IEEE Computational Intelligence Magazine - August 2019 - 62

diversity. Note that in this paper, we
focus on these properties of a solution
set in the objective space rather than in
the decision space, in spite of the equal
importance of the latter one [2].
While many MOEAs aim at enhancing the convergence of a solution set [3],
[4], there has been an increasing number
of MOEAs focusing on preserving the
diversity of a solution set in recent years,
such as the diversity estimation based
MOEAs [5], [6], the decomposition
based MOEAs [7], [8], and the indicator based MOEAs [9], [10]. Diversity
preservation is an important topic in
MOEAs, since a solution set with better
diversity can provide decision makers
more information when choosing their
preferred solutions [11]. Moreover, some
real-world applications naturally require
a solution set with good diversity. For
example, ensemble learning requires
multiple models with sufficient differences [12].
Due to the importance of diversity in
the obtained solution sets, some metrics
have been designed for assessing the
diversity performance of MOEAs, which
can be divided into two categories, i.e.,
metrics assessing only diversity and metrics assessing both convergence and
diversity. However, most of these metrics
have limitations. For the metrics assessing
only diversity, some of them (e.g., Spacing [13]) bias the assessment of evenness,
and some others (e.g., CL n [14]) bias the
assessment of spread, whereas few of
them are able to assess both the evenness

and spread of a solution set. Consider the
three solution sets depicted in Fig. 1,
where S 1 has a good evenness and a
poor spread, S 2 has a poor evenness and
a good spread, and S 3 has a good evenness and a good spread. Obviously, the
diversity of S 3 is significantly better than
S 1 and S 2 . However, for the diversity
metrics biasing the assessment of evenness, they may identify S 1 as the one
with the best diversity; similarly, the
metrics biasing towards the assessment
of spread may identify S 2 as the one
with the best diversity. As for the metrics assessing both convergence and
diversity (e.g., IGD [15]), it is difficult to
merely assess diversity without considering convergence.
To address this issue, this paper proposes a new diversity metric that can
effectively assess both the evenness and
spread of a solution set obtained by
MOEAs. To better compare the diversity
performance of different MOEAs, this
paper also proposes a multi-objective
test suite containing various complex
Pareto fronts, which poses stiff challenges for existing MOEAs in terms of
diversity preservation. Specifically, the
contributions of this paper consist of the
following two aspects:
❏ A performance metric is proposed
for assessing the diversity of a solution set obtained by MOEAs. The
proposed metric assesses both the
evenness and spread of a solution set
by projecting it to the (M -1)dimensional unit hypercube (M

Good Evenness
Good Spread

Poor Evenness
Good Spread

Good Evenness
Poor Spread

denotes the number of objectives),
and calculating the "volume" of the
projected solution set as its diversity.
Both illustrative examples and experimental studies indicate that the proposed metric has better performance
in diversity assessment than existing metrics.
❏ A multi-objective test suite is proposed for comparing the diversity
performance of existing MOEAs,
which contains eight bi- or threeobjective MOPs with simple landscapes but various irregular Pareto
fronts. The proposed test suite poses
stiff challenges to existing MOEAs to
obtain a set of solutions with good
evenness and spread, thereby effectively distinguishing between the
diversity performance of different
MOEAs. According to the experimental results of five popular
MOEAs on the proposed test suite, it
turns out that the compared MOEAs
exhibit significantly different diversity performance, and none of them is
able to perform consistently well on
all the proposed MOPs.
The rest of this paper is organized as
follows. In Section II, existing metrics
for diversity assessment are revisited and
the proposed metric is detailed. In Section III, the effectiveness of the proposed metric is verified by comparing it
with several metrics in assessing the
diversity of solution sets obtained by five
popular MOEAs. In Section IV, existing
multi-objective test suites are reviewed,

Solutions
Solutions
Pareto Front

Pareto Front

Solutions

(a)

f1
(b)

Pareto Front

f1
(c)

FIGURE 1 Three solution sets for a bi-objective optimization problem. (a) Solution set S1, (b) Solution set S2 and (c) Solution set S3.

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2019

IEEE Computational Intelligence Magazine - August 2019

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