1

A Multi- or Many-Objective Evolutionary
Algorithm with Global Loop Update

arXiv:1803.06282v1 [] 25 Jan 2018

Yingyu Zhang

Bing Zeng

Abstract—Multior
many-objective
evolutionary
algorithms(MOEAs),
especially
the
decomposition-based
MOEAs have been widely concerned in recent years. The
decomposition-based MOEAs emphasize convergence and
diversity in a simple model and have made a great success in
dealing with theoretical and practical multi- or many-objective
optimization problems. In this paper, we focus on update
strategies of the decomposition-based MOEAs, and their
criteria for comparing solutions. Three disadvantages of the
decomposition-based MOEAs with local update strategies and
several existing criteria for comparing solutions are analyzed
and discussed. And a global loop update strategy and two
hybrid criteria are suggested. Subsequently, an evolutionary
algorithm with the global loop update is implemented and
compared to several of the best multi- or many-objective
optimization algorithms on two famous unconstraint test suites
with up to 15 objectives. Experimental results demonstrate that
unlike evolutionary algorithms with local update strategies,
the population of our algorithm does not degenerate at any
generation of its evolution, which guarantees the diversity
of the resulting population. In addition, our algorithm wins
in most instances of the two test suites, indicating that it
is very competitive in terms of convergence and diversity.
Running results of our algorithm with different criteria for
comparing solutions are also compared. Their differences
are very significant, indicating that the performance of our
algorithm is affected by the criterion it adopts.
Index Terms—evolutionary algorithms, many-objective optimization, global update strategy, Pareto optimality, decomposition.

I. I NTRODUCTION
lot of real-world problems such as electric power system
reconfiguration problems [1], water distribution system
design or rehabilitation problems [2], automotive engine calibration problems [3], land use management problems [4],
optimal design problems [5]–[7], and problems of balancing
between performance and cost in energy systems [8], etc.,
can be formulated into multi- or many-objective optimization
problems(MOPs) involving more than one objective function.
MOPs have attracted extensive attention in recent years and
different kinds of algorithms for solving them have been

A

This work was supported by the National Natural Science Foundation of
China under Grant 61773192.
Y. Zhang is with the School of Computer Science, Liaocheng University,
Liaocheng 252000, China (e-mail:zhangyingyu@lcu-cs.com).
B. Zeng is with the School of Software Engineering, South China University
of Technology, Guangzhou 510006, China.
Y. Li is with the School of Computer Science, Liaocheng University,
Liaocheng 252000, China.
J. Li is with the School of information science and engineering, Shandong
Normal University, Jinan 250014, China, and also with the School of
Computer Science, Liaocheng University, Liaocheng 252000, China.

Yuanzhen Li

Junqing Li

proposed. Although algorithms based on particle swarm optimization [9] and simulated annealing [10] developed to solve
MOPs are not ignorable, multi- or many-objective evolutionary
algorithms(MOEAs) are more popular and representative in
solving MOPs, such as the non-dominated sorting genetic
algorithm-II (NSGA-II) [11], the strength pareto evolutionary
algorithm 2(SPEA-2) [12], and the multi-objective evolutionary algorithm based on decomposition(MOEA/D) [13],etc. In
General, MOEAs can be divided into three categories [14]. The
first category is known as the indicator-based MOEAs. In an
indication-based MOEA, the fitness of an individual is usually
evaluated by a performance indicator such as hypervolume
[15]. Such a performance indicator is designed to measure the
convergence and diversity of the MOEA, and hence expected
to drive the population of the MOEA to converge to the Pareto
Front(PF) quickly with good distribution. The second category
is the domination-based MOEAs, in which the domination
principle plays a key role. However, in the domination-based
MOEAs, other measures have to be adopted to maintain the
population diversity. In NSGA-II, crowding distances of all
the individuals are calculated at each generation and used to
keep the population diversity , while reference points are used
in NSGA-III [16]. The third category is the decompositionbased MOEAs. In a decomposition based MOEA, a MOP is
decomposed into a set of subproblems and then optimized
simultaneously. A uniformly generated set of weight vectors
associated with a fitness assignment method such as the
weighted sum approach, the Tchebycheff approach and the
penalty-based boundary intersection(PBI) approach, is usually used to decompose a given MOP. Generally, a weight
vector determines a subproblem and defines a neighborhood.
Subproblems in a neighborhood are expected to own similar
solutions and might be updated by a newly generated solution.
The decomposition-based MOEA framework emphasizes the
convergence and diversity of the population in a simple model.
Therefore, it was studied extensively and improved from
different points of view [17]–[23] since it was first proposed
by Zhang and Li in 2007 [13].
Recently, some efforts have been made to blend different
ideas appeared in the domination-based MOEAs and the
decomposition-based MOEAs. For examples, an evolutionary
many-objective optimization algorithm based on dominance
and decomposition(MOEA/DD) is proposed in [21], and a
reference vector guided evolutionary algorithm is proposed
in [20]. In MOEA/DD, each individual is associated with
a subregion uniquely determined by a weight vector, and
each weight vector (or subregion) is assigned to a neighborhood. In an iterative step, mating parents is chosen from

2

the neighboring subregions of the current weight vector with
a given probability δ, or the whole population with a low
probability 1 − δ. In case that no associated individual exists
in the selected subregions, mating parents are randomly chosen
from the whole population. And then serval classical genetic
operators such as the simulated binary crossover(SBX) [24]
and the polynomial mutation [25],etc., are applied on the
chosen parents to generate an offspring. Subsequently, the
offspring is used to update the current population according to
a complicated but well-designed rule based on decomposition
and dominance.
In this paper, we focus on update strategies of the
decomposition-based evolutionary algorithms and the criteria for comparing solutions. Three disadvantages of the
decomposition-based MOEAs with local update strategies and
several existing criteria for comparing solutions are analyzed
and discussed. And a global loop update (GLU) strategy and
two hybrid criteria are suggested. Also, we propose an evolutionary algorithm with the GLU strategy for solving multior many-objective optimization problems(MOEA/GLU). The
GLU strategy is designed to try to avoid the shortcomings of
the decomposition-based MOEAs with local update strategies
and eliminate bad solutions in the initial stage of the evolution,
which is expected to force the population to converge faster
to the PF.
The rest of the paper is organized as follows. In section
II, we provide some preliminaries used in MOEA/GLU and
review serval existing criteria for comparing solutions, i.e., PBI
criterion, dominance criterion and distance criterion. And then
two hybrid criteria for judging the quality of two given solutions are suggested. The disadvantages of the decompositionbased MOEAs with local update strategies are also analyzed
in this section. In section III, the algorithm MOEA/GLU
is proposed. A general framework of it is first presented.
Subsequently, the initialization procedure, the reproduction
procedure, and the GLU procedure are elaborated. Some discussions about advantages and disadvantages of the algorithm
are also made. In section IV, empirical results of MOEA/GLU
on DTLZ1 to DTLZ4 and WFG1 to WFG9 are compared
to those of several other MOEAs, i.e., NSGA-III, MOEA/D,
MOEA/DD and GrEA. Running results of MOEA/GLU with
different criteria are also compared in this section. The paper
is concluded in section V.
II. P RELIMINARIES

AND

M OTIVATIONS

A. MOP
Without loss of generality, a MOP can be formulated as a
minimization problem as follows:
M inimize F (x) = (f1 (x), f2 (x), ..., fM (x))T
Subject to x ∈ Ω,

(1)

where M ≥ 2 is the number of objective functions, x is
a decision vector, Ω is the feasible set of decision vectors,
and F (x) is composed of M conflicting objective functions.
F (x) is usually considered as a many-objective optimization
problems when M is greater than or equal to 4.

A solution x of Eq.(1) is said to dominate the other one
y (x 4 y), if and only if fi (x) ≤ fi (y) for i ∈ (1, ..., M )
and fj (x) < fj (y) for at least one index j ∈ (1, ..., M ). It is
clear that x and y are non-dominated with each other, when
both x 4 y and y 4 x are not satisfied. A solution x is Paretooptimal to Eq.(1) if there is no solution y ∈ Ω such that y 4 x.
F(x) is then called a Pareto-optimal objective vector. The set
of all the Pareto optimal objective vectors is the PF [26]. The
goal of a MOEA is to find a set of solutions, the corresponding
objective vectors of which are approximate to the PF.
B. Criteria for Comparing Solutions
1) Dominance criterion: Dominance is usually used to
judge whether or not one solution is better than the other in
the dominance-based MOEAs. As a criterion for comparing
two given solutions, dominance can be described as follows.
Dominance criterion:A solution x is considered to be better
than the other one y when x 4 y.
As it is discussed in [27], the selection pressure exerted by
the dominance criterion is weak in a dominance-based MOEA,
and becomes weaker as the number of the objective functions
increases. It indicates that such a criterion is too stringent for
MOEAs to choose the better one from two given solutions.
Therefore, in practice, the dominance criterion is usually used
together with other measures.
2) PBI criterion: In a decomposition-based MOEA, approaches used to decompose a MOP into subproblems can
be considered as criteria for comparing two solutions, such as
the weighted sum approach, the Tchebycheff approach and the
PBI approach [13]. Here, we describe the PBI approach as a
criterion for comparing two given solutions.
PBI criterion:A solution x is considered to be better than
the other one y when P BI(x) < P BI(y) , where P BI(•)
is defined as P BI(x) = g P BI (x|w, z ∗ ), ω is a given weight
vector, and z ∗ is the ideal point.
The PBI function can be elaborated as [13]:
M inimize g P BI (x|w, z ∗ ) = d1 + θd2
Subject to x ∈ Ω

(2)

where
(F (x) − z ∗ )T w
kwk


w
,
d2 = F (x) − z ∗ + d1
kwk

d1 =

(3)

and θ is a used-defined constant penalty parameter. In a
decomposition-based MOEA with the PBI criterion, the set
of the weight vectors is usually generated at the initialization
stage by the systematic sampling approach and remains unchanged in the running process of the algorithm. The ideal
point is also set at the initialization stage, but can be updated
by every newly generated offspring.
3) Distance criterion: In [28], a criterion with respect to
the two Euclidean distances d1 and d2 defined by Eq.(3) are
used to judge whether or not a solution is better than the other.
Denote the two Euclidean distances of x and y as {d1x , d2x }
and {d1y , d2y } ,respectively. A criterion for comparing two

3

given solutions with respect to the two distances can be written
as follows.
Distance criterion:A solution x is considered to be better
than the other one y when d2x < d2y . In the case that d2x =
d2y , x is considered to be better than y when d1x < d1y .
4) Two Hybrid Criteria: It has been shown that the dominance criterion can be a good criterion for choosing better
solutions in conjunction with other measures [11], [16] . And
likely, the PBI criterion has achieved great success in MOEAs
[13], [21]. However, there are two facts with respect to these
two criteria respectively can not be ignored. The first one is
that using dominance comparison alone can not exert too much
selection pressure on the current population, and hence, can
not drive the population to converge to the PF of a given
MOP quickly. The second one is that it is not necessarily
P BI(x) < P BI(y) when x 4 y, and vice versa.
Therefore, it might be natural to combine these two criteria
in consideration of the two facts. Here, we suggest two hybrid
criteria.
H1 criterion: One solution x is considered to be better
than the other one y when x 4 y. In the case that the two
solutions do not dominate with each other, x is considered
to be better than y when P BI(x) < P BI(y).
H2 criterion: One solution x is considered to be better
than the other one y when x 4 y. In the case that the two
solutions do not dominate with each other, x is considered
to be better than y when d2x < d2y .
It is clear that the H1 criterion combines dominance with
the PBI criterion, while the H2 criterion associates dominance
with the Euclidean distance d2.

C. The Systematic Sampling Approach
The systematic sampling approach proposed by Das and
Dennis [29] is usually used to generate weight vectors in
MOEAs. In this approach, weight vectors are sampled from
a unit simplex. Let ω = (ω1 , ..., ωM )T is a given weight
vector, ωj (1 6 j 6 M ) is the jth component of ω, δj is
the uniform spacing between two consecutive ωj values, and
1/δj is an integer. The possible values of ωj are sampled
Pj−1
from {0, δj , ..., Kj δj }, where Kj = (1 − i=1 ωi )/δj . In
a special case, all δj are equal to δ. To generate a weight
vector, the systematic sampling approach starts with sampling
from {0, δ, 2δ, ..., 1} to obtain the first component ω1 , and then
from {0, δ, 2δ, ..., K2δ} to get the second component ω2 and
so forth, until the M th component ωM is generated. Repeat
such a process, until a total of


D+M −1
N (D, M ) =
(4)
M −1
different weight vectors are generated, where D > 0 is the
number of divisions considered along each objective coordinate.
The approach can be illustrated by Fig.1, in which each
level represents one component of ω, and each path from the
root to one of the leaves represents a possible weight vector.

0

0

1

0.5

1

0.5

0

0

1

0.5

0.5

0

0

0

0.5

Fig. 1. Generating weight vectors for δ = 0.5 and M = 3 using the
systematic sampling approach.

Therefore, all weight vectors included in the tree can be listed
as follows.
(0,
0,
1)
(0,
0.5, 0.5)
(0,
1,
0)
(5)
(0.5, 0, 0.5)
(0.5, 0.5,
0)
(1,
0,
0)
A recursive algorithm for MOEAs to generate weight vectors using the systematic sampling approach can be found in
section III. Here, we consider two cases of D taking a large
value and a small value respectively. As discussed in [21] and
[29], a large D would add more computational burden to a
MOEA, and a small D would be harmful to the population
diversity. To avoid this dilemma, [16] and [21] present a
two-layer weight vector generation method. At first, a set of
N1 weight vectors in the boundary layer and a set of N2
weight vectors in the inside layer are generated, according to
the systematic sampling approach described above. Then, the
coordinates of weight vectors in the inside layer are shrunk
by a coordinate transformation as
vij =

1−τ
+ τ × ωij ,
M

(6)

where ωij is the ith component of the jth weight vectors in
the inside layer, and τ ∈ [0, 1] is a shrinkage factor set as
τ = 0.5 in [16] and [21]. At last, the two sets of weight vectors
are combined to form the final set of weight vectors. Denote
the numbers of the weight vectors generated in the boundary
layer and the inside layer as D1 and D2 respectively. Then,
the number of the weight vectors generated by the two-layer
weight vector generation method is N (D1, M ) + N (D2, M ).
D. Local update and its advantages
Most of the decomposition-based MOEAs update the population with an offspring generated by the reproduction operators to replace the individuals worse than the offspring in the
current neighborhood. Such an update strategy can be named
as a local update(LU) strategy since it involves only the individuals in the current neighborhood. The decomposition-based
MOEAs with the LU strategy have at least two advantages.
The first one is that the LU strategy can help the algorithms to
converge to the PF faster than other algorithms with non-local
update strategies, which helps them achieve great success on

4

a lot of MOPs in the past ten years. The second one is that
the time complexities of the decomposition-based MOEAs are
usually lower than those of MOEAS with non-local update
strategies. This allows them to have a great advantage in
solving complicated problems or MOPs with many objectives,
since the running time taken by a MOEA to solve a given MOP
becomes much longer as the number of the objective functions
increases.
In spite of the above advantages, the decomposition-based
MOEAs with the LU strategy have their own disadvantages.
The first disadvantage is that when the algorithms deal with
some problems such as DTLZ4, the population may lose
its diversity. As we can see from Fig.2, a running instance
of MOEA/D on DTLZ4 with 3 objectives generates welldistributed results, while the solution set of the other one
degenerates nearly to an arc on a unit circle. What’s worse,
the solution set of some running instances of MOEA/D even
degenerates to a few points on a unit circle in our experiments.

to decide which one individual is to be replaced when there are
multiple individuals worse than the newly generated offspring.
One of the simplest replacement policies is to randomly choose
one individual in the current neighborhood and judge whether
or not the offspring is better than it. If the selected individual
is worse, it will be replaced by the offspring. Or else, the
offspring will be abandoned.
Fig.3 shows the results of the original MOEA/D and its
modified version with the modified LU strategy described
above on DTLZ1 with 3 objectives. As it can been seen from
Fig.3, the modified LU strategy lowers down the convergence
speed of the algorithm, indicating that it is not a good update
strategy.

0.6
0.5
0.4
0.3
0.2

1.2
0.1

1
0.8

0
0

0.2

0.6

0.4

0.6

0.6

0.4

0.2

0

0.4

(a)
0.2
0
0

0
0.5

0.5

0.6

1

1
1.5

1.5
0.5

(a)
0.4

0.3

0.2
1
0.1
0.5
0

0

0

0.2

0.4

0.60

0.2

0.4

0.6

-0.5

(b)
-1
0

0
0.5

0.5
1

1
1.5

1.5

(b)
Fig. 2. Two running instances of MOEA/D on DTLZ4. The first one obtained
a well distributed set of solutions, and the second one obtained a degenerate
set of solutions locating on an arc of a unit circle.

Notice that a call of the LU procedure replaces all individuals worse than the newly generated offspring within the current
neighborhood, which might be the reason resulting in the loss
of the population diversity. Therefore, to avoid the loss of
the population diversity, one can modify the LU procedure to
replace at most one individual at a call. But the problem is how

Fig. 3. Running results of the original MOEA/D and its modified version
with a modified LU on the DTLZ1 problem with 3 objectives. The red dots
are the resulting solutions of the original MOEA/D, and the blue ones are the
resulting solutions of its modified version. Rotate (a) around the Z axis by
about 90 degrees to get (b).

The second disadvantage of the decomposition-based
MOEAs with the LU strategy is that they don’t consider
the individuals beyond the current neighborhood. As we can
see, such a LU strategy allows the MOEAS to update the
population with less time, but it might ignore some important
information leading to better convergence. Fig.4 illustrates this
viewpoint. Although the newly generated individual is better
than individual A, it will be abandoned by the decompositionbased MOEAs with the LU strategy, since it is only compared
to the individuals in the current neighborhood.

5

A

current neighborhood

Fig. 4. Illustration of the second disadvantage of the decomposition-based
MOEAs with the LU strategy. The black dots represents the individuals in
the current population, the red dot is a newly generated individual.

Asafuddoula et al have noticed this disadvantage of the
decomposition-based MOEAs with the LU strategy [28]. The
update strategy of their algorithm involves all of the individuals in the current population, which has been demonstrated
to be effective on the DTLZ and WFG test suites to some
extent. We call such an update strategy a global update(GU)
strategy, since each call of the update procedure considers all
the individuals in the population, and replaces at most one
individual. In Fig.4, individual A will be replaced by the newly
generated individual if a decomposition-based MOEA adopts
the GU strategy instead of the LU strategy.
The third disadvantage of the decomposition-based MOEAs
with the LU strategy relates to the individuals and their
attached weight vectors. As a simple example, consider the
case where an individual x and a newly generated offspring
c are attached to a weight vector ωx , and an individual y
is attached to ωy , so that g P BI (c|wx , z ∗ ) < g P BI (x|wx , z ∗ )
and g P BI (x|wy , z ∗ ) < g P BI (y|wy , z ∗ ) are satisfied. In other
words, c is better than x and x is better than y, when the
weight vector ωx and the weight vector ωy are considered as
the reference weight vector respectively. Therefore, x will be
replaced by c in a typical decomposition-based MOEA. But
so far, there is no decomposition-based MOEA considering x
as a replacement for y.
In order to deal with the three disadvantages of the
decomposition-based MOEAs with the LU strategy, we propose a MOEA with the GLU strategy(i.e. MOEA/GLU) mentioned before, which is presented in Section III.
III. P ROPOSED A LGORITHM -MOEA/GLU
A. Algorithm Framework
The general framework of MOEA/GLU is presented in
Algorithm 1. As it is shown in the general framework, a while
loop is executed after the initiation procedure, in which a f or
loop is included. In the f or loop, the algorithm runs over N
weight vectors ,generates an offspring for each weight vector
in the reproduction procedure, and updates the population with
the offspring in the GLU procedure.
B. Initialization Procedure
The initialization procedure includes four steps. In the
first step, a set of uniformly distributed weight vectors are

Algorithm 1 General Framework of MOEA/GLU
Output: Final Population.
1: Initialization Procedure.
2: while The stop condition is not stisfied do
3:
for i = 1 to N do
4:
Reproduction Procedure.
5:
The GLU procedure
6:
end for
7: end while

generated using the systematic approach proposed in [29].
A recursive algorithm for generating the weight vectors is
presented in algorithm 2 and 3.
Algorithm 2 The systematic sampling approach
Input: D:the number of divisions, M:the number of objectives.
Output: A set of uniform weight vectors.
1: ω = (0, ..., 0);
2: Gen ith Level(ω, 0, 0, D, M );
Algorithm 2 calls the recursive function Gen ith Level
described in algorithm 3 with ω = (0, ..., 0), K = 0, and
i = 0, to generate weight vectors. At the ith level of the
recursive function, the ith component of a weight vector is
generated. As discussed before, the value of each component
of a weight vector ranges from 0 to 1 with the step size 1/D,
and all components of a weight vector sum up to 1. In other
words, all components of a weight vector share D divisions.
Therefore, if K=D(K is the number of divisions that have been
allocated), then the rest of the components are all set to zero.
In addition, if ω[i] is the last component, i.e., i = M − 1,
then all the remaining divisions are assigned to it. Both the
two cases indicate the end of a recursive call, and a generated
weight vector is output.
Algorithm 3 Gen ith Level(ω, K, i, D, M )
1: if k==D then
2:
ω[i], ..., ω[M − 1] ← 0;
3:
output(ω);
4:
return;
5: end if
6: if i==M-1 then
7:
ω[i] ← (D − K)/D;
8:
output(ω);
9:
return;
10: end if
11: for j=0 to D-K do
12:
ω[i] ← j/D;
13:
Gen ith Level(ω, K + j, i + 1, D, M );
14: end for
One of the main ideas of MOEA/GLU is that each individual is attached to a weight vector and a weight vector
owns only one individual. Meanwhile, each weight vector
determines a neighborhood. In the second step, the neighborhoods of all weight vectors are generated by calculating the

6

Euclidean distances of the weight vectors using Eq.(3). Subsequently, a population of N individuals is initialized randomly
and attached to N weight vectors in order of generation in the
third step. Finally, the ideal point is initialized in the fourth
step, which can be updated by every offspring in the course
of evolution.

shortest perpendicular distance. Supposed that the minimum
value of {di1 , di2 , ..., diN } is still dij at a certain generation,
and the offspring c is better than P [i] . Then, P [i] will be
replaced out by c, and considered as a candidate to take the
place of the individual hold by the jth weight vector, i.e.,
P [j].

C. Reproduction Procedure

E. Discussion

The reproduction procedure can be described as follows.
Firstly, a random number r between 0 and 1 is generated. If
r is less than a given selection probability Ps , then choose
two individuals from the neighborhood of the current weight,
or else choose two individuals from the whole population.
Secondly, the SBX operator is applied on the two individuals
to generate two intermediate individuals. Notice that, if both
of the two individuals are evaluated and used to update the
population, then the number of individuals evaluated at each
generation will be twice as many as that of the individuals
in the whole population. However, the number of individuals
evaluated at each generation in many popular MOEAs such
as NSGA-III and MOEADD etc., is usually the same as the
size of the population. Therefore, one of the two intermediate
individuals is abandoned at random for the sake of fairness.
Finally, the polynomial mutation operator is applied on the reserved intermediate individual to generate an offspring, which
will be evaluated and used to update the current population in
the following GLU procedure.

This section gives a simple discussion about the similarities
and differences of MOEA/GLU, MOEA/D, and MOEA/DD.
1) Similarities of MOEA/GLU, MOEA/D, and MOEA/DD.
MOEA/GLU and MOEA/DD can be seen as two variants of MOEA/D to some extent, since all of the three
algorithms employ decomposition technique to deal with
MOPs. In addition, a set of weight vectors is used
to guide the selection procedure, and the concept of
neighborhood plays an important role in all of them.
2) Differences between MOEA/GLU and MOEA/D.
Firstly, MOEA/D uses a LU strategy, and MOEA/GLU
employs the so-called GLU strategy, which considers all
of the individuals in the current population at each call of
the update procedure. Secondly, to judge whether or not
an individual is better than the other, MOEA/D compares
the fitness values of them, while other criteria for
comparing individuals can also be used in MOEA/GLU.
Thirdly, once a individual is generated in MOEA/D, all
the individuals in the current neighborhood that worse
than it will be replaced. However, each individual is
attached to one weight vector in MOEA/GLU, and a
newly generated individual is only compared to the old
one attached to the same weight vector. The replacement
operation occurs only when the new individual is better
than the old one.
3) Differences between MOEA/GLU and MOEA/DD. In
the first place, one weight vector in MOEA/DD not
only defines a subproblem, but also specifies a subregion
that can be used to estimate the local density of a
population. In principle, a subregion owns zero, one,
or more individuals at any generation. In MOEA/GLU,
each individual is attached only to one weight vector,
and a weight vector can hold only one individual. In
the second place, the dominance criterion can be taken
into account in MOEA/GLU, the way that it is used
is different from that of MOEA/DD. In MOEA/GLU,
the dominance between the newly generated individual
and the old one attached to the same weight vector can
be used to judge which of the two is better, while the
dominance criterion is considered among all individuals
within a subregion in MOEA/DD.

D. The GLU procedure
Algorithm 4 The GLU procedure
Input: a new offspring c, the current population P.
1: bFlag=true;
2: while bFlag do
3:
F ind Attathed W eight(c) → i;
4:
if c is better than P[i] then
5:
Swap(c,P[i]);
6:
else
7:
bFlag=false;
8:
end if
9: end while
The GLU procedure is illustrated in Algorithm 4, which can
described as follows. Each individual is attached to a weight
vector, which has the shortest perpendicular distance to the
weight vector. F ind Attathed W eight(c) is designed to find
the attached weight of c, in which the perpendicular distance
is calculated by Eq.(3). Denote the perpendicular distance of
the ith individual P [i] to the jth weight vector as dij . A
given weight vector maintains only one slot to keep the best
individual attached to it generated so far from the beginning of
the algorithm. The minimum value of {di1 , di2 , ..., diN } can
be expected to be dii after the algorithm evolves enough generations. However, in the initialization stage, all the individuals
are generated randomly, and attached to the weight vectors in
order of generation. In other words, the ith weight vector may
not be the one, to which its attached individual P [i] has the

F. Time Complexity
The function Find Attathed Weight in the GLU procedure
runs over all weight vectors, calculates the perpendicular
distances between the newly generated offspring and all weight
vectors, and finds the weight vector, to which the offspring
has the shortest perpendicular distance. Therefore, it takes
O(M N ) times of floating-point calculations for the function

7

Find Attathed Weight to find the attached weight vector of the
offspring, where M is the number of the objective functions
and N is the size of the population.
As it is indicated before, the while loop is designed to
help the individuals in the initial stage of the algorithm to
find their attached weight vectors quickly. The fact that the
individuals at a certain generation do not attach to their corresponding weight vectors causes extra entries into the function
Find Attathed Weight. However, once all of the individuals
are attached to their corresponding weight vectors, the function
Find Attathed Weight will be entered at most two times. Let
the entries into the function Find Attathed Weight be (1+Ni )
times at each call of the GLU procedure,
P and denote the
number of the generations as G. Since i Ni ≤ N and the
GLU procedure is called N G times in the whole process
of MOEA/GLU, the time complexity of the algorithm is
O(M N 2 G), which is the same as that of MOEA/DD, but
worse than that of MOEA/D.
IV. E XPERIMENTAL R ESULTS
A. Performance Metrics
1) Inverted Generational Distance(IGD): Let S be a result
solution set of a MOEA on a given MOP. Let R be a uniformly
distributed representative points of the PF. The IGD value of
S relative to R can be calculated as [30]
P
d(r, S)
(7)
IGD(S, R) = r∈R
|R|
where d(r,S) is the minimum Euclidean distance between r and
the points in S, and |R| is the cardinality of R. Note that, the
points in R should be well distributed and |R| should be large
enough to ensure that the points in R could represent the PF
very well. This guarantees that the IGD value of S is able to
measure the convergence and diversity of the solution set. The
lower the IGD value of S, the better its quality [21].
2) HyperVolume(HV): The HV value of a given solution
set S is defined as [31]
!
[
HV (S) = vol
[f1 (x), z1 ] × . . . × [fM (x), zM ] , (8)
x∈S

where vol(·) is the Lebesgue measure,and z r = (z1 , . . . , zM )T
is a given reference point. As it can be seen that the HV value
of S is a measure of the size of the objective space dominated
by the solutions in S and bounded by z r .
As with [21], an algorithm based on Monte Carlo sampling
proposed in [32] is applied to compute the approximate HV
values for 15-objective test instances, and the WFG algorithm
[33] is adopted to compute the exact HV values for other test
instances for the convenience of comparison. In addition,
QM all
the HV values are normalized to [0, 1] by dividing i=1 zi .
B. Benchmark Problems

1) DTLZ test suite: Problems DTLZ1 to DTLZ4 from the
DTLZ test suite proposed by Deb et al [34] are chosen for
our experimental studies in the first place. One can refer to
[34] to find their definitions. Here, we only summarize some
of their features.

DTLZ1:The
global PF of DTLZ1 is the linear hyper-plane
PM
k
i=1 fi = 0.5. And the search space contains (11 − 1)
local PFs that can hinder a MOEA to converge to the
hyper-plane.
PM 2
• DTLZ2:The global PF of DTLZ2 satisfys
i fi = 1.
Previous studies have shown that this problem is easier
to be solved by existing MOEAs, such as NSGA-III,
MOEADD, etc., than DTLZ1, DTLZ3 and DTLZ4.
• DTLZ3:The definition of the glocal PF of DTLZ3 is the
same as that of DTLZ2. It introduces (3k − 1) local PFs.
All local PFs are parallel to the global PF and a MOEA
can get stuck at any of these local PFs before converging
to the global PF. It can be used to investigate a MOEA’s
ability to converge to the global PF.
• DTLZ4:The definition of the global PF of DTLZ4 is also
the same as that of DTLZ2 and DTLZ3. This problem
can be obtained by modifying DTLZ2 with a different
meta-variable mapping, which is expected to introduce a
biased density of solutions in the search space. Therefore,
it can be used to investigate a MOEA’s ability to maintain
a good distribution of solutions.
To calculate the IGD value of a result set S of a MOEA
running on a MOP, a set R of representative points of the PF
needs to be given in advance. For DTLZ1 to DTLZ4, we take
the set of the intersecting points of weight vectors and the PF
∗
surface as R. Let f ∗ = (f1∗ , ..., fM
) be the intersecting point
of a weight vector w = (w1 , ..., wM )T and the PF surface.
Then fi∗ can be computed as [21]
wi
(9)
fi∗ = 0.5 × PM
j=1 wj
•

for DTLZ1, and

wi
fi∗ = qP
M

j=1

(10)
wj

for DTLZ2, DTLZ3 and DTLZ4.
2) WFG test suite [35], [36]: This test suite allows test
problem designers to construct scalable test problems with
any number of objectives, in which features such as modality
and separability can be customized as required. As discussed
in [35], [36], it exceeds the functionality of the DTLZ test
suite. In particular, one can construct non-separable problems,
deceptive problems, truly degenerative problems, mixed shape
PF problems, problems scalable in the number of positionrelated parameters, and problems with dependencies between
position- and distance-related parameters as well with the
WFG test suite.
In [36], several scalable problems, i.e., WFG1 to WFG9, are
suggested for MOEA designers to test their algoritms, which
can be described as follows.
M inimize F (X) = (f1 (X), ..., fM (X))
fi (X) = xM + 2ihi (x1 , ..., xM−1 )

(11)

T

X = (x1 , ..., xM )

where hi is a problem-dependent shape function determining
the geometry of the fitness space, and X is derived from
a vector of working parameters Z = (z1 , ..., zn )T , zi ∈
[0, 2i] , by employing four problem-dependent transformation

8

functions t1 , t2 , t3 and t4 . Transformation functions must
be designed carefully such that the underlying PF remains
intact with a relatively easy to determine Pareto optimal set.
The WFG Toolkit provides a series of predefined shape and
transformation functions to help ensure this is the case. One
can refer to [35], [36] to see their definitions. Let
′′ T
Z ′′ = (z1′′ , ..., zm
) = t4 (t3 (t2 (t1 (Z ′ ))))

Z ′ = (z1 /2, ..., zn /2n)T .

TABLE I
N UMBER OF P OPULATION S IZE
M
3
5
8
10
15

D1
12
6
3
3
2

D2
2
2
1

Population Size
91
210
156
275
135

(12)

Then xi = zi′′ (zi′′ − 0.5) + 0.5 for problem WFG3, whereas
X = Z ′′ for problems WFG1, WFG2 and WFG4 to WFG9.
The features of WFG1 to WFG9 can be summarized as
follows.
• WFG1:A separable and uni-modal problem with a biased
PF and a convex and mixed geometry.
• WFG2:A non-separable problem with a convex and disconnected geometry, i.e., the PF of WFG2 is composed
of several disconnected convex segments. And all of its
objectives but fM are uni-modal.
• WFG3:A non-separable and uni-modal problem with a
linear and degenerate PF shape, which can be seen as a
connected version of WFG2.
• WFG4:A separable and multi-modal problem with large
”hill sizes”, and a concave geometry.
• WFG5:A separable and deceptive problem with a concave
geometry.
• WFG6:A nonseparable and uni-modal problem with a
concave geometry.
• WFG7:A separable and uni-modal problem with parameter dependency, and a concave geometry.
• WFG8:A nonseparable and uni-modal problem with parameter dependency, and a concave geometry.
• WFG9:A nonseparable, deceptive and uni-modal problem
with parameter dependency, and a concave geometry.
As it can be seen from above, WFG1 and WFG7 are both
separable and uni-modal, and WFG8 and WFG9 have nonseparable property, but the parameter dependency of WFG8
is much harder than that caused of WFG9. In addition, the
deceptiveness of WFG5 is more difficult than that of WFG9,
since WFG9 is only deceptive on its position parameters.
However, when it comes to the nonseparable reduction, WFG6
and WFG9 are more difficult than WFG2 and WFG3. Meanwhile,problems WFG4 to WFG9 share the same EF shape in
the objective space, which is a part of a hyper-ellipse with
radii ri = 2i, where i ∈ {1, ..., M }.
C. Parameter Settings
The parameter settings of MOEA/GLU are listed as follows.
1) Settings for Crossover Operator:The crossover probability is set as pc = 1.0 and the distribution index is
ηc = 30.
2) Settings for Mutation Operator:The mutation probability
is set as pm = 0.6/n, and is different from that of
MOEA/DD, which is 1/n. The distribution index is set
as ηm = 20.
3) Population Size:The population size of MOEA/GLU is
the same as the number of the weight vectors that

TABLE II
N UMBER OF G ENERATIONS
problem
DTLZ1
DTLZ2
DTLZ3
DTLZ4

4)

5)

6)
7)
8)

9)

10)

M =3
400
250
1000
600

M =5
600
350
1000
1000

M =8
750
500
1000
1250

M = 10
1000
750
1500
2000

M = 15
1500
1000
2000
3000

can be calculated by Eq.(4). Since the divisions for 3and 5-objective instances are set to 12 and 6, and the
population sizes of them are 91 and 210, respectively. As
for 8-, 10- and 15-objective instances, two-layer weight
vector generation method is applied. The divisions and
the population sizes of them are listed in Table I.
Number of Runs:The algorithm is independently run 20
times on each test instance, which is the same as that of
other algorithms for comparison.
Number of Generations: All of the algorithms stopped
at a predefined number of generations. The number of
generations for DTLZ1 to DTLZ4 is listed in Table II,
and the number of generations for all the instances of
WFG1 to WFG9 is 3000.
Penalty Parameter in PBI: θ = 5.0.
Neighborhood Size: T = 20.
Selection Probability: The probability of selecting two
mating individuals from the current neighborhood is set
as ps = 0.9.
Settings for DTLZ1 to DTLZ4:As in papers [21],
[28], the number of the objectives are set as M ∈
{3, 5, 8, 10, 15} for comparative purpose. And the number of the decision variables is set as n = M + r − 1,
where r = 5 for DTLZ1, and r = 10 for DTLZ2,
DTLZ3 and DTLZ4. To calculate the HV value we
set the reference point to (1, ..., 1)T for DTLZ1, and
(2, ..., 2)T DTLZ2 to DTLZ4.
Settings for WFG1 to WFG9: The number of the decision variables is set as n = k + l, where k = 2 × (M − 1)
is the position-related variable and l = 20 is the
distance-related variable. To calculate the HV values for
problems WFG1 to WFG9, the reference point is set to
(3, ..., 2M + 1)T .

D. Performance Comparisons on DTLZ1 to DTLZ4
We calculate the IGD values and HV values of the same
solution sets found by MOEA/GLU, and compare the calculation results with those of MOEA/DD, NSGA-III, MOEA/D
and GrEA obtained in [21].

9

3

DTLZ1

5
8
10
15

3

DTLZ2

5
8
10
15

MOEA/GLU
1.073E-4
3.608E-4
1.669E-3
1.732E-4
2.115E-4
2.395E-4
1.457E-3
2.069E-3
3.388E-3
1.429E-3
2.030E-3
3.333E-3
2.261E-3
3.652E-3
6.111E-3
4.418E-4
5.738E-4
7.510E-4
9.513E-4
1.075E-3
1.231E-3
2.553E-3
3.038E-3
3.375E-3
2.917E-3
3.701E-3
4.104E-3
4.394E-3
6.050E-3
7.623E-3

MOEA/DD
3.191E-4
5.848E-4
6.573E-4
2.635E-4
2.916E-4
3.109E-4
1.809E-3
2.589E-3
2.996E-3
1.828E-3
2.225E-3
2.467E-3
2.867E-3
4.203E-3
4.669E-3
6.666E-4
8.073E-4
1.243E-3
1.128E-3
1.291E-3
1.424E-3
2.880E-3
3.291E-3
4.106E-3
3.223E-3
3.752E-3
4.145E-3
4.557E-3
5.863E-3
6.929E-3

NSGA-III
4.880E-4
1.308E-3
4.880E-3
5.116E-4
9.799E-4
1.979E-3
2.044E-3
3.979E-3
8.721E-3
2.215E-3
3.462E-3
6.896E-3
2.649E-3
5.063E-3
1.123E-2
1.262E-3
1.357E-3
2.114E-3
4.254E-3
4.982E-3
5.862E-3
1.371E-2
1.571E-2
1.811E-2
1.350E-2
1.528E-2
1.697E-2
1.360E-2
1.726E-2
2.114E-2

MOEA/D
4.095E-4
1.495E-3
4.743E-3
3.179E-4
6.372E-4
1.635E-3
3.914E-3
6.106E-3
8.537E-3
3.872E-3
5.073E-3
6.130E-3
1.236E-2
1.431E-2
1.692E-2
5.432E-4
6.406E-4
8.006E-4
1.219E-3
1.437E-3
1.727E-3
3.097E-3
3.763E-3
5.198E-3
2.474E-3
2.778E-3
3.235E-3
5.254E-3
6.005E-3
9.409E-3

GrEA
2.759E-2
3.339E-2
1.351E-1
7.369E-2
3.363E-1
4.937E-1
1.023E-1
1.195E-1
3.849E-1
1.176E-1
1.586E-1
5.110E-1
8.061E-1
2.057E+0
6.307E+1
6.884E-2
7.179E-2
7.444E-2
1.411E-1
1.474E-1
1.558E-1
3.453E-1
3.731E-1
4.126E-1
4.107E-1
4.514E-1
5.161E-1
5.087E-1
5.289E-1
5.381E-1

DTLZ4

m

DTLZ3

TABLE III
B EST, M EDIAN AND W ORST IGD VALUES BY MOEA/GLU, MOEA/DD, NSGA-III, MOEA/D AND G R EA ON DTLZ1, DTLZ2, DTLZ3 AND DTLZ4
INSTANCES WITH D IFFERENT N UMBER OF O BJECTIVES . T HE VALUES IN RED ARE THE BEST, AND THE VALUES IN GRAY ARE THE SECOND BEST.
MOEA/IUP
1.598E-4
1.257E-3
8.138E-3
2.965E-4
8.390E-4
2.543E-3
1.987E-3
4.478E-3
1.759E-2
2.173E-3
2.663E-3
4.795E-3
5.299E-3
8.732E-3
1.912E-2
9.111E-5
1.105E-4
1.385E-4
7.218E-5
9.255E-5
1.115E-4
3.540E-4
4.532E-4
5.823E-4
8.397E-4
1.156E-3
1.482E-3
9.325E-4
1.517E-3
2.427E-3

MOEA/DD
5.690E-4
1.892E-3
6.231E-3
6.181E-4
1.181E-3
4.736E-3
3.411E-3
8.079E-3
1.826E-2
1.689E-3
2.164E-3
3.226E-3
5.716E-3
7.461E-3
1.138E-2
1.025E-4
1.429E-4
1.881E-4
1.097E-4
1.296E-4
1.532E-4
5.271E-4
6.699E-4
9.107E-4
1.291E-3
1.615E-3
1.931E-3
1.474E-3
1.881E-3
3.159E-3

NSGA-III
9.751E-4
4.007E-3
6.665E-3
3.086E-3
5.960E-3
1.196E-2
1.244E-2
2.375E-2
9.649E-2
8.849E-3
1.188E-2
2.082E-2
1.401E-2
2.145e-2
4.195E-2
2.915E-4
5.970E-4
4.286E-1
9.849E-4
1.255E-3
1.721E-3
5.079E-3
7.054E-3
6.051E-1
5.694E-3
6.337E-3
1.076E-1
7.110E-3
3.431E-1
1.073E+0

MOEA/D
9.773E-4
3.426E-3
9.113E-3
1.129E-3
2.213E-3
6.147E-3
6.459E-3
1.948E-2
1.123E+0
2.791E-3
4.319E-3
1.010E+0
4.360E-3
1.664E-2
1.260E+0
2.929E-1
4.280E-1
5.234E-1
1.080E-1
5.787E-1
7.348E-1
5.298E-1
8.816E-1
9.723E-1
3.966E-1
9.203E-1
1.077E+0
5.890E-1
1.133E+0
1.249E+0

GrEA
6.770E-2
7.693E-2
4.474E-1
5.331E-1
8.295E-1
1.124E+0
7.518E-1
1.024E+0
1.230E+0
8.656E-1
1.145E+0
1.265E+0
9.391E+1
1.983E+2
3.236E+2
6.869E-2
7.234E-2
9.400E-1
1.422E-1
1.462E-1
1.609E-1
3.229E-1
3.314E-1
3.402E-1
4.191E-1
4.294E-1
4.410E-1
4.975E-1
5.032E-1
5.136E-1

3

DTLZ1

5
8
10
15

3

DTLZ2

5
8
10
15

MOEA/GLU
0.973657
0.973576
0.973279
0.998981
0.998976
0.998970
0.999948
0.999925
0.999888
0.999991
0.999981
0.999971
0.999986
0.999923
0.999826
0.926698
0.926682
0.926652
0.990545
0.990533
0.990513
0.999341
0.999326
0.999296
0.999921
0.999920
0.999919
0.999998
0.999997
0.999993

MOEA/DD
0.973597
0.973510
0.973278
0.998980
0.998975
0.998968
0.999949
0.999919
0.999887
0.999994
0.999990
0.999974
0.999882
0.999797
0.999653
0.926674
0.926653
0.926596
0.990535
0.990527
0.990512
0.999346
0.999337
0.999329
0.999952
0.999932
0.999921
0.999976
0.999954
0.999915

NSGA-III
0.973519
0.973217
0.971931
0.998971
0.998963
0.998673
0.999975
0.993549
0.966432
0.999991
0.999985
0.999969
0.999731
0.999686
0.999574
0.926626
0.926536
0.926359
0.990459
0.990400
0.990328
0.999320
0.978936
0.919680
0.999918
0.999916
0.999915
0.999975
0.999939
0.999887

MOEA/D
0.973541
0.973380
0.972484
0.998978
0.998969
0.998954
0.999943
0.999866
0.999549
0.999983
0.999979
0.999956
0.999695
0.999542
0.999333
0.926666
0.926639
0.926613
0.990529
0.990518
0.990511
0.999341
0.999329
0.999307
0.999922
0.999921
0.999919
0.999967
0.999951
0.999913

GrEA
0.967404
0.964059
0.828008
0.991451
0.844529
0.500179
0.999144
0.997992
0.902697
0.999451
0.998587
0.532348
0.172492
0.000000
0.000000
0.924246
0.923994
0.923675
0.990359
0.990214
0.990064
0.999991
0.999670
0.989264
0.997636
0.996428
0.994729
0.999524
0.999496
0.998431

DTLZ4

m

DTLZ3

TABLE IV
B EST, M EDIAN AND W ORST HV VALUES BY MOEA/GLU, MOEA/DD, NSGA-III, MOEA/D AND G R EA ON DTLZ1, DTLZ2, DTLZ3 AND DTLZ4
INSTANCES WITH D IFFERENT N UMBER OF O BJECTIVES . T HE VALUES IN RED ARE THE BEST, AND THE VALUES IN GRAY ARE THE SECOND BEST.
MOEA/GLU
0.926717
0.926457
0.924931
0.990565
0.990532
0.990451
0.999345
0.999322
0.999252
0.999922
0.999921
0.999919
0.999996
0.999994
0.999990
0.926731
0.926729
0.926725
0.990570
0.990570
0.990569
0.999364
0.999363
0.999362
0.999919
0.999914
0.999910
0.999990
0.999979
0.999959

MOEA/DD
0.926617
0.926346
0.924901
0.990558
0.990515
0.990349
0.999343
0.999311
0.999248
0.999923
0.999922
0.999921
0.999982
0.999951
0.999915
0.926731
0.926729
0.926725
0.990575
0.990573
0.990570
0.999364
0.999363
0.998360
0.999921
0.999920
0.999917
0.999915
0.999762
0.999680

NSGA-III
0.926480
0.925805
0.924234
0.990453
0.990344
0.989510
0.999300
0.924059
0.904182
0.999921
0.999918
0.999910
0.999910
0.999793
0.999780
0.926659
0.926705
0.799572
0.991102
0.990413
0.990156
0.999363
0.999361
0.994784
0.999915
0.999910
0.999827
0.999910
0.999581
0.617313

MOEA/D
0.926598
0.925855
0.923858
0.990543
0.990444
0.990258
0.999328
0.999303
0.508355
0.999922
0.999920
0.999915
0.999918
0.999792
0.999628
0.926729
0.926725
0.500000
0.990569
0.990568
0.973811
0.999363
0.998497
0.995753
0.999918
0.999907
0.999472
0.999813
0.546405
0.502115

GrEA
0.924652
0.922650
0.621155
0.963021
0.808084
0.499908
0.953478
0.791184
0.498580
0.962168
0.735934
0.499676
0.000000
0.000000
0.000000
0.924613
0.924094
0.500000
0.990514
0.990409
0.990221
0.999102
0.999039
0.998955
0.999653
0.999608
0.999547
0.999561
0.999539
0.999521

10

1) DTLZ1:From the calculation results listed in Table III
and Table IV, it can be seen that MOEA/GLU and
MOEA/DD perform better than the other three algorithms on all of the IGD values and most of the HV
values. Specifically, MOEA/GLU wins in the best and
median IGD values of the 3-, 8-, 10- and 15-objective
instances, and MOEA/DD wins in the worst IGD values
of the 3-, 8-, 10- and 15-objective instances. As for the
5-objective instance, MOEA/GLU wins in all of the IGD
values. When it comes to the HV values, MOEA/GLU
performs the best on the 3-, 5- and 15-objective instances, and MOEA/DD shows the best performance
on the 10-objective instance as listed in Table IV. In
addition, MOEA/GLU wins in the median and worst
HV values of the 8-objective instance, and NSGA-III
wins in the best HV value of it. Although all of the
values obtained by MOEA/GLU and MOEA/DD are
close, MOEA/GLU wins in most of the IGD and HV
values. Therefore, MOEA/GLU can be considered as the
best optimizer for DTLZ1.
2) DTLZ2:As it can be seen from Table III, MOEA/D,
MOEA/GLU and MOEA/DD are significantly better
than the other two on all of the IGD values of DTLZ2.
As for the IGD values, MOEA/GLU performs the best
on the 3-, 5- and 8-objective instances, and MOEA/D
performs the best on the 10-objective instance. In addition, MOEA/GLU wins in the best value of the 15objective instance, and MOEA/DD wins in the median
and worst values of it. When it comes to the HV values,
MOEA/GLU performs the best on the 3-, 5- and 15objective instances, MOEA/DD performs the best on the
10-objective instance and wins in the worst value of the
8-objective instance, and GrEA wins in the best and median values of the 8-objective instance. On the whole, the
differences of MOEA/GLU, MOEA/DD and MOEA/D
are not significant on DTLZ2, but MOEA/GLU wins
in more values than both MOEA/D and MOEA/DD.
Therefore, MOEA/GLU can also be considered as the
best optimizer for DTLZ2.
3) DTLZ3:Again, MOEA/GLU and MOEA/DD are the
best two optimizer for DTLZ3, and their performances
are also close. As for the IGDvalues, MOEA/GLU
performs the best on the 5- and 8-objective instances,
MOEA/DD performs the best on the 10-objective instance, MOEA/GLU wins in the best and median values
of the 3-objective instance and the best value of the 15objective instance, MOEA/DD wins in the median and
worst values of the 15-objective instance, and the worst
value of the 3-objective. As far as the HV values are
concerned, MOEA/GLU performs the best on the 3-, 5-,
8- and 15-objective instances, and MOEA/DD performs
the best on the 10-objective instance. Since MOEA/GLU
wins in more values than the other four algorithms, it
can be considered as the best optimizer for DTLZ3.
4) DTLZ4:It is clear that MOEA/GLU performs the best
on all of the IGD values of DTLZ4. However, it is
hard to distinguish the better one from MOEA/GLU and
MOEA/DD when it comes to the HV values. Interest-

ingly, the performance of MOEA/GLU and MOEA/DD
are so close that all of the HV values of the 3-objective
instance, the best and median HV values of the 8objective instance obtained by them are equal in terms
of 6 significant digits. But taking the performance on
the IGD values into consideration, MOEA/GLU is the
best optimizer for DTLZ4.
E. Performance Comparisons on WFG1 to WFG9
The HV values of MOEA/GLU, MOEA/DD, MOEA/D and
GrEA on WFG1 to WFG5 are listed in Table V, and the
HV values on WFG6 to WFG9 are listed in Table VI. The
comparison results can be concluded as follows.
1) WFG1:MOEA/DD wins in all the values of WFG1
except the worst value of the 3-objective instance, and
hence be regarded as the best optimizer for WFG1.
2) WFG2:MOEA/GLU shows the best performance on the
3-objective instance, while MOEA/DD performs the best
on the 8-objective instance. In addition, MOEA/GLU
wins in the best and median values of 5- and 10objective instances, while MOEA/DD wins in the worst
values of them. Obviously, MOEA/GLU and MOEA/DD
are the best two optimizer for WFG2, but it is hard to
tell which one is better, since the differences between
them are not significant.
3) WFG3:MOEA/GLU performs the best on the 10objective instance, MOEA/DD shows the best performance on the 3-objective instance, and GrEA wins in
the 5- and 8-objective instances. The values obtained by
the three algorithms are very close. They all have their
own advantages.
4) WFG4:MOEA/GLU shows the best in all values of
WFG4, and is considered as the winner.
5) WFG5:Like in WFG4, MOEA/GLU is the winner of
WFG5, since it wins in all values except the median
and worst values of the 3-objective instance.
6) WFG6:MOEA/GLU and GrEA are the best two optimizer of WFG. The values obtained by them are not
significant with ups and downs on both sides. Specifically, MOEA/GLU wins in the 3-objective instance,
the best values of the 5- and 8-objective instances, the
median and worst values of the 10-objective instance.
GrEA wins in all the other values.
7) WFG7:MOEA/GLU wins in all the values of WFG7,
and is considered as the best optimizer.
8) WFG8:MOEA/GLU wins in most of the values of
WFG8 except the best value of the 5-objective instance
and the median value of the 8-objective instance. Therefore, it can also be regarded as the best optimizer for
WFG8.
9) WFG9:The situation of WFG9 is a little bit complicated,
but it is clear that MOEA/GLU, MOEA/DD and GrEA
are all better than MOEA/D. To be specific, GrEA wins
in the 8-objective instance, and it might be said that
MOEA/DD performs the best on the 3- and 5-objective
instance although the worst value of it on the 3-objective
instance is slightly worse than that of MOEA/GLU. In

11

TABLE V
B EST, M EDIAN AND W ORST HV VALUES BY MOEA/GLU, MOEA/DD,
MOEA/D AND G R EA ON WFG1 TO WFG5 INSTANCES WITH D IFFERENT
N UMBER OF O BJECTIVES . T HE VALUES IN RED ARE THE BEST, AND THE
VALUES IN GRAY ARE THE SECOND BEST.

TABLE VI
B EST, M EDIAN AND W ORST HV VALUES BY MOEA/GLU, MOEA/DD,
MOEA/D AND G R EA ON WFG6 TO WFG9 INSTANCES WITH D IFFERENT
N UMBER OF O BJECTIVES . T HE VALUES IN RED ARE THE BEST, AND THE
VALUES IN GRAY ARE THE SECOND BEST.

m

8
10

WFG2

3
5
8
10

WFG3

3
5
8
10

WFG4

3
5
8
10

WFG5

3
5
8
10

GrEA
0.794748
0.692567
0.627963
0.876644
0.831814
0.790367
0.811760
0.681959
0.616006
0.866298
0.832016
0.757841
0.950084
0.942908
0.800186
0.980806
0.976837
0.808125
0.980012
0.840293
0.778291
0.964235
0.959740
0.956533
0.699502
0.672221
0.662046
0.695221
0.684583
0.671553
0.657744
0.649020
0.638147
0.543352
0.513261
0.501210
0.723403
0.722997
0.722629
0.881161
0.879484
0.877642
0.787287
0.784141
0.679178
0.896261
0.843257
0.840257
0.689784
0.689177
0.688885
0.836232
0.834726
0.832212
0.838183
0.641973
0.571933
0.791725
0.725198
0.685882

3

WFG6

MOEA/D
0.932609
0.929839
0.815356
0.918652
0.915737
0.912213
0.918252
0.911586
0.808931
0.922484
0.915715
0.813928
0.951685
0.803246
0.796567
0.982796
0.978832
0.807951
0.963691
0.800333
0.787271
0.962841
0.957434
0.773474
0.697968
0.692355
0.679281
0.669009
0.662925
0.654729
0.529698
0.457703
0.439274
0.382068
0.337978
0.262496
0.724682
0.723945
0.723219
0.870868
0.862132
0.844219
0.784340
0.737386
0.718648
0.747485
0.712680
0.649713
0.693135
0.687378
0.681305
0.829696
0.826739
0.812225
0.779091
0.753486
0.705938
0.730990
0.715161
0.673789

5
8
10

3

WFG7

5

MOEA/DD
0.937694
0.933402
0.899253
0.963464
0.960897
0.959840
0.922284
0.913024
0.877784
0.926815
0.919789
0.864689
0.958287
0.952467
0.803397
0.986572
0.985129
0.980035
0.981673
0.967265
0.789739
0.968201
0.965345
0.961400
0.703664
0.702964
0.701624
0.673031
0.668938
0.662951
0.598892
0.565609
0.556725
0.552713
0.532897
0.504943
0.727060
0.726927
0.726700
0.876181
0.875836
0.875517
0.920869
0.910146
0.902710
0.913018
0.907040
0.888885
0.693665
0.693544
0.691173
0.833159
0.832710
0.830367
0.852838
0.846736
0.830338
0.848321
0.841118
0.829547

5
8
10

3

WFG8

WFG1

3

MOEA/GLU
0.937116
0.928797
0.915136
0.906874
0.899351
0.862874
0.839662
0.831208
0.781919
0.887565
0.843225
0.794202
0.959834
0.958155
0.808454
0.995169
0.993049
0.813859
0.978775
0.795215
0.778920
0.981398
0.978021
0.779176
0.700589
0.695748
0.689587
0.679497
0.675726
0.662165
0.572932
0.554256
0.526689
0.572593
0.554042
0.531208
0.731535
0.731180
0.730558
0.883419
0.881701
0.880210
0.939271
0.933853
0.926261
0.967623
0.963674
0.951068
0.698469
0.692607
0.685518
0.844325
0.841781
0.838402
0.892830
0.889458
0.884971
0.919163
0.916148
0.911875

5
8
10

3

WFG9

m

5
8
10

MOEA/GLU
0.710228
0.701988
0.698358
0.858096
0.846655
0.840335
0.912150
0.901300
0.880581
0.938343
0.927854
0.914464
0.731908
0.731809
0.731691
0.888158
0.887856
0.887592
0.948854
0.947862
0.946082
0.976171
0.975644
0.974641
0.678825
0.677146
0.674987
0.806626
0.805050
0.803366
0.895652
0.845761
0.823666
0.961919
0.923244
0.881384
0.695369
0.642755
0.642240
0.809717
0.751592
0.749481
0.828505
0.809564
0.746497
0.843321
0.830062
0.803744

MOEA/DD
0.708910
0.699663
0.689125
0.850531
0.838329
0.828315
0.876310
0.863087
0.844535
0.884394
0.859986
0.832299
0.727069
0.727012
0.726907
0.876409
0.876297
0.874909
0.920763
0.917584
0.906219
0.927666
0.923441
0.917141
0.672022
0.670558
0.668593
0.818663
0.795215
0.792900
0.876929
0.845975
0.730348
0.896317
0.844036
0.715250
0.707269
0.687401
0.638194
0.834616
0.797185
0.764723
0.772671
0.759369
0.689923
0.717168
0.717081
0.696061

MOEA/D
0.702840
0.695081
0.684334
0.846015
0.813844
0.754054
0.692409
0.661156
0.567108
0.643198
0.582342
0.409210
0.725252
0.724517
0.723449
0.859727
0.843424
0.811292
0.729953
0.708701
0.605900
0.706473
0.625828
0.596189
0.671355
0.669927
0.664120
0.808204
0.793773
0.771763
0.537772
0.446544
0.347990
0.508652
0.350409
0.270931
0.688940
0.681725
0.636355
0.798069
0.789998
0.727728
0.633476
0.604016
0.548119
0.572925
0.546451
0.516309

GrEA
0.699876
0.693984
0.685599
0.855839
0.847137
0.840637
0.912095
0.902638
0.885712
0.943454
0.927443
0.884145
0.723229
0.722843
0.722524
0.884174
0.883079
0.881305
0.918742
0.910023
0.901292
0.937582
0.902343
0.901477
0.671845
0.669762
0.667948
0.797496
0.792692
0.790693
0.803050
0.799986
0.775434
0.841704
0.838256
0.830394
0.702489
0.638103
0.636575
0.823916
0.753683
0.747315
0.842953
0.831775
0.765730
0.860676
0.706632
0.686917

addition, the median and worst values of MOEA/GLU
on the 10-objective instance are far better than those of
other algorithms, while the best value is sightly worse
than that of GrEA.
On the whole, MOEA/GLU shows a very competitive
performance on the WFG test suite, especially WFG4, WFG5,
WFG7 and WFG8, of which MOEA/GLU wins almost all the
HV values.

12

F. Performance Comparisons of Algorithms with Different
Criteria for Comparison
In this subsection, we compare the HV values of
MOEA/GLU with different criteria for comparing solutions on
WFG1 to WFG9 with different objectives. The HV values are
listed in Table VII. The comparison results can be concluded as
follows. For the sake of convenience, we denote MOEA/GLU
with the PBI, H1, and H2 criteria as MOEA/GLU-PBI,
MOEA/GLU-H1, and MOEA/GLU-H2, respectively.
•

•

•

•

•

•

•

•

•

WFG1:MOEA/GLU-H2 is the best optimizer since it
performs the best on all instances of WFG1.
WFG2:MOEA/GLU-H2 wins in all values of WFG2 except the best value of the 3-objective instance. Therefore,
it is considered to be the best optimizer for WFG2.
WFG3:The situation is a little bit complicated for WFG3.
Specifically, MOEA/GLU-PBI wins on the worst value
of the 5-objective instance and the best value of the
8-objective instance. And MOEA/GLU-H1 wins in the
best and median values of the 3-objective instance, while
MOEA/GLU-H2 is the best on other values. Therefore,
MOEA/GLU-H2 can be considered the best optimizer for
WFG3.
WFG4: MOEA/GLU-H1 is the best optimizer that wins
in all the values of WFG4.
WFG5: MOEA/GLU-PBI has the best median value for
the 3-objective instance of WFG5, and MOEA/GLUH2 wins in the worst value of the 3-objective instance,
while MOEA/GLU-H1 performs the best on all the other
values. Therefore, MOEA/GLU-H1 is considered the best
optimizer for WFG5.
WFG6: MOEA/GLU-PBI has the best median value for
the 5-objective instance of WFG6, and MOEA/GLUH2 wins in the best value of the 8-objective instance
and the worst value of the 10-objective instance, while
MOEA/GLU-H1 performs the best on all other values.
Therefore, MOEA/GLU-H1 is considered the best optimizer for WFG6.
WFG7: Since MOEA/GLU-H1 wins in all the values of
WFG7 except the worst value of its 3-objective instance,
it is considered the best optimizer.
WFG8: MOEA/GLU-PBI performs the worst on all
the values of WFG8. MOEA/GLU-H1 wins in its 3objective instance, while MOEA/GLU-H2 wins in the
10-objective instance. As for 5- and 8-objective instance,
MOEA/GLU-H1 wins in the median and worst value,
and MOEA/GLU-H2 wins in the best value. It is clear
that MOEA/GLU-PBI is the worst optimizer for WFG8.
However, as for MOEA/GLU-H1 and MOEA/GLU-H2,
it is still hard to say which one of the two is better for
WFG8.
WFG9: MOEA/GLU-PBI wins in the best value of the
5-objective instance, the worst value of the 8-objective
instance , and the median value of 10-objective instance
of WFG9. MOEA/GLU-H2 only wins in the worst value
of the 10-objective instance, while MOEA/GLU-H1 wins
in all the other values of WFG9. Therefore, MOEA/GLUH1 can be considered the best optimizer for WFG9.

On the whole, MOEA/GLU-H1 is the best optimizer for
WFG4 to WFG7, and MOEA/GLU-H2 is the best for WFG1
to WFG3, and WFG9. As for WFG8, Both MOEA/GLU-H1
and MOEA/GLU-H2 are better than MOEA/GLU-PBI, but it
is hard to say which one of the two is better. These indicate
that the running results of MOEA/GLU are affected by the
criterion for comparing solutions it adopts.
V. C ONCLUSION
In this paper, we propose a MOEA with the so-called GLU
strategy, i.e., MOEA/GLU. The main ideas of MOEA/GLU
can be concluded as follows. Firstly, MOEA/GLU employs a
set of weight vectors to decompose a given MOP into a set
of subproblems and optimizes them simultaneously, which is
similar to other decomposition-based MOEAs. Secondly, each
individual is attached to a weight vector and a weight vector
owns only one individual in MOEA/GLU, which is the same
as that in MOEA/D, but different from that in MOEA/DD.
Thirdly, MOEA/GLU adopts a global update strategy, i.e. the
GLU strategy. Our experiments indicate that the GLU strategy
can overcome the disadvantages of MOEAs with local update
strategies discussed in section II, although it makes the time
complexity of the algorithm higher than that of MOEA/D.
These three main ideas make MOEA/GLU a different algorithm from other MOEAs, such as MOEA/D, MOEA/DD,
and NSGA-III, etc. Additionally, the GLU strategy is simpler
than the update strategies of MOEA/DD and NSGA-III. And
the time complexity of MOEA/GLU is the same as that of
MOEA/DD, but worse than that of MOEA/D.
Our algorithm is compared to several other MOEAs, i.e.,
MOEA/D, MOEA/DD, NSGA-III, GrEA on 3, 5, 8, 10, 15objective instances of DTLZ1 to DTLZ4, and 3, 5, 8, 10objective instances of WFG1 to WFG9. The experimental
results show that our algorithm wins in most of the instances.
In addition, we suggest two hybrid criteria for comparing
solutions, and compare them with the PBI criterion. The
empirical results show that the two hybrid criteria is very
competitive in 3, 5, 8, 10-objective instances of WFG1 to
WFG9.
Our future work can be carried out in the following three
aspects. Firstly, it is interesting to study the performances of
MOEA/GLU on other MOPs, such as the ZDT test problems,
the CEC2009 test problems, combinatorial optimization problems appeared in [37], [38], and especially some real-world
problems with a large number of objectives. Secondly, it might
be valuable to apply the two hybrid criteria for comparing
solutions to other MOEAs. Thirdly, improve MOEA/GLU to
overcome its shortcomings. As we can see, the algorithm
contains at least two shortcomings. One is that all of its experimental results on WFG1 are worse than those of MOEA/DD
except for the best HV value of the 3-objective instance.
The other is that its time complexity is worse than that of
MOEA/D. Further research is necessary to be carried out to
try to overcome these two shortcomings.
ACKNOWLEDGMENT
The authors would like to thank Qingfu Zhang and Ke Li
for their generously giving the java codes of MOEA/D and

13

10

WFG4

3
5
8
10

WFG7

3
5
8
10

MOEA/DD.
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[4] O. Chikumbo, E. Goodman, and K. Deb, “Approximating a multidimensional pareto front for a land use management problem: A
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Hypervolume-Driven Analytical Programming for Solar-Powered Irrigation System Optimization. Heidelberg: Springer International Publishing, 2013, pp. 147–154.

H1
0.958855
0.884332
0.810262
0.994851
0.993575
0.815756
0.976819
0.970539
0.773388
0.980088
0.788371
0.770895
0.698434
0.695167
0.685562
0.845047
0.844616
0.843364
0.895639
0.893686
0.887629
0.920249
0.918867
0.916739
0.678718
0.676451
0.674706
0.808181
0.805537
0.804091
0.889190
0.844953
0.8248850
0.934571
0.908858
0.885137

H2
0.958309
0.955588
0.811369
0.997409
0.996668
0.816768
0.992986
0.983890
0.800754
0.995465
0.993171
0.801428
0.698400
0.693015
0.692078
0.844654
0.842033
0.838127
0.892399
0.888761
0.882556
0.917481
0.915163
0.911581
0.677100
0.675703
0.671803
0.821287
0.804396
0.801026
0.902643
0.842148
0.819766
0.948360
0.924340
0.886750

WFG3

PBI
0.957614
0.955452
0.810162
0.994983
0.993170
0.814648
0.978952
0.961105
0.776725
0.982992
0.978054
0.781965
0.697911
0.695548
0.690822
0.843774
0.840175
0.833143
0.891617
0.888071
0.882510
0.917412
0.916424
0.909849
0.672998
0.671628
0.666242
0.805790
0.804308
0.803095
0.877491
0.844134
0.811508
0.942152
0.905199
0.877100

WFG6

8

H2
0.944690
0.939554
0.922585
0.931487
0.922682
0.895859
0.918863
0.867385
0.851531
0.918984
0.877223
0.869973
0.731411
0.730869
0.730343
0.883788
0.881128
0.878642
0.938517
0.928525
0.920050
0.963828
0.959950
0.953305
0.731854
0.731742
0.731573
0.887789
0.887662
0.887201
0.947672
0.946120
0.944323
0.976109
0.975063
0.974119

WFG9

5

H1
0.937488
0.924179
0.903350
0.908070
0.898754
0.873925
0.844850
0.787144
0.706517
0.876852
0.846479
0.746061
0.731777
0.731550
0.731255
0.885457
0.884489
0.882801
0.940762
0.937541
0.930123
0.971182
0.968851
0.964127
0.731874
0.731773
0.731506
0.888112
0.888013
0.887752
0.948562
0.947616
0.945989
0.976717
0.975950
0.975258

WFG5

WFG1

3

PBI
0.932478
0.919064
0.907255
0.909324
0.897364
0.860576
0.843381
0.831511
0.743512
0.849143
0.844050
0.785389
0.727187
0.726180
0.724841
0.881501
0.879884
0.876368
0.934740
0.930307
0.921402
0.967500
0.961761
0.951755
0.729737
0.728050
0.727450
0.887632
0.887157
0.886747
0.946957
0.945634
0.944550
0.976190
0.975073
0.974186

WFG8

m

WFG2

TABLE VII
B EST, M EDIAN AND W ORST HV VALUES BY MOEA/GLU WITH T HREE D IFFERENT C RITERIA : PBI, H1 AND H2 ON INSTANCES OF WFG1 TO WFG9
WITH 3,5,8 AND 10 O BJECTIVES . T HE VALUES IN RED FONT ARE THE BEST.

PBI
0.690390
0.685735
0.675661
0.679563
0.673978
0.669298
0.586041
0.550607
0.532548
0.574988
0.549540
0.529822
0.708928
0.699913
0.692277
0.859456
0.847747
0.832130
0.916292
0.900157
0.883341
0.937329
0.923721
0.911045
0.689751
0.641612
0.638641
0.806133
0.750934
0.748504
0.816273
0.803022
0.761689
0.837137
0.825754
0.762914

H1
0.700632
0.696145
0.684853
0.681048
0.676910
0.666466
0.578472
0.552667
0.540043
0.565746
0.548960
0.508638
0.713367
0.703288
0.699125
0.856809
0.849112
0.840610
0.913827
0.902256
0.893311
0.941969
0.928654
0.915935
0.697593
0.642693
0.642119
0.799793
0.753767
0.749479
0.824253
0.803785
0.740563
0.839769
0.825240
0.757464

H2
0.699646
0.692045
0.685778
0.682644
0.676994
0.666298
0.605893
0.592805
0.581368
0.574100
0.555640
0.534696
0.713228
0.702914
0.694282
0.858598
0.848262
0.838766
0.916850
0.897580
0.884989
0.938397
0.926283
0.916056
0.693049
0.642552
0.641827
0.790106
0.750096
0.746307
0.823161
0.797496
0.741695
0.835042
0.823727
0.764740

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Yingyu Zhang received the B.Eng. degree in computer science and technology from Changsha University of Science and Technology, Changsha, China,
in 2002, and the M.Eng. and Ph.D. degrees in
computer science from Huazhong University of Science and Technology, Wuhan, China, in 2007, and
2011, respectively. He is now a lecturer with the
School of Computer Science, Liaocheng University,
Liaocheng, China. His research interests includes
quantum optimization, evolutionary multi-objective
optimization, machine learning, and cloud computing.

Bing Zeng received the B.Econ. degree in economics from Huazhong Agricultural University,
Wuhan, China, in 2004, and received the B.Eng.
degree in computer science and technology, the
M.Eng. and Ph.D. degrees in information security
from Huazhong University of Science and Technology, Wuhan, China, in 2004, 2007 and 2012,
respectively. He is currently an Assistant Professor at
South China University of Technology, Guangzhou,
China. His research interests are in cryptography and
network security, with a focus on secure multiparty
computation.

Yuanzhen Li received his PhD degree from Beijing University of Posts and Telecommunications
in 2010. He is now an associate professor in the
Department of Computer Science and Technology,
Liaocheng University, China. His research interests
include wireless communications, evolutionary computation and Multi-objective optimization.

Junqing Li received the B.Sc. and Ph.D. degrees
from Shandong Economic University, Northeastern
University in 2004 and 2016, respectively. Since
2004, he has been with the School of Computer
Science, Liaocheng University, Liaocheng, China,
where he became an Associate Professor in 2008.
He also works with the School of information and
Engineering, Shandong Normal University, Jinan,
Shandong, China, where he became the doctoral
supervisor. His current research interests include
intelligent optimization and scheduling.

