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Journal of Intelligent & Fuzzy Systems 26 (2014) 115–125
DOI:10.3233/IFS-120719
IOS Press
115
Multi-criteria group decision making method
based on intuitionistic linguistic aggregation
operators
Xin-Fan Wanga,b, Jian-Qiang Wanga,∗and Wu-E Yanga
aSchool of Business, Central South University, Changsha, Hunan, China
bSchool of Science, Hunan University of Technology, Zhuzhou, Hunan, China
Abstract. For multi-criteria group decision making problems with intuitionistic linguistic information, we define a new score
function and a new accuracy function of intuitionistic linguistic numbers, and propose a simple approach for the comparison
between two intuitionistic linguistic numbers. Based on the intuitionistic linguistic weighted arithmetic averaging (ILWAA)
operator, we define two new intuitionistic linguistic aggregation operators, such as the intuitionistic linguistic ordered weighted
averaging (ILOWA) operator and the intuitionistic linguistic hybrid aggregation (ILHA) operator, and establish various properties
of these operators. The ILOWA operator weights the ordered positions of the intuitionistic linguistic numbers instead of weighting
the arguments themselves. The ILHA operator generalizes both the ILWAA operator and the ILOWA operator at the same time,
and reflects the importance degrees of both the given intuitionistic linguistic numbers and the ordered positions of these arguments.
Furthermore, based on the ILHA operator and the ILWAA operator, we develop a multi-criteria group decision making approach,
in which the criteria values are intuitionistic linguistic numbers and the criteria weight information is known completely. Finally,
an example is given to illustrate the feasibility and effectiveness of the developed method.
Keywords: Multi-criteria group decision making, intuitionistic linguistic number, intuitionistic linguistic ordered weighted aver-
aging (ILOWA) operator, intuitionistic linguistic hybrid aggregation (ILHA) operator
1. Introduction
In the socio-economic activities, there are a lot
of multi-criteria decision making problems. A multi-
criteria decision making problem is to find a desirable
solution from a finite number of feasible alternatives
assessed on multiple criteria, both quantitative and qual-
itative. Depending on quantitative aspects presented by
each decision making problem we can handle different
types of precise numerical values, but in other cases, the
∗Corresponding author. Jian-Qiang Wang, School of Business,
Central South University, Changsha, Hunan 410083, China. Tel.: +86
731 88830594; E-mail: jqwang@csu.edu.cn.
problems present qualitative aspects that are complex
to assess by means of exact values. In the latter case, the
use of the fuzzy linguistic approaches [1-14] has pro-
vided very good results. It handles qualitative aspects
that are represented in qualitative terms by means of lin-
guistic variables, that is, variables whose values are not
numbers but linguistic terms, such as “poor”, “slightly
poor”, “fair”, “slightly good”, “good”, etc.
In the last decades, a number of linguistic multi-
criteria decision making problems were studied and
many linguistic aggregation operators were presented
[2, 5, 15–24]. Herrera et al. [2] proposed a consen-
sus model for group decision making under linguistic
assessments information. Herrera et al. [15] developed
1064-1246/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved
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116 X.-F. Wang et al. / GMCDM method based on intuitionistic linguistic aggregation operators
several group decision making processes using lin-
guistic ordered weighted averaging (LOWA) operator.
Herrera and Herrera-Viedma [5] presented three aggre-
gation operators for weighted linguistic information,
such as the linguistic weighted disjunction (LWD) oper-
ator, linguistic weighted conjunction (LWC) operator
and linguistic weighted averaging (LWA) operator. Her-
rera and Herrera-Viedma [16] proposed three steps
to follow in the linguistic decision analysis of group
decision making problems with linguistic information.
Xu [17] presented the linguistic hybrid aggregation
(LHA) operator and applied it to group decision mak-
ing. Tang and Zheng [18] developed a new linguistic
modeling technique based on semantic similarity rela-
tion among linguistic labels. Wei [19] proposed some
2-tuple linguistic aggregation operators, such as the 2-
tuple linguistic weighted harmonic averaging (TWHA)
operator, 2-tuple linguistic ordered weighted harmonic
averaging (TOWHA) operator and 2-tuple linguistic
combined weighted harmonic averaging (TCWHA)
operator, and developed a multi-criteria group deci-
sion making method based on the TWHA and TCWHA
operators. Wei [20] proposed a grey relational analy-
sis method for 2-tuple linguistic multi-criteria group
decision making with incomplete weight information.
Atanassov [25] extended Zadeh’ fuzzy set [26] and
introduced the notion of intuitionistic fuzzy set (IFS),
which is characterized by a membership function and a
non-membership function. Gau and Buehrer [27] pre-
sented the concept of vague set, but Bustine and Burillo
[28] pointed out that vague sets are intuitionistic fuzzy
sets. The IFS is more useful and effective in deal-
ing with vagueness and uncertainty than the traditional
fuzzy set [29–32], and has already been applied in many
fields, such as decision analysis [33, 34], cluster anal-
ysis [35–37], pattern recognition [38, 39], forecasting
[40], manufacturing grid [41] and so on.
Aggregation operators of intuitionistic fuzzy infor-
mation and their applications to multi-criteria decision
making is a new branch of IFS theory, which has
attracted significant interest from researchers in recent
years. Xu [34] proposed some intiutionistic fuzzy arith-
metic aggregation operators, such as the intuitionistic
fuzzy weighted averaging (IFWA) operator, intu-
itionistic fuzzy ordered weighted averaging (IFOWA)
operator and intuitionistic fuzzy hybrid aggregation
(IFHA) operator. Zhao et al. [42] extended the IFWA,
IFOWA and IFHA operators to provide a new class of
operators referred to as the generalized IFWA (GIFWA)
operator, generalized IFOWA (GIFOWA) operator and
generalized IFHA (GIFHA) operator. Xia and Xu [43]
developed various generalized intuitionistic fuzzy point
aggregation operators, such as the generalized intu-
itionistic fuzzy point weighted averaging (GIFPWA)
operators, generalized intuitionistic fuzzy point ordered
weighted averaging (GIFPOWA) operators and gen-
eralized intuitionistic fuzzy point hybrid averaging
(GIFPHA) operators. Xu and Xia [44] proposed some
induced generalized intuitionistic fuzzy aggregation
operators, including the induced generalized intuition-
istic fuzzy Choquet integral operators and induced
generalized intuitionistic fuzzy Dempster-Shafer oper-
ators. In addition, they established various properties
of these operators and applied them in multi-criteria
decision making fields.
Shu, Wang, et al. extended the IFS and defined
the concepts of intuitionistic triangular fuzzy number
[45], intuitionistic trapezoidal fuzzy number [46] and
intuitionistic linguistic number [47], respectively. Shu
et al. [45] defined four operations of the intuitionistic
triangular fuzzy numbers and used them in fault tree
analysis. Wang and Li [47] proposed the intuitionis-
tic linguistic weighted arithmetic averaging (ILWAA)
operator and developed a multi-criteria decision making
approach in which the criteria values are intuitionis-
tic linguistic numbers. Wang and Zhang [48], Wan and
Dong [49] introduced some intuitionistic trapezoidal
arithmetic aggregation operators, such as the intuition-
istic trapezoidal weighted averaging (ITWA) operator,
intuitionistic trapezoidal ordered weighted averaging
(ITOWA) operator and intuitionistic trapezoidal hybrid
aggregation (ITHA) operator, and developed several
multi-criteria decision making approaches in which
the criteria values are intuitionistic trapezoidal fuzzy
numbers.
From the [47], we know that an intuitionistic lin-
guistic number, characterized by a linguistic term, a
membership function and a non-membership function,
is a generalization of linguistic term and intuitionis-
tic fuzzy number. In processes of cognition of things,
people may not possess a precise or sufficient level of
knowledge of the problem domain, due to the increas-
ing complexity of the socio-economic environments.
In such a case, they usually have some hesitation and
indeterminacy in providing their linguistic evaluation
values over the objects considered, which makes the
results of cognitive performance reveal the character-
istics of affirmation, negation and hesitation. As the
linguistic term and intuitionistic fuzzy number cannot
be used to completely express all the information in
a situation as such, their applications are limited. The
intuitionistic linguistic number can describe the fuzzy
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X.-F. Wang et al. / GMCDM method based on intuitionistic linguistic aggregation operators 117
characters of things more detailedly and comprehen-
sively, therefore, it is more suitable and reasonable to
express decision making information taking the form
of intuitionistic linguistic numbers rather than linguis-
tic terms and intuitionistic fuzzy numbers. In addition,
the ILWAA operator presented in this reference weights
only the intuitionistic linguistic numbers. To solve the
drawback, we shall propose an intuitionistic linguis-
tic ordered weighted averaging (ILOWA) operator. The
ILOWA operator first reorders all the given intuition-
istic linguistic numbers in descending order and then
weights these ordered arguments, and finally aggre-
gates all these ordered weighted arguments into a
collective one. The fundamental characteristic of the
ILOWA operator is to weight the ordered positions of
the intuitionistic linguistic numbers instead of weight-
ing the arguments themselves. Furthermore, weights
represent different aspects in both the ILWAA opera-
tor and the ILOWA operator, and both the operators
consider only one of them. Thus, in the following we
shall propose another new aggregation operator called
the intuitionistic linguistic hybrid aggregation (ILHA)
operator. The ILHA operator first weights the given
intuitionistic linguistic numbers, and then reorders the
weighted arguments in descending order and weights
these ordered arguments by the ILHA weights, and
finally aggregates all the weighted arguments into a col-
lective one. Obviously, the ILHA operator generalizes
both the ILWAA operator and the ILOWA operator at
the same time, and reflects the importance degrees of
both the given intuitionistic linguistic numbers and the
ordered positions of these arguments. In order to do
so, we organize the paper as follows. In Section 2, we
define a new score function and a new accuracy function
of intuitionistic linguistic numbers, and based on these
two functions, propose a simple approach for the com-
parison between two intuitionistic linguistic numbers.
In Section 3, we propose two new aggregation opera-
tors called the intuitionistic linguistic ordered weighted
averaging (ILOWA) operator and the intuitionistic lin-
guistic hybrid aggregation (ILHA) operator, and study
some desirable properties of these two operators. In
Section 4, based on the ILHA operator and the ILWAA
operator, we develop a multi-criteria group decision
making approach, in which the criteria values are intu-
itionistic linguistic numbers and the criteria weight
information is known completely. In Section 5, an
illustrative example is given to verify the developed
approach and to demonstrate its feasibility and effec-
tiveness. Finally, conclusions of this paper are presented
in Section 6.
2. Preliminaries
Definition 1 [2, 14]. Let S={sθ|θ=0,1,··· ,2l},in
which lis a positive integer, sθrepresents a possible
value for a linguistic variable, and it should satisfy the
following characteristics:
1) The set is ordered: sa>s
bif a>b;
2) There is the negation operator: neg(sa)=sbsuch
that a+b=2l.
then we call Sa discrete linguistic term set.
For example, let l=4, then Scan be defined as:
S={s0=extremely poor, s1=very poor,
s2=poor, s3=slightly poor, s4=fair, s5=slightly
good, s6=good, s7=very good, s8=extremely
good}.
To preserve all the given information, the discrete
linguistic term set Sshould be extended to a continuous
linguistic term set ˜
S={sθ|θ∈[0,q]}, in which sa>s
b
if a>b, and q(q>2l) is a sufficiently large positive
integer. If sθ∈S, then we call sθthe original linguistic
term, otherwise, we call sθthe virtual linguistic term.
Let sa,s
b,s
θ∈˜
S, then the following operational laws
are valid [14]:
1) sa+sb=sa+b;
2) λsθ=sλθ,λ∈[0,1].
Definition 2 [25]. Let Xbe a universe of discourse, then
an IFS Vin Xis given by:
V={(x, µV(x),ν
V(x))|x∈X}
where the functions µV:X→[0,1] and νV:X→
[0,1] determine the degree of membership and the
degree of non-membership of the element xto V,
respectively, and for every x∈X:
0≤µV(x)+νV(x)≤1
Let πV(x)=1−µV(x)−νV(x) for all x∈X, then
πV(x) is called the degree of hesitancy or indeterminacy
of xto V.
Generally, for convenience, α=(µα,ν
α) is called
an intuitionistic fuzzy number, where µα∈[0,1], να∈
[0,1] and µα+να≤1.
Based on the linguistic term set and the IFS, Wang
and Li [47] defined the intuitionistic linguistic number
set as follows.
Definition 3 [47]. Let Xbe a universe of discourse,
sθ(x)∈˜
S, then an intuitionistic linguistic number set A
in Xis an object having the following form:
A=x, sθ(x),µ
A(x),ν
A(x)|x∈X
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118 X.-F. Wang et al. / GMCDM method based on intuitionistic linguistic aggregation operators
which is characterized by a linguistic term sθ(x), a mem-
bership function µAand a non-membership function νA
of the element xto sθ(x), where
µA:X→˜
S→[0,1],x→sθ(x)→µA(x)
νA:X→˜
S→[0,1],x→sθ(x)→νA(x)
with the condition
0≤µA(x)+νA(x)≤1,x∈X
Let πA(x)=1−µA(x)−νA(x) for all x∈X, then
πA(x) is called the degree of hesitancy of xto sθ(x).
When µA(x)=1, ν
A(x)=0, the intuitionistic ling-
uistic number set is reduced to the linguistic term set.
For convenience, β=sθ(β),µ(β),ν(β)is called an
intuitionistic linguistic number, where sθ(β)is a linguis-
tic term, µ(β)∈[0,1], ν(β)∈[0,1], µ(β)+ν(β)≤1,
and let be the set of all intuitionistic linguistic
numbers.
For example, β=s5,0.6,0.3is an intuitionistic
linguistic number, and from it, we know that the degree
that the evaluation object belongs to s5is 0.6, the degree
that the evaluation object doesn’t belong to s5is 0.3 and
the degree of decision maker’s hesitancy is 0.1.
Obviously, the intuitionistic linguistic number gen-
eralizes both the linguistic term and the intuitionistic
fuzzy number at the same time, and it is more practical
and reasonable to express decision making informa-
tion taking the form of intuitionistic linguistic numbers
rather than linguistic terms and intuitionistic fuzzy
numbers.
In the following, we introduce some operational laws
of the intuitionistic linguistic numbers.
Definition 4 [47]. Let β1=sθ(β1),µ(β1),ν(β1)and
β2=sθ(β2),µ(β2),ν(β2)be two intuitionistic linguis-
tic numbers, then
1) β1⊕β2=sθ(β1)+θ(β2),
θ(β1)µ(β1)+θ(β2)µ(β2)
θ(β1)+θ(β2),θ(β1)ν(β1)+θ(β2)ν(β2)
θ(β1)+θ(β2);
2) λβ1=sλθ(β1),µ(β1),ν(β1),λ∈[0,1].
It can be easily proven that all the above results are
also intuitionistic linguistic numbers. Based on Defini-
tion 4, we can further obtain the following relation.
1) β1⊕β2=β2⊕β1;
2) (β1⊕β2)⊕β3=β1⊕(β2⊕β3);
3) λ(β1⊕β2)=λβ1⊕λβ2,λ∈[0,1];
4) λ1β1⊕λ2β1=(λ1+λ2)β1,λ1,λ
2∈[0,1].
The basis of the concepts of maximum expected values,
minimum expected values and compromise expected
values, Wang and Li [47] defined a score function and an
accuracy function for the comparison between two intu-
itionistic linguistic numbers, but these two functions are
cumbersome. In this paper, we define a new score func-
tion and a new accuracy function of the intuitionistic
linguistic numbers.
Definition 5. Let β=sθ(β),µ(β),ν(β)be an intu-
itionistic linguistic number, the score of βcan be
evaluated by a new score function h(β) shown as
h(β)=θ(β)(µ(β)−ν(β))(1)
where h(β)∈[−q, q].
Definition 6. Let β=sθ(β),µ(β),ν(β)be an intu-
itionistic linguistic number, the degree of accuracy of
βcan be evaluated by a new accuracy function H(β)
shown as
H(β)=θ(β)(µ(β)+ν(β))(2)
where H(β)∈[0,q].
Hong and Choi [33] showed that the relation between
the score function and the accuracy function is similar
to the relation between mean and variance in statistics.
So, by using expectation-variance principle, we pro-
pose a simple method for the comparison between two
intuitionistic linguistic numbers, which is based on the
score function h(β) and the accuracy function H(β) and
defined as follows.
Definition 7. Let β1=sθ(β1),µ(β1),ν(β1)and β2=
sθ(β2),µ(β2),ν(β2)be two intuitionistic linguistic
numbers, then
1) if h(β1)<h(β2), then β1is smaller than β2, denoted
by β1<β
2;
2) if h(β1)=h(β2), then
(a) if H(β1)=H(β2), then β1is equal to β2,
denoted by β1=β2;
(b) if H(β1)<H(β2), then β1is smaller than β2,
denoted by β1<β
2;
(c) if H(β1)>H(β2), then β1is bigger than β2,
denoted by β1>β
2.
3. Intuitionistic linguistic aggregation operators
Based on Definition 4, Wang and Li [47] presented
the ILWAA operator, but the ILWAA operator weights
only the intuitionistic linguistic numbers. In the follow-
ing, we shall propose two new intuitionistic linguistic
aggregation operators, such as the ILOWA operator and
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X.-F. Wang et al. / GMCDM method based on intuitionistic linguistic aggregation operators 119
the ILHA operator, and investigate various properties
of these operators.
Definition 8 [47]. Let βj=sθ(βj),µ(βj),ν(βj)(j=
1,2,··· ,n) be a collection of intuitionistic linguistic
numbers and let ILWAA: n→,if
ILWAAw(β1,β
2,··· ,β
n)
=w1β1⊕w2β2⊕···⊕wnβn(3)
then ILWAA is called an intuitionistic linguistic
weighted arithmetic averaging operator of dimension
n, where w=(w1,w
2,··· ,w
n)Tis the weight vector
of βj(j=1,2,··· ,n), with wj∈[0,1] and
n
j=1
wj=
1. Especially, if w=(1n, 1n, ··· ,1n)T, then the
ILWAA operator is reduced to the intuitionistic linguis-
tic arithmetic averaging (ILAA) operator of dimension
n, which is defined as follows:
ILAAw(β1,β
2,··· ,β
n)
=1
n(β1⊕β2⊕···⊕βn)(4)
Theorem 1. Let βj=sθ(βj),µ(βj),ν(βj)(j=
1,2,··· ,n)be a collection of intuitionistic linguistic
numbers, and w=(w1,w
2,··· ,w
n)Tbe the weight
vector of βj(j=1,2,··· ,n), with wj∈[0,1] and
n
j=1
wj=1, then their aggregated value by using the
ILWAA operator is also an intuitionistic linguistic num-
ber, and
ILWAAw(β1,β
2,··· ,β
n)
=sn
j=1
wjθ(βj)
,
n
j=1
wjθ(βj)µ(βj)
n
j=1
wjθ(βj)
,
n
j=1
wjθ(βj)ν(βj)
n
j=1
wjθ(βj)(5)
Proof. Obviously, from Definition 4, the aggregated
value by using the ILWAA operator is also an intu-
itionistic linguistic number. In the following, we prove
Equation (5) by using mathematical induction on n.
1) For n=2: since
w1β1=sw1θ(β1),µ(β1),ν(β1)
w2β2=sw2θ(β2),µ(β2),ν(β2)
then
ILWAAw(β1,β
2)=w1β1⊕w2β2
=sw1θ(β1)+w2θ(β2),
w1θ(β1)µ(β1)+w2θ(β2)µ(β2)
w1θ(β1)+w2θ(β2),
w1θ(β1)ν(β1)+w2θ(β2)ν(β2)
w1θ(β1)+w2θ(β2)
=s2
j=1
wjθ(βj)
,
2
j=1
wjθ(βj)µ(βj)
2
j=1
wjθ(βj)
,
2
j=1
wjθ(βj)ν(βj)
2
j=1
wjθ(βj).
2) If equation (5) holds for n=k, that is
ILWAAw(β1,β
2,··· ,β
k)
=sk
j=1
wjθ(βj)
,
k
j=1
wjθ(βj)µ(βj)
k
j=1
wjθ(βj)
,
k
j=1
wjθ(βj)ν(βj)
k
j=1
wjθ(βj).
Then, when n=k+1, by Definition 4, we have
ILWAAw(β1,β
2,··· ,β
k,β
k+1)
=sk
j=1
wjθ(βj)
,
k
j=1
wjθ(βj)µ(βj)
k
j=1
wjθ(βj)
,
k
j=1
wjθ(βj)ν(βj)
k
j=1
wjθ(βj)⊕
wk+1sθ(βk+1),µ(βk+1),ν(βk+1)
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120 X.-F. Wang et al. / GMCDM method based on intuitionistic linguistic aggregation operators
=sk
j=1
wjθ(βj)+wk+1θ(βk+1)
,
k
j=1
wjθ(βj)µ(βj)+wk+1θ(βk+1)µ(βk+1)
k
j=1
wjθ(βj)+wk+1θ(βk+1)
,
k
j=1
wjθ(βj)ν(βj)+wk+1θ(βk+1)ν(βk+1)
k
j=1
wjθ(βj)+wk+1θ(βk+1)
=sk+1
j=1
wjθ(βj)
,
k+1
j=1
wjθ(βj)µ(βj)
k+1
j=1
wjθ(βj)
,
k+1
j=1
wjθ(βj)ν(βj)
k+1
j=1
wjθ(βj).
i.e. (5) holds for n=k+1.
Thus, based on 1) and 2), (5) holds for all n∈N,
which completes the proof of Theorem 1.
Definition 9. Let βj=sθ(βj),µ(βj),ν(βj)(j=
1,2,··· ,n) be a collection of intuitionistic linguistic
numbers. An intuitionistic linguistic ordered weighted
averaging (ILOWA) operator of dimension nis a map-
ping ILOWA: n→, that has an associated weight
vector ω=(ω1,ω
2,··· ,ω
n)Tsuch that ωj∈[0,1]
and
n
j=1
ωj=1. Furthermore,
ILOWAω(β1,β
2,··· ,β
n)
=ω1βτ(1) ⊕ω2βτ(2) ⊕···⊕ωnβτ(n)(6)
where (τ(1),τ(2),··· ,τ(n))is a permutation of
(1,2,··· ,n) such that βτ(j−1) ≥βτ(j)for all j. Espe-
cially, if ω=(1n, 1n, ··· ,1n)T, then the ILOWA
operator is reduced to the ILAA operator.
Similar to Theorem 1, we have the following.
Theorem 2. Let βj=sθ(βj),µ(βj),ν(βj)(j=
1,2,··· ,n)be a collection of intuitionistic linguis-
tic numbers, then their aggregated value by using the
ILOWA operator is also an intuitionistic linguistic num-
ber, and
ILOWAω(β1,β
2,··· ,β
n)
=sn
j=1
ωjθ(βτ(j))
,
n
j=1
ωjθ(βτ(j))µ(βτ(j))
n
j=1
ωjθ(βτ(j))
,
n
j=1
ωjθ(βτ(j))ν(βτ(j))
n
j=1
ωjθ(βτ(j))(7)
where ω=(ω1,ω
2,··· ,ω
n)Tis the weight vector
related to the ILOWA operator, with ωj∈[0,1] and
n
j=1
ωj=1, which can be determined similar to the
OWA weights (for example, we can use the normal
distribution based method [50]).
The ILOWA operator has the following properties.
Theorem 3. Let βj=sθ(βj),µ(βj),ν(βj)(j=
1,2,··· ,n)be a collection of intuitionistic linguistic
numbers and ω=(ω1,ω
2,··· ,ω
n)Tis the weight
vector related to the ILOWA operator, with ωj∈[0,1]
and
n
j=1
ωj=1, then
1) (Idempotency) If all βj(j=1,2,··· ,n)are equal,
i.e. βj=βfor all j, then
ILOWAω(β1,β
2,··· ,β
n)=β.
2) (Boundary)
β−≤ILOWAω(β1,β
2,··· ,β
n)≤β+.
where
β−=min
j{sθ(βj)},min
j{µ(βj)},max
j{ν(βj)},
β+=max
j{sθ(βj)},max
j{µ(βj)},min
j{ν(βj)}.
3) (Monotonicity) Let β∗
j=sθ(β∗
j),µ(β∗
j),ν(β∗
j)
(j=1,2,··· ,n)be a collection of intuitionis-
tic linguistic numbers, if sθ(βj)≤sθ(β∗
j),µ(βj)≤
µ(β∗
j),ν(βj)≥ν(β∗
j), for all j, then
ILOWAω(β1,β
2,··· ,β
n)
≤ILOWAω(β∗
1,β
∗
2,··· ,β
∗
n).
4) (Commutativity) Let ˜
βj=sθ(˜
βj),µ(˜
βj),ν(˜
βj)
(j=1,2,··· ,n)be a collection of intuitionistic
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X.-F. Wang et al. / GMCDM method based on intuitionistic linguistic aggregation operators 121
linguistic numbers, then
ILOWAω(β1,β
2,··· ,β
n)
=ILOWAω(˜
β1,˜
β2,··· ,˜
βn).
where ˜
β1,˜
β2,··· ,˜
βnis any permutation of
(β1,β
2,··· ,β
n).
Theorem 4. Let βj=sθ(βj),µ(βj),ν(βj)(j=
1,2,··· ,n)be a collection of intuitionistic linguistic
numbers and ω=(ω1,ω
2,··· ,ω
n)Tis the weight vec-
tor related to the ILOWA operator, with ωj∈[0,1] and
n
j=1
ωj=1, then
1) If ω=(1,0,··· ,0)T, then
ILOWAω(β1,β
2,··· ,β
n)=max
j{βj}.
2) If ω=(0,0,··· ,1)T, then
ILOWAω(β1,β
2,··· ,β
n)=min
j{βj}.
3) If ωj=1,ωi=0, and i/=j, then
ILOWAω(β1,β
2,··· ,β
n)=βτ(j)
where βτ(j)is the largest of βj(j=1,2,··· ,n).
From Definition 8 and 9, we know that the ILWAA
operator weights only the intuitionistic linguistic
numbers, whereas the ILOWA operator weights only
the ordered positions of the intuitionistic linguistic
numbers instead of weighting the arguments them-
selves. To overcome this limitation, in what follows,
we develop an ILHA operator, which weights both
the given intuitionistic linguistic numbers and their
ordered positions.
Definition 10. Let βj=sθ(βj),µ(βj),ν(βj)(j=
1,2,··· ,n) be a collection of intuitionistic lin-
guistic numbers. An intuitionistic linguistic hybrid
aggregation (ILHA) operator of dimension nis a
mapping ILHA: n→, which has an associated
vector ω=(ω1,ω
2,··· ,ω
n)Twith ωj∈[0,1] and
n
j=1
ωj=1, such that
ILHAw,ω(β1,β
2,··· ,β
n)
=ω1β
τ(1) ⊕ω2β
τ(2) ⊕···⊕ωnβ
τ(n)(8)
where β
τ(j)is the jth largest of weighted intuitionistic
linguistic numbers (nw1β1,nw
2β2,··· ,nw
nβn), w=
(w1,w
2,··· ,w
n)Tis the weight vector of βj, with
wj∈[0,1] and
n
j=1
wj=1, and nis the balancing
coefficient, which plays a role of balance.
Theorem 5. Let βj=sθ(βj),µ(βj),ν(βj)(j=
1,2,··· ,n)be a collection of intuitionistic linguistic
numbers and β
τ(j)=sθ(β
τ(j)),µ(β
τ(j)),ν(β
τ(j))(j=
1,2,··· ,n), then
ILHAw,ω(β1,β
2,··· ,β
n)
=sn
j=1
ωjθ(β
τ(j))
,
n
j=1
ωjθ(β
τ(j))µ(β
τ(j))
n
j=1
ωjθ(β
τ(j))
,
n
j=1
ωjθ(β
τ(j))ν(β
τ(j))
n
j=1
ωjθ(β
τ(j))(9)
and the aggregated value derived by using the ILHA
operator is also an intuitionistic linguistic number.
Theorem 6. If ω=(1n, 1n, ··· ,1n)T, the
ILHA operator is reduced to the ILWAA operator.
Theorem 7. If w=(1n, 1n, ··· ,1n)T, the
ILHA operator is reduced to the ILOWA operator.
Obviously, the ILHA operator generalizes both the
ILWAA operator and the ILOWA operator at the same
time, and reflects the importance degrees of both the
given intuitionistic linguistic numbers and the ordered
positions of these arguments.
Example. Suppose β1=s4,0.8,0.1,β2=
s6,0.7,0.2,β3=s5,0.9,0.1,β4=s3,0.7,0.1
and β5=s3,0.8,0.2are five intuitionistic linguistic
numbers, and w=(0.22,0.26,0.15,0.17,0.20)Tis
the weight vector of βj(j=1,2,··· ,5), then by the
operational law in Definition 4, we get the weighted
intuitionistic linguistic numbers as
β
1=s4.40,0.8,0.1,β
2=s7.80,0.7,0.2,
β
3=s3.75,0.9,0.1,β
4=s2.55,0.7,0.1,
β
5=s3.00,0.8,0.2.
By Equation (1) in Definition 5, we calculate the
scores of β
j(j=1,2,··· ,5):
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122 X.-F. Wang et al. / GMCDM method based on intuitionistic linguistic aggregation operators
h(β
1)=3.08,h(β
2)=3.90,h(β
3)=3.00,
h(β
4)=1.53,h(β
5)=1.80.
Since
h(β
2)>h(β
1)>h(β
3)>h(β
5)>h(β
4)
then
β
τ(1) =s7.80,0.7,0.2,β
τ(2) =s4.40,0.8,0.1,
β
τ(3) =s3.75,0.9,0.1,β
τ(4) =s3.00,0.8,0.2,
β
τ(5) =s2.55,0.7,0.1.
Suppose that ω=(0.1117,0.2365,0.3036,0.2365,
0.1117)Tis the weight vector related to the ILHA oper-
ator (derived by the normal distribution based method
[50]), then, by (9), it follows that
ILHAw,ω(β1,β
2,β
3,β
4,β
5)
=s4.0447,0.7996,0.1391.
4. A group decision making method based on
the ILHA and ILWAA operator
In the following, based on the ILHA and ILWAA
operator, we develop a multi-criteria group decision
making method, in which the criteria values are intu-
itionistic linguistic numbers and the criteria weight
information is known completely.
Let A={A1,A
2,··· ,A
m}be a set of alternatives,
and B={B1,B
2,··· ,B
n}be a set of criteria, w=
(w1,w
2,··· ,w
n)Tis the weight vector of Bj(j=
1,2,··· ,n), where wj∈[0,1],
n
j=1
wj=1. Let D=
{d1,d
2,··· ,d
t}be a set of decision makers, and
e=(e1,e
2,··· ,e
t)Tbe the weight vector of dk(k=
1,2,··· ,t) with ek∈[0,1] and
t
k=1
ek=1. Assume
that decision makers D={d1,d
2,··· ,d
t}repre-
sent the characteristics of the alternatives Ai(i=
1,2,··· ,m) by the intuitionistic linguistic numbers
β(k)
ij =sθ(β(k)
ij ),µ(β(k)
ij ),ν(β(k)
ij )with respect to the cri-
teria Bj(j=1,2,··· ,n), and derive the decision
matrix Rk=(β(k)
ij )m×n(k=1,2,··· ,t), where sθ(β(k)
ij )
are the linguistic evaluation values of Aiwith respect to
Bj,µ(β(k)
ij ) and ν(β(k)
ij ) indicate, respectively, the degree
of membership and the degree of non-membership
that Aiattach to sθ(β(k)
ij )with respect to Bj(i=
1,2,··· ,m;j=1,2,··· ,n;k=1,2,··· ,t).
To get the best alternative(s), the group decision mak-
ing procedure is given as follows.
Step 1. Normalize the decision making information
in the matrix Rk(k=1,2,··· ,t). For the benefit-
type criteria, we do nothing; for the cost-type criteria,
we utilize the linguistic negation operator ˜
sθ(β(k)
ij )=
neg sθ(β(k)
ij )=s2l−θ(β(k)
ij )to make linguistic evalua-
tion values be normalized.
For convenience, the normalized criteria values of
the alternatives Ai(i=1,2,··· ,m) with respect to the
criteria Bj(j=1,2,··· ,n) are also denoted by β(k)
ij =
sθ(β(k)
ij ),µ(β(k)
ij ),ν(β(k)
ij ).
Step 2. Utilize the ILWAA operator
β(k)
i=ILWAAw(β(k)
i1,β
(k)
i2,··· ,β
(k)
in ) (10)
to aggregate the criteria values of the ith row of the
decision matrix Rkand derive the individual overall
values β(k)
iof the alternatives Ai, given by the decision
makers dk(i=1,2,··· ,m;k=1,2,··· ,t).
Step 3. Utilize the ILHA operator
βi=ILHAe,v(β(1)
i,β
(2)
i,··· ,β
(t)
i)=
v1β(τ(1))
i⊕v2β(τ(2))
i⊕···⊕vtβ(τ(t))
i(11)
to aggregate the individual overall values β(1)
i,
β(2)
i,··· ,β
(t)
iand derive the collective overall values
βiof the alternatives Ai(i=1,2,··· ,m), where v=
(v1,v
2,··· ,v
t)Tis the weighting vector of the ILHA
operator with vk∈[0,1] and
t
k=1
vk=1, and β(τ(k))
i
is the kth largest of the weighted intuitionistic lin-
guistic numbers (te1β(1)
i,te
2β(2)
i,··· ,te
tβ(t)
i), where
(τ(1),τ(2),··· ,τ(n))is a permutation of (1,2,··· ,n)
and tis the balancing coefficient.
Step 4. Utilize equation (1) to calculate the scores h(βi)
of the collective overall values βiof the alternatives
Ai(i=1,2,··· ,m).
Step 5. By Definition 7, utilize the scores
h(βi)(i=1,2,··· ,m) to rank the alternatives
Ai(i=1,2,··· ,m), and then select the best one(s) (if
there is no difference between two scores h(βi) and
h(βp), then we need to calculate the accuracy degrees
H(βi) and H(βp) of the collective overall values βi
and βpby equation (2), respectively, and then rank
the alternatives Aiand Ap, in accordance with the
accuracy degrees H(βi) and H(βp)).
AUTHOR COPY
X.-F. Wang et al. / GMCDM method based on intuitionistic linguistic aggregation operators 123
5. Illustrative example
In this section, a multi-criteria group decision mak-
ing problem involving a company’s making a project
investment is used to illustrate the developed method
using the ILHA and ILWAA operator.
Consider that a risk investment company wants
to make a project investment, which will be chosen
from four alternative enterprises Ai(i=1,2,3,4) by
three decision makers whose weight vector is e=
(0.3,0.4,0.3)T. In assessing the potential contribu-
tion of each enterprise, three factors are considered,
B1−profitability, B2−competitiveness, and B3−risk
affordability. Suppose that the weight vector of Bj(j=
1,2,3) is w=(0.3727, 0.3500, 0.2773)T, and the
characteristic information of the alternatives Ai(i=
1,2,3,4) with respect to the criteria Bj(j=1,2,3)
given by decision makers dk(k=1,2,3) are repre-
sented by the intuitionistic linguistic numbers β(k)
ij =
sθ(β(k)
ij ),µ(β(k)
ij ),ν(β(k)
ij )listed in Tables 1, 2 and 3.
Because all criteria Bj(j=1,2,3) are benefit-type,
their values need not to be normalized. In the follow-
ing, we utilize the developed approach to get the best
investment enterprise.
Step 1. Utilize Equation (10) to aggregate the criteria
values of the ith row of the decision matrix Rkand
derive the individual overall values β(k)
iof the alterna-
tives Aiby the decision makers dk(i=1,2,3,4; k=
1,2,3).
β(1)
1=s4.2773,0.7349,0.1976,
β(2)
1=s4.7000,0.7870,0.1683,
β(3)
1=s4.9773,0.7557,0.2443,
β(1)
2=s4.3727,0.8000,0.1680,
β(2)
2=s4.2546,0.7915,0.1263,
β(3)
2=s4.5319,0.7753,0.1614,
β(1)
3=s5.2773,0.7631,0.1663,
β(2)
3=s5.3273,0.7460,0.1720,
β(3)
3=s5.7000,0.8187,0.1570,
β(1)
4=s4.2773,0.8327,0.1000,
β(2)
4=s4.7227,0.8371,0.1395,
β(3)
4=s4.5319,0.8228,0.1386.
Table 1
Decision matrix R1
B1B2B3
A1s4,0.8,0.1s4,0.7,0.2s5,0.7,0.3
A2s5,0.8,0.2s4,0.8,0.1s4,0.8,0.2
A3s5,0.7,0.1s5,0.7,0.3s6,0.9,0.1
A4s4,0.8,0.1s4,0.9,0.1s5,0.8,0.1
Table 2
Decision matrix R2
B1B2B3
A1s4,0.9,0.1s6,0.7,0.2s4,0.8,0.2
A2s3,0.8,0.2s5,0.7,0.1s5,0.9,0.1
A3s4,0.7,0.1s7,0.8,0.2s5,0.7,0.2
A4s5,0.8,0.2s5,0.9,0.1s4,0.8,0.1
Table 3
Decision matrix R3
B1B2B3
A1s4,0.7,0.3s6,0.7,0.3s5,0.9,0.1
A2s3,0.7,0.2s5,0.8,0.1s6,0.8,0.2
A3s5,0.8,0.2s7,0.9,0.1s5,0.7,0.2
A4s3,0.9,0.1s5,0.7,0.2s6,0.9,0.1
Step 2. Utilize Equation (11) to aggregate all the
individual overall values β(1)
i,β
(2)
i,β
(3)
iand derive the
collective overall values βiof the alternatives Ai(i=
1,2,3,4), where the weight vector of ILHA operator
is ω=(0.2429,0.5142,0.2429)Twhich is determined
by the weighting method based on normal distribution
[50].
β1=s4.6084,0.7608,0.2122,
β2=s4.2933,0.7855,0.1527,
β3=s5.3924,0.7854,0.1634,
β4=s4.3467,0.8318,0.1213.
Step 3. By Equation (1), we calculate the scores h(βi)
of alternatives Ai(i=1,2,3,4).
h(β1)=2.5280,h(β2)=2.7166,
h(β3)=3.3540,h(β4)=3.0885.
Step 4. By Definition 7, we obtain that
h(β3)>h(β4)>h(β2)>h(β1)
then
A3A4A2A1.
So the best investment enterprise is A3.
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124 X.-F. Wang et al. / GMCDM method based on intuitionistic linguistic aggregation operators
6. Conclusions
In this paper, we have presented a simple approach
for the comparison between two intuitionistic linguistic
numbers depending on a new score function and a new
accuracy function. Based on the ILWAA operator, we
have proposed two new intuitionistic linguistic aggre-
gation operators, such as the ILOWA operator and the
ILHA operator, and established various properties of
these operators. The ILWAA operator weights only the
intuitionistic linguistic numbers, whereas the ILOWA
operator weights only the ordered positions of the intu-
itionistic linguistic numbers instead of weighting the
arguments themselves, the ILHA operator weights both
the given intuitionistic linguistic numbers and their
ordered positions, and generalizes both the ILWAA
operator and the ILOWA operator at the same time.
In addition, either of the ILOWA weights or the ILHA
weights can be derived from the normal distribution
based method, which can relieve the influence of unfair
arguments on the final results by assigning low weights
to those unduly high or unduly low ones. Furthermore,
based on the ILHA operator and the ILWAA opera-
tor, we have developed a multi-criteria group decision
making method under intuitionistic linguistic environ-
ment. The work in this paper develops the theories of
the aggregation operators and fuzzy decision making
theory, and it can be widely applied in the supply chain
selection, investment decisions, project evaluation and
economic benefits comprehensive evaluation and other
related decision making.
Acknowledgment
This work was supported by the National Natural Sci-
ence Foundation of China (No. 71271218, 71221061
and 61174075), the Humanities and Social Science
Foundation of the Ministry of Education of China
(No. 12YJA630114 and 10YJC630338), and the Nat-
ural Science Foundation of Hunan Province of China
(No. 11JJ6068). The authors are very grateful to the
Editor-in-Chief and the anonymous reviewers for their
constructive comments and suggestions that have led to
an improved version of this paper.
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