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Journal of Intelligent & Fuzzy Systems 37 (2019) 8097–8111
DOI:10.3233/JIFS-190566
IOS Press
8097
Some spherical linguistic Muirhead
mean operators with their application
to multi-attribute decision making
Hongfei Caoa, Runtong Zhanga,∗and Jun Wangb
aSchool of Economics and Management, Beijing Jiaotong University, Beijing, China
bSchool of Economics and Management, Beijing University of Chemical Technology, Beijing, China
Abstract. This paper aims to propose a new tool to express decision makers’ preference information in multi-attribute decision
making (MADM) producers. By taking advantages of spherical fuzzy sets (SFSs) and linguistic variables (LVs), we give the
definition of spherical linguistic sets (SLSs) and provide operations of spherical linguistic numbers (SLNs). Based on the
proposed operations, we incorporate Muirhead mean (MM) into SLSs and introduce novel spherical linguistic aggregation
operators. These proposed operators adsorb the inherent advantages of MM, i.e., taking the interrelationship among any
numbers of aggregated inputs into account and producing flexible information fusion process. Furthermore, we apply the
proposed method in MADM and present the main steps of a new method. In order to show its effectiveness, we use the
method to solve an actual MADM problem. The advantages and superiorities of the proposed method are also studied.
Keywords: Spherical fuzzy set, spherical linguistic set, spherical linguistic muirhead mean, multi-attribute decision making
1. Introduction
We always encounter decision-making problems in
real-life and mostly, the decision-making problems
refer to multi-attribute decision making (MADM)
[1–5]. For example, when we buy a car we usu-
ally evaluate all potential alternatives from multiple
aspects, such as degree of comfort, price, brand rep-
utation, overall performance, etc. In recent years, the
MADM methods and related decision making tech-
niques have been a very active research field, which
has attracted many scholars’ interests [6–15]. In the
framework of MADM theory, there are four funda-
mental elements, i.e., candidate alternatives, multiple
attributes, evaluation values and decision-making
∗Corresponding author. Runtong Zhang, School of Economics
and Management, Beijing Jiaotong University, Beijing, China.
Tel.: +86 01051683854; E-mail: rtzhang@bjtu.edu.cn.
methods. Evidently, in advance of the procedure of
determining the most optimal alternative, the possi-
ble alternatives and attributes that are evaluated have
been confirmed already. Hence, the most important
two parts are the evaluation values and decision-
making methods. For evaluation values it is widely
known that decision makers (DMs) can hardly get
all information of alternatives within a limited time.
Therefore, DMs would like to use fuzzy numbers
rather than crisp numbers to express their evaluation
values. With the development of fuzzy sets theories,
DMs have more choices to select an appropriate tool
to denote their preferences. Recently, Prof. Yager [16]
provided a new methodology for dealing with com-
plicated MADM problems, called Pythagorean fuzzy
sets (PFSs). Since the introduction of PFSs, more
and more scholars and scientists have shifted their
attention to research on PFSs based MADM. Zhang
and Xu [17] gave operations of Pythagorean fuzzy
ISSN1064-1246/19/$35.00 © 2019 – IOS Press and the authors. All rights reserved
8098 H. Cao et al. / Some spherical linguistic Muirhead mean operators
numbers (PFNs) as well we Pythagorean fuzzy TOP-
SIS method. Zhang [18] introduced the Pythagorean
fuzzy QUAIFLEX approach based on a new com-
parison method for PFNs. Garg [19] extended PFSs
to interval-valued PFSs and investigated their appli-
cations in MADM. Zhang [20] studied similarity
measure for PFSs. Ren et al. [21] proposed a novel
Pythagorean fuzzy MADM approach from the point
of view of TODIM. Gou et al. [22] focused on con-
tinuous Pythagorean fuzzy information, investigated
their important properties and introduced a method-
ology for integrating it. Garg [23] introduced new
Pythagorean fuzzy correlation coefficients between
PFSs and investigated their applications in pattern
recognition and medical diagnosis. More scholars
and scientists shifted their attention to aggregation
operators for Pythagorean fuzzy information and
some operators, such as Pythagorean fuzzy Bonfer-
roni means [24, 25], Pythagorean fuzzy Maclaurin
symmetric means [26], Pythagorean fuzzy Muirhead
mean [27, 28], Pythagorean fuzzy power mean [29],
and Pythagorean fuzzy point operators [30] have been
developed. Some scholars devoted themselves to the
investigation of Pythagorean fuzzy operations and a
few operational rules of PFNs, such as Einstein oper-
ations [31–33] symmetric operations [34], interaction
operations [35], Hamacher operations [36], exponen-
tial operations [37], and Frank operations [38] were
introduced.
In intuitionistic fuzzy sets (IFSs) and PFSs we have
membership grades (MGs) and non-membership
grades (NGs) and the indeterminacy grades (IGs) are
a default once the MGS and NGs are determined.
For example, in IFSs if the MG is 0.2 and NG is 0.3,
then the IG is 1 −0.2−0.3=0.5 and in PFSs the IG
is √1−0.22−0.32=0.93. However, in Coung’s
[39, 40] opinion the IGs should be determined by
DMs rather than a default. Hence, Coung [39, 40]
proposed a new computational intelligence tool, i.e.
PIcture Fuzzy Sets (PIFSs). In the framework, DMs
are allowed to express not only the positive and neg-
ative information, but also the neutral ideas. Hence,
PIFSs can describe more information and are more
effective to deal with human opinion involving more
answers of types: yes, abstain, no, and refusal, such
as voting. To distinguish IFSs and PIFSs, Cuong
called the three information functions the positive
membership degree (PMD), the negative member-
ship degree (ND), and the neutral membership degree
(NMD). It is not difficult to find out that PIFSs express
more information than IFSs and are more powerful
to deal with complicated decision making systems.
Since the appearance, PIFSs have gained extensive
attention from scholars all over the world. Basically,
recent researches on PIFs mainly focus on similarity
measures [41–43], outranking methods [44, 45] and
picture fuzzy aggregation operators [46–51].
Recently, motivated by PFSs Ashraf et al. [52]
relaxed the restraint of PIFs and proposed the concept
of spherical fuzzy set (SFS), whose constraint is that
the sum of squares of PMD, ND, and NMD is less than
or equal to 1. Evidently, SFSs take full advantages of
both PFSs and PIFSs. Compared with PFSs, SFSs
contains more decision information and are better
to represent DMs’ opinions. Compared with PIFSs,
SPFSs give DMs more freedom and consequently less
information distortion is lead. Afterwards, scholars
started to study on SFSs based on MADM meth-
ods and quite a few achievements have been reported
[53, 54]. However, SFSs still have drawbacks as they
only express DMs’ quantitative evaluation informa-
tion and neglect their qualitative assessments. More
and more scholars have started to know the high
necessity to take DMs’ qualitative decision making
information into account before determining the opti-
mal choices [55–57]. Therefore, this paper tries to
extend the power SFSs theory by additionally taking
DMs’ qualitative evaluation into consideration and
propose the spherical linguistic sets (SLSs). In the
framework of SLSs, DMs utilize LVs to express their
evaluation information and additionally, they are also
allowed to provide the PMDs, NDs, and NMDs of
LVs, such that the sum of the square of three degrees
is less than or equal to one. Hence, the proposed SLSs
inherit the advantages and superiorities of LVs and
SFSs, i.e. they not only describe DMs’ preference
information quantificationally and qualitatively, but
also give DMs enough freedom to express their eval-
uations. SLSs are parallel to picture fuzzy linguistic
sets (PFLSs) [58], but are more powerful as they have
a laxer constraint, which gives experts more freedom
to express their preferences. In addition, SLSs are also
stronger than Pythagorean linguistic sets (PLSs) [59]
as they additionally captures the DMs’ neutral grades.
When fusing spherical linguistic (SL) information,
the SL aggregation operators are needed. Recently the
good performance of the Muirhead mean (MM) [60]
in capturing the interrelationship among any numbers
of input variables has deeply impressed us [61–63].
Therefore, we extend MM to SLSs and propose new
spherical linguistic operators. Finally, we use the pro-
posed operators to introduce a new MADM method.
For easy description, the following of this paper is
organized as follows. Section 2 proposes the SLSs
H. Cao et al. / Some spherical linguistic Muirhead mean operators 8099
and related concepts. Section 3 describes the SL
aggregation operators and their properties. Section 4
gives a new algorithm for MADM in which attributes
are in the form of SL information. Section 5 shows
the effectiveness of the proposed method.
2. Preliminaries
Concepts that will be used in the paper are dis-
cussed in this section.
2.1. Spherical fuzzy sets
Definition 1. [52] Let Xbe an ordinary fixed set, a
spherical fuzzy set (SFS) Adefined on Xis given by
A={(x, μA(x),η
A(x),v
A(x)) |x∈X}(1)
Where μA(x), ηA(x) and vA(x) represent
the positive membership degree, the neutral
membership degree, and the negative mem-
bership degree respectively, satisfying the
condition that μA(x),η
A(x),v
A(x)∈[0,1] and
0≤μA(x)2+ηA(x)2+vA(x)2≤1. The refusal
degree of Ais expressed as πA(x)=
1−μA(x)2−ηA(x)2−vA(x)2. For conve-
nience, (μA(x),η
A(x),v
A(x)) is called a spherical
fuzzy number (SFN), which can be simple denoted
by α=(μ, η, v).
It is easy to find out that the SFS only considers
DMs’ quantitative evaluation information. Basically,
DMs usually give their evaluations both quantita-
tively and qualitatively. Hence, we combine SFSs and
linguistic variables (LVs) to more comprehensively
describe DMs’ assessment information.
Definition 2. Let Xbe an ordinary fixed set, a spher-
ical linguistic set (SLS) defined on Xcan be given
as
A=sθ(x),μ
A(x),η
A(x),v
A(x)|x∈X(2)
Where sθ(x)∈S, and μA(x), ηA(x), and vA(x)
represent the positive membership degree, the
neutral membership degree, and the negative
membership degree, satisfying the condi-
tions that 0 ≤μA(x),η
A(x),v
A(x)≤1 and
0≤μA(x)2+ηA(x)2+vA(x)2≤1. Then for
each x∈X, the refusal degree of xto sθ(x)is
πA(x)=1−μA(x)2−ηA(x)2−vA(x)2. For con-
venience, we call α=(sθ(x),μ
A(x),η
A(x),v
A(x))
a spherical linguistic number (SLN), which can be
denoted asα=(sθ,μ,η,v).
Based on the operations of SFNs proposed by
Ashraf [52], we proposed operational rules of SLNs.
Definition 3. Let α1=(sθ1,μ
1,η
1,v
1), α2=
(sθ2,μ
2,η
2,v
2) and α=(sθ,μ,η,v) be any three
SLNs, and λbe positive real number, then
(1) α1⊕α2=sθ1+θ2,μ2
1+μ2
2−μ2
1μ2
2,
η1η2,v
1v2;
(2) α1⊗α2=sθ1×θ2,μ
1μ2,η2
1+η2
2−η2
1η2
2,
v2
1+v2
2−v2
1v2
2;
(3) λα =sλ×θ,1−1−μ2λ,η
λ,v
λ;
(4) αλ=sθλ,μ
λ,1−1−η2λ,
1−1−v2λ.
Definition 4. Let α=(sθ,μ,η,v) be a SLN, then the
sore is defined as follows
S(α)=1
4μ2−v2+2×sθ=s1
4(μ2−v2+2)×θ
(3)
For any two SLNs α1=(sθ1,μ
1,η
1,v
1) and α2=
(sθ2,μ
2,η
2,v
2), if S(α1)>S(α2), then α1>α
2and
if S(α1)>S(α2), then α1=α2.
2.2. Muirhead mean and dual Muirhead mean
The MM is an aggregation operator introduced by
Muirhead [60] for crisp numbers. This operator can
capture the interrelationship among all the aggregated
arguments.
Definition 5. [60] Let aj(j=1,2, ..., n) be a collec-
tion of crisp numbers and Q=(q1,q
2, ..., qn)∈Rn
be a vector of parameter, then the MM operator is
defined as
MMQ(a1,a
2, ..., an)=⎛
⎝1
n!
ϑ∈Tn
n
j=1
aqj
ϑ(j)⎞
⎠
1
n
j=1
qj
(4)
Where ϑ(j)(j=1,2, ..., n) is any permutation of
(1,2, ..., n), and Tnis the collection of all permuta-
tions of (1,2, ..., n).
8100 H. Cao et al. / Some spherical linguistic Muirhead mean operators
The dual form of MM is called the dual Muirhead
mean (DMM) and its definition is given as follows.
Definition 6. [64] Let aj(j=1,2, ..., n) be a collec-
tion of crisp numbers and Q=(q1,q
2, ..., qn)∈Rn
be a vector of parameter, then the DMM operator is
defined as
DMMQ(a1,a
2, ..., an)
=1
n
j=1
qj⎛
⎝
ϑ∈Tn
n
j=1qjaϑ(j)⎞
⎠
1
n!
(5)
where ϑ(j)(j=1,2, ..., n) is any permutation of
(1,2, ..., n), and Tnis the collection of all permu-
tations of (1,2, ..., n).
3. Some spherical linguistic aggregation
operators
3.1. The spherical linguistic Muirhead mean
(SLMM) operator
Definition 7. Let αj=(sθj,μ
j,η
j,v
j) be a collec-
tion of SLNs, and Q=(q1,q
2, ..., qn)∈Rnbe a set
of parameters, then the spherical linguistic Muirhead
mean (SLMM) operator is defined as
SLMMQ(α1,α
2, ..., αn)
=⎛
⎝1
n!
ϑ∈Tn
n
j=1
αqj
ϑ(j)⎞
⎠
1
n
j=1
qj
(6)
Where ϑ(j)(j=1,2, ..., n) is any permutation of
(1,2, ..., n), and Tnis the collection of all permuta-
tions of (1,2, ..., n).
Theorem 1. Let αj=(sθj,μ
j,η
j,v
j)(j=1,2,
..., n)be a collection of SLNs,Q=(q1,q
2,
..., qn)∈Rnbe a vector of parameters, then the
aggregated value by the SLMM operator is still a
SLNs and
SLMMQ(α1,α
2, ..., αn)=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
s
1
n!
ϑ∈Tn
n
j=1
θqj
ϑ(j)
1
n
j=1
qj
,
⎛
⎜
⎜
⎜
⎝
1−⎛
⎝
ϑ∈Tn⎛
⎝1−
n
j=1
μ2qj
ϑ(j)⎞
⎠⎞
⎠
1
n!⎞
⎟
⎟
⎟
⎠
1
n
j=1
qj
,
1−⎛
⎝1−
ϑ∈Tn1−
n
j=11−η2
ϑ(j)qj1
n!⎞
⎠
1
n
j=1
qj
,
1−⎛
⎝1−
ϑ∈Tn1−
n
j=11−v2
ϑ(j)qj1
n!⎞
⎠
1
n
j=1
qj⎞
⎟
⎟
⎟
⎟
⎠
(7)
Proof. According to the operations of Definition 3,
we can obtain
αqj
ϑ(j)=sθqj
ϑ(j)
,μ
qj
ϑ(j),1−1−η2
ϑ(j)qj,
1−1−v2
ϑ(j)qj,
and
n
j=1
αqj
ϑ(j)=⎛
⎜
⎝sn
j=1
θqj
ϑ(j)
,
n
j=1
μqj
ϑ(j),
1−
n
j=11−η2
ϑ(j)qj,
1−
n
j=11−v2
ϑ(j)qj⎞
⎠
Then,
ϑ∈Tn
n
j=1
αqj
ϑ(j)=⎛
⎜
⎝s
ϑ∈T
n
j=1
θqj
ϑ(j)
,
1−
ϑ∈Tn⎛
⎝1−
n
j=1
μ2qj
ϑ(j)⎞
⎠,
H. Cao et al. / Some spherical linguistic Muirhead mean operators 8101
ϑ∈Tn
1−
n
j=11−η2
ϑ(j)qj,
ϑ∈Tn
1−
n
j=11−v2
ϑ(j)qj⎞
⎠
Further,
1
n!
ϑ∈Tn
n
j=1
αqj
ϑ(j)=⎛
⎜
⎝s1
n!
ϑ∈Tn
n
j=1
θqj
ϑ(j)
,
1−⎛
⎝
ϑ∈Tn⎛
⎝1−
n
j=1
μ2qj
ϑ(j)⎞
⎠⎞
⎠
1
n!
,
⎛
⎝
ϑ∈Tn
1−
n
j=11−η2
ϑ(j)qj⎞
⎠
1
n!
,
⎛
⎝
ϑ∈Tn
1−
n
j=11−v2
ϑ(j)qj⎞
⎠
1
n!⎞
⎟
⎠
Consequently, we have
1
n!
ϑ∈Tn
n
j=1
αqj
ϑ(j)
1
n
j=1
qj
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
s
1
n!
ϑ∈Tn
n
j=1
θ
qj
ϑ(j)
1
n
j=1
qj
,
⎛
⎜
⎜
⎝
1−
ϑ∈Tn1−
n
j=1
μ2qj
ϑ(j)1
n!⎞
⎟
⎟
⎠
1
n
j=1
qj
,
1−⎛
⎝1−
ϑ∈Tn1−
n
j=11−η2
ϑ(j)qj1
n!⎞
⎠
1
n
j=1
qj
,
1−⎛
⎝1−
ϑ∈Tn1−
n
j=11−ν2
ϑ(j)qj1
n!⎞
⎠
1
n
j=1
qj⎞
⎟
⎟
⎟
⎟
⎠
which completes the proof of Theorem 1.
In what follows, we will explore some properties
of the SLMM operator.
Property 1 (Idempotency). Let αj=(sθj,μ
j,
ηj,v
j)(j=1,2, ..., n)be a collection of SLNs, if all
SLNs are equal, i.e.,αj=α=(sθj,μ
j,η
j,v
j)for all
j,then
SLMMQ(α1,α
2,···,α
n)=α=(sθ,μ,η,v
)(8)
Proof. Since αj=α=(sθ,μ,η,v), then we have
SLMMQ(α1,α
2,···,α
n)=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
s
1
n!
ϑ∈Tn
n
j=1
θ
qj
ϑ(j)1n
j=1
qj
,
⎛
⎜
⎜
⎝
1−
ϑ∈Tn1−
n
j=1
μ2qj
ϑ(j)1
n!⎞
⎟
⎟
⎠
1
n
j=1
qj
,
1−⎛
⎝1−
ϑ∈Tn1−
n
j=11−η2
ϑ(j)qj1
n!⎞
⎠
1
n
j=1
qj
,
1−⎛
⎝1−
ϑ∈Tn1−
n
j=11−v2
ϑ(j)qj1
n!⎞
⎠
1
n
j=1
qj⎞
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
s
⎛
⎜
⎝1
n!
θ∈Tn
θ
n
j=1
qj⎞
⎟
⎠
1n
j=1
qj
,
⎛
⎜
⎜
⎜
⎜
⎝
1−⎛
⎜
⎜
⎝
ϑ∈Tn
⎛
⎜
⎜
⎝1−μ
2
n
j=1
qj
ϑ(j)⎞
⎟
⎟
⎠⎞
⎟
⎟
⎠
1
n!⎞
⎟
⎟
⎟
⎟
⎠
1
n
j=1
qj
,
8102 H. Cao et al. / Some spherical linguistic Muirhead mean operators
1−⎛
⎜
⎜
⎝1−⎛
⎜
⎝
ϑ∈Tn
⎛
⎜
⎝1−1−η2
ϑ(j)
n
j=1
qj⎞
⎟
⎠⎞
⎟
⎠
1
n!⎞
⎟
⎟
⎠
1
n
j=1
qj
,
1−⎛
⎜
⎜
⎝1−⎛
⎜
⎝
ϑ∈Tn
⎛
⎜
⎝1−1−v2
ϑ(j)
n
j=1
qj⎞
⎟
⎠⎞
⎟
⎠
1
n!⎞
⎟
⎟
⎠
1
n
j=1
qj⎞
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
s
⎛
⎜
⎝θ
n
j=1
qj⎞
⎟
⎠
1n
j=1
qj
,
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1−⎛
⎜
⎜
⎝⎛
⎜
⎜
⎝1−μ
2
n
j=1
qj
ϑ(j)⎞
⎟
⎟
⎠
n!⎞
⎟
⎟
⎠
1
n!⎞
⎟
⎟
⎟
⎟
⎟
⎠
1
n
j=1
qj
,
1−⎛
⎜
⎜
⎜
⎝
1−⎛
⎜
⎜
⎝⎛
⎜
⎝1−1−η2
ϑ(j)
n
j=1
qj⎞
⎟
⎠
n!⎞
⎟
⎟
⎠
1
n!⎞
⎟
⎟
⎟
⎠
1
n
j=1
qj
,
1−⎛
⎜
⎜
⎜
⎝
1−⎛
⎜
⎜
⎝⎛
⎜
⎝1−1−v2
ϑ(j)
n
j=1
qj⎞
⎟
⎠
n!⎞
⎟
⎟
⎠
1
n!⎞
⎟
⎟
⎟
⎠
1
n
j=1
qj⎞
⎟
⎟
⎟
⎟
⎟
⎠
=(sθ,μ,η,v
)
Property 2 (Monotonicity). Let αjand βj(j=
1,2,...,n)be any two collections of SLNs. If αj≥
βjfor all j,then
SLMMQ(α1,α
2,···,α
n)≥SLMMQ(β1,β
2,···,β
n)
(9)
Proof. As αj≥βjholds for all j, then
αqj
ϑ(j)≥βqj
ϑ(j),
and
n
j=1
αqj
ϑ(j)≥
n
j=1
βqj
ϑ(j)
Hence,
ϑ∈Tn
n
j=1
αqj
ϑ(j)≥
ϑ∈Tn
n
j=1
βqj
ϑ(j),
and
1
n!
ϑ∈Tn
n
j=1
αqj
ϑ(j)≥1
n!
ϑ∈Tn
n
j=1
βqj
ϑ(j)
Therefore,
1
n!
ϑ∈Tn
n
j=1
αqj
ϑ(j)
1
n
j=1
qj
≥1
n!
ϑ∈Tn
n
j=1
βqj
ϑ(j)
1
n
j=1
qj
,
which completes the proof.
Property 3 (Boundedness). Let αj(j=1,2, .., n)be
a collection of SLNs, if
α−=sθj, min μj,maxηj,max vj,
and
α+=sθj,maxμj, min ηj, min vj,
α+≥SLMMQ(α1,α
2,···,α
n)≥α+,(10)
Proof. Based on Properties 1 and 2,
SLMMQ(α1,α
2,···,α
n)≤
SLMMQα+,α
+,···,α
+=α+,
and
SLMMQ(α1,α
2,···,α
n)≥
SLMMQα−,α
−,···,α
−=α−.
So, we can get α+≥SLMMQ(α1,α
2,···,α
n)≥
α−.
The SLMM operator is proposed based on the MM
operator. Hence, SLMM inherits the powerfulness
and flexibility of MM. To better illustrate the flex-
ibility of the SLMM operator, in the followings we
discuss special cases of SLMM with respect to the
parameter vector Q.
Special case 1: when Q=(1,0,···,0), then the
SLMM operator reduces to the spherical linguistic
average (SLA) operator, i.e.
H. Cao et al. / Some spherical linguistic Muirhead mean operators 8103
SLMM(1,0,··· ,0)(α1,α
2,···,α
n)=⎛
⎜
⎝s1
n
n
j=1
θj
,
1−
n
j=11−μ2
j1/n
,
n
j=1
η1/n
j,
n
j=1
v1/n
j⎞
⎠(11)
Special case 2: when Q=(1,1,0,0,···,0), then
the SLMM operator reduces to the spherical linguistic
Bonferroni mean (BM) operator, i.e.
SLMM(1,1,0,0,··· ,0)(α1,α
2,···,α
n)
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
s⎛
⎜
⎝1
n(n−1)
n
i,j=1
i/=j
(θiθj)⎞
⎟
⎠
1
2,
⎛
⎜
⎜
⎜
⎜
⎝
1−⎛
⎜
⎜
⎝
n
i,j =1
i/=j1−μ2
iμ2
j⎞
⎟
⎟
⎠
1
n(n−1)⎞
⎟
⎟
⎟
⎟
⎠
1
2
,
1−⎛
⎜
⎜
⎝1−
n
i,j =1
i/=j1−1−η2
i1−η2
j 1
n(n−1)⎞
⎟
⎟
⎠
1
2
,
1−⎛
⎜
⎜
⎝1−
n
i,j =1
i/=j1−1−v2
i1−v2
j 1
n(n−1)⎞
⎟
⎟
⎠
1
2⎞
⎟
⎟
⎟
⎟
⎠
(12)
Special case 3: when Q=
k
(1,1,···1,
n−k
0,0,···0),
then the SLMM operator reduces to the spherical lin-
guistic Maclaurin symmetric mean (MSM) operator,
i.e.
SLMM
k
(1,1,···1,
n−k
0,0,···0)(α1,α
2,···,α
n)
=⎛
⎜
⎜
⎜
⎜
⎝
s1
Ck
n
1≤i1<...<ik≤nk
j=1
θij1/k,
⎛
⎜
⎜
⎜
⎝
1−⎛
⎝
1≤i1<···<ik≤n⎛
⎝1−
k
j=1
μ2
ij⎞
⎠⎞
⎠
1!Ck
n⎞
⎟
⎟
⎟
⎠
1/k
,
1−⎛
⎜
⎝1−
1≤i1<...<ik≤n1−
k
j=11−η2
ij1!Ck
n⎞
⎟
⎠
1/k
,
1−⎛
⎜
⎝1−
1≤i1<...<ik≤n1−
k
j=11−v2
ij1!Ck
n⎞
⎟
⎠
1/k⎞
⎟
⎟
⎠
(13)
Special case 4: when Q=(1,1,···,1) or Q=
(1!n, 1!n, ···,1!n), then the SLMM operator
reduces spherical linguistic geometric (SLG) oper-
ator, i.e.
SLMM(1,1,··· ,1)or (1/n,1/n,··· ,1/n)(α1,α
2,···,α
n)
=⎛
⎜
⎝sn
j=1
θ1/n
j
,
n
j=1
μ1/n
j,
1−
n
j=11−η2
j1/n
,
1−
n
j=11−v2
j1/n⎞
⎠(14)
3.2. The spherical linguistic weighted Muirhead
mean (SLWMM) operator
Definition 8. Let αj=(sθj,μ
j,η
j,v
j)(j=1,2,
...,n) be a collection of SLNs and Q=(q1,q
2,
..., qn)∈Rnbe a vector of parameter. Let w=
(w1,w
2, ..., wn)Tbe the weight vector, satisfying
wj∈[0,1] and
n
j=1
wj=1. If
SLWMMQ(α1,α
2, ..., αn)
=⎛
⎝1
n!
ϑ∈Tn
n
j=1nwϑ(j)αϑ(j)qj⎞
⎠
1
n
j=1
qj
(15)
8104 H. Cao et al. / Some spherical linguistic Muirhead mean operators
Then SLWMMQis the SLWMM operator,
where ϑ(j)(j=1,2, ..., n) is any permutation of
(1,2, ..., n), and Tnis the collection of all permu-
tations of (1,2, ..., n).
Theorem 2. Let αj=(sθj,μ
j,η
j,v
j)(j=1,2,
..., n)be a collection of SLNs and Q=(q1,q
2,
..., qn)∈Rnbe a vector of parameter. Then, the
aggregated value by using the SLWMM operator is
still a SLN and
SLWMMQ(α1,α
2, ..., αn)
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
s
1
n!
ϑ∈Tn
n
j=1nwϑ(j)θϑ(j)qj
1
n
j=1
qj
,
⎛
⎜
⎝
1−
ϑ∈Tn1−
n
j=11−1−μ2qj
ϑ(j)nwϑ(j)qj1
n!⎞
⎟
⎠
1
n
j=1
qj
,
1−⎛
⎝1−
ϑ∈Tn1−
n
j=11−η2nwϑ(j)
ϑ(j)qj1
n!⎞
⎠
1
n
j=1
qj
,
1−⎛
⎝1−
ϑ∈Tn1−
n
j=11−v2nwϑ(j)
ϑ(j)qj1
n!⎞
⎠
1
n
j=1
qj⎞
⎟
⎟
⎟
⎟
⎠
(16)
The proof of Theorem 2 is similar to that of Theo-
rem 1. Similarly, the SLWMM operator also has the
properties of monotonicity and boundedness.
3.3. The spherical linguistic dual Muirhead
mean (SLDMM) operator
Definition 9. Let αj=(sθj,μ
j,η
j,v
j)(j=1,2,
...,n) be a collection of SLNs and Q=
(q1,q
2, ..., qn)∈Rnbe a vector of parameter. Then
the SLDMM operator is defined as
SLDMMQ(α1,α
2, ..., αn)
=1
n
j=1
qj⎛
⎝
ϑ∈Tn
n
j=1qjαϑ(j)⎞
⎠
1
n!
(17)
Where ϑ(j)(
j=1,2, ..., n)is any permutation of
(1,2, ..., n), and Snis the collection of all permuta-
tions of (1,2, ..., n).
Theorem 4. Let αj=sθj,μ
j,η
j,v
j(j=1,2,
..., n)be a collection of SLNs and
Q=(q1,q
2, ..., qn)∈Rnbe a vector of param-
eter. Then, the aggregated value by using the
SLDMM operator is also a SLN, and
SLDMMQ(α1,α
2, ..., αn)
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
s
1
n
j=1
qj
ϑ∈Tn
n
j=1
(qjθϑ(j))1/n!,
1−⎛
⎝1−
ϑ∈Tn1−
n
j=11−μ2
ϑ(j)qj1
n!⎞
⎠
1
n
j=1
qj
,
⎛
⎜
⎜
⎝
1−
ϑ∈Tn1−
n
j=1
η2qj
ϑ(j)1
n!⎞
⎟
⎟
⎠
1
n
j=1
qj
,
⎛
⎜
⎜
⎝
1−
ϑ∈Tn1−
n
j=1
v2qj
ϑ(j)1
n!⎞
⎟
⎟
⎠
1
n
j=1
qj⎞
⎟
⎟
⎟
⎠
(18)
The proof of Theorem 4 is similar to that of Theo-
rem 1. In what follows, we will discuss some special
cases of the SLDMM operator through changing the
values of parameter vector Q.
Special case 1. when Q=(1,0,···,0), then the
SLMM operator reduces to the SLG operator, which
is shown as Equation (14)
Special case 2. when Q=(1,1,0,0,···,0), then
the SLMM operator reduces to the spherical linguistic
geometric BM operator, i.e.
SLDMM(1,1,0,0,··· ,0)(α1,α
2,···,α
n)
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
s
1
2⎛
⎜
⎝
n
i,j=1
i/=j
(θi+θj)⎞
⎟
⎠
1
n(n+1),
H. Cao et al. / Some spherical linguistic Muirhead mean operators 8105
1−⎛
⎜
⎜
⎝1−
n
i,j =1
i/=j1−1−μ2
i1−μ2
j 1
n(n−1)⎞
⎟
⎟
⎠
1
2
,
⎛
⎜
⎜
⎜
⎜
⎝
1−⎛
⎜
⎜
⎝
n
i,j =1
i/=j1−η2
iη2
j⎞
⎟
⎟
⎠
1
n(n−1)⎞
⎟
⎟
⎟
⎟
⎠
1
2
,
⎛
⎜
⎜
⎜
⎜
⎝
1−⎛
⎜
⎜
⎝
n
i,j =1
i/=j1−v2
iv2
j⎞
⎟
⎟
⎠
1
n(n−1)⎞
⎟
⎟
⎟
⎟
⎠
1
2⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(19)
Special case 3. when Q=
k
(1,1,···1,
n−k
0,0,···0),
then the SLMM operator reduces to the spherical
linguistic dual MSM operator, i.e.
SLDMM
k
(1,1,···1,
n−k
0,0,···0)(α1,α
2,···,α
n)
=⎛
⎜
⎜
⎜
⎝
s
1
k
1≤i1<...ik≤nk
j=1
θij1
Ck
n
,
1−⎛
⎜
⎝1−
1≤i1<...<ik≤n1−
k
j=11−μ2
ij1!Ck
n⎞
⎟
⎠
1/k
,
⎛
⎜
⎜
⎝
1−
1≤i1<...<ik≤n1−
k
j=1
η2
ij1!Ck
n⎞
⎟
⎟
⎠
1/k
,
⎛
⎜
⎜
⎝
1−
1≤i1<...<ik≤n1−
k
j=1
v2
ij1!Ck
n⎞
⎟
⎟
⎠
1/k⎞
⎟
⎟
⎠(20)
Special case 5. when Q=(1,1,···,1) or Q=
(1!n, 1!n, ···,1!n), then the SLMM operator
reduces SLA operator, which is shown as Equation
(11).
3.4. The spherical linguistic weighted dual
Muirhead mean (SLWDMM) operator
Definition 10. Let αj=(sθj,μ
j,η
j,v
j)(j=1,2,
...,n) be a collection of SLNs and Q=
(q1,q
2, ..., qn)∈Rnbe a vector of parameter. Let
w=(w1,w
2, ..., wn)Tbe the weight vector, satisfy-
ing wj∈[0,1] and
n
j=1
wj=1. If
SLWDMMQ(α1,α
2, ..., αn)
=1
n
j=1
qj⎛
⎝
ϑ∈Tn
n
j=1qjαnwϑ(j)
ϑ(j)⎞
⎠
1
n!
(21)
Then SLWDMMQis the SLWDMM operator,
where ϑ(j)(j=1,2, ..., n) is any permutation of
(1,2, ..., n), and Tnis the collection of all permu-
tations of (1,2, ..., n).
Theorem 5. Let αj=(sθj,μ
j,η
j,v
j)(j=1,2,
..., n)be a collection of SLNs and Q=(q1,q
2,
..., qn)∈Rnbe a vector of parameter. Then, the
aggregated value by using the SLWDMM operator
is still a SLN and
SLWDMMQ(α1,α
2, ..., αn)
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
s
1
n
j=1
qj
ϑ∈Tn
n
j=1qjθnwϑ(j)
ϑ(j)1
n!
,
1−⎛
⎝1−
ϑ∈Tn1−
n
j=11−μ2nwϑ(j)
ϑ(j)qj1
n!⎞
⎠
1
n
j=1
qj
,
⎛
⎜
⎝
1−
ϑ∈Tn1−
n
j=11−1−η2
ϑ(j)nwϑ(j)qj1
n!⎞
⎟
⎠
1
n
j=1
qj
,
⎛
⎜
⎝
1−
ϑ∈Tn1−
n
j=1
(1 −(1 −v2
ϑ(j))nwϑ(j))qj1
n!⎞
⎟
⎠
1
n
j=1
qj⎞
⎟
⎟
⎟
⎠
(22)
8106 H. Cao et al. / Some spherical linguistic Muirhead mean operators
The proof of Theorem 5 is similar to that of The-
orem 1. In addition, SLWDMM has the properties of
monotonicity and boundedness.
4. A novel approach to MADM based on the
proposed operators
In this section, we present a new MADM method
based on the proposed operators.
4.1. Description of a typical MADM problem
with spherical linguistic information
A typical MADM problem with spherical linguis-
tic information can be described as follows. Suppose
there are malternatives A={A1,A
2, ..., Am}with
nattributes G={G1,G
2, ..., Gn}. Weight vector
of attributes is w=(w1,w
2, ..., wn)T, satisfy-
ing 0 ≤wj≤1 and n
j=1wj=1. For attribute
Gj(j=1,2, ..., n) of alternative Ai(i=1,2, ..., m),
decision makers are required to utilize a SLN
α=(sθij ,μ
ij ,η
ij ,v
ij ) to express their evaluation
value. Hence, a decision matrix can be denoted by
R=(αij )m×n. In the followings, we present a method
for solving such a MADM based on the proposed
operators.
4.2. An algorithm to spherical linguistic MADM
problems
Step 1. Normalize original decision matrix according
to the following formula.
αij ="sθij ,μ
ij ,η
ij ,v
ij Gjis benefit type
sθij ,v
ij ,μ
ij ,η
ij Gjis cos ttype
(23)
Step 2. For alternative Ai(i=1,2, ..., m), utilize the
SLWMM operator
αi=SLWMMQ(αi1,α
i2, ..., αin)(24)
or the SLWDMM operator
αi=SLWDMMQ(αi1,α
i2, ..., αin)(25)
to compute the comprehensive evaluation value.
Step 3. According to Definition 4, compute the score
values of overall evaluation values.
Step 4. Rank alternatives according to their score
values and select the optimal one.
5. An application of the proposed method in
investment project selection
In this section, we apply the proposed method
in an investment selection problem to demonstrate
the effectiveness of our proposed method. Now an
investment company’s wants to invest its money to
a project. In order to obtain a stable return, this
company invites a set of experts to evaluate four pos-
sible projects from four aspects. Alternatives can be
denoted as {A1,A
2,A
3,A
4}and the attributes are
{G1,G
2,G
3,G
4}, wherein G1is the reputation of
project, G2denotes the ability of risk tolerance, G3
represents the socio-economic influence, and G4is
the environmental friendliness. The weight vector of
attributes is w=(0.32,0.26,0.18,0.24)T. Experts
are required to use SLNs to express their evaluation
information. Hence, a spherical linguistic decision
matrix is shown in Table 1. In the followings, we
determine the best alternative on the basis of the pro-
posed method.
5.1. Decision-making process
Step 1. As all attributes are benefit, the original spher-
ical linguistic decision matrix does not need to be
normalized.
Step 2. Utilize the SLWMM operator to calculate the
overall evaluation values of alternatives. Without loss
if generality, suppose Q=(1,1,1,1) and we can get
α1=(s3.7932,0.1664,0.2319,0.5755);
α2=(s4.1424,0.2799,0.2120,0.5405);
α3=(s3.8549,0.1549,0.2319,0.5695);
α4=(s3.7547,0.1549,0.2693,0.5978).
Step 3. Calculate the scores of the overall values and
we can derive
S(α1)=1.6088,S(α2)=1.8498,
S(α3)=1.6380,S(α4)=1.5644
Step 4. Rank the suppliers according to the rank of
overall values. That is A2>A
3>A
1>A
4, and A2
is the best alternatives.
In Step 2, if we utilize the SLWDMM opera-
tor (suppose Q=(1,1,1,1)) to aggregate decision
makers’ preference information for each alternative,
we can get
H. Cao et al. / Some spherical linguistic Muirhead mean operators 8107
Table 1
Spherical linguistic decision matrix
G1G2G3G4
A1(s5,0.2,0.1,0.7)(
s3,0.2,0.2,0.5)(
s5,0.1,0.3,0.5)(
s3,0.2,0.2,0.5)
A2(s4,0.2,0.2,0.6)(
s5,0.4,0.1,0.5)(
s4,0.2,0.3,0.5)(
s4,0.4,0.1,0.5)
A3(s3,0.2,0.1,0.7)(
s4,0.1,0.2,0.5)(
s4,0.1,0.2,0.6)(
s5,0.3,0.3,0.3)
A4(s6,0.3,0.2,0.5)(
s3,0.1,0.2,0.7)(
s4,0.2,0.1,0.6)(
s3,0.1,0.4,0.5)
Table 2
Scores and ranking orders with different parameter vector Qin the SLWMM operator
QScore function S(αi)(
i=1,2,3,4)Ranking orders
Q=(1,0,0,0)S(α1)=2.3439 S(α2)=2.6358 S(α3)=2.4039 S(α4)=2.3195 A2A3A1A4
Q=(1,1,0,0)S(α1)=2.0298 S(α2)=2.2985 S(α3)=2.0557 S(α4)=1.9885 A2A3A1A4
Q=(1,1,1,0)S(α1)=1.6448 S(α2)=1.8773 S(α3)=1.6698 S(α4)=1.6025 A2A3A1A4
Q=(1,1,1,1)S(α1)=1.6088 S(α2)=1.8498 S(α3)=1.6380 S(α4)=1.5644 A2A3A1A4
Table 3
Scores and ranking orders with different parameter vector Qin the SLWDMM operator
QScore function S(αi)(
i=1,2,3,4)Ranking orders
Q=(1,0,0,0)S(α1)=2.2946 S(α2)=2.6873 S(α3)=2.3498 S(α4)=2.2201 A2A3A1A4
Q=(1,1,0,0)S(α1)=2.1034 S(α2)=2.3844 S(α3)=2.1385 S(α4)=2.0528 A2A3A1A4
Q=(1,1,1,0)S(α1)=1.8612 S(α2)=2.1008 S(α3)=1.8793 S(α4)=1.8285 A2A3A1A4
Q=(1,1,1,1)S(α1)=1.8198 S(α2)=2.0430 S(α3)=1.8695 S(α4)=1.7952 A2A3A1A4
α1=(s4.1579,0.2014,0.1842,0.5384);
α2=(s4.4306,0.3358,0.1549,0.5180);
α3=(s4.1568,0.2120,0.1842,0.4959);
α4=(s4.1570,0.2120,0.1979,0.5635);
Hence, the scores of alternatives are
S(α1)=1.8198,S(α2)=2.0430,
S(α3)=1.8695,S(α4)=1.7952
Thus, the ranking order of the alternatives is A2>
A3>A
1>A
4, which means that A2is the best alter-
natives.
5.2. The influence of the parameters on the
results
In this subsection, we analyze the impacts of the
parameter vector Qon the scores and ranking orders.
Obviously, the parameter vector Qis very impor-
tant for the aggregation results by the SLWMM
and SLWDMM operators. To better investigate how
the parameter Qinfluents the aggregation results,
we assign different values to Qin the SLWMM
and SLWDMM operators, and present the results in
Tables 2 and 3.
As seen from Tables 2 and 3, an apparent fact
is that with different parameter vector Qin the
SLWMM and SLWDMM operators, different scores
can be obtained. However, it is not difficult to find
out that the ranking orders of alternatives are the
same with different parameter vector Q. This phe-
nomena demonstrates the robustness of our proposed
method. In addition, the interrelationship among
more attributes is considered, the decrease of the
scores will become. Therefore, the parameter vec-
tor Qcan be viewed as decision makers’ appetite
towards to risk. Generally, decision makers can
choose the value of Qaccording tom actual need.
In addition, as discussed above the main flexibil-
ity of proposed method is that it has the power
to capture any numbers of attributes. For instance,
when Q=(1,1,0,0), then our proposed method
play the same role as the Bonferroni mean does,
i.e. they consider the interrelationship among any
two attributes. When Q=(1,1,1,0), the proposed
operators can capture the interrelationship among
multiple attributes, which is similar to the Maclau-
rin symmetric mean operator. When Q=(1,1,1,1),
then the proposed method reflect the interrelation-
ship among all attributes, which is the most distinct
characteristic of the proposed method.
5.3. Advantages of the proposed method
To better illustrate the advantages and superiori-
ties of our proposed method, we conduct comparative
8108 H. Cao et al. / Some spherical linguistic Muirhead mean operators
Table 4
Scores and ranking orders of alternatives by different MADM methods
Method Score function S(αi)(
i=1,2,3,4)Ranking orders
The method proposed by Liu and Zhang [58] based
on the A-PFLWAA operator
S(α1)=1.4565 S(α2)=1.7081 A2A3A4A1
(suppose g(x)=−log x)S(α3)=1.5976 S(α4)=1.4674
The method based on the SLWMM operator
presented in this paper
S(α1)=1.6088 S(α2)=1.8498 A2A3A1A4
(suppose Q=(1,1,1,1))S(α3)=1.6380 S(α4)=1.5644
Table 5
Scores and ranking orders of alternatives by different MADM methods based on revised decision matrix
Method Score function S(αi)(
i=1,2,3,4)Ranking orders
The method proposed by Liu and Zhang [58] based
on the A-PFLWAA operator
Cannot be calculated None
The method based on the SLWMM operator
presented in this paper
S(α1)=1.6088 S(α2)=1.8758 A2A3A1A4
(suppose Q=(1,1,1,1))S(α3)=1.6380 S(α4)=1.5644
Table 6
Characteristics of different methods
Method Whether consider Whether consider Whether allows the sum Whether considers the
the PMD and ND the NMD of information functions interrelationship
to be greater than one among attributes
The method proposed by Liu and Zhang [58] Yes Yes No No
The method presented by Peng and Yang [59] Yes No No No
The proposed method in this paper Yes Yes Yes No
analysis. Details can be found in the following sub-
sections.
5.3.1. The ability of capturing the
interrelationship among attributes
We utilize the method proposed by Liu and Zhang
[58] based on the Archimedean picture fuzzy lin-
guistic weighted arithmetic averaging (A-PFLWAA)
operator and our proposed method based on the
SLWMM operator to solve the above investment
project selection problem and present the decision
results in Table 4.
As seen from Table 4, the best alternatives obtained
by the method introduced by Liu and Zhang [58] and
our developed method are the same, which proves
the effectiveness of the newly presented method.
Nonetheless, the ranking order of alternatives derived
by Liu and Zhang’s [58] method is slightly dif-
ferent with that gained by our method. This is
because Liu and Zhang’s [58] method is based on
the assumption that attributes are independent. How-
ever, this assumption is usually unreliable in real
MADM problem. As a matter of fact, instead of
independence there always exists weak or strong
interrelationship between attributes. In the above
MADM problem, obviously there is interrelationship
among the attributes G1and G2, i.e. the increase of
the ability of risk tolerance leads to the increase of
projects’ reputation. Our proposed method is based
on the SLWMM operator, which is capable to cap-
ture such kind of interrelationship between attributes,
making it more suitable to deal with real decision
making problems. Hence, our method is more pow-
erful, suitable and flexible than Liu and Zhang’s [58]
MADM method.
5.3.2. The greater freedom that provides for
DMs
The constraint of PFLSs is that the sum of PDM,
ND, and NMD of a LV should be less than or equal
to one, i.e. μ+η+v≤1. However, this restriction
cannot be always strictly satisfied in practical MADM
problems. Hence, PFLSs cannot comprehensively
express attribute values proposed by DMs. To better
demonstrate this situation, we replace the evaluation
value of G1of alternative A2with (s4,0.4,0.4,0.6).
Afterwards, we can compare the proposed method
with that developed by Liu and Zhang [58]. The deci-
sion results are listed in Table 5.
H. Cao et al. / Some spherical linguistic Muirhead mean operators 8109
As seen in Table 5, the method introduced by Liu
and Zhang [58] cannot deal with the revised decision
matrix. This is due to the rigorous restriction and nar-
row scope of application of PFLSs. While our method
can effectively deal with cases wherein μ+η+v≥
1 and μ2+η2+v2≤1. Hence, our method is more
flexible and can handle more complicated MADM
problems and results in less information loss.
5.3.3. The capability of considering the NMD
Additionally, our proposed method is also more
powerful than the method proposed by Peng and Yang
[59]. This is because the Pythagorean fuzzy linguistic
sets do not consider the degrees that DMs can-
not express their information value. In other words,
the Pythagorean fuzzy linguistic sets only consider
PMDs, NDs while neglect the NMDs of LVs. The
proposed SLSs not only consider the PMDs and NDs
but also take the NMDs into account. Hence, SLS
is more general and flexible than Pythagorean fuzzy
linguistic set and Pythagorean fuzzy linguistic set can
be regarded as a special case of SLS.
To sum up, the advantages of the proposed method
are: (1) It considers the interrelationship among any
numbers of attributes; (2) It takes into account not
only DMs’ PMDs and NDs but also their NMDs,
which can comprehensively describe DMs’ prefer-
ence information; (3) It permits the square sum of
the three information functions to be less than or
equal to one, giving DMs more freedom to express
their evaluation values and results in less information
distortion. Hence, our method is more powerful and
suitable to deal with real MADM problems. To bet-
ter illustrate the advantages and superiorities of our
proposed method, we list the main characteristics of
some existing methods in Table 6.
6. Conclusions
The paper proposed the SLSs by combing SFSs
with LVs, which are more effective to deal with
both DMs’ quantitative and qualitative evaluation
information. Afterwards, we proposed a set of SL
aggregation operators to deal with the interrelation-
ship among SLNs. To make our proposed method
more convenient to use, we gave an algorithm to solve
MADM problems based on our proposed aggregation
operators. An application in an investment selection
problem has shown that our proposed method has
good effectiveness. Given the good of performance
of SLSs in representing DMs’ evaluation values, in
the future we shall investigate more aggregation oper-
ators for SLNs.
Acknowledgments
This research is supported by National Natural Sci-
ence Foundation of China (71532002) and a major
project of the National Social Science Foundation of
China (18ZDA086).
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