3. The Intuitionistic Linguistic Weighted Bonferroni Mean Operators
The Bonferroni mean (BM) has a significant advantage of capturing the interrelationship of the individual arguments; however, the traditional BM can only process the crisp numbers and cannot deal with intuitionistic linguistic. In this section, we will extend BM to deal with intuitionistic linguistic information and develop an intuitionistic linguistic Bonferroni mean (ILBM) operator and an intuitionistic linguistic weighted Bonferroni mean (ILWBM) operator. Further, we will discuss some desirable characteristics of them and some special cases with respect to the parameters
p
and
q
in Bonferroni.
Definition 7.
Let
a
~
j
=
〈
s
θ
j
,
(
u
j
,
v
j
)
〉
(
j
=
1,2
,
…
,
n
)
be a collection of ILNs, and
ILB
:
Ω
n
→
Ω
; if
(14)
IL
B
p
,
q
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
=
(
1
n
(
n

1
)
∑
i
,
j
=
1
i
≠
j
n
a
~
i
p
a
~
j
q
)
1
/
(
p
+
q
)
,
where
Ω
is the set of all intuitionistic linguistic numbers and for any
p
,
q
≥
0
, then
ILB
p
,
q
is called the intuitionistic linguistic Bonferroni mean (ILB).
According to the operations of ILNs, we can get the following result.
Theorem 8.
Let
p
,
q
≥
0
, and let
a
~
j
=
〈
s
θ
j
,
(
u
j
,
v
j
)
〉
(
j
=
1,2
,
…
,
n
)
be a collection of ILNs. Then, the aggregated result by formula (14) is also an ILN, and
(15)
I
L
B
p
,
q
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
=
〈
∏
i
,
j
=
1
i
≠
j
n
(
1

v
j
1

(
1

v
i
)
p
s
(
(
1
/
n
(
n

1
)
)
∑
i
,
j
=
1
,
i
≠
j
n
θ
i
p
θ
j
q
)
1
/
(
p
+
q
)
,
(
(
1

(
∏
i
,
j
=
1
i
≠
j
n
(
1

u
i
p
u
j
q
)
)
1
/
n
(
n

1
)
)
1
/
(
p
+
q
)
,
1

(
1

(
∏
i
,
j
=
1
i
≠
j
n
(
1

v
j
1

(
1

v
i
)
p
×
(
1

v
j
)
q
)
∏
i
,
j
=
1
i
≠
j
n
(
1

v
j
1

(
1

v
i
)
p
)
1
/
n
(
n

1
)
)
1
/
(
p
+
q
)
)
〉
.
We use mathematical induction to prove this theorem shown as follows.
Proof.
(1) Firstly, we need to prove that
(16)
∑
i
,
j
=
1
i
≠
j
n
a
~
i
p
a
~
j
q
=
〈
∏
i
,
j
=
1
i
≠
j
n
(
1

(
1

v
i
)
p
(
1

v
j
)
q
)
s
∑
i
,
j
=
1
,
i
≠
j
n
θ
i
p
θ
j
q
,
(
1

(
∏
i
,
j
=
1
i
≠
j
n
(
1

u
i
p
u
j
q
)
)
,
∏
i
,
j
=
1
i
≠
j
n
(
1

(
1

v
i
)
p
(
1

v
j
)
q
)
)
〉
.
By the operations of ILNs defined in (3)–(6), we have
(17)
a
~
i
p
=
〈
s
θ
i
p
,
(
u
i
p
,
1

(
1

v
i
)
p
)
〉
,
a
~
j
q
=
〈
s
θ
j
q
,
(
u
j
q
,
1

(
1

v
j
)
q
)
〉
,
(18)
a
~
i
p
a
~
j
q
=
〈
s
θ
i
p
,
(
u
i
p
,
1

(
1

v
i
)
p
)
〉
⊗
〈
s
θ
j
q
,
(
u
j
q
,
1

(
1

v
j
)
q
)
〉
=
〈
s
θ
i
p
θ
j
q
,
(
u
i
p
u
j
q
,
1

(
1

v
i
)
p
(
1

v
j
)
q
)
〉
.
(a) When
n
=
2
, by formulas (18) and (3), we can get
(19)
∑
i
,
j
=
1
i
≠
j
2
a
~
i
p
a
~
j
q
=
a
~
1
p
a
~
2
q
+
a
~
2
p
a
~
1
q
=
〈
s
θ
1
p
θ
2
q
,
(
u
1
p
u
2
q
,
1

(
1

v
1
)
p
(
1

v
2
)
q
)
〉
+
〈
s
θ
2
p
θ
1
q
,
(
u
2
p
u
1
q
,
1

(
1

v
2
)
p
(
1

v
1
)
q
)
〉
=
〈
s
θ
1
p
θ
2
q
+
θ
2
p
θ
1
q
,
(
1

(
1

u
1
p
u
2
q
)
(
1

u
2
p
u
1
q
)
,
(
1

(
1

v
1
)
p
(
1

v
2
)
q
)
×
(
1

(
1

v
2
)
p
(
1

v
1
)
q
)
)
s
θ
1
p
θ
2
q
+
θ
2
p
θ
1
q
,
(
1

(
1

u
1
p
u
2
q
)
(
1

u
2
p
u
1
q
)
,
〉
=
〈
∏
i
,
j
=
1
i
≠
j
2
(
1

u
i
p
u
j
q
)
s
∑
i
,
j
=
1
,
i
≠
j
2
θ
i
p
θ
j
q
,
(
1

∏
i
,
j
=
1
i
≠
j
2
(
1

u
i
p
u
j
q
)
,
∏
i
,
j
=
1
i
≠
j
2
(
1

(
1

v
i
)
p
(
1

v
j
)
q
)
)
〉
;
that is, when
n
=
2
, formula (16) is right.
(b) Suppose that when
n
=
k
, formula (16) is right; that is,
(20)
∑
i
,
j
=
1
i
≠
j
k
a
~
i
p
a
~
j
q
=
〈
∏
i
,
j
=
1
i
≠
j
k
(
1

(
1

v
i
)
p
(
1

v
j
)
q
)
s
∑
i
,
j
=
1
,
i
≠
j
k
θ
i
p
θ
j
q
,
(
1

(
∏
i
,
j
=
1
i
≠
j
k
(
1

u
i
p
u
j
q
)
)
,
∏
i
,
j
=
1
i
≠
j
k
(
1

(
1

v
i
)
p
(
1

v
j
)
q
)
)
〉
;
then, when
n
=
k
+
1
, we have
(21)
∑
i
,
j
=
1
i
≠
j
k
+
1
a
~
i
p
a
~
j
q
=
∑
i
,
j
=
1
i
≠
j
k
a
~
i
p
a
~
j
q
+
∑
i
=
1
k
a
~
i
p
a
~
k
+
1
q
+
∑
j
=
1
k
a
~
k
+
1
p
a
~
j
q
.
Firstly, we prove that
(22)
∑
i
=
1
k
a
~
i
p
a
~
k
+
1
q
=
〈
∏
i
=
1
k
(
1

(
1

v
i
)
p
(
1

v
k
+
1
)
q
)
s
∑
i
=
1
k
θ
i
p
θ
k
+
1
q
,
(
1

∏
i
=
1
k
(
1

u
i
p
u
k
+
1
q
)
,
∏
i
=
1
k
(
1

(
1

v
i
)
p
(
1

v
k
+
1
)
q
)
)
〉
.
We also use the mathematical induction on
k
as follows.
(i) When
k
=
2
, we have
(23)
a
~
i
p
a
~
3
q
=
〈
s
θ
i
p
θ
3
q
,
(
u
i
p
u
3
q
,
1

(
1

v
i
)
p
(
1

v
3
)
q
)
〉
∑
i
=
1
2
a
~
i
p
a
~
k
+
1
q
=
a
~
1
p
a
~
3
q
+
a
~
2
p
a
~
3
q
=
〈
s
θ
1
p
θ
3
q
,
(
u
1
p
u
3
q
,
1

(
1

v
1
)
p
(
1

v
3
)
q
)
〉
+
〈
s
θ
2
p
θ
3
q
,
(
u
2
p
u
3
q
,
1

(
1

v
2
)
p
(
1

v
3
)
q
)
〉
=
〈
s
θ
1
p
θ
3
q
+
θ
2
p
θ
3
q
,
(
1

(
1

u
1
p
u
3
q
)
(
1

u
2
p
u
3
q
)
,
(
1

(
1

v
1
)
p
(
1

v
3
)
q
)
×
(
1

(
1

v
2
)
p
(
1

v
3
)
q
)
)
s
θ
1
p
θ
3
q
+
θ
2
p
θ
3
q
,
(
1

(
1

u
1
p
u
3
q
)
(
1

u
2
p
u
3
q
)
,
〉
=
〈
s
∑
i
=
1
2
θ
i
p
θ
3
q
,
(
1

∏
i
=
1
2
(
1

u
i
p
u
3
q
)
,
∏
i
=
1
2
(
1

(
1

v
i
)
p
(
1

v
3
)
q
)
)
〉
.
(ii) Suppose that
k
=
l
, then formula (22) is right; that is,
(24)
∑
i
=
1
l
a
~
i
p
a
~
l
+
1
q
=
〈
∏
i
=
1
l
(
1

(
1

v
i
)
p
(
1

v
l
+
1
)
q
)
s
∑
i
=
1
l
θ
i
p
θ
l
+
1
q
,
(
1

∏
i
=
1
l
(
1

u
i
p
u
l
+
1
q
)
,
∏
i
=
1
l
(
1

(
1

v
i
)
p
(
1

v
l
+
1
)
q
)
)
〉
.
Then, when
k
=
l
+
1
, we have
(25)
∑
i
=
1
l
+
1
a
~
i
p
a
~
l
+
2
q
=
∑
i
=
1
l
a
~
i
p
a
~
l
+
2
q
+
a
~
l
+
1
p
a
~
l
+
2
q
=
〈
∏
i
=
1
l
(
1

(
1

v
i
)
p
(
1

v
l
+
2
)
q
)
s
∑
i
=
1
l
θ
i
p
θ
l
+
2
q
,
(
1

∏
i
=
1
l
(
1

u
i
p
u
l
+
2
q
)
,
∏
i
=
1
l
(
1

(
1

v
i
)
p
(
1

v
l
+
2
)
q
)
)
〉
+
〈
s
θ
l
+
1
p
θ
l
+
2
q
,
(
u
l
+
1
p
u
l
+
2
q
,
1

(
1

v
l
+
1
)
p
(
1

v
l
+
2
)
q
)
〉
=
〈
s
∑
i
=
1
l
+
1
θ
i
p
θ
l
+
2
q
,
(
1

∏
i
=
1
l
+
1
(
1

u
i
p
u
l
+
2
q
)
,
∏
i
=
1
l
+
1
(
1

(
1

v
i
)
p
(
1

v
l
+
2
)
q
)
)
〉
;
that is, for
k
=
l
+
1
, formula (22) is also right.
(iii) So, for all
k
, formula (22) is right.
Similarly, we can prove that
(26)
∑
j
=
1
k
a
~
k
+
1
p
a
~
j
q
=
〈
∏
j
=
1
k
(
1

(
1

v
k
+
1
)
p
(
1

v
j
)
q
)
s
∑
j
=
1
k
θ
k
+
1
p
θ
j
q
,
(
1

∏
j
=
1
k
(
1

u
k
+
1
p
u
j
q
)
,
∏
j
=
1
k
(
1

(
1

v
k
+
1
)
p
(
1

v
j
)
q
)
)
〉
.
So, by formulas (20), (22), and (26), formula (21) can be transformed as
(27)
∑
i
,
j
=
1
i
≠
j
k
+
1
a
~
i
p
a
~
j
q
=
∑
i
,
j
=
1
i
≠
j
k
a
~
i
p
a
~
j
q
+
∑
i
=
1
k
a
~
i
p
a
~
k
+
1
q
+
∑
j
=
1
k
a
~
k
+
1
p
a
~
j
q
=
〈
∏
i
,
j
=
1
i
≠
j
k
(
1

(
1

v
i
)
p
(
1

v
j
)
q
)
s
∑
i
,
j
=
1
,
i
≠
j
k
θ
i
p
θ
j
q
,
(
1

(
∏
i
,
j
=
1
i
≠
j
k
(
1

u
i
p
u
j
q
)
)
,
∏
i
,
j
=
1
i
≠
j
k
(
1

(
1

v
i
)
p
(
1

v
j
)
q
)
)
〉
+
〈
∏
i
=
1
k
(
1

(
1

v
i
)
p
(
1

v
k
+
1
)
q
)
s
∑
i
=
1
k
θ
i
p
θ
k
+
1
q
,
(
1

∏
i
=
1
k
(
1

u
i
p
u
k
+
1
q
)
,
∏
i
=
1
k
(
1

(
1

v
i
)
p
(
1

v
k
+
1
)
q
)
)
〉
+
〈
∏
j
=
1
k
(
1

(
1

v
k
+
1
)
p
(
1

v
j
)
q
)
s
∑
j
=
1
k
θ
k
+
1
p
θ
j
q
,
(
1

∏
j
=
1
k
(
1

u
k
+
1
p
u
j
q
)
,
∏
j
=
1
k
(
1

(
1

v
k
+
1
)
p
(
1

v
j
)
q
)
)
〉
=
〈
∏
i
,
j
=
1
i
≠
j
k
+
1
(
1

(
1

v
i
)
p
(
1

v
j
)
q
)
s
∑
i
,
j
=
1
,
i
≠
j
k
+
1
θ
i
p
θ
j
q
,
(
1

(
∏
i
,
j
=
1
i
≠
j
k
+
1
(
1

u
i
p
u
j
q
)
)
,
∏
i
,
j
=
1
i
≠
j
k
+
1
(
1

(
1

v
i
)
p
(
1

v
j
)
q
)
)
〉
.
So, when
n
=
k
+
1
, formula (16) is also right.
Thus, formula (16) is right, for all
n
.
(2) Then, we can prove that formula (15) is right.
By formula (16), we can get
(28)
ILB
p
,
q
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
=
(
1
n
(
n

1
)
∑
i
,
j
=
1
i
≠
j
n
a
~
i
p
a
~
j
q
)
1
/
(
p
+
q
)
=
(
1
n
(
n

1
)
〈
∏
i
,
j
=
1
i
≠
j
n
(
1

(
1

v
i
)
p
(
1

v
j
)
q
)
s
∑
i
,
j
=
1
,
i
≠
j
n
θ
i
p
θ
j
q
,
(
1

(
∏
i
,
j
=
1
i
≠
j
n
(
1

u
i
p
u
j
q
)
)
,
∏
i
,
j
=
1
i
≠
j
n
(
1

(
1

v
i
)
p
(
1

v
j
)
q
)
)
〉
)
1
/
(
p
+
q
)
=
(
〈
(
∏
i
,
j
=
1
i
≠
j
n
(
1

(
1

v
i
)
p
(
1

v
j
)
q
)
)
1
/
n
(
n

1
)
s
(
1
/
(
n
(
n

1
)
)
)
∑
i
,
j
=
1
,
i
≠
j
n
θ
i
p
θ
j
q
,
(
1

(
∏
i
,
j
=
1
i
≠
j
n
(
1

u
i
p
u
j
q
)
)
1
/
n
(
n

1
)
,
(
∏
i
,
j
=
1
i
≠
j
n
(
1

(
1

v
i
)
p
×
(
1

v
j
)
q
)
∏
i
,
j
=
1
i
≠
j
n
(
1

(
1

v
i
)
p
)
1
/
n
(
n

1
)
)
〉
)
1
/
(
p
+
q
)
=
〈
(
1

(
∏
i
,
j
=
1
i
≠
j
n
(
1

u
i
p
u
j
q
)
)
1
/
n
(
n

1
)
)
1
/
p
+
q
s
(
(
1
/
n
(
n

1
)
)
∑
i
,
j
=
1
,
i
≠
j
n
θ
i
p
θ
j
q
)
1
/
(
p
+
q
)
,
(
(
1

(
∏
i
,
j
=
1
i
≠
j
n
(
1

u
i
p
u
j
q
)
)
1
/
n
(
n

1
)
)
1
/
(
p
+
q
)
,
1

(
1

(
∏
i
,
j
=
1
i
≠
j
n
(
1

(
1

v
i
)
p
×
(
1

v
j
)
q
)
∏
i
,
j
=
1
i
≠
j
n
(
1

(
1

v
i
)
p
)
1
/
n
(
n

1
)
)
1
/
(
p
+
q
)
)
〉
.
Example 9.
Suppose that there are three intuitionistic linguistic numbers
a
~
1
=
〈
s
2
,
(
0.6,0.1
)
〉
,
a
~
2
=
〈
s
4
,
(
0.4,0.3
)
〉
, and
a
~
3
=
〈
s
1
,
(
0.8,0.2
)
〉
, and suppose that
p
=
1
and
q
=
2
; then, we can calculate the
IL
B
1,2
(
a
~
1
,
a
~
2
,
a
~
3
)
shown as follows:
(29)
IL
B
1,2
(
a
~
1
,
a
~
2
,
a
~
3
)
=
〈
1

(
1

(
v

)
(
1
/
(
3
×
2
)
)
)
1
/
(
1
+
2
)
s
(
(
1
/
(
3
×
2
)
)
(
θ
1
θ
2
2
+
θ
1
θ
3
2
+
θ
2
θ
1
2
+
θ
2
θ
3
2
+
θ
3
θ
1
2
+
θ
3
θ
2
2
)
)
1
/
(
1
+
2
)
,
(
(
(
1

u
2
u
3
2
)
(
1

u
3
u
1
2
)
(
1

u
3
u
2
2
)
)
(
1
/
(
3
×
2
)
)
(
(
1

u
3
u
2
2
)
)
(
1
/
(
3
×
2
)
)
1

(
(
1

u
1
u
2
2
)
(
1

u
1
u
3
2
)
×
(
1

u
2
u
1
2
)
(
1

u
2
u
3
2
)
(
1

u
3
u
1
2
)
×
(
1

u
3
u
2
2
)
)
1
/
(
3
×
2
)
)
1
/
(
1
+
2
)
,
1

(
1

(
v

)
1
/
(
3
×
2
)
)
1
/
(
1
+
2
)
)
s
(
(
1
/
(
3
×
2
)
)
(
θ
1
θ
2
2
+
θ
1
θ
3
2
+
θ
2
θ
1
2
+
θ
2
θ
3
2
+
θ
3
θ
1
2
+
θ
3
θ
2
2
)
)
1
/
1
+
2
〉
,
where
(30)
v

=
(
1

(
1

v
1
)
(
1

v
2
)
2
)
×
(
1

(
1

v
1
)
(
1

v
3
)
2
)
(
1

(
1

v
2
)
(
1

v
1
)
2
)
×
(
1

(
1

v
2
)
(
1

v
3
)
2
)
(
1

(
1

v
3
)
(
1

v
1
)
2
)
×
(
1

(
1

v
3
)
(
1

v
2
)
2
)
.
Replace the data of
a
~
1
,
a
~
2
, and
a
~
3
; we can get
(31)
v

=
(
1

(
1

0.1
)
*
(
1

0.3
)
2
)
*
(
1

(
1

0.1
)
*
(
1

0.2
)
2
)
*
(
1

(
1

0.3
)
*
(
1

0.1
)
2
)
*
(
1

(
1

0.3
)
*
(
1

0.2
)
2
)
*
(
1

(
1

0.2
)
*
(
1

0.1
)
2
)
*
(
1

(
1

0.2
)
*
(
1

0.3
)
2
)
=
0.01212
,
IL
B
1,2
(
a
~
1
,
a
~
2
,
a
~
3
)
=
〈
(
1

(
0.01212
)
1
/
(
3
×
2
)
)
1
/
(
1
+
2
)
s
(
(
1
/
(
3
×
2
)
)
(
(
2
*
4
2
+
2
*
1
2
+
4
*
2
2
+
4
*
1
2
+
1
*
2
2
+
1
*
4
2
)
)
)
1
/
(
1
+
2
)
,
(
*
(
1

0.8
*
0
.
6
2
)
*
(
1

0.8
*
0
.
4
2
)
)
(
1
/
6
)
(
*
(
1

0.8
*
0
.
6
2
)
*
(
1

0.8
*
0
.
4
2
)
1

(
(
1

0.6
*
0
.
4
2
)
*
(
1

0.6
*
0
.
8
2
)
)
(
1
/
6
)
1

(
(
1

0.6
*
0
.
4
2
)
*
(
1

0.6
*
0
.
8
2
)
*
(
1

0.4
*
0
.
6
2
)
*
(
1

0.4
*
0
.
8
2
)
*
(
1

0.8
*
0
.
6
2
)
*
(
1

0.8
*
0
.
4
2
)
1

(
(
1

0.6
*
0
.
4
2
)
*
(
1

0.6
*
0
.
8
2
)
)
(
1
/
6
)
)
(
1
/
3
)
,
1

(
1

(
0.01212
)
1
/
(
3
×
2
)
)
1
/
(
1
+
2
)
)
s
(
(
1
/
(
3
×
2
)
)
(
(
2
*
4
2
+
2
*
1
2
+
4
*
2
2
+
4
*
1
2
+
1
*
2
2
+
1
*
4
2
)
)
)
1
/
1
+
2
〉
=
〈
s
2.310
,
(
0.606,0.195
)
〉
.
In the following, we will discuss some special cases of the
IL
B
p
,
q
operator shown as follows.
(1) When
q
=
0
, formula (15) reduces to an intuitionistic linguistic generalized mean operator; it follows that
(32)
IL
B
p
,
0
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
=
〈
(
1

(
∏
i
=
1
n
(
1

(
1

v
i
)
p
)
)
1
/
n
)
1
/
p
s
(
(
1
/
n
)
∑
i
=
1
n
θ
i
p
)
1
/
p
,
(
(
1

(
∏
i
=
1
n
(
1

u
i
p
)
)
1
/
n
)
1
/
p
,
1

(
1

(
∏
i
=
1
n
(
1

(
1

v
i
)
p
)
)
1
/
n
)
1
/
p
)
〉
.
(2) If
p
=
1
and
q
=
0
, then (15) reduces to an intuitionistic linguistic average operator
(33)
B
1,0
(
a
1
,
a
2
,
…
,
a
n
)
=
1
n
∑
i
=
1
n
a
i
IL
B
1,0
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
=
〈
(
∏
i
=
1
n
v
i
)
1
/
n
s
(
1
/
n
)
∑
i
=
1
n
θ
i
,
(
(
1

(
∏
i
=
1
n
(
1

u
i
)
)
1
/
n
)
,
(
∏
i
=
1
n
v
i
)
1
/
n
)
〉
.
(3) If
p
→
0
and
q
=
0
, then (15) reduces to an intuitionistic linguistic geometric mean operator
(34)
lim
p
→
0
B
p
,
0
(
a
1
,
a
2
,
…
,
a
n
)
=
〈
(
(
∏
i
=
1
n
u
i
)
1
/
n
,
1

(
∏
i
=
1
n
(
1

v
i
)
)
1
/
n
)
s
(
∏
i
=
1
n
θ
i
)
1
/
n
,
(
(
∏
i
=
1
n
u
i
)
1
/
n
,
1

(
∏
i
=
1
n
(
1

v
i
)
)
1
/
n
)
〉
.
The traditional BM has the properties of commutativity, idempotency, monotonicity, and boundedness; in the following, we will prove that ILB also has these properties.
Theorem 10 (commutativity).
Let
(
a
~
1
′
,
a
~
2
′
,
…
,
a
~
n
′
)
be any permutation of
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
; then,
(35)
I
L
B
p
,
q
(
a
~
1
′
,
a
~
2
′
,
…
,
a
~
n
′
)
=
I
L
B
p
,
q
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
.
Proof.
Let
(36)
ILB
p
,
q
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
=
(
1
n
(
n

1
)
∑
i
,
j
=
1
i
≠
j
n
a
~
i
p
a
~
j
q
)
1
/
(
p
+
q
)
ILB
p
,
q
(
a
~
1
′
,
a
~
2
′
,
…
,
a
~
n
′
)
=
(
1
n
(
n

1
)
∑
i
,
j
=
1
i
≠
j
n
(
a
~
i
′
)
p
(
a
~
j
′
)
q
)
1
/
(
p
+
q
)
.
Since
(
a
~
1
′
,
a
~
2
′
,
…
,
a
~
n
′
)
is any permutation of
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
, we have
(37)
(
1
n
(
n

1
)
∑
i
,
j
=
1
i
≠
j
n
a
~
i
p
a
~
j
q
)
1
/
(
p
+
q
)
=
(
1
n
(
n

1
)
∑
i
,
j
=
1
i
≠
j
n
(
a
~
i
′
)
p
(
a
~
j
′
)
q
)
1
/
(
p
+
q
)
;
thus,
(38)
IL
B
p
,
q
(
a
~
1
′
,
a
~
2
′
,
…
,
a
~
n
′
)
=
IL
B
p
,
q
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
.
Theorem 11 (idempotency).
Let
a
~
j
=
a
~
,
j
=
1,2
,
…
,
n
; then
I
L
B
p
,
q
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
=
a
~
.
Proof.
Since
a
~
j
=
a
~
, for all
j
, we have
(39)
IL
B
p
,
q
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
=
(
1
n
(
n

1
)
∑
i
,
j
=
1
i
≠
j
n
a
~
i
p
a
~
j
q
)
1
/
(
p
+
q
)
=
(
1
n
(
n

1
)
∑
i
,
j
=
1
i
≠
j
n
a
~
p
+
q
)
1
/
(
p
+
q
)
=
(
a
~
p
+
q
)
1
/
(
p
+
q
)
=
a
~
.
Theorem 12 (monotonicity).
Let
a
~
i
(
i
=
1,2
,
…
,
n
)
and
b
~
i
(
i
=
1,2
,
…
,
n
)
be two collections of IFNs. If
a
~
i
≥
b
~
i
, for all
i
, then
(40)
I
L
B
p
,
q
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
≥
I
L
B
p
,
q
(
b
~
1
,
b
~
2
,
…
,
b
~
n
)
.
Proof.
Since
a
~
i
≥
b
~
i
for all
i
, we have
(41)
a
~
i
p
a
~
j
q
≥
b
~
i
p
b
~
j
q
,
∑
i
,
j
=
1
i
≠
j
n
a
~
i
p
a
~
j
q
≥
∑
i
,
j
=
1
i
≠
j
n
b
~
i
p
b
~
j
q
.
So,
(42)
(
1
n
(
n

1
)
∑
i
,
j
=
1
i
≠
j
n
a
~
i
p
a
~
j
q
)
1
/
(
p
+
q
)
≥
(
1
n
(
n

1
)
∑
i
,
j
=
1
i
≠
j
n
b
~
i
p
b
~
j
q
)
1
/
(
p
+
q
)
;
that is,
IL
B
p
,
q
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
≥
IL
B
p
,
q
(
b
~
1
,
b
~
2
,
…
,
b
~
n
)
.
Theorem 13 (boundedness).
The
I
L
B
p
,
q
operator lies between the
max
and
min
operators:
(43)
min
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
≤
I
L
B
p
,
q
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
≤
max
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
.
Proof.
Let
a
~
=
min
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
and
b
~
=
max
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
.
Since
a
~
≤
a
~
j
≤
b
~
, then
(44)
(
1
n
(
n

1
)
∑
i
,
j
=
1
i
≠
j
n
a
~
p
a
~
q
)
1
/
(
p
+
q
)
≤
(
1
n
(
n

1
)
∑
i
,
j
=
1
i
≠
j
n
a
~
i
p
a
~
j
q
)
1
/
(
p
+
q
)
≤
(
1
n
(
n

1
)
∑
i
,
j
=
1
i
≠
j
n
b
~
p
b
~
q
)
1
/
(
p
+
q
)
.
That is,
(45)
a
~
≤
(
∑
j
=
1
n
s
(
a
~
j
,
x
~
)
a
~
j
λ
∑
j
=
1
n
s
(
a
~
j
,
x
~
)
)
1
/
λ
≤
b
~
;
that is,
min
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
≤
IL
B
p
,
q
≤
max
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
.
In
IL
B
p
,
q
operator, we only consider the input parameters and their interrelationships and do not consider the importance of each input parameter itself. However, in many practical situations, the weight of input data is also an important parameter. So, we can define an intuitionistic linguistic weighted Bonferroni mean (ILWBM) operator.
Definition 14.
Let
a
~
j
=
〈
s
θ
(
a
~
j
)
,
(
u
(
a
~
j
)
,
v
(
a
~
j
)
)
〉
(
j
=
1,2
,
…
,
n
)
be a collection of ILNs, and
ILWB
:
Ω
n
→
Ω
, if
(46)
ILWB
ω
p
,
q
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
=
(
1
n
(
n

1
)
∑
i
,
j
=
1
i
≠
j
n
b
~
i
p
b
~
j
q
)
1
/
(
p
+
q
)
,
where
Ω
is the set of all intuitionistic linguistic numbers and
ω
=
(
ω
1
,
ω
2
,
…
,
ω
n
)
T
is the weight vector of
a
~
j
(
j
=
1,2
,
…
,
n
)
,
ω
j
∈
[
0,1
]
,
∑
j
=
1
n
ω
j
=
1
.
b
~
j
=
n
ω
j
a
~
j
;
n
is a balance parameter. Then, ILWB is called the intuitionistic linguistic weighted Bonferroni mean (ILWBM) operator.
Theorem 15.
ILB operator is a special case of the ILWB operator.
Proof.
If
ω
=
(
1
/
n
,
1
/
n
,
…
,
1
/
n
)
T
, then
b
~
j
=
n
ω
j
a
~
j
=
n
(
1
/
n
)
a
~
j
=
a
~
j
:
(47)
ILW
B
ω
p
,
q
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
=
(
1
n
(
n

1
)
∑
i
,
j
=
1
i
≠
j
n
b
~
i
p
b
~
j
q
)
1
/
(
p
+
q
)
=
(
1
n
(
n

1
)
∑
i
,
j
=
1
i
≠
j
n
a
~
i
p
a
~
j
q
)
1
/
(
p
+
q
)
=
IL
B
p
,
q
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
.
Theorem 16.
Let
p
,
q
≥
0
and
a
~
j
=
〈
s
θ
j
,
(
u
j
,
v
j
)
〉
(
j
=
1,2
,
…
,
n
)
be a collection of ILNs, and
ω
=
(
ω
1
,
ω
2
,
…
,
ω
n
)
T
is the weight vector of
a
~
j
(
j
=
1,2
,
…
,
n
)
,
ω
j
∈
[
0,1
]
,
∑
j
=
1
n
ω
j
=
1
. Then, the aggregated result by formula (46) is also an ILN, and
(48)
I
L
W
B
ω
p
,
q
(
a
~
1
,
a
~
2
,
…
,
a
~
n
)
=
〈
∏
i
,
j
=
1
i
≠
j
n
(
1

(
1

(
1

u
i
)
n
ω
i
)
p
s
(
(
n
p
+
q

1
/
(
n

1
)
)
∑
i
,
j
=
1
,
i
≠
j
n
ω
i
p
ω
j
q
θ
i
p
θ
j
q
)
1
/
(
p
+
q
)
,
(
(
1

(
∏
i
,
j
=
1
i
≠
j
n
(
1

(
1

(
1

u
i
)
n
ω
i
)
p
×
(
1

(
1

u
j
)
n
ω
j
)
q
)
∏
i
,
j
=
1
i
≠
j
n
(
1

(
1

(
1

u
i
)
n
ω
i
)
p
)
1
/
n
(
n

1
)
)
1
/
(
p
+
q
)
,
1

(
1

(
∏
i
,
j
=
1
i
≠
j
n
(
1

(
1

v
i
n
ω
i
)
p
×
(
1

v
j
n
ω
j
)
q
)
∏
i
,
j
=
1
i
≠
j
n
(
1

(
1

(
1

u
i
)
n
ω
i
)
p
)
1
/
n
(
n

1
)
)
1
/
(
p
+
q
)
)
s
(
(
n
p
+
q

1
/
(
n

1
)
)
∑
i
,
j
=
1
,
i
≠
j
n
ω
i
p
ω
j
q
θ
i
p
θ
j
q
)
1
/
p
+
q
〉
.
Similar to Theorem 8, it can be proved by using mathematical induction on
n
.
Example 17.
Suppose that there are three intuitionistic linguistic numbers
a
~
1
=
〈
s
2
,
(
0.6,0.1
)
〉
,
a
~
2
=
〈
s
4
,
(
0.4,0.3
)
〉
, and
a
~
3
=
〈
s
1
,
(
0.8,0.2
)
〉
, and the weight vector
ω
=
(
0.40,0.35,0.25
)
(suppose
p
=
1
and
q
=
2
); then, we can calculate the
ILW
B
1,2
(
a
~
1
,
a
~
2
,
a
~
3
)
shown as follows:
(49)
ILW
B
1,2
(
a
~
1
,
a
~
2
,
a
~
3
)
=
〈
(
1

(
v

)
1
/
(
3
×
2
)
)
1
/
(
1
+
2
)
s
(
(
3
1
+
2

1
/
(
3

1
)
)
(
θ

)
)
1
/
(
1
+
2
)
,
(
(
1

(
u

)
1
/
(
3
×
2
)
)
1
/
(
1
+
2
)
,
1

(
1

(
v

)
1
/
(
3
×
2
)
)
1
/
(
1
+
2
)
)
s
(
(
3
1
+
2

1
/
(
3

1
)
)
(
θ

)
)
1
/
1
+
2
〉
,
where
(50)
θ

=
ω
1
ω
2
2
θ
1
θ
2
2
+
ω
1
ω
3
2
θ
1
θ
3
2
+
ω
2
ω
1
2
θ
2
θ
1
2
+
ω
2
ω
3
2
θ
2
θ
3
2
+
ω
3
1
ω
1
2
θ
3
θ
1
2
+
ω
3
1
ω
2
2
θ
3
θ
2
2
,
u

=
∏
i
,
j
=
1
i
≠
j
3
(
1

(
1

(
1

u
i
)
3
ω
i
)
(
1

(
1

u
j
)
3
ω
j
)
2
)
=
(
1

(
1

(
1

u
1
)
3
*
ω
1
)
*
(
1

(
1

u
2
)
3
*
ω
2
)
2
)
*
(
1

(
1

(
1

u
1
)
3
*
ω
1
)
*
(
1

(
1

u
3
)
3
*
ω
3
)
2
)
*
(
1

(
1

(
1

u
2
)
3
*
ω
2
)
*
(
1

(
1

u
1
)
3
*
ω
1
)
2
)
*
(
1

(
1

(
1

u
2
)
3
*
ω
2
)
*
(
1

(
1

u
3
)
3
*
ω
3
)
2
)
*
(
1

(
1

(
1

u
3
)
3
*
ω
3
)
*
(
1

(
1

u
1
)
3
*
ω
1
)
2
)
*
(
1

(
1

(
1

u
3
)
3
*
ω
3
)
*
(
1

(
1

u
2
)
3
*
ω
2
)
2
)
,
v

=
∏
i
,
j
=
1
i
≠
j
3
(
1

(
1

v
i
3
ω
i
)
(
1

v
j
3
ω
j
)
2
)
=
(
1

(
1

v
1
3
*
ω
1
)
*
(
1

v
2
3
*
ω
2
)
2
)
*
(
1

(
1

v
1
3
*
ω
1
)
*
(
1

v
3
3
*
ω
3
)
2
)
*
(
1

(
1

v
2
3
*
ω
2
)
*
(
1

v
1
3
*
ω
1
)
2
)
*
(
1

(
1

v
2
3
*
ω
2
)
*
(
1

v
3
3
*
ω
3
)
2
)
*
(
1

(
1

v
3
3
*
ω
3
)
*
(
1

v
1
3
*
ω
1
)
2
)
*
(
1

(
1

v
3
3
*
ω
3
)
*
(
1

v
2
3
*
ω
2
)
2
)
.
Replace the data of
a
~
1
,
a
~
2
, and
a
~
3
; we can get
(51)
θ

=
0.4
*
0.35
2
*
2
*
4
2
+
0.4
*
0.25
*
2
*
1
2
+
0.35
*
0.4
2
*
4
*
2
2
+
0.35
*
0.25
2
*
4
*
1
2
+
0.25
*
0.4
2
*
1
*
2
2
+
0.25
*
0.35
2
*
1
*
4
2
=
3.4015
,
u

=
(
(
1

(
1

0.4
)
(
3
*
0.35
)
)
2
1

(
1

(
1

0.6
)
(
3
*
0.4
)
)
*
(
1

(
1

0.4
)
(
3
*
0.35
)
)
2
)
*
(
1

(
1

(
1

0.6
)
(
3
*
0.4
)
)
*
(
1

(
1

0.8
)
(
3
*
0.25
)
)
2
)
*
(
1

(
1

(
1

0.4
)
(
3
*
0.35
)
)
*
(
1

(
1

0.6
)
(
3
*
0.4
)
)
2
)
*
(
(
1

(
1

0.8
)
∧
(
3
*
0.25
)
)
∧
2
1

(
1

(
1

0.4
)
(
3
*
0.35
)
)
*
(
1

(
1

0.8
)
(
3
*
0.25
)
)
2
)
*
(
1

(
1

(
1

0.8
)
(
3
*
0.25
)
)
*
(
1

(
1

0.6
)
(
3
*
0.4
)
)
2
)
*
(
1

(
1

(
1

0.8
)
(
3
*
0.25
)
)
*
(
1

(
1

0.4
)
(
3
*
0.35
)
)
2
)
=
0.23367
,
v

=
(
1

(
1

0.1
(
3
*
0.4
)
)
*
(
1

0.3
(
3
*
0.35
)
)
2
)
*
(
1

(
1

0.1
(
3
*
0.4
)
)
*
(
1

0.2
(
3
*
0.25
)
)
2
)
*
(
1

(
1

0.3
(
3
*
0.35
)
)
*
(
1

0.1
(
3
*
0.4
)
)
2
)
*
(
1

(
1

0.3
(
3
*
0.35
)
)
*
(
1

0.2
(
3
*
0.25
)
)
2
)
*
(
1

(
1

0.2
(
3
*
0.25
)
)
*
(
1

0.1
(
3
*
0.4
)
)
2
)
*
(
1

(
1

0.2
(
3
*
0.25
)
)
*
(
1

0.3
(
3
*
0.35
)
)
2
)
=
0.01646
;
then,
(52)
ILW
B
1,2
(
a
~
1
,
a
~
2
,
a
~
3
)
=
〈
(
1

(
v

)
1
/
(
3
×
2
)
)
1
/
1
+
2
s
(
(
3
1
+
2

1
/
(
3

1
)
)
(
θ

)
)
1
/
(
1
+
2
)
,
(
(
1

(
u

)
1
/
(
3
×
2
)
)
1
/
(
1
+
2
)
,
1

(
1

(
v

)
1
/
(
3
×
2
)
)
1
/
(
1
+
2
)
)
s
(
(
3
1
+
2

1
/
(
3

1
)
)
(
θ

)
)
1
/
1
+
2
〉
=
〈
s
2.4829
,
(
0.5992,0.2086
)
〉
.
It is easy to prove that the
ILW
B
p
,
q
operator hasthe properties of commutativity and monotonicity, but it has not the property of idempotency.