Computational Intelligence - November 2017 - 90

Parallel coordinates can clearly reflect the
Pareto dominance relation between two
solutions (such as polyline a being dominated by polyline b in Figure 1, assuming a minimization problem scenario) if

the solution polylines are not overcrowded. It is worth mentioning that one can
remove dominated solutions in parallel
coordinates if they are only interested in
non-dominated ones. This may make the

20
Objective Value

Objective Value

20
15
10
5
0

1

2

15
10
5
0

3 4 5 6 7 8 9 10
Number of Objective
(a)

1

2

3 4 5 6 7 8 9 10
Number of Objective
(b)

1.0

1.0

0.8

0.8

Objective Value

Objective Value

FIGURE 3 The solution sets obtained by NSGA-II and GrEA on the 10-objective WFG7, and
their evaluation results on the convergence metric GD + (the smaller the better). (a) NSGA-II
(GD + = 3.63E - 1) and (b) GrEA (GD + = 6.32E - 2).

0.6
0.4
0.2
0.0

1

2

3 4 5 6 7 8 9 10
Number of Objective
(a)

0.6
0.4
0.2
0.0

1

2

3 4 5 6 7 8 9 10
Number of Objective
(b)

FIGURE 4 The solution sets obtained by AR and IBEA on the 10-objective DTLZ2. (a) AR and
(b) IBEA.

1

The setting of the population size and maximum
evaluations was 100 and 30,000, respectively. This setting was used in all conducted experiments in this
paper, unless explicitly mentioned otherwise. In addition, the grid division in GrEA was set to 8.

1.0

1.0

1.0

0.8

0.8

0.8

0.6

0.6

0.6

0.4 f3

0.4 f3

0.4 f3

0.2
0.0
0.0
0.2
0.4
0.6 f1
0.8
1.0

0.0
0.2
0.4
0.6
f2 0.81.0

plot clearer when comparing the quality
of solution sets.
In addition to reflecting the Pareto
dominance relation, parallel coordinates
can largely imply the convergence of
solution sets by their range. Figure 2 is
such an example, where the parallel
coordinates representation of two
solution sets obtained by one run1 of
two EMO algorithms, NSGA-II [33]
and GrEA [34], on the 10-objective DTLZ2 problem [35] is shown. As
can be seen, NSGA-II has an inferior
convergence, with its solution set ranging from 0 to around 3.5 in contrast to
the problem's Pareto front ranging from
0 to 1. GrEA has a good convergence
on this problem and its solution set has
the same range as the Pareto front.
These observations can be confirmed
by the results of the convergence metric
GD + [36] shown in the figure. GD + is
a modified version of the original GD
[37], which makes it compatible with
Pareto dominance.
However, we may not be able to
accurately know the convergence of
solution sets by their range shown in
parallel coordinates. That is, even if two
solution sets are located in the same
range, they can perform considerably
differently in terms of convergence.

0.2
0.0
0.0
0.2
0.4
0.6 f1
0.8
1.0

0.0
0.2
0.4
0.6
f2 0.81.0

(a)

(b)

0.2
0.0
0.0
0.2
0.4
0.6 f1
0.8
1.0

0.0
0.2
0.4
0.6
f2 0.81.0
(c)

FIGURE 5 The solution sets obtained by IBEA, SMS-EMOA and MOEA/D on the 3-objective DTLZ2, shown in Cartesian coordinates. (a) IBEA, (b)
SMS-EMOA and (c) MOEA/D.

90

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2017
Table of Contents for the Digital Edition of Computational Intelligence - November 2017
