Computational Intelligence - November 2017 - 97
50
50
40
Objective Value
Objective Value
60
40
30
20
10
0
1
c
20
10
1
2
3
Number of Objective
(b)
4
50
Objective Value
Objective Value
50
b
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0
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Number of Objective
(a)
a
40
30
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0
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Number of Objective
(c)
4
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(d)
4
FIGURE 23 Examples of the parallel coordinates plot in MS Excel, MATLAB, LaTeX, and Origin.
(a) MS Excel, (b) MATLAB, (c) LaTex and (d) Origin.
features of a many-objective solution set
into easily seen 2D patterns.
❏ The order of objective axes matters in
parallel coordinates. To better present
the relationship between objectives, it
may need to be rearranged according
to features of the solution set at hand.
Our subsequent study is towards overcoming/alleviating the difficulties of
interpreting the parallel coordinates plot
presented in this paper. Particularly, how to
arrange the order of objectives will be the
focus of our future work as it had presented
its usefulness in the paper. In this regard, a
straightforward thought is to place the most
conflicting objectives or the most harmonious objectives together so that people could
see some meaningful patterns (such as the
examples in Figures 21 and 22). Another
thought is to consider the coverage of the
lines between objectives in a parallel coordinates plot; people may acquire more
information from less coverage of the lines,
for example, after exchanging the order of
objectives f1 and f2 in Figure 8.
Acknowledgment
The authors would like to thank the Associate Editor and the three reviewers for
their thoughtful suggestions and constructive comments. This work was supported
in part by the Engineering and Physical
Sciences Research Council (EPSRC) of
U.K. under Grants EP/K001523/1 and
EP/J017515/1, and National Natural Science Foundation of China (NSFC) under
Grants 61329302 and 61403326. X. Yao
was also supported by a Royal Society
Wolfson Research Merit Award.
References
[1] A. Inselberg, "The plane with parallel coordinates,"
Visual Comput., vol. 1, no. 2, pp. 69-91, Aug. 1985.
[2] E. J. Wegman, "Hyperdimensional data analysis using
parallel coordinates," J. Am. Stat. Assoc., vol. 85, no. 411,
pp. 664-675, Sept. 1990.
[3] A. Inselberg, Parallel Coordinates. Boston, MA: Springer, 2009, pp. 2018-2024.
[4] K. Miettinen, "Survey of methods to visualize alternatives in multiple criteria decision making problems,"
OR Spectr., vol. 36, no. 1, pp. 3-37, Jan. 2014.
[5] C. M. Fonseca and P. J. Fleming, "Multiobjective
optimization and multiple constraint handling with evolutionary algorithms. I. A unified formulation," IEEE
Trans. Syst., Man, Cybernet. A: Syst. Humans, vol. 28, no.
1, pp. 26-37, Jan. 1998.
[6] K. Deb, Multi-Objective Optimization Using Evolutionary
Algorithms. New York: Wiley, 2001.
[7] H. Ishibuchi, N. Tsukamoto, and Y. Nojima, "Evolutionary many-objective optimization: A short review," in
Proc. IEEE Congr. Evolutionary Computation, Hong Kong,
China, June 2008, pp. 2419-2426.
[8] R. C. Purshouse and P. J. Fleming, "Evolutionary
many-objective optimisation: An exploratory analysis,"
in Proc. IEEE Congr. Evolutionary Computation, Canberra,
Australia, Dec. 2003, vol. 3, pp. 2066-2073.
[9] P. Fleming, R. Purshouse, and R. Lygoe, "Manyobjective optimization: An engineering design perspective," in Proc. 3rd Int. Conf. Evolutionary Multi-Criterion
Optimization, Guanajuato, Mexico, Mar. 2005, pp. 14-32.
[10] D. J. Walker, R. Everson, and J. E. Fieldsend, "Visualizing mutually nondominating solution sets in manyobjective optimization," IEEE Trans. Evol. Comput., vol.
17, no. 2, pp. 165-184, Apr. 2013.
[11] T. Tuar and B. Filipi, "Visualization of Pareto front approximations in evolutionary multiobjective optimization:
A critical review and the prosection method," IEEE Trans.
Evol. Comput., vol. 19, no. 2, pp. 225-245, Apr. 2015.
[12] Z. He and G. G. Yen, "Visualization and performance metric in many-objective optimization," IEEE
Trans. Evol. Comput., vol. 20, no. 3, pp. 386-402, June 2016.
[13] T. Wagner, N. Beume, and B. Naujoks, "Pareto-, aggregation-, and indicator-based methods in many-objective
optimization," in Proc. 4th Int. Conf. Evolutionary MultiCriterion Optimization, Matsushima, Japan, Mar. 2007,
pp. 742-756.
[14] H. K. Singh, A. Isaacs, and T. Ray, "A Pareto corner
search evolutionary algorithm and dimensionality reduction in many-objective optimization problems," IEEE
Trans Evol. Comput., vol. 15, no. 4, pp. 539-556, Aug. 2011.
[15] M. Li, S. Yang, X. Liu, and R. Shen, "A comparative
study on evolutionary algorithms for many-objective optimization," in Proc. 7th Int. Conf. Evolutionary Multi-Criterion
Optimization, Sheffield, U.K., Mar. 2013, pp. 261-275.
[16] U. K. Wickramasinghe and X. Li, "Using a distance
metric to guide PSO algorithms for many-objective
optimization," in Proc. Conf. Genetic and Evolutionary
Computation, Montreal, Quebec, Canada, July 2009,
pp. 667-674.
[17] R. J. Lygoe, M. Cary, and P. J. Fleming, "A manyobjective optimisation decision-making process applied
to automotive diesel engine calibration," in Proc. 8th Int.
Conf. Simulated Evolution and Learning, Kanpur, India,
Dec. 2010, pp. 638-646.
[18] R. Wang, R. C. Purshouse, and P. J. Fleming,
"Whatever works best for you" - A new method for a
priori and progressive multi-objective optimisation," in
Proc. 7th Int. Conf. Evolutionary Multi-Criterion Optimization, Sheffield, U.K., Mar. 2013, pp. 337-351.
[19] Y. Yuan, H. Xu, B. Wang, and X. Yao, "A new dominance relation-based evolutionary algorithm for manyobjective optimization," IEEE Trans. Evol. Comput., vol.
20, no. 1, pp. 16-37, Feb. 2016.
[20] R. Cheng, Y. Jin, M. Olhofer, and B. Sendhoff, "A
reference vector guided evolutionary algorithm for many-objective optimization," IEEE Trans. Evol. Comput.,
vol. 20, no. 5, pp. 773-791, Oct. 2016.
[21] Y. Xiang, Y. Zhou, M. Li, and Z. Chen, "A vector
angle-based evolutionary algorithm for unconstrained
many-objective optimization," IEEE Trans. Evol. Comput., vol. 21, no. 1, pp. 131-152, Feb. 2017.
[22] H. Ishibuchi, H. Masuda, Y. Tanigaki, and Y. Nojima, "Difficulties in specifying reference points to calculate the inverted generational distance for many-objective
optimization problems," in Proc. IEEE Symp. Computational Intelligence Multi-Criteria Decision-Making, Orlando,
FL, Dec. 2014, pp. 170-177.
[23] M. Li, S. Yang, and X. Liu, "A performance comparison indicator for Pareto front approximations in many-objective optimization," in Proc. Conf. Genetic and Evolutionary Computation, Madrid, Spain, July 2015, pp. 703-710.
[24] H. Wang, Y. Jin, and X. Yao, "Diversity assessment
in many-objective optimization," IEEE Trans. Cybernet.,
vol. 47, no. 6, pp. 1510-1522, June 2017.
[25] W. Hu and G. G. Yen, "Adaptive multiobjective particle swarm optimization based on parallel cell coordinate
system," IEEE Trans. Evol. Comput., vol. 19, no. 1, pp. 1-18,
Feb. 2015.
[26] R. Hernández Gómez, C. A. Coello Coello, and E.
Alba Torres, "A multi-objective evolutionary algorithm
based on parallel coordinates," in Proc. Genetic and Evolutionary Computation Conf., Denver, CO, July 2016.
[27] J. Cheng, G. Yen, and G. Zhang, "A many-objective
evolutionary algorithm with enhanced mating and environmental selections," IEEE Trans. Evol. Comput., vol. 19,
no. 4, pp. 592-605, Aug. 2015.
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NOVEMBER 2017 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE
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