Computational Intelligence - November 2017 - 91

1.0

0.8

0.8

0.8

0.6
0.4
0.2
0.0

1

2
Number of Objective
(a)

3

Objective Value

1.0
Objective Value

Objective Value

1.0

0.6
0.4
0.2
0.0

1

2
Number of Objective
(b)

0.6
0.4
0.2
0.0

3

1

2
Number of Objective
(c)

3

FIGURE 6 The corresponding parallel coordinates of the solution sets in Figure 5. (a) IBEA, (b) SMS-EMOA and (c) MOEA/D.

B. Coverage

In parallel coordinates, it is straightforward to see which region a solution set
does not reach on any objective2. For
example, in Figure 4 the solution set
obtained by the AR method [39] concentrates in one tiny area and the set by
2

Note that for real-world problems whose Pareto front
is unknown, we cannot tell if a solution set reaches the
optimal region of objectives or not.

2.0

2.0

1.5
1.0

1.5

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

1.0

f3

0.5

0.5

f3

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)

FIGURE 7 An artificial example of two solution sets (A and B) having the same parallel coordinates plots shown in Figure 8. (a) Solution set A and (b) Solution set B.

2.0

2.0
Objective Value

Objective Value

Figure 3 gives such an example, where
solution sets obtained by one run of
NSGA-II and GrEA on the 10-objective WFG7 problem [38] are shown. As
seen, both algorithms virtually reach the
range of the Pareto front (from 0 to 2i
where i is the objective index of the
problem), but they have different GD +
results. NSGA-II is returned a significantly higher (worse) GD + value than
GrEA. This occurence can be from two
possibilities. One is that the solution
set  of NSGA-II is not actually close to
the Pareto front. The other is that
most of solutions in the set converge
already while a small portion of the
set is far away (but still in the range of
the Pareto front).
In addition, it is worth mentioning
that even if the "height" of two solution
sets in the parallel coordinate plot is different, we may also not be able to tell
the convergence difference between
them if the range of a problem's Pareto
front is unknown. This is because different solution sets may converge into different parts of the Pareto front, especially
in the situation where the Pareto front is
highly convex.

1.5
1.0
0.5
0.0

1.5
1.0
0.5
0.0

1

2
Number of Objective
(a)

3

1

2
Number of Objective
(b)

3

FIGURE 8 The parallel coordinates plots of the solution sets in Figure 7. (a) Solution set A and
(b) Solution set B.

IBEA [40] fails to cover the first six
objectives on the 10-objective DTLZ2.
Moreover, we can conjecture some distribution features of solution sets from
their parallel coordinates representation.
Take the solution sets in Figure 5 as an

example; their parallel coordinates representation is shown in Figure 6. From
Figure 6, we can know that the solution
sets of IBEA and SMS-EMOA [41] fail
to cover the region between 0.0 and 0.2
on all three objectives. Also, most of the

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Table of Contents for the Digital Edition of Computational Intelligence - November 2017
