Z Linguistic-induced ordered weighted averaging operator for multiple attribute group decision-making: XIAN et al.

Abstract

To solve the problems of making decision with uncertain and imprecise information, Zadeh proposed the concept of Z‐number as an ordered pair, the first component of which is a restriction of variable, and the second one is a measure of reliability of the first component. But the decision‐makers’ confidence in decision‐making was neglected. In this paper, firstly, we present a new method to evaluate and rank ‐numbers based on the operations of trapezoidal Type 2 fuzzy numbers and generalized trapezoidal fuzzy numbers. Then, linguistic‐induced ordered weighted averaging operator and linguistic combined weighted averaging aggregation operator are developed to solve multiple attribute group decision‐making problems. And we analyze the main properties of them by utilizing some operational laws of fuzzy linguistic variables. Finally, a numerical example is provided to illustrate the rationality of the proposed method.

Full text

DOI: 10.1002/int.22050
RESEARCH ARTICLE

Linguistic‐induced ordered weighted
averaging operator for multiple attribute
group decision‐making
Sidong Xian
|
Jiahui Chai
|
Hailin Guo
Laboratory of Intelligent Analysis and
Decision on Complex Systems,
Chongqing University of Posts and
Telecommunications, Chongqing, China
Correspondence
Sidong Xian, Laboratory of Intelligent
Analysis and Decision on Complex
Systems, Chongqing University of Posts and
Telecommunications, Chongqing 400065,
China.
Email: sidx@163.com
Funding information
Major entrustment projects of the
Chongqing Bureau of quality and
technology supervision, P.R. China,
Grant/Award Number: CQZJZD2018001;
Graduate Teaching Reform Research
Program of Chongqing Municipal
Education Commission, P.R. China,
Grant/Award Number: YJG183074;
Chongqing research and innovation
project of graduate students, P.R. China,
Grant/Award Numbers: CYS17227,
CYS18252
Abstract
To solve the problems of making decision with uncertain
and imprecise information, Zadeh proposed the concept of
Z‐number as an ordered pair, the first component of which
is a restriction of variable, and the second one is a measure
of reliability of the first component. But the decision‐
makers’confidence in decision‐making was neglected. In
this paper, firstly, we present a new method to evaluate and
rank

‐numbers based on the operations of trapezoidal
Type 2 fuzzy numbers and generalized trapezoidal fuzzy
numbers. Then,

linguistic‐induced ordered weighted
averaging operator and

linguistic combined weighted
averaging aggregation operator are developed to solve
multiple attribute group decision‐making problems. And
we analyze the main properties of them by utilizing some
operational laws of fuzzy linguistic variables. Finally, a
numerical example is provided to illustrate the rationality
of the proposed method.
KEYWORDS
trapezoidal type 2 fuzzy number, ranking, Z-number,
Zlinguistic-induced ordered weighted averaging operator, multiple
attribute group decision-making
1
|
INTRODUCTION
In the real world, much decision information is described with uncertain and imprecise natural
language, which results it is hard to make explicit decision. To solve this issue, many math methods
were presented. For example, Zadeh
1
proposed the fuzzy set theory to deal with uncertain decision
Int J Intell Syst. 2018;1-26. wileyonlinelibrary.com/journal/int © 2018 Wiley Periodicals, Inc.
|
1

environment. The intuitionistic fuzzy sets was developed by Atanassov
2
to deal with fuzziness and it
is greater than Zadeh’s. For solving group decision‐making problems under the combined weighted
averaging aggregation situation with intuitionistic fuzzy linguistic variables, Xian
3
introduced the
new intuitionistic fuzzy linguistic hybrid aggregation (NIFLHA) operator, the interval‐valued
intuitionistic fuzzy combined weighted averaging (IVIFCWA)
4
operator, and the intuitionistic fuzzy
interval‐valued linguistic entropic combined weighted averaging (IFIVLECWA) operator.
5
Castillo
et al
6–8
also attempted to deal with imprecise Type 2 fuzzy sets intervals. However, these methods
did not consider the reality of the decision information. Then, a new concept of Z‐number was
developed by Zadeh.
9
AZ‐number is an ordered pair, which not only has the first component,
A
,be
a restriction on the value, but has the second component,
B
, be a reliability of the first one. Z‐
number makes it possible to express the uncertainty information, and it is widely used in the reams
of economics, decision‐making, prediction, and risk assessment.
Since Zadeh opens the door to a new field, there are many questions only mentioned but
did not answer. As Zadeh says in the article,
9
the problems involving computation with
Z‐numbers are easy to state but far from easy to solve, the ranking and the computation of
Z‐number is an issue, but the method of which is rarely proposed in the existing articles.
Zadeh
9
proposed a general computation framework and arithmetic operations, but they are
too complex to realize in practical application. Kang et al
10
, Bakar Gegov
16
converted
Z‐number to fuzzy number, which made the loss of original information. One approach
related to converting Z‐number to crisp number introduced by Gardashova
11
also has this
disadvantage. Aliev et al
12,15
proposed an approach based on linear programming and
developed basic arithmetic operations with continuous and discrete Z‐numbers. It has an
advantage that avoids the information missing. A method proposed by Jiang et al
14
overcomed
the issue as well. Aliev et al
15
introduced a human‐like fundamental approach for ranking of
Z‐numbers which ranks Z‐numbers by computing optimism degrees and pessimism degrees
of them.
We usually use Z‐number to express two information, but decision‐maker’s(DM’s) confidence is
also very crucial in decision‐making.Wecanseethiscircumstancewhenwemakedecision‐making:
It's credible that the employee is sure this man will be good at the job. “Thismanwillbegoodatthe
job”is a restriction of variable. “Not sure”is the employee’s(DM’s) confidence, which is DM's
subjective evaluation and based on DM's experience. And “credible”is the reliability of “the
employee is sure this man will be good at the job,”which is an objective evaluation of it and related
to the DM's preference, the ability of the decision object and so on. How can we express these three
information with Z‐number? It's just the question we will discuss in this study.
To solve the problem, we proposed the trapezoidal Type 2 fuzzy number, which can reflect
the confidence of DM well. The Type 2 fuzzy number has two elements, including the
restriction of variable and the DM’s confidence. For example, the referred question above
“the employee is sure this man will be good at the job”can expressed as (be good at the
job, sure) by Type 2 fuzzy number. Then adding “credible,”the second component of Z‐number
be the reliability of “the employee is sure this man will be good at the job,”which is of the
Z‐number. Therefore, the new Z‐number,

‐number can expressed as (be good at the job, sure)
credible. For the outstanding applicability
18–20
of the linguistic fuzziness expressing, we make
trapezoidal Type 2 fuzzy number and trapezoidal fuzzy number be the elements of

‐number.
In this way, semantics of triplex linguistic variables can be represented by

‐number.
Moreover, some operations of

‐numbers such as add, subtract, and multiply are developed
definitely.
2
|
XIAN ET AL.

The rest of the paper is as follows: In Section 2, we describe some basic concepts about
trapezoidal Type 2 fuzzy number, generalized trapezoidal fuzzy number, and so on. In Section 3,

‐number, the

linguistic variables, and a new method for ranking

‐numbers are developed
in this section. The

linguistic‐induced ordered weighted averaging (LIOWA) and

linguistic
combined weighted averaging aggregation (LCWAA) operator are introduced, respectively, in
Section 4. In Section 5, a method based on the

LIOWA operator and the

LCWAA operator for
multiple attribute group decision‐making (MAGDM) is proposed. Section 6 is a numerical
example and discussion. Section 7 is conclusion.
2
|
PRELIMINARIES
In the following, we briefly review some concepts. These basic concepts are used through the
paper unless stated otherwise.
2.1
|
A trapezoidal type 2 fuzzy number and generalized trapezoidal
fuzzy number
Inspired by the idea of Han et al
17
, we expand the triangular Type 2 fuzzy number (TT2FN) into
trapezoidal Type 2 fuzzy number. And Chen
21,22
proposed the generalized trapezoidal fuzzy number.
Definition 1

is a trapezoidal Type 2 fuzzy number, and it can be defined
as

〈〉≤abcd μ μ μ μ a b=[, ,, ];[ , , , ],(
abcd
≤≤ ≤ ≤ ≤ ≤ ≤cd μ μ μ μ,0 1)
abcd
. The
linguistic evaluation is related to the primary trapezoidal membership function, and the
linguistic confidence is related to the secondary trapezoidal membership function. Its
primary membership function can be defined as:

⎜⎟
⎜⎟
⎧
⎨
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
∈∞
∈
∈
∈
∈∞
μ
x
xa
ba
xa
ba
μμμμ x ab
xbc
dc
xd
dc
μμμμ x cd
xd
()=
0if(−,),
1
−−−[, , , ] if [,),
1if[,),
1
−−−[, , , ] if [,),
0if[,+).
abcd
abcd
(1)
Its secondary membership function can be defined as:

⎧
⎨
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
∈∞
∈
∈
∈
∈∞
γ
xμx
μx μx
μμx
μx μx xμxμx
xμxμx
μx μx
μμx
μx μx xμxμx
xμx
=
0if(−,()),
1
()− () −()
()− () if [ ( ), ( )),
1if[(),()),
1
()− () −()
()− () if [ ( ), ( )),
0if[(),+).
μx
a
ba
a
ba
ab
bc
dc
d
dc
cd
d
() (2)
XIAN ET AL.
|
3

Clearly, there is 
≤≤γ
01
μx()
.
Remark 1If

〈〉abcd μ μ μ μ b
c
=[, ,, ];[ , , , ], =
abcd and
μ
μ=
b
c
, then

is
a TT2FN.
17
Definition 2 (Chen
22
; Chen and Hsieh
23
)

is a generalized trapezoidal fuzzy number,
and it can be defined as


ef ghω=(, , , ;
)
, where


ef ghω=(, , , ;
)
.
2.2
|
Z‐number
Zadeh
9
proposed Z‐number to model uncertain information. It can be defined as an ordered
pair of fuzzy number as follows.
Definition 3 (Zadeh
9
)AZ‐number is an ordered pair of fuzzy numbers denoted as
Z
AB=( , )
. The first component,
A
, is a restriction (constraint) on the values which is a
real‐valued uncertain variable. The second component,
B
, is a measure of reliability
(certainty) of the first component.
2.3
|
Linguistic variables
Zadeh
26
introduced the concept of linguistic variable as “a variable whose values are not
numbers but words or sentences in a natural or artificial language.”A linguistic value is less
precise than a number, but it is closer to human cognitive processes that are used to solve
uncertain problems successfully. Supposing that ∣
S
si T={ =0,1,2,…, }
ibe a totally ordered
discrete linguistic term set, where
s
i
denotes the
i
th linguistic variable of
S
and
T
represents the
cardinality of
S
. For example, a set of seven term
S
could be given as follows:
S s Very Poor VP s Poor P s Medium Poor MP
s Fair F s Medium Good MG s Good G
s Very Good VG
= { = ( ), = ( ), = ( ),
= ( ), = ( ), = ( ),
=()},
012
34 5
6
in which
ss<
i
j
iff
i
j<.
Similarly, a set of seven term
F
could be given as follows:
F f Very Low VL f Low L f Medium Low ML
f Medium M f Medium High MH f High H
fVeryHighVH
= { = ( ), = ( ), = ( ),
= ( ), = ( ), = ( ),
=()},
012
34 5
6
in which
f
f<
ij
iff
i
j<.
And a set of five term
R
could be given as follows:
R r Unlikely U r Not Very Likely NVL r Likely L
r Very Likely VL r Extremely Likely EL
= { = ( ), = ( ), = ( ),
=(),= ()},
01 2
34
in which rr<
ij
iff
i
j<.
4
|
XIAN ET AL.

Definition 4 Let

〈
〉
abcd μ μ μ μ=[, ,, ];[ , , , ]
abcd
be a trapezoidal Type 2 fuzzy
linguistic variable, where
∈≤≤≤abcd Sa b c d
[
,,,] ,
and
∈μμμμ F
[
,,,]
,
abcd
≤≤≤≤≤μμμμ0
1
abcd
.
Definition 5 Let

〈
〉
abcd μ μ μ μ=[, , , ];[ , , , ]
abcd
11111
1111and

〈abcd=[ , , , ]
;
22222
〉μμμμ
[
,,,]
abcd
2222
be two trapezoidal Type 2 fuzzy linguistic variables. The operation
rules are proposed:
(1)
 ⎡
⎣
⎢
⎤
⎦
⎥
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
aabbccdd αμ αμ
αα
αμ αμ
αα
αμ αμ
αα
αμ αμ
αα
+=[+,+,+,+]; +
+,
+
+,+
+,+
+
,
aa
bbccdd
12 12121212
1122
12
1122
12
1122
12
1122
12
where
∥
∥α=
iabcd+++
4
iiii
.
(2)
 ⎡
⎣
⎢
⎤
⎦
⎥
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
aabbccdd αμ αμ
αα
αμ αμ
αα
αμ αμ
αα
αμ αμ
αα
−=[−,−,−,−]; +
+,
+
+,+
+,+
+
,
aa
bbccdd
12 12121212
1122
12
1122
12
1122
12
1122
12
where
∥
∥α=
iabcd+++
4
iiii
.
(3) 〈〉≥
λ
λa λb λc λd μ μ μ μ λ=[ , , , ];[ , , , ]( 0)
abcd
11111
1111 ,
(4)

⋅〈⋅⋅⋅⋅ ⋅ ⋅ ⋅aabb c cdd μ μ μ μ μ=[ , , , ];[ , ,
aabbc
1 2 1 2 1 2121 2 12121
⋅
〉
μμμ,]
cdd
212
.
Remark 2If
μ
μμμ====
1
abcd
, the operation rules are as follows:
(1)

aabbccdd+=[+,+,+,+]
1212121212
,
(2)

aabbccdd−=[−,−,−,−]
1212121212
,
(3)
≥
λ
λa λb λc λd λ=[ , , , ]( 0)
21111
,
(4)

⋅⋅⋅⋅⋅aabbccdd=[ , , , ]
1 2 111 212 11
.
Example 1 Let

〈〉= [1, 5, 6, 8]; [0.2, 0.4, 0.6, 0.8]
1,

〈= [2, 3, 6, 9]; [0.1, 0.2
,
2
〉0
.6, 1.0] and
λ
=
2
, then
(1)

〈
〉
+ = [3,8,12,17];[0.15,0.3,0.6,0.9]
12 ,
(2)

〈
〉
− = [−1, 2, 0, −1]; [0.15, 0.3, 0.6, 0.9]
12 ,
(3) 〈
〉λ
= [2, 10, 12, 16]; [0.2, 0.4, 0.6, 0.8]
1,
(4)

⋅〈
〉
= [2, 15, 36, 72]; [0.02, 0.08, 0.36, 0.8]
12 .
XIAN ET AL.
|
5

Definition 6 Let


ef ghω=(, , , ;
)
be a generalized trapezoidal fuzzy linguistic
variable, where
∈≤≤≤ef gh R e f g h
[
,,,] ,
, the value 
ω
represents the maximum
degree of the four membership, and 
≤≤ω
01
.
Definition 7 Let


efghω=( , , , ;
)
11
111
1
and


efghω=( , , , ;
)
22
222
2
, the operation
rules are proposed
22,23
:
(1)
 

efghω efghω
eeffgghh ωω
+=(,,,;)+(,,,;),
=( + , + , + , + ;min( , )).
12 1
1112
222
12
121 2
12
12
12
(2)
 

efghω efghω
eeffgghh ωω
−=(,,,;)−(,,,;),
=( − , − , − , − ;min( , )).
12 1
1112
222
12
121 2
12
12
12
(3)


ef gh ω ω×=(,,,;min(,)),
12 12
e e ee hh eh h
f f ff gg fg g
g f ff gg fg g
h e ee hh eh h
where =min(×, ×, ×, × ),
=min(×,×,×,×),
=min(×,×,×,×),
=min(×,×,×,×).
121 2121 2
121 2121 2
121 2121 2
121 2121 2
(4) Let efghefg,,, ,,,
11112
22
, and h
2
be positive numbers, then



∕∕
∕∕∕ ∕
efghω efghω
eeffgghh ω ω
=(, , , ; )(, , , ; ),
=( , , , ;min( , )).
12 1
1112
222
12
12 1 2 12
12
12
(5)


λ
λe λf λg λh ω λ=( , , , ; ), >
0
.
2.4
|
IOWA operator
Yager
27
introduced the induced ordered weighted averaging (IOWA) operator as follows:
Definition 8 A IOWA operator of dimension nis a function ⋅→
φ
RR R:nn
n
IOWA ,to
which a weighting vector
ω
i
is associated to aggregate the set of second arguments of a list
of npairs
μp μp μ p{( , ), ( , ),…,( , )}
11 22 nn
according to the following expressing:
∑
⋅
φ
up up u p ωp(( , ), ( , ),…,( , )) =
,
i
n
iσiIOWA 1122nn
=1
()
(3)
where
→→
σ
nn(1, 2, …, ) (1, 2,…,
)
is a permutation, and
p
σi()
is

ivalue of the pair
up
(
,)
σi σi
() () having the
i
th largest
u
i
,
u
i
is the order variable and

iis the argument
variable represented in the form of individual variable,
∈
ω
[0, 1]
i
and
∑
ω=
1
i
ni
=1 .
6
|
XIAN ET AL.

3
|

LINGUISTIC VARIABLES AND ITS RANKING
In this section, we propose

‐number to express the uncertain information. Then its linguistic
variable and ranking are also introduced.
3.1
|

‐number
Inspired by Z‐number proposed by Zadeh,
9
we propose

‐number as follows:
Definition 9 A

‐number,

=( , )
, has two components,

, is a restriction on
the values. The second component,

, is a measure of reliability of the first component.
Among this,

〈
〉
abcd μ μ μ μ=[, ,, ];[ , , , ]
abcd
,


ef ghω=(, , , ;
)
.So

=( , )=
〈
〉
abcd μ μ μ μ
(
[, , , ];[ , , , ]
abcd

ef ghω,( , , , ; ))
.
3.2
|

linguistic variables and its ranking
Inspired by Han et al
17
and Wan
24
, we use

‐number to present the semantics of triple
linguistic variables.
Definition 10 Let
X
be a finite universal set. A triple linguistic set
D
ˆforms as follows:
〈〉∣∈DsfrxX
ˆ={( , , ) }
,
θx σx ρx() () () (4)
where
→↦∈ → ↦∈sX Sx s SfX Fx f Fr:, ,: , ,
θθx
σσx
ρ() ()
→↦∈XRxr R:,
.
ρx()
S
F,
, and
R
are three linguistic ordered scales,
f
σx()
is the linguistic membership degree of
s
θx()
, and
r
ρx()
is the linguistic membership degree of
〈
〉
sf,
θx σx
() ()
. The semantics of
〈〉Dsfr
ˆ={( , , )}
θx σx ρx() () ()
can be presented as

‐number if the linguistic values are
translated into trapezoidal Type 2 fuzzy numbers and generalized trapezoidal fuzzy
numbers. If
s
θx() and
f
θx(
)
are represented by trapezoidal Type 2 fuzzy numbers
〈
〉
abcd μ μ μ μ[, , , ],[ , , , ]
abcdas well as
r
ρx()
is represented by generalized trapezoidal
fuzzy numbers

ef ghω
(
,,, ;
)
, the semantics can be expressed as

‐number,

 〈abcd μ μ μ μ=( , )=( [ , , , ];[ , , , ]
abcd 
〉ef ghω,( , , , ; ))
.
3.3
|
Computation with

‐numbers
Zadeh
9
proposed the problem about computation with Z‐numbers, but it's easy to state but far
from easy to solve. There is an example to explain computation with

‐numbers. It is likely that
it usually takes ten minutes from home to school. It is likely that it usually takes twenty minutes
from school to the supermarket. How long will it take me from home to the supermarket? It is
the sum of ((10 minutes, usually),likely) and ((20 minutes, usually),likely).
Definition 11 Let

=( , )
111
,

=( ,
)
222
, the operation rules are proposed:
(1)



+=(+,
)
12 1 2
+
2
12
,
XIAN ET AL.
|
7

(2)



−=(−,
)
12 1 2
+
2
12
,
(3) 
λ
λ=( , )
111
,
(4)

⋅⋅⋅=( ,
)
12 1212
.
Example 2 Let

 〈=( , )=([1,5,6,8];[0.2,0.4,0.6,0.8]
111 〉, (0.3, 0.5, 0.7, 0.8;
1
))
,

 〈=( , )=([2,3,6,9];[0.1,0.2,0.6,1.0]
222 〉λ, (0.2, 0.5, 0.6, 0.7; 1) ), =
2
,
then
(1)

〈〉+ = ( [3, 8, 12, 17]; [0.15, 0.3, 0.6, 0.9] , (0.25, 0.5, 0.65, 0.75; 1))
,
12
(2)

〈〉− =([−1,2,0,−1];[0.15,0.3,0.6,0.9] ,(0.25,0.5,0.65,0.75;1))
,
12
(3) 〈〉
λ
= ( [2, 10, 12, 16]; [0.2, 0.4, 0.6, 0.8] , (0.3, 0.5, 0.7, 0.8; 1))
,
1
(4)

⋅〈 〉= ( [2, 15, 36, 72]; [0.02, 0.08, 0.36, 0.8] , (0.06, 0.25, 0.42, 0.56; 1))
.
12
3.4
|
A new method for ranking

‐numbers
We propose a method for ranking

‐numbers. It includes two parts: One calculates the score of

and

, respectively, the other calculates the score of

. Then, rank the score of to

‐
numbers rank

‐numbers. Assuming there are g

‐numbers,

,,…,
g12 , where

〈
〉
abcd μ μ μ μ=[, , , ];[ , , , ]
iiiii
aibicidi,

〈
〉
abcd μ μ μ μ=[, , , ];[ , , , ]
jjjjj
ajbjcjdj,

ef=( ,
,
ii
i

ghω,;
)
ii
i
,
≤≤ij g
1
,
,
≤≤ ≤≤≤≤≤ωefgh
k0
1, 0 iiii
iand
k
is a real value. The new
method is proposed as follows:
1. Calculate the score of

by fuzzy matrix.
17
≥
p
Plμ lμ c d μ a b μ
lμ lμ
=( )=
min{ + , max ( ( + ) − ( + ) , 0)}
+
,
ij ij
iijjii
ijj
j
iijj
and
l
daμ i g= − , = , = 1, 2,…
iii
iμμμμ+++
4
a
i
b
i
c
i
d
i
. Then, the fuzzy matrix is represented as
follows:
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⋯
⋯
⋮
⋯
P
pp p
pp p
pp p
=
,
j
j
jj jj
11 12 1
21 22 2
12
where
≥
p
0
ij
,
p
p+=
1
ij ji , and if
b
c=
ii
, then
p
=1/
2
ii . Let a new vector 
S
core ()
iexpress
the priority of

i
,

⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
∑
Score gg pg
()= 1
(−1) +2−1 .
i
j
g
ij
=1
2. Calculate the score of

i
with the approach presented by Chen.
21
First, translate


efghω=( , , , ; )
ii
iii
i
into a standardized generalized trapezoidal fuzzy number

*
j.
8
|
XIAN ET AL.



efghω ω=( , , , ; ) =( , , , ; ).
*** * *
iiiiie
k
f
k
g
k
h
k
B
*
i
iiii
i
Second, calculate the centroid point 
xy
(
ˆ,ˆ)
**
ii.





⎟
⎧
⎨
⎪
⎪
⎩
⎪
⎪
⎞
⎠≠≤
≤
(
y
ω
ehand ω
ωehand ω
ˆ=
×+2
6if 0 < 1,
2if = 0 < 1.
**
**
gf
gf
ii
ii
−
−
*
**
**
i
i
ii
ii
i
i
i
*
*
*
*



̃
xyf g h eω y
ω
ˆ=ˆ(+)+( +)( −
ˆ)
2
.
** **
Bii ii
*
**
*
*
i
iii
i
Third, calculate the standard deviation

seq f q gq hq
eq f q gq hq
=(−
̄)+( − ̄)+( − ̄)+( − ̄)
4−1 ,
=(−
̄)+( − ̄)+( − ̄)+( − ̄)
3,
** * *
** **
iiiiiiii
iiiiiiji
222 2
2222
*
i
where
q
efgh
̄=( + + + )/
4
** * *
iiiii
, and
≤≤ig
1
. Fourth, use the standard deviation

s
*
i
and

y
ˆ
*
ito derive a new value y
ˆs
i
B*.



yωys
ˆ=2−ˆ×,
s
i
i
ii
*
*
**
where ≤y
0
<ˆ0.5
s
i
*and
≤≤ig
1
, to obtain a new point 
()
xy
ˆ,ˆs
ii
**. Fifth, calculate the
ranking value 
S
core ()
*
iwith the new point 
xy
(
ˆ,ˆ
)
s
**
ii.




S
core x x y x x y()=(
ˆ−min[
ˆ]) + (ˆ−0),= (
ˆ−min[
ˆ]) + (ˆ)
.
*
iig
s
ig
s
=1,2, …,
22
=1,2, …,
22
**
***
*
ii
iii
i
3. For the first membership of

‐number is the restriction of value, and the second
membership is a measure of the first one, so

has more impact on decision‐making
than

. Therefore, a new variable
β
is introduced, which indicates the degree of
impact on decision‐making that

achieved β
(
0< <1)
.

Score ()= .
i
Score βScore()+ ( )
2
*
ii

S
core ()
iis the size of

‐number and the size of
β
can influence the result. The bigger the

S
core ()
iis, the bigger the

‐number is.
XIAN ET AL.
|
9

Example 3 Let

 〈=( , )=([1,5,6,8];[0.2,0.4,0.6,0.8]
111 〉, (0.3, 0.5, 0.7, 0.8;
1
))
,

 〈=( , )=([2,3,6,9];[0.2,0.5,0.7,1.0]
222 〉, (0.2, 0.5, 0.6, 0.7; 1)),
β
=0.
6
, then
we have
1. Calculate 
S
core ()
i. The fuzzy matrix is as follows:

()
P=0.571 0.519
0.779 0.714
.


⎜⎟
⎛
⎝
⎞
⎠
Score v
Score v
()= = 1
2×(2−1) 0.571 + 0.519 + 2
2−1 =0.545,
( ) = = 0.747.
11
22
2. Calculate

S
core ()
i
.













̃
()
yωgfhe
x
yf g h e ω y
ω
x
seq f q gq hq
yωys
Score x x y
ˆ=× ((( *−*)/( *−*)) + 2)
6=1 × (( (0.7 − 0.5)/(0.8 − 0.3) ) + 2)
6=0,
ˆ=
ˆ(*+*)+( *+*)−
ˆ
2=0.4 × (0.5 + 0.7) + (0.8 + 0.3) × (1 − 0.4)
2×1 = 0.585,
ˆ= 0.523,
=(*− ̄)+(
*− ̄)+(
*− ̄)+(*− ̄ )
3,
=(0.3 − 0.575) + (0.5 − 0.575) + (0.7 − 0.575) + (0.8 − 0.575)
3= 0.222,
ˆ=2−ˆ×=
1
2− 0.4 × 0.222 = 0.411,
(*)= (
ˆ−min[
ˆ]) + (ˆ),
= (0.585 − 0.523) − 0.411 = 0.416.
*
*
*
*
*
**
*
B
jjj j
s
j
s
1111
11 11
12
1
2
1
212
2222
1=1,2
22
22
j
1
1
*
1
*
1
*11
*
1
2
1
1
*
1
*
1
*1
*
11
With the same method, calculate

S
core ()=0.34
2
*
2
.
3. Calculate the score of

.


 ∴
Score Score βScore
Score Score βScore
Score Score
()= ()+ ()
2=0.545 + 0.6 × 0.416
2= 0.397,
()= ()+ ()
2=0.747 + 0.6 × 0.342
2= 0.476,
()< ( ) < .
*
*
111
222
12 12
4
|
THE

LIOWA OPERATOR AND THE

LCWAA
OPERATOR
The concept of the IOWA operator was proposed by Yager and Filev. When we make decision‐
making, it's normal that the available information is uncertain and imprecise. In this case, we
10
|
XIAN ET AL.

cannot represent the uncertain information as exact numbers. So, it is more suitable for

‐numbers to assess the uncertainty. Inspired by Xian’s work, we will introduce the

LIOWA
operator as follows:
Definition 12 Let

#be a set of

variables,

 ∈in=( , ) ( =1,2,…, )
iii#,a

LIOWA operator of dimension nis a function 
⋅→ϕR:
LOWA nn
I, to which a
weighting vector
ω
i
is associated to aggregate the set of second arguments of a list of n
pairs
 μμ μ{( , ), ( , ),…,( , )}
nn
1122
according to the following expressing:
  
∑⋅ϕμμ μ ω(( , ), ( , ),…,( , )) = ,
LIOWA n n
i
n
iσi
1122
=1
() (5)
where
→
σ
nn: (1, 2,…, ) (1, 2, …, )
is a permutation, and

σi()
is

ivalue of the pair
u
(
,)
σi σi() ()
having the
i
th largest
u
i
,
u
i
is the order variable and

iis the argument
variable represented in the form of individual

variable,
∈
ω
[0, 1]
i
, and
∑
ω=
1
i
ni
=1 .
Theorem 1 Let
ui n
(
,)(=1,2,…,
)
ii
be a collection of

LIOWA pairs,

iin u
(
,)
ii
is
referred to as the

variable donated by

 〈abcd=( , )=([ , , , ];
iii iiii

〉μμμμ efghω
[
,,,],([,,,]; )
)
aibicidiiiii
i, then their aggregated value by using the

LIOWA operator is also a

variable, and
  


⎛
⎝
⎜
⎜
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎞
⎠
⎟
⎟
∑
∑∑∑∑
⋅
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∑∑∑
ϕμμ μ ω
ωa ωb ωc ωd
ωα μ
ωα
ωα μ
ωα
ωα μ
ωα
ωα μ
ωα
e
n
f
n
g
n
h
nω
(( , ), ( , ),…,( , )) =
=,,,;
,,
,,
,, , ;min ,
[]
LIOWA n n
i
n
iσi
i
k
iσi
i
k
iσi
i
k
iσi
i
k
iσi
i
niσi aσi
i
niσi
i
niσi bσi
i
niσi
i
niσi cσi
i
niσi
i
niσi dσi
i
niσi
i
n
i
n
i
n
i
n
in
1122
=1
()
=1
()
=1
()
=1
()
=1
()
=1 () ()
=1 ()
=1 () ()
=1 ()
=1 () ()
=1 ()
=1 () ()
=1 ()
=1 =1 =1 =1
=1,2,…
σi σi σi σi
σi
() () () ()
()
(6)
where

σi()
〈abcd μ μ μ μ=( [ , , , ];[ , , , ]
σi σi σi σi aσi bσi cσi dσi
() () () () () () () ()
〉efgh,( , , , ;
σi σi σi σi() () () ()

ω
)
)
σi() ,
∥
∥α=
σi
abcd
()
+++
4
σi σi σi σi() () () (), and
W
ωω ω=( , ,…, )
n
T
12 is an associated weight-
ing,
∈
ω
[0, 1]
i
.
Proof For
n=
2
, since


〈〉abcd μ μ μ μ
efghω
=( , )=([ , , , ];[ , , , ],
(, , , ; )),
σσσ σσσσ
aσbσcσdσ
σσσσ
(1) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1)
(1) (1) (1) (1) σ(1)
XIAN ET AL.
|
11







⎛
⎝
⎜⎞
⎠
⎟
⎛
⎝
⎜
⎜
⎡
⎣
⎢
⎤
⎦
⎥
⎛
⎝
⎜⎞
⎠
⎟
⎞
⎠
⎟
⎟
〈〉
⋅⋅
⋅⋅
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
∥∥ ∥∥
abcd μ μ μ μ
efghω
ωω
ωω
ωa ωa ωb ωb ωc ωc ωd ωd
ωα μ ωα μ
ωα ωα
ωα μ ωα μ
ωα ωα
ωα μ ωα μ
ωα ωα
ωα μ ωα μ
ωα ωα
ee
ffgg
hh ωω
=( , )=([ , , , ];[ , , , ],
(, , , ; )),
+
=+,
+
2,
=[ + , + , + , + ];
+
+,+
+,
+
+,+
+,
+
2,+
2,+
2,+
2;min( , ) .
σσσ σσσσ
aσbσcσdσ
σσσσ
σσ
σσ
σσ
σσσσσσσσ
σaσσaσ
σσ
σbσσbσ
σσ
σcσσcσ
σσ
σdσσdσ
σσ
σσσσσ σσ σ
(2) (2) (2) (2) (2) (2) (2) (2) (2) (2) (2)
(2) (2) (2) (2)
1 (2) 2 (2)
1(2)2(2
(2) (2)
1 (1) 2 (2) 1 (1) 2 (2) 1 (1) 2 (2) 1 (1) 2 (2)
1(1) (1) 2(2) (2)
1 (1) 2 (2)
1(1) (1) 2(2) (2)
1 (1) 2 (2)
1(1) (1) 2(2) (2)
1 (1) 2 (2)
1(1) (1) 2(2) (2)
1 (1) 2 (2)
(1) (2) (1) (2) (1) (2) (1) (2)
σ
σσ
(2)
(1) (2)
Suppose that if equation holds for
∈nkk
N
=,
, that is,



⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎞
⎠
⎟
⎟
∑∑ ∑∑∑∑
⋅∑
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∑∑ ∑
ωkωa ωb ωc ωd
ωα μ
ωα
ωα μ
ωα
ωα μ
ωα
ωα μ
ωα
e
n
f
n
g
n
h
n
ω
=, = ,,, ;
,, ,
,,,,;
min [ ] .
i
k
iσi
i
k
σi i
kσi
i
k
iσi
i
k
iσi
i
k
iσi
i
k
iσi
i
kiσi aσi
i
kiσi
i
kiσi bσi
i
kiσi
i
kiσi cσi
i
kiσi
i
kiσi dσi
i
kiσi
i
kσi i
k
σi i
k
σi i
kσi
ik
=1
()
=1
() =1 ()
=1
()
=1
()
=1
()
=1
()
=1 () ()
=1 ()
=1 () ()
=1 ()
=1 () ()
=1 ()
=1 () ()
=1 ()
=1 () =1 () =1 () =1 ()
=1,2,… σi()
Then, when
nk=+
1
, using the computation laws in Definition 11, we have
  

⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
∑
∑∑∑∑
⋅⋅ ⋅ ⋅
∑
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
ωω ωω
k
ωa ωb ωc ωd ωα μ
ωα
ωα μ
ωα
ωα μ
ωα
ωα μ
ωα
+ + ··· + +
=, +(,)
=,,,; ,
,, ,
σσkσkkσk
i
k
σi i
kσi
σk σk
i
k
iσi
i
k
iσi
i
k
iσi
i
k
iσi i
kiσi aσi
i
kiσi
i
kiσi bσi
i
kiσi
i
kiσi cσi
i
kiσi
i
kiσi dσi
i
kiσi
1 (1) 2 (2) ( ) +1 ( +1)
=1
() =1 ()
(+1) (+1)
=1
()
=1
()
=1
()
=1
() =1 () ()
=1 ()
=1 () ()
=1 ()
=1 () ()
=1 ()
=1 () ()
=1 ()
12
|
XIAN ET AL.




⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎞
⎠
⎟
⎟
∑∑∑∑
∑∑∑ ∑
〈〉
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∑∑ ∑
e
n
f
n
g
n
h
nω
abcd μ μ μ μ
efghω
ωa ωb ωc ωd ωα μ
ωα
ωα μ
ωα
ωα μ
ωα
ωα μ
ωα
e
n
f
n
g
n
h
nω
,,, ;min[]
+( [ , , , ]; [ , , , ] ,
(, , , ; )),
=,,,; ,
,, ,
,,, ;min[].
i
kσi i
k
σi i
k
σi i
kσi
ik
σk σk σk σk aσk bσk cσk dσk
σk σk σk σk
i
k
iσi
i
k
iσi
i
k
iσi
i
k
iσi i
kiσi aσi
i
kiσi
i
kiσi bσi
i
kiσi
i
kiσi cσi
i
kiσi
i
kiσi dσi
i
kiσi
i
kσi i
k
σi i
k
σi i
kσi
ik
=1 () =1 () =1 () =1 ()
=1,2,…
(+1) (+1) (+1) (+1) (+1) (+1) (+1) (+1)
(+1) (+1) (+1) (+1)
=1
+1
()
=1
+1
()
=1
+1
()
=1
+1
() =1
+1
() ()
=1
+1
()
=1
+1
() ()
=1
+1
()
=1
+1
() ()
=1
+1
()
=1
+1
() ()
=1
+1
()
=1
+1
() =1
+1
() =1
+1
() =1
+1
()
=1,2,… +1
σi
σk
σi
()
(+1)
()
Thus, the equation holds for all n.
□
Theorem 2 (Commutativity)
 uu u
(
(, ),(, ),…,( , ))
******
nn1122
is any permutation of
the

variable vector  uu u
(
(, ),(, ),…,( , ))
nn11 22 , then
   

ϕu u u ϕuu u(( *,*), ( *,*), …, ( *,*)) = (( , ), ( , ),…,( , ))
.
LOWA nn LOWA nn
I1122I11 22
(7)
Proof Let
  
  


∑
∑
⋅
⋅
ϕu u u ω
ϕuu u ω
((,),(,),…,(, ))= .
(( , ), ( , ),…,( , )) = .
****** *
LOWA nn
i
n
iσi
LOWA nn
i
n
iσi
I1122
=1
()
I11 22
=1
()
Since
 uu u
(
(, ),(, ),…,( , ))
******
nn1122
is any permutation of uu
(
(, ),(, ),…
,
11 22
u(, ))
nn
, then we have

=
*
σi σi() (
)
for all
i
in(=1,2,…, )
,so 

ϕu(( , ),
**
LOWAI11
uu
(
, ),…,( , )
)
****
nn22
 
∑⋅ϕuu u ω=((,),(,),…,(,))=
LOWA nn i
ni
I11 22 =1

σi()
.
□
Theorem 3 [Monotonicity] Let  uu u
(
(, ),(, ),…,( , ))
******
nn
1122and
u
(
(, ),
11
uu
(
, ),…,( , ))
nn22
be two

variable vectors, if

<
*
iifor all
i
in(=1,2,…, )
,
then
 
 


ϕuu u
ϕu u u
(( , ), ( , ),…,( , ))
<((,),(,),…,(,)).
******
LOWA nn
LOWA nn
I11 22
I1122(8)
XIAN ET AL.
|
13

Proof Let
  
  


∑⋅
∑⋅
ϕuu u ω
ϕu u u ω
(( , ), ( , ),…,( , )) = .
((,),(,),…,(, ))= .
****** *
LOWA nn
i
n
iσi
LOWA nn
i
n
iσi
I11 22
=1
()
I1122
=1
()
Since

<
*
iifor all
i
in(=1,2,…, )
, it follows that

<
*
σi σi() ()
,
i
in(=1,2,…, )
, then
 

ϕuu u(( , ), ( , ),…,( , )
)
LOWA nn
I11 22  

ϕu u u<((,),(,),…,(,)
)
******
LOWA nn
I1122.
□
Theorem 4 [Idempotency] If

∈,
i
#
and

=
i
for all
i
in(=1,2,…, )
, where


〈〉abcd μ μ μ μ ef ghω=([ , , , ];[ , , , ] ,(, , , ; )
)
abcd
, then
 

ϕuu u(( , ), ( , ),…,( , )) =
.
LOWA nn
I11 22 (9)
Proof Since

=
i
for all
i
in(=1,2,…, )
, let
    
  
⋅⋅ ⋅
⋅⋅ ⋅ ⋅
ϕuu u ωω ω
ωω ω ωωω
(( , ), ( , ),…, ( , ) ) = + + ··· + ,
= + + ··· + , = ( + + ··· + )
.
LOWA nn nn
nn
I11 2 2 11 2 2
12 12
According to Theorem 1 and
∑
ω=
1
i
ni
=1 , we have




⎛
⎝
⎜
⎜
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎞
⎠
⎟
⎟
∑∑∑∑⋅
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
〈〉
ωω ω ωa ωb ωc ωd
ωαμ
ωα
ωαμ
ωα
ωαμ
ωα
ωαμ
ωα ef ghω
abcd μ μ μ μ ef ghω
(++…+) = , , , ;
,, ,
,(, , , ; )
=([, , , ];[ , , , ],(, , , ; )=
n
i
n
i
i
n
i
i
n
i
i
n
i
i
nia
i
ni
i
nib
i
ni
i
nic
i
ni
i
nid
i
ni
abcd
12
=1 =1 =1 =1
=1
=1
=1
=1
=1
=1
=1
=1
So,
 

ϕuu u(( , ), ( , ),…,( , )) =
LIOWA nn11 22 .
□
Theorem 5 (Boundedness) Let

 =min( , ,…, )
min12 ,

 =max( , ,…,
)
Min12 ,
then

 

≤≤ϕuu u(( , ), ( , ),…,( , ))
.
mLOWA nn M
I11 22
(10)
Proof Since

≤≤
mi
M
for all
i
in(=1,2,…, )
and
∑
ω=
1
i
ni
=1 , using Theorems 1,
2, 3, 4, then
14
|
XIAN ET AL.

    
 

    
 



⋅⋅ ⋅
≥⋅ ⋅ ⋅
⋅
⋅⋅ ⋅
≤⋅ ⋅ ⋅
⋅
ϕuu u ωω ω
ωω ω
ωω ω
ϕuu u ωω ω
ωω ω
ωω ω
(( , ), ( , ),…,( , )) = + + ··· + ,
+ + ··· + ,
= ( + + ··· + ) = .
(( , ), ( , ),…,( , )) = + + ··· + ,
+ + ··· + ,
=( + +…+ ) = .
LIOWA nn nn
mmnm
nm m
LIOWA nn nn
MMnM
nM M
11 2 2 11 2 2
12
12
11 2 2 11 2 2
12
12


≤≤ϕuu u
S
o (( , ), ( , ),…,( , ))
mLIOWA nn
M
11 22
.
□
Remark 3If
μ
μμμ====
1
abcd
, then, we get the

LIOWA operator as follows:
  


⎛
⎝
⎜
⎜
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎞
⎠
⎟
⎟
∑
∑∑∑∑
⋅
∑
ϕuu u ω
ωa ωb ωc ωd n
(( , ), ( , ),…,( , )) =
=,,,,
,
LOWA nn
i
n
iσi
i
n
iσi
i
n
iσi
i
n
iσi
i
n
iσi i
ni
I11 22
=1
()
=1
()
=1
()
=1
()
=1
() =1 (11)
where

σi()
is the
i
th largest of the
μ
i
variable

i.
Remark 4If
≥
μ
μ
ii+
1
for all
i
, the ordered position of
μ
i
is the same as the ordered
position of

i,the

LIOWA operator reduced to the

LCWAA operator.
  

∑
⋅ϕuu u ω(( , ), ( , ),…,( , )) =
.
LOWA nn
i
n
ii
I11 22
=1
(12)
Similarly, we can develop the

LCWAA operator.
Definition 13 Let

#be a set of

variables,

 ∈in=( , ) ( =1,2,…, )
iii#,
a
LCWA
A
operator of dimension nis a function


⋅→ϕR:
LCWAA nn
, to which a
weighting vector
ω
i
is associated to aggregate the set of second arguments  {, ,…, }
n12
according to the following expressing:
  
∑⋅ϕω(, ,…, )=
,
LCWAA n
i
n
ii12
=1
(13)
where
∈
ω
[0, 1]
i
and
∑
ω=
1
i
ni
=1 .
Theorem 6

iis referred to as the

variable donated by

=( , )
=
iii
〈abcd μ μ μ μ
(
[, , , ];[ , , , ]
iiii aibicidi
〉efgh ω,([ , , , ]; ))
iiiii, then their aggregated value by
using the

LCWAA operator is also a

variable, and
XIAN ET AL.
|
15

  


⎛
⎝
⎜
⎜
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎞
⎠
⎟
⎟
∑∑∑∑∑⋅
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∥∥
∑∑
∑
∑
ϕ ω ωa ωb ωc ωd
ωαμ
ωα
ωαμ
ωα
ωαμ
ωα
ωαμ
ωα
e
n
f
n
g
n
h
nω
(, ,…, )= = , , ,
,, ,
,,,,
;min[ ] ,
LCWAA n
i
n
ii
i
n
ii
i
n
ii
i
n
ii
i
n
ii
i
niiai
i
nii
i
niibi
i
nii
i
niici
i
nii
i
niidi
i
nii
i
nii
n
ii
n
i
ni
in
12
=1 =1 =1 =1 =1
=1
=1
=1
=1
=1
=1
=1
=1
=1 =1 =1
=1
=1,2,···
i
i(14)
where

〈abcd μ μ μ μ=( [ , , , ];[ , , , ]
iiiii
aibicidi〉efghω,( , , , ; )
)
iiiii
,
∥
∥αabc=( + +
+
iiii
d
)/
4
i
and
W
ωω ω= ( , , ···, )
n
T
12 is an associated weighting,
∈
ω
[0, 1]
i
.
Theorem 7 (Monotonicity) Let
 
(
,,…,
)
** *
n
12
and
 
(
,,…,)
n12
be two

variable
vectors, if

<
*
iifor all
i
in(=1,2,…, )
, then
    

ϕϕ(, ,…, )< ( , ,…, )
.
** *
LCWAA nLCWAA n12 12
(15)
Theorem 8 (Idempotency) If

∈,
i
#
and

=
i
for all
i
in(=1,2,…, )
, where


〈〉abcd μ μ μ μ ef ghω=([ , , , ];[ , , , ] ,(, , , ; )
)
abcd
, then
  

ϕ(, ,…, )= .
LCWAA n12
(16)
Theorem 9 (Boundedness) Let

 =min( , ,…, )
min12 ,

 =max( , ,…,
)
Min12 ,
then

  

≤≤ϕ(, ,…, ) .
mLCWAA nM12 (17)
Remark 5If
μμμμ=== =
1
abcd
, then, we get the

LCWAA operator as follows:
   

⎛
⎝
⎜
⎜
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎞
⎠
⎟
⎟
∑∑∑∑∑
⋅∑
ϕ ω ωa ωb ωc ωd n
(, ,…, )= = , , , ,
.
LCWAA n
i
n
ii
i
n
ii
i
n
ii
i
n
ii
i
n
ii i
ni
12
=1 =1 =1 =1 =1
=1 (18)
5
|
AN APPROACH TO MULTI‐ATTRIBUTE GROUP
DECISION‐MAKING WITH

‐NUMBER
In this section, we consider the MAGDM problems based on the

LIOWA operator and

LCWAA operator.
Let
X
xx x={ , ,…, }
m12
be a discrete set of alternatives, and Ccc c={ , ,…, }
n12 be the set of
attributes, whose weight vector is
W
ωω ω={ , ,…, }
n
T
12
,where ∈
ω
jn[0,1], =1,2,…,
j,and
16
|
XIAN ET AL.

∑
ϖ=
1
k
pk
=1 .LetDddd={ , ,…, }
p12 be the set of DMs, ϖϖϖϖ={ , ,…, }
n12 be the weighting vector
of DMs, with ∈ϖk
p
[0,1], =1,2,…,
kand
∑
ϖ=
1
k
pk
=1 . We suppose that the DMs
dk
provide
the

linguistic values

ij
k() of the alternatives
∈xX
i
with respect to the attribute ∈c
C
j,and
construct the

decision matrices, where

=( , )=
ij kij kij k() () ()
〈
(
abcd[, , , ];
ij kij kij kij k() () () ()
μμμμ
[
,,,]
aij k bij k cij k dij k() () () ()
〉())
efghω,,,,;
ij kij kij kij k() () () () Bij k() be a

linguistic variable. In the
next, we shall apply the

LIOWA operator to MAGDM based on

linguistic information. The
method involves the following steps:
1. Calculate the overall preference

linguistic values imkp
(
=1,2,…, , =1,2,…, ) of
the alternative
x
i
by utilizing the decision information given in the

linguistic
decision matrix

k(
)
and the

LIOWA operator under the order vari-
ables ∈uUj n,=(1,2,…,)
j
.

 

〈〉
()
)
abcd μkμkμkμk
efghω
ϕu u u
=( , )=([ , , , ];[ ( ), ( ), ( ), ( )],
,,, ;
=((,),(,),…,(,)),
ikikikikikikikaibicidi
ikikikik
ZL OWA kk
nn
k
() () () () () () ()
() () () ()
I11
() 22
() ()
ik()
(19)
where ∈
W
ωω ω ω j n={ , ,…, } , [0,1], =1,2,…,
nTj12 and
∑
ω=
1
j
nj
=1 .
2. Utilize the

LIOWA operator:

  

∑
〈〉
⋅
abcd μ μ μ μ efghω
ϕϖ
=( , )=( [ , , , ];[ , , , ] ,( , , , ; )
=((),(),…,())=
iii iiii
aibicidiiiii
LOWA n
p
k
p
ki
σk
I1
(1) 2
(2) ()
=1
()
i
(20)
to derive the collective overall preference

fuzzy linguistic variables of the alternative
X
i
,
where ϖϖϖϖ={ , ,…, }
n12 Tbe the weighting vector of DMs, with
∈ϖk[0, 1], = 1,
k
∑
pϖ2,…, , =
1
k
pk
=1 and be the
k
th largest of the

linguistic variable

ik()
.
3. Calculate the

S
core ()
iof the overall linguistic variables

im,(=1,2,…,
)
i
.
4. Rank all the alternatives xi m(=1,2,…,
)
i
and select the best one(s).
5. End.
Then, we shall apply the

LCWAA operator to MAGDM based on

linguistic information.
The method involves the following steps:
1. Calculate the overall preference

linguistic values

ik()
,imkp
(
=1,2,…, , =1,2,…, )
of the alternative
x
i
by utilizing the decision information given in the

linguistic
decision matrix

k(
)
and the

LCWAA operator under the order vari-
ables ∈uUj n,=(1,2,…,)
j
.

 

〈〉
)
()
abcd μkμkμkμk
efghω ϕ
=( , )=([ , , , ];[ ( ), ( ), ( ), ( )] ,
,,, ; = (, ,…, ),
ikikikikikikikaibicidi
ikikikikCLWAA
kk nk
() () () () () () ()
() () () () 1
() 2
() ()
ik() (21)
where ∈
W
ωω ω ω j n={ , ,…, } , [0,1], =1,2,…,
nTj12 and
∑
ω=
1
j
nj
=1 .
XIAN ET AL.
|
17

2. Utilize the

LCWAA operator:

  

∑
〈〉
⋅
abcd μ μ μ μ efghω
ϕϖ
=( , )=( [ , , , ];[ , , , ],( , , , ; ))
=(,,…,)= ,
iii iiii
aibicidiiiii
CLWAA np
k
p
ki
k
1
(1) 2
(2) ()
=1
i
(22)
to derive the collective overall preference

fuzzy linguistic variables

im(=1,2,…, )
i
of
the alternative
X
i
, where ϖϖϖϖ={ , ,…, }
p12 Tbe the weighting vector of DMs,
with ∈∑
ϖkpϖ[0,1]( =1,2,…, ), =
1
kk
pk
=1 .
3. Calculate the

S
core ()
iof the overall linguistic variables

im,( =1,2,…,
)
i
.
4. Rank all the alternatives xi m(=1,2,…,
)
i
and select the best one(s).
5. End.
6
|
ILLUSTRATIVE EXAMPLE AND DISCUSSION
In this section, an example modified from Chen
25
is used to illustrate the proposed method. In
the example, a software company needs an engineer. There are three candidates
A
1
,
A
2
, and
A
3
.
Three DMs
d
1
,
d
2
, and
d
3interview them. Five benefit criteria are considered:
(1) emotional steadiness (
C
1
),
(2) oral communication skill (C
2
),
(3) personality (C
3
),
(4) past experience (
C
4
),
(5) self‐confidence (C
5
).
The weight vector of the five attributes is
ω
= (0.2, 0.25, 0.3, 0.15, 0.1)
. And the weight vector
of the three DMs is
ϖ= (0.32, 0.35, 0.33)
. The degree of the impact on decision‐making that

achieved,
β
=0.
6
. The decision matrices are listed in the Tables 1, 2, 3.
1. Translate the linguistic values into trapezoidal fuzzy numbers, which are shown in
Tables 4, 5, and 6. Based on it, a transformation from linguistic numbers to

‐number
is designed. The

‐number decision matrices are show in Tables 7, 8, and 9.
2. Calculate the decision matrices given by DMs with

LIOWA operator. The
comprehensive decision matrix is shown in Table 10.
3. Calculate the
i
th row in the comprehensive decision matrix by utilizing the

LIOWA
operator.



〈〉
〈〉
〈〉
= ( [5.56, 7.25, 7.25, 8.77]; [0.59, 0, 84, 0.88, 0.96] , (0.29, 0.32, 0.39, 0.55; 1) ),
= ( [8.08, 8.80, 8.80, 9.58]; [0.78, 0.84, 0.92, 0.94] , (0.39, 0.41, 0.53, 0.67; 1)),
= ( [6.77, 8.27, 8.27, 9.54]; [0.81, 0.84, 0.91, 0.97] , (0.37, 0.39, 0.52, 0.67; 1)).
A
A
A
1
2
3
4. Calculate the size of

‐number,

S
core i( ), = 1, 2, 3
A
i()
. And rank all the
alternatives with 
S
core i( ), = 1, 2, 3
Ai() .
18
|
XIAN ET AL.

TABLE 1 The linguistic decision table by decision‐maker
d
1
C1
C2
C
3
A
1
〈〉MG H L
(
12, ( ; , )
)
〈〉GVH VL
(
13, ( ; , )) 〈〉FVH EL
(
17, ( ; , )
)
A
2
〈〉GH EL
(
12, ( ; , )
)
〈〉VG VH VL
(
15, ( ; , )) 〈〉VG VH L
(
13, ( ; , ) )
A
3
〈〉VG H EL
(
13, ( ; , )
)
〈〉MG VH L
(
11, ( ; , )
)
〈〉GVH L
(
14, ( ; , )
)
C4
C
5
A
1
〈〉VG VH L
(
14, ( ; , ) ) 〈〉FM U
(
20, ( ; , ) )
A
2
〈〉VG VH L
(
11, ( ; , ) ) 〈〉VG M L
(
16, ( ; , ) )
A
3
〈〉GVH L
(
16, ( ; , )
)
〈〉GM NVL
(
17, ( ; , ) )
Abbreviations: EL, Extremely Likely; F, Fair; G, Good; H, High; L, Likely; L, Low; M, Medium; MG, Medium Good; MH,
Medium High; ML, Medium Low; MP, Medium Poor; NVL, Not Very Likely; P, Poor; U, Unlikely; VG, Very Good; VH, Very
High; VL, Very Likely; VL, Very Low; VP, Very Poor.
TABLE 2 The linguistic decision table by decision‐maker
d
2
C1
C2
C
3
A
1
〈〉GVH L
(
17, ( ; , )
)
〈〉MG VH EL
(
13, ( ; , )) 〈〉GH UL
(
16, ( ; , ) )
A
2
〈〉GVH U
(
12, ( ; , )
)
〈〉VG VH EL
(
14, ( ; , ) ) 〈〉VG H L
(
13, ( ; , )
)
A
3
〈〉GVH L
(
13, ( ; , )
)
〈〉GVH EL
(
15, ( ; , )
)
〈〉MG H L
(
12, ( ; , )
)
C4
C
5
A
1
〈〉GVH NVL
(
15, ( ; , )
)
〈〉FMH U
(
12, ( ; , ))
A
2
〈〉VG VH U
(
16, ( ; , )
)
〈〉MG MH L
(
20, ( ; , )
)
A
3
〈〉VG VH EL
(
17, ( ; , ) ) 〈〉GMH VL
(
16, ( ; , )
)
Abbreviations: EL, Extremely Likely; F, Fair; G, Good; H, High; L, Likely; L, Low; M, Medium; MG, Medium Good; MH,
Medium High; ML, Medium Low; MP, Medium Poor; NVL, Not Very Likely; P, Poor; U, Unlikely; VG, Very Good; VH, Very
High; VL, Very Likely; VL, Very Low; VP, Very Poor.
TABLE 3 The linguistic decision table by decision‐maker
d
3
C1
C2
C
3
A
1
〈〉MG MH VL
(
16, ( ; , ) ) 〈〉FVH L
(
17, ( ; , )
)
〈〉GH NVL
(
11, ( ; , )
)
A
2
〈〉MG MH L
(
12, ( ; , )
)
〈〉VG VH U
(
15, ( ; , )
)
〈〉GH VL
(
16, ( ; , )
)
A
3
〈〉FMH U
(
13, ( ; , )) 〈〉VG VH L
(
12, ( ; , ) ) 〈〉VG VH L
(
15, ( ; , ) )
C4
C
5
A
1
〈〉VG VH L
(
12, ( ; , ) ) 〈〉FMH U
(
13, ( ; , ))
A
2
〈〉VG VH U
(
16, ( ; , )
)
〈〉GMH L
(
18, ( ; , )
)
A
3
〈〉MG VH U
(
16, ( ; , )
)
〈〉MG MH L
(
11, ( ; , )
)
Abbreviations: EL, Extremely Likely; F, Fair; G, Good; H, High; L, Likely; L, Low; M, Medium; MG, Medium Good; MH,
Medium High; ML, Medium Low; MP, Medium Poor; NVL, Not Very Likely; P, Poor; U, Unlikely; VG, Very Good; VH, Very
High; VL, Very Likely; VL, Very Low; VP, Very Poor.
XIAN ET AL.
|
19


〉〉 ∴〉〉
Score Score Score
Score Score Score A A A
( ) = 0.22 ( ) = 0.36 ( ) = 0.34,
() () () .
AAA
AAA231
123
231
The most desired one is
A
2
.
Then, we utilize the

LIOWA operator, the

LCWAA operator and the TT2OWA
operator
17
to calculate the collective overall preference fuzzy linguistic information values

Ai( )( = 1, 2, 3
)
i
and
vi( = 1, 2, 3
)
i
in Table 11.
In the next, we are going to present the ordering of the engineers in Table 12.
TABLE 4 Linguistic variables for the ratings
V
eryPoor VP()
[
0, 0, 0, 1]
P
oor P()
[
0, 1, 1, 3]
M
ediumPoor MP(
)
[
1, 3, 3, 5]
Fair F(
)
[
3, 5, 5, 7]
M
ediumGood MG(
)
[
5, 7, 7, 9]
G
ood G()
[
7, 9, 9, 10]
V
eryGood VG(
)
[
9, 9, 9, 10]
TABLE 5 Linguistic variables for the confidence
V
eryLow VL(
)
[
0, 0, 0, 0.1]
Low L()
[
0, 0.1, 0.1, 0.3]
M
ediumLow ML(
)
[
0.1, 0.3, 0.3, 0.5]
M
edium M()
[
0.3, 0.3, 0.5, 0.7]
M
ediumHigh MH()
[
0.5, 0.7, 0.7, 0.9]
High H(
)
[
0.7, 0.9, 0.9, 1.0]
V
eryHigh VH(
)
[
0.9, 0.9, 1.0, 1.0]
TABLE 6 Linguistic variables for the reliability
Unlikely U(
)
[
0, 0, 0, 0.1]
N
otVeryLikely NVL(
)
[
0, 0.1, 0.1, 0.3]
Likely L()
[
0.3, 0.3, 0.5, 0.7]
V
eryLikely VL()
[
0.5, 0.7, 0.7, 0.9]
E
xtremelyLikely EL(
)
[
0.9, 0.9, 1.0, 1.0]
20
|
XIAN ET AL.

TABLE 7 The transformed fuzzy decision table by decision‐maker
d
1
C
1
C
2
A
1
〈〉
(
12,([5,7,7,9];[0.7,0.9,0.9,1.0],(0.3,0.3,0.5,0.7;1))
)
〈〉
(
13, ( [7, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0.5, 0.7, 0.7, 0.9; 1))
)
A
2
〈〉
(
12, ( [7, 9, 9, 10]; [0.7, 0.9, 0.9, 1.0] , (0.9, 0.9, 1.0, 1.0; 1))
)
〈〉
(
15, ( [9, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0.5, 0.7, 0.7, 0.9; 1))
)
A
3
〈〉
(
13, ( [9, 9, 9, 10]; [0.7, 0.9, 0.9, 1.0] ,(0.9, 0.9, 1.0, 1.0; 1))) 〈〉
(
11,([5,7,7,9];[0.9,0.9,1.0,1.0],(0.3,0.3,0.5,0.7;1))
)
C
3
C
4
A
1
〈〉
(
17,([3,5,5,7];[0.9,0.9,1.0,1.0],(0.9,0.9,1.0,1.0;1))
)
〈〉
(
14, ( [9, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0.3, 0.3, 0.5, 0.7; 1))
)
A
2
〈〉
(
13, ( [9, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0.3, 0.3, 0.5, 0.7; 1))
)
〈〉
(
11, ( [9, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0.3, 0.3, 0.5, 0.7; 1))
)
A
3
〈〉
(
14, ( [7, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0.3, 0.3, 0.5, 0.7; 1))
)
〈〉
(
17,([3,5,5,7];[0.9,0.9,1.0,1.0],(0.9,0.9,1.0,1.0;1))
)
C
5
A
1
〈〉
(
20,([3,5,5,7];[0.3,0.3,0.5,0.7],(0,0,0,0.1;1))
)
A
2
〈〉
(
16, ( [9, 9, 9, 10]; [0.3, 0.3, 0.5, 0.7] , (0.3, 0.3, 0.5, 0.7; 1))
)
A
3
〈〉
(
17, ( [7, 9, 9, 10]; [0.3, 0.3, 0.5, 0.7] , (0, 0.1, 0.1, 0.3; 1))
)
XIAN ET AL.
|
21

TABLE 8 The transformed fuzzy decision table by decision‐maker
d
2
C
1
C
2
A1〈〉
(
17, ( [7, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0.3, 0.3, 0.5, 0.7; 1))
)
〈〉
(
13,([5,7,7,9];[0.9,0.9,1.0,1.0],(0.9,0.9,1.0,1.0;1)))
A2〈〉
(
12, ( [7, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0, 0, 0, 0.1; 1))
)
〈〉
(
14, ( [9, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0.9, 0.9, 1.0, 1.0; 1)))
A
3
〈〉
(
13, ( [7, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0.3, 0.3, 0.5, 0.7; 1))
)
〈〉
(
15, ( [7, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0.9, 0.9, 1.0, 1.0; 1)))
C
3
C
4
A1〈〉
(
16, ( [7, 9, 9, 10]; [0.7, 0.9, 0.9, 1.0] , (0, 0, 0, 0.1; 1))
)
〈〉
(
15, ( [7, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0, 0.1, 0.1, 0.3; 1)))
A2〈〉
(
13, ( [9, 9, 9, 10]; [0.7, 0.9, 0.9, 1.0] , (0.3, 0.3, 0.5, 0.7; 1))
)
〈〉
(
16, ( [9, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0, 0, 0, 0.1; 1)))
A
3
〈〉
(
12,([5,7,7,9];[0.7,0.9,0.9,1.0],(0.3,0.3,0.5,0.7;1))
)
〈〉
(
17, ( [9, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0.9, 0.9, 1.0, 1.0; 1)))
C
5
A1〈〉
(
12,([3,5,5,7];[0.5,0.7,0.7,0.9],(0,0,0,0.1;1))
)
A2〈〉
(
20,([5,7,7,9];[0.5,0.7,0.7,0.9],(0.3,0.3,0.5,0.7;1))
)
A
3
〈〉
(
16, ( [7, 9, 9, 10]; [0.5, 0.7, 0.7, 0.9] , (0.5, 0.7, 0.7, 0.9; 1))
)
22
|
XIAN ET AL.

TABLE 9 The transformed fuzzy decision table by decision‐maker
d
3
C
1
C
2
A
1
〈〉
(
16,([5,7,7,9];[0.5,0.7,0.7,0.9],(0.5,0.7,0.7,0.9;1))
)
〈〉
(
17,([3,5,5,7];[0.9,0.9,1.0,1.0],(0.3,0.3,0.5,0.7;1)))
A
2
〈〉
(
12,([5,7,7,9];[0.5,0.7,0.7,0.9],(0.3,0.3,0.5,0.7;1))
)
〈〉
(
15, ( [9, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0, 0, 0, 0.1; 1)))
A
3
〈〉
(
13,([3,5,5,7];[0.5,0.7,0.7,0.9],(0,0,0,0.1;1))
)
〈〉
(
12, ( [9, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0.3, 0.3, 0.5, 0.7; 1)))
C
3
C
4
A
1
〈〉
(
11, ( [7, 9, 9, 10]; [0.7, 0.9, 0.9, 1.0] , (0, 0.1, 0.1, 0.3; 1))) 〈〉
(
12, ( [9, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0.3, 0.3, 0.5, 0.7; 1)))
A
2
〈〉
(
16, ( [7, 9, 9, 10]; [0.7, 0.9, 0.9, 1.0] , (0.5, 0.7, 0.7, 0.9; 1))) 〈〉
(
17, ( [9, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0.9, 0.9, 1.0, 1.0; 1)))
A
3
〈〉
(
15, ( [9, 9, 9, 10]; [0.9, 0.9, 1.0, 1.0] , (0.3, 0.3, 0.5, 0.7; 1))) 〈〉
(
16,([5,7,7,9];[0.9,0.9,1.0,1.0],(0,0,0,0.1;1)))
C
5
A
1
〈〉
(
13,([3,5,5,7];[0.5,0.7,0.7,0.9],(0,0,0,0.1;1))
)
A
2
〈〉
(
18, ( [7, 9, 9, 10]; [0.5, 0.7, 0.7, 0.9] , (0.3, 0.3, 0.5, 0.7; 1)))
A
3
〈〉
(
11,([5,7,7,9];[0.5,0.7,0.7,0.9],(0.3,0.3,0.5,0.7;1))
)
XIAN ET AL.
|
23

Definition 14 Let

i
A be a group of trapezoidal type‐2 fuzzy numbers(TT2FNs),

abcd μμ μμ=<[ , , , ];[ , , , ]
>
iiiii
a
i
b
i
c
i
d
i, a trapezoidal type‐2 fuzzy OWA operator
(TT2OWA) of dimension nis a function
→ϕ:Ω Ω
TT OWA
n
2
+
, where
Ω+
is the set of
TABLE 10 Comprehensive decision table
d
1
A
1
〈〉
(
[5.6, 7, 7, 8.55]; [0.82, 0.84, 0.84, 0.94] , (0.4, 0.42, 0.54, 0.68; 1))
A
2
〈〉
(
[8.7, 9, 9, 10]; [0.8, 0.83, 0.93, 0.96] , (0.46, 0.5, 0.64, 0.8; 1))
A
3
〈〉
(
[7.2, 8.8, 8.8, 9.9]; [0.86, 0.86, 0.93, 0.98] ; (0.58, 0.62, 0.74, 0.86; 1)
)
d2
A
1
〈〉
(
[6.3, 8.3, 8.3, 9.55]; [0.81, 0.88, 0.95, 0.99] , (0.24, 0.26, 0.34, 0.44; 1)
)
A
2
〈〉
(
[8, 8.6, 8.6, 9.8]; [0.81, 0.83, 0.93, 0.87] , (0.3, 0.3, 0.4, 0.52; 1))
A
3
〈〉
(
[7.2, 8.8, 8.8, 9.9]; [0.86, 0.86, 0.93, 0.98] ; (0.58, 0.62, 0.74, 0.86; 1)
)
d
3
A
1
〈〉
(
[4.8, 6.5, 6.5, 8.25]; [0.69, 0.80, 0.85, 0.95] , (0.22, 0.28, 0.3, 0.54; 1)
)
A
2
〈〉
(
[7.6, 8.8, 8.8, 9.9]; [0.74, 0.85, 0.90, 0.98] , (0.4, 0.44, 0.54, 0.68; 1)
)
A
3
〈〉
(
[6, 7.2, 7.2, 8.8]; [0.77, 0.83, 0.85, 0.97] , (0.18, 0.18, 0.3, 0.46; 1)
)
TABLE 11 Comprehensive decision table

LIOWA

A(
)
1
〈〉
(
[5.56, 7.25, 7.25, 8.77]; [0.59, 0, 84, 0.88, 0.96] , (0.29, 0.32, 0.39, 0.55; 1) )

A()
2
〈〉
(
[8.08, 8.80, 8.80, 9.58]; [0.78, 0.84, 0.92, 0.94] , (0.39, 0.41, 0.53, 0.67; 1
)

A(
)
3
〈〉
(
[6.77, 8.27, 8.27, 9.54]; [0.81, 0.84, 0.91, 0.97] , (0.37, 0.39, 0.52, 0.67; 1)
)

LOWA

A(
)
1
〈〉
(
[6.49, 7.88, 7.88, 9.13]; [0.74, 0.79, 0.89, 0.94] , (0.47, 0.48, 0.6, 0.71; 1)
)

A()
2
〈〉
(
[7.82, 8.58, 8.58, 9.71]; [0.77, 0.81, 0.89, 0.89] , (0.33, 0.35, 0.45, 0.61; 1)
)

A(
)
3
〈〉
(
[6.89, 8.04, 8.04, 9.74]; [0.73, 0.84, 0.88, 0.96] , (0.26, 0.3, 0.39, 0.46; 1)
)

LCWAA

A(
)
1
〈〉
(
[6.79, 8.8, 8.8, 9]; [0.68, 0.77, 0.87, 0.93] , (0.45, 0.48, 0.6, 0.74; 1)
)

A()
2
〈〉
(
[7.03, 8.44, 8.44, 9.69]; [0.59, 0.65, 0.7, 0.74] , (0.41, 0.44, 0.53, 0.66; 1)
)

A(
)
3
〈〉
(
[5.25, 7.86, 7.86, 9.25]; [0.67, 0.68, 0.79, 0.88] , (0.27, 0.3, 0.4, 0.63; 1)
)
TT2OWA
v
1
0
.213
v
2
0
.435
v
3
0
.38
4
Abbreviations: LIOWA, linguistic‐induced ordered weighted averaging; LOWA, linguistic ordered weighted averaging;
LCWAA, linguistic combined weighted averaging aggregation.
24
|
XIAN ET AL.

TT2FNs. And a weighting vector
ω
i
is associated to aggregate the set of second arguments
according to the following expressing:
  
∑
⋅ϕω(, ,…, )=
TT OWA n
i
n
iσi
212
=1
()
(23)
where
→
σ
nn: (1, 2, …, ) (1, 2, …, )
is a permutation,

σi(
)
and is the
i
th largest one of all
numerical values

∈snωin( =1,2,…, ), [0,1]( =1,2,…, )
si and
∑
ω=
1
i
n
i
=1 .
From Table 12, we see that the ranking of the engineers is different when calculating with
different operators. That is because every criterion and DM has different weight in different
operators. Particularly, the engineers ranking of the

LOWA operator and the TT2OWA
operator are different shows

‐number is more comprehensive.
7
|
CONCLUSION
There are many uncertain and imprecise information appear in decision‐making, so Zadeh
proposed the concept of

‐number to overcome this issue. In this paper, we first develop the
triplex linguistic variables to express

‐number, so that more information can be covered
simultaneously. Without the issue of missing information that other methods often has, a new
method of ranking

‐numbers and the operations of them are developed. Based on

LIOWA
operator and

LCWAA operator, a method of MAGDM is also proposed. In the future, we shall
continue working in the application of the developed operators to other domains, such as risk
profile and trend analysis.
ACKNOWLEDGMENTS
The authors express their gratitude to Prof. Ronald R. Yager, the editor, and the anonymous
reviewers for their valuable and constructive comments. And this study was supported by the
Major entrustment projects of the Chongqing Bureau of quality and technology supervision
(CQZJZD2018001), Graduate Teaching Reform Research Program of Chongqing Municipal
Education Commission (YJG183074), Chongqing Basic and Frontier Research Project, P.R.
China (cstc 2015jcyjBX0090) and Chongqing research and innovation project of graduate
students (CYS18252, CYS17227).
TABLE 12 Comprehensive decision table

LIOWA
AAA>>
23
1

LOWA
AAA>>
312

LCWAA
AAA>>
213
TT2OWA
AAA>>
23
1
Abbreviations: LIOWA, linguistic‐induced ordered weighted averaging; LOWA, linguistic ordered weighted averaging;
LCWAA, linguistic combined weighted averaging aggregation.
XIAN ET AL.
|
25

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How to cite this article: Xian S, Chai J, Guo H.

Linguistic‐induced ordered weighted
averaging operator for multiple attribute group decision‐making. Int J Intell Syst. 2018;
1‐26. https://doi.org/10.1002/int.22050
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