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S. Zapotecas-Martínez, A. García-Nájera and A. López-Jaimes / Expert Systems With Applications 120 (2019) 357–371 359
depth ( Srinivas & Deb, 1994 ). However, a good approximation of
the Pareto front has to fulfill two goals simultaneously: conver-
gence and diversity. Therefore, to distribute the solutions along
the entire trade-off curve, Pareto dominance has to be used in co-
operation with a second criterion. Some methods that have been
proposed to distribute solutions along the Pareto optimal front in-
clude fitness sharing and niching ( Deb & Goldberg, 1989 ), cluster-
ing ( Zitzler et al., 2002 ), crowding distance ( Deb, Pratap, Agarwal,
& Meyarivan, 2002 ), among many others. In the early days of the
20 0 0s, Pareto-based MOEAs became one of the most used strate-
gies to achieve a relatively good approximation of the real PF. How-
ever, their use has decreased because of the difficulty to achieve
a proper spread of solutions ( Farhang-Mehr & Azarm, 2002; Gee,
Tan, Shim, & Pal, 2015; Hallam, Blanchfield, & Kendall, 2005 ) and
the loss of their discriminant property in high-dimensional ob-
jective spaces ( Brockhoff & Zitzler, 2006; López Jaimes & Coello
Coello, 2015 ). Examples of recent Pareto-based approaches are
SPEA2+SDE ( Li, Yang, & Liu, 2014 ), -DEA ( Yuan, Xu, Wang, & Yao,
2016 ), and SPEA/R ( Jiang & Yang, 2017 ) .
Indicator-based Approaches. A strategy widely employed by MOEAs
to achieve a relatively good representation of the PF is related
to the use of performance indicators ( Farhang-Mehr & Azarm,
2002; Gee et al., 2015; Hallam et al., 2005 ). The indicator-based
evolutionary algorithm (IBEA) ( Zitzler & Künzli, 2004 ) posed the
possibility to optimize a performance indicator in the evolu-
tionary process of MOEAs. As it is well known, there exists a
large number of indicators to assess the performance of MOEAs,
see for example the studies reported in ( Jiang, Ong, Zhang, &
Feng, 2014; Okabe, Jin, & Sendhoff, 2003; Zitzler, Thiele, Lau-
manns, Fonseca, & da Fonseca, 2003 ). Such indicators can as-
sess, in different ways, convergence and diversity, or both of them
at the same time. In particular, a good representation of the
real PF of a MOP can be achieved by using performance indica-
tors such as hypervolume ( Zitzler & Thiele, 1998 ), R 2 ( Hansen &
Jaszkiewicz, 1998 ), IGD ( Coello Coello & Reyes Sierra, 2004 ), among
others. However, the use of indicator-based MOEAs is limited by
the high computational cost to compute a particular indicator,
such as the hypervolume which increases as the number of objec-
tives increases. Recent indicator-based approaches are RIB-EMOA
( Zapotecas-Martínez, Sosa Hernández, Aguirre, Tanaka, & Coello
Coello, 2014 ), MOMBI-II ( Hernández Gómez & Coello Coello, 2015 ),
MOEA/IGD-NS ( Tian, Zhang, Cheng, & Jin, 2016 ), BCE-IBEA ( Li, Yang,
& Liu, 2016 ), and LIBEA ( Zapotecas-Martínez, López-Jaimes, &
García-Nájera, 2018 ).
Decomposition-based Approaches. In the last decade, scalarizing
functions have been employed by several evolutionary approaches,
giving rise to the so-called MOEAs based on decomposition.
Decomposition-based approaches rely on solving a number of
scalarizing functions which are formulated by the same number
of weight vectors. This strategy to solve MOPs have been found
to be very effective to deal with complicated test problems (see
for instance ( Li & Zhang, 2009; Zhang & Li, 2007; Zhang, Liu, & Li,
2009 )). In addition, by having a well-distributed set of weight vec-
tors, a well representative sample of the real PF can be obtained.
For this reason, decomposition-based MOEAs have become an ex-
cellent alternative to deal with multi-objective problems. Some
variants or improvements of MOEA/D can be found in Li and
Zhang (2009) , Zhang, Zhou, and Jin (2008a) , Li, Deb, Zhang, and
Kwong (2015) . On the other hand, different metaheuristics based
on decomposition are for instance MOPSO/D ( Peng & Zhang, 2008 ),
MOABC/D ( Medina, Das, Coello, & Ramírez, 2014 ), and dMOPSO
( Zapotecas-Martínez & Coello Coello, 2011 ).
The MOEAs mentioned above, are a flavor of distinct evolu-
tionary methodologies based on different principles. However, a
more comprehensive review of this type of approaches can be
found in ( Coello Coello, 1999; Nedjah & de Macedo Mourelle, 2015;
Trivedi, Srinivasan, Sanyal, & Ghosh, 2017; Zhou et al., 2011 ).
2.3. Decomposition of a multi-objective problem
It is well-known that a Pareto optimal solution to the problem
in Eq. (1) is an optimal solution of a scalar optimization problem in
which the objective is an aggregation of all the objective functions
f
i
’s ( Miettinen, 1999 ). Many scalar approaches have been proposed
to aggregate the objectives of a MOP. Among them, the Tchebycheff
approach
( Bowman Jr., 1976 ) is one of the most used methods. This
technique transforms the vector of function values F into a scalar
minimization problem which is of the form:
minimize : g
tch
(x | λ, z ) = max
1 ≤ j≤M
{ λ
j
| z
j
− f
j
(x ) |} (2)
s.t. x ∈ ⊂ R
n
where is the feasible region, z = (z
1
, . . . , z
k
)
is a reference point
such that z
j
= max { f
j
(x ) | x ∈ } for each i = { 1 , . . . , M} , and λ =
(λ
1
, . . . , λ
M
)
is a weight vector, i.e., λ
l
≥ 0 for all j = 1 , . . . , M and
M
j=1
λ
j
= 1 .
For each Pareto optimal point x
, there exits a weight vector λ
such that x
is the optimum solution of Eq. (2) and each optimal
solution of Eq. (2) is a Pareto optimal solution of the problem in
Eq. (1) .
Therefore, an appropriate representation of the Pareto front
could be reached by solving different scalarizing problems. Such
problems can be defined by a set of well-distributed weight vec-
tors, which establish the search direction in the optimization pro-
cess. This is the principal idea behind mathematical programming
methods for multi-objective problems and current multi-objective
evolutionary approaches based on decomposition, see for example
the works reported in ( Peng & Zhang, 2008; Zhang & Li, 2007 ).
3. Related work
3.1. Grey wolf optimizer
Mirjalili et al. (2014) proposed the Grey Wolf Optimizer (GWO),
which is inspired by the canis lupus (grey wolf). According to Mir-
jalili et al., grey wolves live in a pack from five to twelve members,
and they have a very strict social dominant hierarchy: the alpha
wolves, the beta wolves, the delta wolves, and the omega wolves.
The alpha is mostly responsible for making decisions about hunt-
ing, sleeping place and time to wake. The beta wolves are subor-
dinate wolves that help the alpha in making decisions and other
pack activities. Therefore, beta wolves are probably the best can-
didates to be the alpha in case one of the alpha wolves passes
away or becomes very old. Delta wolves have to submit to alphas
and betas, but they dominate the omega. Scouts, sentinels, elders,
hunters, and caretakers belong to this category. Finally, the lowest
ranking grey wolves are the omegas, which play the role of scape-
goats and always have to submit to all the other dominant wolves.
In addition to the social hierarchy of wolves, GWO also consid-
ers group hunting as a social behavior of grey wolves. The main
phases of grey wolf hunting are: (a) tracking, chasing, and ap-
proaching the prey, (b) pursuing, encircling, and harassing the prey
until it stops moving, and (c) attacking the prey.
The mathematical model of the social hierarchy of grey wolves
considered by the GWO is as follows: The fittest solution as the
alpha ( α). Consequently, the second and third best solutions are
named beta ( β) and delta ( δ), respectively. The rest of the candi-
date solutions are assumed to be omega ( ω). In GWO, the hunting
(optimization) is guided by α, β, and δ. The ω wolves follow these