IEEE Computational Intelligence Magazine - November 2019 - 57

NSGA-III is similar to those reported in
the original study [32].
A Wilcoxon rank-sum test on eight
low-dimensional DTLZ problems
reveals that M2M-v2 performs the best,
followed by M2M-v3 and v12. Importantly, the original M2M (v1) and
NSGA-III fail to perform well on these
problems. On 12 high-dimensional
DTLZ problems, MOEA/D performs
reasonably well, but the best performance comes from the M2M-v2 variant
with selection completely restrictive
within subpopulations and recombination restrictive to subpopulations only
67% of the time. More elaborate results
can be found at [39].
C. Simulation Results on
WFG Problems

WFG problems produce a non-uniform
scaling of objective values, thereby requiring an appropriate normalization of
objectives for an EMO algorithm to
work well. Table IV also shows the median IGD values of top three M2M variants
and other original methods - NSGA-III,
MOEA/D and M2M - on WFG problems. As before, MOEA/D and M2M
algorithms are applied without any objective normalization, as they were originally
proposed. M2M variants are also applied
without any normalization here.
We observe a completely different outcome compared to those ob served in DTLZ problems. Now,
NSGA-III works the best and it works
so well compared to other algorithms
that all other methods are significantly
worse (with p 1 0.05) in the Wilcoxon
rank-sum test.
Overall, for solving scaled WFG
problems, NSGA-III outperfor ms
MOEA/D, M2M and M2M-variants on
both low and high-dimensional problems. However, M2M-v2, v3, and v4 are
much better than original M2M and
other M2M-variants and also better
than the MOEA/D algorithm. While
NSGA-III's super-performance is due to
its internal normalization procedure
which handles WFG problems better,
certain M2M-variants with a controlled
implicit parallelism built in perform better than MOEA/D and the original

TABLE IV Median IGD values of NSGA-III, MOEA/D, original M2M, and top three
M2M variants on DTLZ (marked with initial D) and WFG (marked with initial W)
problems are shown. For each problem, the best performing method is shown in
bold and other methods having statistically insignificant performance difference
from the best are shown with gray backgrounds.
METHOD

NSGA-III

MOEA/D

M2M

M2M-V2

M2M-V12

M2M-V13

D1-3

0.00185

0.00160

0.00578

0.00098

0.00155

0.00131

D1-5

0.00119

0.00046

0.00158

0.00046

0.00059

0.00056

D2-3

0.00266

0.00075

0.00110

0.00075

0.00076

0.00072

D2-5

0.00640

0.00091

0.00196

0.00182

0.00188

0.00162

D3-3

0.00374

0.00419

0.03603

0.00203

0.00324

0.00292

D3-5

0.00971

0.00233

0.01889

0.00158

0.00353

0.00286

D4-3

0.00084

0.95033

0.00016

0.00015

0.00015

0.00014

D4-5

0.00111

0.34324

0.00044

0.00033

0.00025

0.00024

D1-8

0.00496

0.00595

0.01013

0.00363

0.00245

0.00293

D1-10

0.00407

0.00734

0.00972

0.00381

0.00231

0.00257

D1-15

0.00503

0.02529

0.03594

0.01092

0.00393

0.00355

D2-8

0.01755

0.00236

0.02813

0.02320

0.01454

0.01390

D2-10

0.01756

0.00334

0.00659

0.00540

0.00517

0.00490

D2-15

0.02022

0.00629

0.008587

0.00610

0.00572

0.00571

D3-8

0.03679

0.01651

0.03822

0.03009

0.02966

0.03382

D3-10

0.01444

0.00521

0.01500

0.00498

0.00729

0.00627

D3-15

0.03086

1.24061

0.02801

0.00668

0.00645

0.00569

D4-8

0.00318

0.58025

0.00851

0.00641

0.00508

0.00525

D4-10

0.00444

0.33428

0.00458

0.00270

0.00231

0.00235

D4-15

0.00696

0.50060

0.01289

0.00345

0.00287

0.00333

METHOD

NSGA-III

MOEA/D

M2M

M2M-V2

M2M-V3

M2M-V4

W5-3

0.03050

0.06753

0.06966

0.06902

0.06887

0.06907

W5-5

0.03211

0.22495

0.17842

0.17852

0.17865

0.17865

W6-3

0.03120

0.06671

0.06712

0.07447

0.07348

0.07348

W6-5

0.03254

0.30685

0.18024

0.18039

0.18030

0.18065

W7-3

0.00731

0.06113

0.08279

0.06995

0.06887

0.06717

W7-5

0.00902

0.30683

0.17892

0.17892

0.17883

0.17889

W8-3

0.05545

0.07803

0.08776

0.08085

0.08051

0.08191

W8-5

0.08915

0.29803

0.19507

0.19471

0.19435

0.19450

W5-8

0.03167

0.57438

0.44775

0.44401

0.44074

0.44169

W5-10

0.03206

0.71380

0.57248

0.56765

0.56321

0.56396

W5-15

0.03303

0.91872

0.82185

0.80408

0.80325

0.80368

W6-8

0.03096

0.85795

0.58616

0.57624

0.57249

0.57414

W6-10

0.03308

0.94410

0.69693

0.69009

0.69330

0.70661

W6-15

0.03982

1.22729

0.99066

0.99749

1.01479

1.00532

W7-8

0.00776

0.83858

0.57651

0.56881

0.55003

0.56387

W7-10

0.00837

0.93456

0.69693

0.69009

0.69330

0.70661

W7-15

0.01148

1.20531

0.92340

0.93531

0.94147

0.96234

W8-8

0.15502

0.88407

0.49582

0.51177

0.51509

0.54198

W8-10

0.19963

0.92550

0.71983

0.70690

0.70509

0.71303

W8-15

0.25047

1.13228

0.98375

0.98898

1.00979

1.00891

NOVEMBER 2019 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE

57
IEEE Computational Intelligence Magazine - November 2019

Table of Contents for the Digital Edition of IEEE Computational Intelligence Magazine - November 2019
