IEEE Computational Intelligence Magazine - February 2022 - 90

Fig. 5. If each algorithm is terminated at
the 100th generation in Fig. 4, all algorithms
are evaluated as having the same
performance (whereas they show different
performance after the 5000th generation).
In Fig. 5, MOEA/D-WS can be evaluated
as being the best and the worst depending
on the termination condition. Since
any specification is not fair, it is needed to
use various termination conditions (at least
two or three specifications). In this case,
EMO algorithms are compared as anytime
algorithms [27]. In some studies (e.g.,
Tanabe et al. [28]), EMO algorithms are
compared as anytime algorithms under various
termination conditions. Since it is not
likely that a single algorithm is always the
best over a wide range of generations, the
choice of an appropriate algorithm from a
large algorithm pool depending on the termination
condition seems to be a promising
research direction. Of course, it is an
interesting challenge to try to develop an
EMO algorithm with robust performance
over a wide range of termination conditions.
In real-world applications, it is also
important to develop a specialized EMO
algorithm which works well under a specific
termination condition (e.g., under a limited
number of solution evaluations such as
500 solution evaluations).
When an anytime EMO algorithm is
designed, an important issue is the performance
oscillation of the current population
through generation update. Some EMO
algorithms with a steady state ()1n + generation
update mechanism such as SMSEMOA
[4] carefully improve the current
population by creating only a single new
solution at each generation. Thus, usually
they do not show any severe performance
oscillation of the current population
through generation update. However, some
1.5
1.6
1.7
1.8
1.9
other EMO algorithms with a ()
nn+
generation update mechanism such as
NSGA-II [2] replace many solutions with
new solutions. As a result, they often show
performance oscillation through generation
update. That is, the performance of the
current population can be worse than that
of the previous population. Moreover, the
final populations of those algorithms often
include solutions that are dominated by old
solutions (which were generated and deleted
in previous generations) as shown in
[29]. For designing a good anytime EMO
algorithm with stable performance, it is
important to carefully store good solutions
in the current or archive population to
achieve a stable performance with no
severe performance oscillation.
III. Specification of Population
Size
MOEA/DD
MOEA/D-Tch
NSGA-III
θ-DEA
102
103
Population Size
FIGURE 10 Average HV results of the final
population for five-objective WFG3.
104
In general, performance indicator values
are influenced by the solution set size. The
hypervolume value is improved by adding
any non-dominated solution to a solution
set. Thus, larger solution sets are likely to
have larger hypervolume values than
smaller ones. For example, in NSGA-II,
the final population is constructed by
removing half solutions from a merged
population of parents and offspring. It is
likely that a larger (i.e., better) hypervolume
value can be obtained by using the
merged population instead of the final
population. These discussions explain why
the size of the final population should be
the same for fair comparison (i.e., why
the population size should be the same in
all the compared EMO algorithms).
However, it is likely that an appropriate
population size is different in each
EMO algorithm. Some algorithms work
well with a large population while others
are efficient with a small population.
Thus, any specification is not fair. This is
because different performance comparison
results will be obtained from different
specifications. Depending on the
population size, different algorithms will
be evaluated as good algorithms.
For demonstrating large effects of the
population size on performance comparison
results, MOEA/D-Tch [3], NSGAIII
[8], MOEA/DD [9] and DEA
are applied to the five-objective WFG3
i[11]
90
IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | FEBRUARY 2022
problem in a similar manner to [30]
under the following setting:
Population size: 210, 1001, 5985.
Neighborhood size: 20.
Polynomial mutation: Index 20,
Probability 1/n.
SBX crossover: Index 30, Probability
1.0.
Termination condition: 210,000
solution evaluations.
Each algorithm is evaluated under
three population size specifications: standard
(210), large (1001), and very large
(5985). The termination condition corresponds
to 1000 generations with the
standard population size 210. Average
hypervolume values are calculated using
the final population in the normalized
objective space over 31 runs.
Figure 10 summarizes experimental
results for the three population size specifications
in each EMO algorithm. The
population size has large effects on comparison
results in Fig. 10. While MOEA/
D-Tch with the very large population
works very well, its performance is not
good under the standard setting. The largest
average hypervolume value is obtained
in Fig. 10 by MOEA/D-Tch with the
very large population. However, it cannot
be concluded that MOEA/D-Tch is the
best algorithm in Fig. 10. This is because
the calculated average hypervolume value
of 5985 solutions cannot be compared in
a fair manner with that of much fewer
solutions obtained by the other algorithms
(e.g., 210 solutions by NSGA-III).
One idea to compare all results in
Fig. 10 in a fair manner is to select a
solution subset of the same size (e.g., 210
solutions) from the final population.
Here, a hypervolume-based greedy forward
selection method [31], [32] is used.
Solutions are selected one by one in a
greedy manner for hypervolume maximization
of the selected solution subset.
That is, each solution is evaluated in the
greedy algorithm by its contribution to
the hypervolume of the selected solution
subset. From the final population, 210
solutions are selected using the greedy
algorithm. After that, their hypervolume
is calculated. Then, the average result is
calculated over 31 runs. Figure 11 shows
experimental results where the largest
Average HV Value
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