IEEE Computational Intelligence Magazine - August 2019 - 72

IMOP1

1

IMOP2

1

IMOP3

0.9

IMOP4

0

f1

1

0

1

f1

IMOP5

3

f3
0
0

0
f2

0.5 0.5 f1

1 1

0
0

f1

1 1

f1

IMOP8

1

f3

f3

-0.5

1.2

f1

0
f2

IMOP7

1

f2

0
-0.1

IMOP6

1.2

-0.2
-0.5

f3

f2

0

0
0

f3

0

f2

f2

1

0
f2

11

-0.9
0

f1

f2

1 1

f1

0

FIGURE 9 The Pareto fronts of the proposed multi-objective test suite.

TABLE V CPF values obtained by SPEA2, IBEA, NSGA-III, BCE-IBEA, and SMS-EMOA on IMOP1-IMOP8, averaged over 30 runs. The
best result in each row is highlighted.
PROBLEM

SPEA2

IBEA

NSGA-III

BCE-IBEA

SMS-EMOA

IMOP1

8.3863e-1 (1.62e-2) +

2.5286e-1 (1.95e-2) −

2.7417e-1 (2.33e-2) −

7.2631e-1 (4.92e-2) +

5.5286e-1 (7.05e-3)

IMOP2

5.6706e-1 (1.39e-1) +

2.3659e-1 (9.93e-2) +

2.1624e-1 (8.26e-2) ≈

2.3160e-1 (6.43e-2) +

1.7326e-1 (5.62e-2)

IMOP3

5.3281e-1 (8.27e-2) +

3.9632e-1 (1.21e-1) +

1.6827e-1 (4.74e-2) −

5.9031e-1 (6.15e-2) +

3.1527e-1 (8.46e-2)

IMOP4

9.2925e-1 (1.58e-1) +

8.4805e-1 (8.78e-2) +

4.1299e-1 (5.17e-2) +

9.3858e-1 (1.24e-1) +

2.3181e-1 (3.33e-2)
3.7208e-1 (2.80e-2)

IMOP5

8.7751e-1 (1.94e-2) +

7.3104e-1 (3.75e-2) +

3.4081e-1 (3.92e-2) −

8.5648e-1 (2.90e-2) +

IMOP6

8.9772e-1 (2.00e-1) +

7.3277e-1 (3.00e-1) +

3.7190e-1 (1.31e-1) ≈

9.1744e-1 (3.97e-2) +

3.7061e-1 (1.46e-1)

IMOP7

1.3776e-1 (2.66e-1) ≈

5.5467e-2 (1.69e-1) ≈

1.6552e-2 (5.17e-2) ≈

7.8012e-2 (1.78e-1) +

8.5742e-4 (1.11e-3)

IMOP8

6.3370e-1 (3.17e-1) +

9.4194e-1 (2.07e-2) +

4.2660e-1 (5.69e-2) −

9.3241e-1 (2.34e-2) +

4.7037e-1 (1.99e-1)

distributed reference points on irregular
Pareto fronts, and so far no much
research has been done to address this
issue [41], [53]-[55]. In order to better
use the proposed test suite in assessing
the performance of MOEAs, the mathematical formulations of the Pareto fronts
as well as the methods for sampling a set
of uniformly distributed reference
points on the Pareto fronts are given in
Supplementary Materials II.
C. Experimental Results on the
Proposed Test Suite

This subsection verifies the effectiveness
of the proposed test suite in distinguishing between the diversity performance of MOEAs. To this end, the
five MOEAs compared in Section III
are tested on the proposed test suite

72

with the same parameter settings. Besides, the parameters K, L, a 1, a 2 and
a 3 in IMOP1-IMOP8 are set to 5, 5,
0.05, 0.05 and 10, respectively.
The non-dominated solution sets
obtained by SPEA2, IBEA, NSGA-III,
BCE-IBEA, and SMS-EMOA on
IMOP1-IMOP8 are presented in Fig. 3
in the Supplementary Materials. From
the figure, it can be observed that the
five compared MOEAs exhibit significantly different diversity performances
on the eight MOPs. To be specific, for
IMOP1-IMOP4 whose Pareto fronts
are irregular curves in bi- or threedimensional space, SPEA2 shows the
best diversity performance on IMOP1
and IMOP2, while the solution set
obtained by BCE-IBEA has the best
diversity on IMOP3 and IMOP4. For

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2019

IMOP5 whose Pareto front contains
eight circles, all the compared MOEAs
are able to find a set of solutions covering the whole Pareto front, but the
solutions obtained by NSGA-III and
SMS-EMOA are of poor distribution.
For IMOP6 whose Pareto front consists
of several grids, SPEA2, IBEA, and
BCE-IBEA show better diversity performance than the other MOEAs.
IMOP7 is quite challenging to the
compared MOEAs, such that none of
solution sets obtained by all the MOEAs
can cover the whole Pareto front. As for
IMOP8, the diversity performance of
IBEA and BCE-IBEA is significantly
better than that of the other MOEAs.
Table V lists the CPF values of the
obtained solution sets, and the box plots
of the CPF values are depicted in Fig. 4
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