IEEE Computational Intelligence Magazine - August 2019 - 70

obtain the best CL n values on DTLZ5
and DTLZ6, respectively, as they obtained
fewer solutions than the other MOEAs
on DTLZ5 and DTLZ6. However, it is
obvious that the solution sets obtained by

SPEA2 and BCE-IBEA have better
diversity. Similarly, IBEA obtains the best
PD value on DTLZ5 and DTLZ6, which
is due to the outliers (i.e., extreme solutions) in the solution sets. NSGA-III

TABLE IV Definition of the proposed multi-objective test suite, where K and L are
parameters controlling the length of decision variables, and a1, a2, and a3 are
parameters controlling the difficulty in diversity preservation.
PROBLEM

DEFINITION

COMMON

x = (x 1, f, x K, x K +1, f, x K +L) ! [0, 1] K +L
a1

K

y1 = e 1 / xi o
K i =1
y2 = e
y3 = e
g=

1
^K/2h
1
6K/2@

K +L

/

^K/2h

/

i =1

a2

xi o

a3

K

/

i =^K/2h +1

xi o

(x i - 0.5) 2

i = K +1

IMOP1

f1 (x) = g + cos 8 ` r y 1 j
2
f2 (x) = g + sin 8 ` r y 1 j
2

IMOP2

IMOP3

f1 (x) = g + cos 0.5 ` r y 1 j
2

f2 (x) = g + sin 0.5 ` r y 1 j
2
1
f1 (x) = g + 1 + cos (10ry 1) - y 1
5
f2 (x) = g + y 1

IMOP4

f1 (x) = (1 + g) y 1
f2 (x) = ^1 + gh` y 1 + 1 sin ^10ry 1hj
10
f3 (x) = (1 + g) (1 - y 1)

IMOP5

h 1 = 0.4 cos ` r ^8y 2hj + 0.1y 3 cos ^16ry 2h
4
h 2 = 0.4 sin ` r ^8y 2hj + 0.1y 3 sin ^16ry 2h
4

f1 (x) = g + h 1
f2 (x) = g + h 2
f3 (x) = g + 0.5 - h 1 - h 2
IMOP6

r = max {0, min {sin 2 (3ry 2), sin 2 (3ry 3)} - 0.05}
f1 (x) = (1 + g) y 2 + ^ r h
f2 (x) = (1 + g) y 3 + ^ r h

f3 (x) = (0.5 + g) (2 - y 2 - y 3) + ^ r h

IMOP7

h 1 = ^1 + gh cos ` r y 2 j cos ` r y 3 j
2
2

h 2 = ^1 + gh cos ` r y 2 j sin ` r y 3 j
2
2
h 3 = ^1 + gh sin ` r y 2 j
2

r = min {min {| h 1 - h 2 | , | h 2 - h 3 |}, | h 3 - h 1 |}
f1 (x) = h 1 + 10 max {0, r - 0.1}
f2 (x) = h 2 + 10 max {0, r - 0.1}
f3 (x) = h 3 + 10 max {0, r - 0.1}
IMOP8

f1 (x) = y 2
f2 (x) = y 3
3

f3 (x) = (1 + g) =3 - /

i =2

70

y i (1 + sin (19ry i))
G
1 +g

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2019

obtains the best IGD and T p values on
DTLZ2 and DTLZ4, since the reference
points have the same distribution as the
solution set obtained by NSGA-III. In
fact, the solution sets obtained by SPEA2
and BCE-IBEA have better evenness than
those obtained by NSGA-III on DTLZ2
and DTLZ4. NSGA-III also obtains the
best HV values on DTLZ2 and DTLZ4,
which is attributed to the fact that most
solutions obtained by it are on the border,
which are biased by the HV metric. To
summarize, none of the above metrics can
accurately reflect the diversity performance of the compared MOEAs on
DTLZ1-DTLZ7, CWDV, and ML-DMP.
By contrast, according to Table III,
it can be found that the CPF values
are consistent with the observations
obtained from Fig. 1 in the Supplementary Materials. Specifically, NSGA-III
has the best diversity performance on
DTLZ1, SPEA2 has the best diversity
perfor mance on DTLZ2-DTLZ4,
SPEA2 and BCE-IBEA have similar
diversity performance on DTLZ5-
DTLZ6, BCE-IBEA has the best diversity perfor mance on DTLZ7 and
CWDV, and IBEA has the best diversity
performance on ML-DMP. Moreover,
the box plots of the above seven metric
values are shown in Fig. 2 in the Supplementary Materials. It can be seen that
the variance of the CPF values is low,
and each MOEA has significantly different CPF values. By contrast, the variance of other metric values are relatively
high on some test instances, and all the
MOEAs have very similar metric values
on some other test instances. Therefore,
the effectiveness of the proposed CPF in
diversity assessment is confirmed.
On the other hand, it can be observed from Fig. 1 in the Supplementary Mater ials that the difference
between the diversity performance of
SPEA2 and BCE-IBEA is negligible on
most MOPs. This is mainly due to the
fact that most of the test MOPs have
simplex-like Pareto fronts, making it
easy for MOEAs to obtain a diverse
solution set once they converge to the
Pareto front. In order to better compare
the diversity performance of different
MOEAs, a multi-objective test suite
IEEE Computational Intelligence Magazine - August 2019

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