Computing, 10 (2010), 468-480.   Google Scholar

[20]

N. Srinivas and K. Deb, Muiltiobjective optimization using nondominated sorting in genetic algorithms, Evolutionary Computation, 2 (1994), 221-248.   Google Scholar

[21]

S. Q. Tang, Z. X. Cai and J. H. Zheng, A fast method of constructing the non-dominated set: Arena's principle, The Fourth International Conference on Natural Computation, 2008. Google Scholar

[22]

C. K. Vira and Y. Y. Haimes, Multiobjective Decision Making: Theory and Methodology, Courier Dover Publications, 2008. Google Scholar

[23]

H. D. Wang and Y. Xin, Corner sort for Pareto-based many-objective optimization, IEEE Transactions on Cybernetics, 44 (2014), 92-102.   Google Scholar

[24]

J. XiongZ. B. ZhouK. TianT. J. Liao and J. M. Shi, A multi-objective approach for weapon selection and planning problems in dynamic environments, Journal of Industrial and Management Optimization, 13 (2017), 1189-1211.  doi: 10.3934/jimo.2016068.  Google Scholar

[25]

X. Y. ZhangY. TianR. Cheng and Y. C. jin, An efficient approach to nondominated sorting for evolutionary multiobjective optimization, IEEE Transactions on Evolutionary Computation, 19 (2015), 201-213.   Google Scholar

[26]

X. Y. Zhang, Y. Tian, R. Cheng and Y. C. Jin, Empirical analysis of a tree-based efficient non-dominated sorting approach for many-objective optimization, Computational Intelligence, IEEE, 2016. Google Scholar

[27]

L. ZhangJ. Zhang and Y. Zhang, Second-order optimality conditions for cone constrained multi-objective optimization, Journal of Industrial and Management Optimization, 14 (2018), 1041-1054.  doi: 10.3934/jimo.2017089.  Google Scholar

[28]

Y. R. ZhouZ. F. Chen and J. Zhang, Ranking vectors by means of the dominance degree matrix, IEEE Transactions on Evolutionary Computation, 21 (2017), 34-51.   Google Scholar

[29]

E. Zitzler and L. Thiele, Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach, IEEE Transations Evolutionary Computation, 3 (1999), 257-271.   Google Scholar

[30]

E. Zitzler, M. Laumanns and L. Thiele, SPEA2: Improving the strength pareto evolutionary algorithm for multiobjective optimization., Fifth Conference on Evolutionary Methods for Design, Optimization and Control with Applications to Industrial Problems, (2001), 95–100. Google Scholar

show all references

References:
[1]

A. Cheng and L. Cheng-Chew, Optimizing System-On-Chip verifications with multi-objective genetic evolutionary algorithms, Journal of Industrial and Management Optimization, 10 (2014), 383-396.  doi: 10.3934/jimo.2014.10.383.  Google Scholar

[2]

D. W. Corne, J. D. Knowles and M. J. Oates, The pareto envelope-based selection algorithm for multiobjective optimization, Parallel Problem Solving from Nature PPSN VI. Springer, (2000), 839–848. Google Scholar

[3]

D. W. Corne, N. R. Jerram, J. D. Knowles and M. J. Oates, PESA-II: Region-based selection in evolutionary multi-objective optimization, Genetic and Evolutionary Computation Conference, (2001), 283–290. Google Scholar

[4]

K. Deb and J. Himanshu, An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: solving problems with box constraints, IEEE Transactions Evolutionary Computation, 18 (2014), 577-601.   Google Scholar

[5]

K. DebA. PratapS. Agarwal and T. A. M. T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197.   Google Scholar

[6]

M. DrozdikA. YouheiA. Hernan and T. Kiyoshi, Computational cost reduction of nondominated sorting using the M-front, IEEE Transactions on Evolutionary Computation, 19 (2015), 659-678.   Google Scholar

[7]

H. B. FangQ. WangY. C. Tu and M. F. Horstemeyer, An efficient non-dominated sorting method for evolutionary algorithms, Evolutionary Computation, 16 (2008), 355-384.   Google Scholar

[8]

F. A. Fortin, S. Grenier and M. Parizeau, Generalizing the Improved Run-time Complexity Algorithm for Non-Dominated Sorting, Proceedings of the 15th annual conference on Genetic and evolutionary computation, 2013. Google Scholar

[9]

J. Himanshu and K. Deb, An evolutionary many-objective optimization algorithm using reference-point based nondominated sorting approach, part II: Handling constraints and extending to an adaptive approach, IEEE Transations Evolutionary Computation, 18 (2014), 602-622.   Google Scholar

[10]

M. T. Jensen, Reducing the run-time complexity of multiobjective EAs: The NSGA-II and other algorithms, IEEE Transactions on Evolutionary Computation, 7 (2003), 503-515.   Google Scholar

[11]

M. Kent and E. Keedwell, Deductive sort and climbing sort: New methods for non-dominated sorting, Evolutionary Computation, 20 (2012), 1-26.   Google Scholar

[12]

T. C. Koopmans and others, Activity Analysis of Production and Allocation, , Wiley New York, 1951. Google Scholar

[13]

H. T. KungL. Fabrizio and F. P. Franco, On finding the maxima of a set of vectors, Journal of the ACM (JACM), 22 (1975), 469-476.  doi: 10.1145/321906.321910.  Google Scholar

[14]

Q. Long, W. N. Xu and K. Q. Zhao, Dynamic sorting genetic algorithm for multi-objective optimization, Swarm and Evolutionary Computation, (2017). Google Scholar

[15]

B. Maxim and A. Shalyto, A provably asymptotically fast version of the generalized Jensen algorithm for non-dominated sorting, International Conference on Parallel Problem Solving from Nature, Springer, Cham, 2014. Google Scholar

[16]

S. Nidamarthi and K. Deb, Muiltiobjective optimization using nondominated sorting in genetic algorithms, Evolutionary Computation, 2 (1994), 221-248.   Google Scholar

[17]

G. Patrik and A. Syberfeldt, A new algorithm using the non-dominated tree to improve non-dominated sorting, Evolutionary Computation, 26 (2018), 89-116.   Google Scholar

[18]

K. Samuel and J. Gillis, Mathematical methods and theory in games, programming, and economics, Physics Today, 13 (1960), 54. Google Scholar

[19]

C. ShiZ. Y. YanZ. Z. Shi and L. Zhang, A fast multi-objective evolutionary algorithm based on a tree structure, Applied Soft Computing, 10 (2010), 468-480.   Google Scholar

[20]

N. Srinivas and K. Deb, Muiltiobjective optimization using nondominated sorting in genetic algorithms, Evolutionary Computation, 2 (1994), 221-248.   Google Scholar

[21]

S. Q. Tang, Z. X. Cai and J. H. Zheng, A fast method of constructing the non-dominated set: Arena's principle, The Fourth International Conference on Natural Computation, 2008. Google Scholar

[22]

C. K. Vira and Y. Y. Haimes, Multiobjective Decision Making: Theory and Methodology, Courier Dover Publications, 2008. Google Scholar

[23]

H. D. Wang and Y. Xin, Corner sort for Pareto-based many-objective optimization, IEEE Transactions on Cybernetics, 44 (2014), 92-102.   Google Scholar

[24]

J. XiongZ. B. ZhouK. TianT. J. Liao and J. M. Shi, A multi-objective approach for weapon selection and planning problems in dynamic environments, Journal of Industrial and Management Optimization, 13 (2017), 1189-1211.  doi: 10.3934/jimo.2016068.  Google Scholar

[25]

X. Y. ZhangY. TianR. Cheng and Y. C. jin, An efficient approach to nondominated sorting for evolutionary multiobjective optimization, IEEE Transactions on Evolutionary Computation, 19 (2015), 201-213.   Google Scholar

[26]

X. Y. Zhang, Y. Tian, R. Cheng and Y. C. Jin, Empirical analysis of a tree-based efficient non-dominated sorting approach for many-objective optimization, Computational Intelligence, IEEE, 2016. Google Scholar

[27]

L. ZhangJ. Zhang and Y. Zhang, Second-order optimality conditions for cone constrained multi-objective optimization, Journal of Industrial and Management Optimization, 14 (2018), 1041-1054.  doi: 10.3934/jimo.2017089.  Google Scholar

[28]

Y. R. ZhouZ. F. Chen and J. Zhang, Ranking vectors by means of the dominance degree matrix, IEEE Transactions on Evolutionary Computation, 21 (2017), 34-51.   Google Scholar

[29]

E. Zitzler and L. Thiele, Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach, IEEE Transations Evolutionary Computation, 3 (1999), 257-271.   Google Scholar

[30]

E. Zitzler, M. Laumanns and L. Thiele, SPEA2: Improving the strength pareto evolutionary algorithm for multiobjective optimization., Fifth Conference on Evolutionary Methods for Design, Optimization and Control with Applications to Industrial Problems, (2001), 95–100. Google Scholar

Figure 1.  Cases of dominance comparisons
Figure 2.  Generate a point belonging to $ \mathcal{F}_2 $
Figure 3.  An example of fixed features population generator
Figure 4.  Time consumption for series (ⅰ)
Figure 5.  Number of comparisons for series (ⅰ)
Figure 6.  Time consumption for series (ⅱ)
Figure 7.  Number of comparisons for series (ⅱ)
Figure 8.  Time consumption for series (ⅲ)
Figure 9.  Number of comparisons for series (ⅲ)
Figure 10.  Time consumption for series (ⅳ)
Figure 11.  Number of comparisons for series (ⅳ)
Figure 12.  Time consumption for series (ⅴ)
Figure 13.  Number of comparisons for series (ⅴ)
Figure 14.  Average time consumption for algorithms
Figure 15.  Average number of comparison for algorithms
Figure 16.  Average Comparison efficiency for algorithms
Figure 17.  Objective function value space
Figure 18.  Numerical performance on SCH
Figure 19.  Numerical performance on FON
Figure 20.  Numerical performance on KUR
Table 1.  Five series of populations
Series No. Description $ m $ $ k $ $ N $
Series (ⅰ) fixed $ m $ 3 1 $ N=(200) $
various $ k $ 3 2 $ N=(100,100) $
$ \sum N=200 $ 3 3 $ N=(70,70,60) $
3 4 $ N=(50,50,50,50) $
3 5 $ N=(40,40,40,40,40) $
3 6 $ N=(33,33,33,33,33,35) $
Series (ⅱ) fixed $ m $ 3 5 $ N=(10,10,10,10,10) $
fixed $ k $ 3 5 $ N=(20,20,20,20,20) $
various $ N $ 3 5 $ N=(30,30,30,30,30) $
3 5 $ N=(40,40,40,40,40) $
3 5 $ N=(50,50,50,50,50) $
3 5 $ N=(60,60,60,60,60) $
Series (ⅲ) various $ m $ 2 5 $ N=(20,20,20,20,20) $
fixed $ k $ 3 5 $ N=(20,20,20,20,20) $
fixed $ N $ 4 5 $ N=(20,20,20,20,20) $
5 5 $ N=(20,20,20,20,20) $
6 5 $ N=(20,20,20,20,20) $
7 5 $ N=(20,20,20,20,20) $
Series (ⅳ) fixed $ m $ 3 1 $ N=50 $
fixed $ k $ 3 1 $ N=100 $
various $ N $ 3 1 $ N=150 $
3 1 $ N=200 $
3 1 $ N=250 $
3 1 $ N=300 $
Series (ⅴ) fixed $ m $ 3 10 $ N_i=1,\; i=1,\cdots,k $
various $ k $ 3 20 $ N_i=1,\; i=1,\cdots,k $
various $ N $ 3 30 $ N_i=1,\; i=1,\cdots,k $
3 40 $ N_i=1,\; i=1,\cdots,k $
3 50 $ N_i=1,\; i=1,\cdots,k $
3 60 $ N_i=1,\; i=1,\cdots,k $
Series (vi) fixed $ m $ 3 5 $ N_i $ is a
fixed $ k $ 3 5 random integer
various $ N $ 3 5 between 1 and 50
Series No. Description $ m $ $ k $ $ N $
Series (ⅰ) fixed $ m $ 3 1 $ N=(200) $
various $ k $ 3 2 $ N=(100,100) $
$ \sum N=200 $ 3 3 $ N=(70,70,60) $
3 4 $ N=(50,50,50,50) $
3 5 $ N=(40,40,40,40,40) $
3 6 $ N=(33,33,33,33,33,35) $
Series (ⅱ) fixed $ m $ 3 5 $ N=(10,10,10,10,10) $
fixed $ k $ 3 5 $ N=(20,20,20,20,20) $
various $ N $ 3 5 $ N=(30,30,30,30,30) $
3 5 $ N=(40,40,40,40,40) $
3 5 $ N=(50,50,50,50,50) $
3 5 $ N=(60,60,60,60,60) $
Series (ⅲ) various $ m $ 2 5 $ N=(20,20,20,20,20) $
fixed $ k $ 3 5 $ N=(20,20,20,20,20) $
fixed $ N $ 4 5 $ N=(20,20,20,20,20) $
5 5 $ N=(20,20,20,20,20) $
6 5 $ N=(20,20,20,20,20) $
7 5 $ N=(20,20,20,20,20) $
Series (ⅳ) fixed $ m $ 3 1 $ N=50 $
fixed $ k $ 3 1 $ N=100 $
various $ N $ 3 1 $ N=150 $
3 1 $ N=200 $
3 1 $ N=250 $
3 1 $ N=300 $
Series (ⅴ) fixed $ m $ 3 10 $ N_i=1,\; i=1,\cdots,k $
various $ k $ 3 20 $ N_i=1,\; i=1,\cdots,k $
various $ N $ 3 30 $ N_i=1,\; i=1,\cdots,k $
3 40 $ N_i=1,\; i=1,\cdots,k $
3 50 $ N_i=1,\; i=1,\cdots,k $
3 60 $ N_i=1,\; i=1,\cdots,k $
Series (vi) fixed $ m $ 3 5 $ N_i $ is a
fixed $ k $ 3 5 random integer
various $ N $ 3 5 between 1 and 50
Table 2.  Multi-objective test problems
Pro. $ n $ Variable Objective
bounds functions
SCH 1 $ [-5,10] $ $ \begin{array}{l}f_1(x)=x^2 \\f_2(x)=(x-2)^2\end{array} $
FON 3 $ [-4,4] $ $ \begin{array}{l}f_1(x)=1-\exp(-\sum_{i=1}^3(x_i-\frac{1}{\sqrt{3}})^2)\\f_2(x)=1-\exp(-\sum_{i=1}^3(x_i+\frac{1}{\sqrt{3}})^2)\end{array} $
KUR 3 $ [-5,5] $ $ \begin{array}{l}f_1(x)=\sum_{i=1}^{n-1}(-10\exp(-0.2\sqrt{x_i^2+x_{i+1}^2}\; ))\\ f_2(x)=\sum_{i=1}^n(|x_i|^{0.8}+5\sin^3(x_i))\end{array} $
Pro. $ n $ Variable Objective
bounds functions
SCH 1 $ [-5,10] $ $ \begin{array}{l}f_1(x)=x^2 \\f_2(x)=(x-2)^2\end{array} $
FON 3 $ [-4,4] $ $ \begin{array}{l}f_1(x)=1-\exp(-\sum_{i=1}^3(x_i-\frac{1}{\sqrt{3}})^2)\\f_2(x)=1-\exp(-\sum_{i=1}^3(x_i+\frac{1}{\sqrt{3}})^2)\end{array} $
KUR 3 $ [-5,5] $ $ \begin{array}{l}f_1(x)=\sum_{i=1}^{n-1}(-10\exp(-0.2\sqrt{x_i^2+x_{i+1}^2}\; ))\\ f_2(x)=\sum_{i=1}^n(|x_i|^{0.8}+5\sin^3(x_i))\end{array} $
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