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(3) Since
Z
1
is an intuitionistic fuzzy similarity matrix, then we have
z
(
1
)
z
(
1
)
ki
=
.
ik
Thereby
n
∨
z
(
1
)
jk
z
(
1
)
ki
z
ji
=
1
(
∧
)
=
(
ma
k
{
{
μ
z
(
1
)
jk
,μ
z
(
1
)
ki
}
,
mi
k
{
{
,
}}
)
min
max
v
z
(
1
)
jk
v
z
(
1
)
ki
k
=
=
(
ma
k
{
min
{
μ
z
(
1
)
jk
,μ
z
(
1
)
ik
}
,
mi
k
{
max
{
v
z
(
1
)
jk
,
v
z
(
1
)
ik
}}
)
=
(
ma
k
{
min
{
μ
z
(
1
)
ik
,μ
z
(
1
)
jk
}
,
mi
k
{
max
{
v
z
(
1
)
ik
,
v
z
(
1
)
jk
}}
)
n
k
=
1
(
z
(
1
)
ik
z
(
1
)
kj
=
∧
)
=
z
ij
(2.31)
z
(
1
)
ij
z
(
2
)
ij
Theorem 2.5
(Zhang et al. 2007) Let
Z
1
=
(
)
n
×
n
,
Z
2
=
(
)
n
×
n
and
Z
3
=
z
(
3
)
(
ij
)
n
×
n
be three intuitionistic fuzzy similarity matrices. Then their composition
operation satisfies the associative law:
(
Z
1
◦
Z
2
)
◦
Z
3
=
Z
1
◦
(
Z
2
◦
Z
3
)
(2.32)
z
it
)
n
×
n
. Then by Theorem
Proof
Let
(
Z
1
◦
Z
2
)
◦
Z
3
=
(
z
it
)
n
×
n
and
Z
1
◦
(
Z
2
◦
Z
3
)
=
(
2.1, we have
n
∨
n
∨
n
∨
n
∨
z
(
1
)
ij
z
(
2
)
jk
z
(
3
)
kt
z
(
1
)
ij
z
(
2
)
jk
z
(
3
)
kt
z
it
=
j
=
1
(
∧
)
∧
)
=
j
=
1
((
∧
)
∧
)
k
=
1
k
=
1
n
k
=
1
n
∨
n
∨
n
k
=
1
(
z
(
1
)
ij
z
(
2
)
jk
z
(
3
)
kt
z
(
1
)
ij
z
(
2
)
jk
z
(
3
)
kt
=
1
(
∧
(
∧
))
=
∧
(
∧
))
j
=
j
=
1
z
(
1
)
ij
n
∨
n
∨
z
(
2
)
jk
z
(
3
)
kt
=
∧
1
(
∧
)
j
=
1
k
=
z
it
,
=
i
,
t
=
1
,
2
,...,
n
Hence, Eq. (
2.32
) holds, which completes the proof.
Corollary 2.2
(Zhang et al. 2007) Let
Z
be an intuitionistic fuzzy similarity matrix.
Then for any positive integers
m
1
and
m
2
,wehave
Z
m
1
+
m
2
Z
m
1
Z
m
2
=
◦
where
Z
m
1
and
Z
m
2
are the
m
1
and
m
2
compositions of
Z
, respectively. Furthermore,
Z
m
1
,
Z
m
2
and their composition matrix
Z
m
1
+
m
2
are the intuitionistic fuzzy similarity
matrix.
Definition 2.5
(Zhang et al. 2007) If the intuitionistic fuzzy matrix
Z
=
(
z
ij
)
n
×
n
satisfies the following condition:
(1) Reflexivity:
z
ii
=
(
1
,
0
)
,
i
=
1
,
2
,...,
n
.
(2) Symmetry:
z
ij
=
μ
z
ij
=
μ
z
ji
,
v
z
ij
=
,
=
,
,...,
z
ji
, i.e.,
v
z
ji
,
i
j
1
2
n
.
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