IEEE Computational Intelligence Magazine - November 2019 - 62

VII. Discussion on Explicit
Control of Implicit Parallelism
on EMO Algorithms

popular evolutionary many-objective
optimization algorithms. We argued in
the figure that NSGA-III constitutes a
broad implicit parallelism, while M2M
provides a restrictive implicit parallelism.
Our extensive simulation studies on 36

In Figure 2, we started with a sketch of
our initial idea of the levels of explicit
control of implicit parallelism present in

TABLE VIII Wilcoxon rank-sum test values for the variants of MOEA/D on DTLZ and
WFG test problems.
NEIGHBORHOOD UPDATE AND MATING RESTR. PROB., b
PROBLEM
INSTANCE

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

DTLZ1-3

1

1

1

1

0

1

0

0

0

0

0

DTLZ1-5

0

0

0

0

0

0

0

0

0

0

0

DTLZ1-8

1

1

0

0

0

0

0

0

0

0

0

DTLZ1-10

0

0

0

0

0

0

1

1

1

1

1

DTLZ1-15

0

0

0

0

0

0

0

0

0

0

0

DTLZ2-3

0

1

0

1

1

0

0

0

0

0

0

DTLZ2-5

1

1

1

1

1

1

1

1

1

0

0

DTLZ2-8

1

1

1

1

1

1

1

1

1

0

0

DTLZ2-10

1

1

0

0

0

0

0

0

0

0

1

DTLZ2-15

0

0

0

0

0

0

0

0

1

1

1

DTLZ3-3

1

1

0

0

0

0

0

0

0

0

1

DTLZ3-5

1

1

1

1

1

1

1

1

1

0

0

DTLZ3-8

0

0

0

0

0

0

1

0

0

0

1

DTLZ3-10

0

0

0

0

0

0

0

0

0

0

0

DTLZ3-15

0

0

0

0

0

0

0

0

0

0

0

DTLZ4-3

0

0

0

0

0

0

0

0

0

0

0

DTLZ4-5

0

0

0

0

0

0

0

0

0

0

0

DTLZ4-8

0

0

0

0

0

0

0

0

0

0

0

DTLZ4-10

0

0

0

0

0

0

0

0

0

0

0

DTLZ4-15

0

1

1

1

1

0

1

1

1

1

0

RANKING

7

9

5

6

5

4

6

5

6

3

5

6

5

8

11

15

WFG PROBLEMS

0

7

4

7

3

MOEA /D-0.9
NSGA-III-0.9
MOEA /D

7

M2M-NS-v13
M2M, M2M-S-v11, v12
NSGA-III

7

MOEA /D-N

RANKING

0.2
0.4
0.6
0.8
Proportion of Success in DTLZ Problems

1

FIGURE 3 Proportion of successful performance of various EMO algorithms over 20, three to
15-objective DTLZ problems.

62

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2019

algorithms of two sets of widely-used test
problems - unscaled DTLZ and scaled
WFG problems - and having three to 15
objectives have not only revealed that our
intuition about the extent of parallelism
in these algorithms was right, but the statistical Wilcoxon rank-sum tests have
allowed us a way to rank these algorithms. Tables V to VI are used to count
the number of times an algorithm performs similar (with p 1 0.0001) compared to the best performing algorithm
for each problem. Figures 3 and 4 shows
the relative ranking of different algorithms on DTLZ and WFG problems,
respectively. It is interesting to observe
that the original MOEA/D performs the
best on DTLZ problems due to uniform
scaling nature of objective functions in
these problems. Any normalization
(MOEA/D-N indicates an average performance with a simple or a more
sophisticated NSGA-III normalization)
procedure deteriorates its performance.
These normalization procedures depend
on the population diversity at each generation, which, in principle, can vary
from start to the end of a run. Since
DTLZ problems do not require any
additional normalization of objectives to
make meaningful computation of
Euclidean distances (d 1 and d 2) needed
for its PBI or other objective distance
computations, any normalization based
on statistics of changing populations
makes the evaluation of these distance
measures noise-prone. MOEA/D with
90% neighborhood update and recombination within the same subpopulation
(MOEA/D-0.9) performs slightly worse
than the original MOEA/D. The figure
also shows that NSGA-III's performance
gets enhanced about three times by the
use of a restricted recombination (90%
recombination within a subpopulation,
NSGA-III-0.9).This is a remarkable finding of this study, which the original NSGAIII study did not investigate. However, the
performance of the original M2M and its
variants with normalization are almost
unchanged on DTLZ problems.
For WFG problems (Figure 4), the
observations are different. Since for realworld problems, a non-uniform scaling
of objectives is likely, WFG problems
IEEE Computational Intelligence Magazine - November 2019

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