IEEE Computational Intelligence Magazine - August 2022 - 43

2) Operators
To generate offspring solutions with real
variables, NSGA-II, LMEA, LSMOFNSGA-II,
SparseEA, MOEA/PSL, and
PM-MOEA use simulated binary crossover
(SBX) [54] and polynomial mutation
(PM) [55], MOEA/DVA and
MOEA/D-DE use differential evolution
and PM, WOF-SMPSO uses particle
swarm optimization and PM, IMMOEA
uses a Gaussian process based
inverse model and PM, and LMOCSO
uses an enhanced competitive swarm
optimizer and PM. To generate offspring
solutions with binary variables, NSGAII
and MOEA/PSL use uniform crossover
and bit-flip mutation, SparseEA
and PM-MOEA use their own crossover
and mutation operators. Besides,
MOEA/D-DE, MOEA/DVA, LMEA,
WOF-SMPSO, LSMOF-NSGA-II, IMMOEA,
and LMOCSO generate real
variables within [, ]01 D
and round them
to obtain binary solutions. The distribution
index of SBX and PM is set to 20,
the probability of SBX and uniform
crossover is set to 1, the probability of
PM and bit-flip mutation is set to
D
1 , and the parameters CR and F in
differential evolution are set to 1 and
0.5, respectively.
3) Number of evaluations
and population size
For fair comparisons, the number of
function evaluations available to all the
MOEAs is set to
100 D ,# and the population
size of all the MOEAs is set
to 100.
4) Problems
The MOEAs are compared on eight
benchmark problems (i.e., SMOP1-
SMOP8) and seven real-world applications
(i.e., neural network training,
feature selection, pattern mining, community
detection, critical node detection,
portfolio optimization, and the
knapsack problem). The definitions of
the eight benchmark problems can be
found in [20], the definitions of the first
six real-world applications can be found
in [22], and the definition of the knapsack
problem is referred to [52]. For the
knapsack problem, the capacity of each
knapsack is set to one tenth of the total
weight of all items, and the constraint
violations (i.e., caused by overweight
items) are added to the objectives to
avoid explicit constraints. For all the
problems, the number of objectives is
set to 2, the number of decision variables
varies from 100 to 5000, and the
other parameter settings are listed
in Table II.
B. Comparisons on
Benchmark Problems
Table III lists the CSD values of the
solution sets obtained by the compared
MOEAs on benchmark problems
SMOP1-SMOP8 with 100 to 5000
decision variables, averaged over 30
runs. The best CSD value in each row
is shown in bold, and all the results statistically
similar to the best one in
terms of the Wilcoxon rank sum test
[56] with a significance level of 0.05
are highlighted. According to the table,
five observations can be made: Firstly,
genetic algorithms have better average
rankings than differential evolution
(i.e., NSGA-II vs. MOEA/D-DE and
LMEA vs. MOEA/DVA), which is
mainly due to the powerful exploration
ability of genetic operators on
multimodal landscapes. Secondly, the
variable grouping based MOEAs (i.e.,
MOEA/DVA and LMEA) and novel
variation operator based MOEAs (i.e.,
IM-MOEA and LMOCSO) exhibit
worse performance than classical
MOEAs (i.e., NSGA-II and MOEA/
D-DE), which indicates that these
SMOP1
SMOP5
2.5
3
1.5
2
0.5
1
0 0.5 1 1.5 2 2.5 3
f1
NSGA-II
3.5
2.5
3
1.5
2
0.5
1
0 0.5 1 1.5 2 2.5 3
f1
IM-MOEA LMOCSO
MOEA/D-DE MOEA/DVA LMEA
SparseEA
2.5
3
1.5
2
0.5
1
0.5 1 1.5 2 2.5 3 3.5
f1
WOF-SMPSO LSMOF-NSGA-II
MOEA/PSL PM-MOEA
SMOP7
FIGURE 3 Solution sets (in objective space) with median CSD obtained by 11 MOEAs on SMOP1, SMOP5, and SMOP7 with
5000 decision variables.
AUGUST 2022 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 43
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