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(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 .
Then Z is called an intuitionistic fuzzy similarity matrix.
Based on Theorem 2.3 and Definition 2.4, we have
Corollary 2.1 (Zhang et al. 2007) The composition matrix of two intuitionistic
fuzzy similarity matrices is an intuitionistic fuzzy matrix. However, the composition
matrix of two intuitionistic fuzzy similarity matrices may not be an intuitionistic
fuzzy similarity matrix. For example, let
(
,
)
(
.
,
.
)(
.
,
.
)
1
0
0
2
0
3
0
5
0
2
(
.
,
.
)
(
,
)
(
.
,
.
)
Z 1 =
0
2
0
3
1
0
0
1
0
7
(
0
.
5
,
0
.
2
)(
0
.
1
,
0
.
7
)
(
1
,
0
)
(
1
,
0
)
(
0
.
4
,
0
.
4
)(
0
.
9
,
0
.
1
)
Z 2 =
(
0
.
4
,
0
.
4
)
(
1
,
0
)
(
0
.
3
,
0
.
3
)
(
0
.
9
,
0
.
1
)(
0
.
3
,
0
.
3
)
(
1
,
0
)
Obviously, both Z 1 and Z 2 are intuitionistic fuzzy similarity matrices, but the
composition matrix of Z 1 and Z 2 is as follows:
(
1
,
0
)
(
0
.
4
,
0
.
3
)(
0
.
9
,
0
.
1
)
Z
=
Z 1
Z 2 =
(
0
.
4
,
0
.
3
)
(
1
,
0
)
(
0
.
3
,
0
.
3
)
(
0
.
9
,
0
.
1
)(
0
.
4
,
0
.
3
)
(
1
,
0
)
where z 23 =
z 32 , i.e., Z does not satisfy symmetry property. Thus, Z is not an intu-
itionistic fuzzy similaritymatrix. But when the compositionmatrix of an intuitionistic
fuzzy similarity matrix and itself is an intuitionistic fuzzy similarity matrix:
z ( 1 )
ij
Theorem 2.4 (Zhang et al. 2007) Let Z 1
= (
) n × n be an intuitionistic fuzzy
similarity matrix. Then the composition matrix Z
=
Z 1
Z 1
= (
z ij ) n × n is also an
intuitionistic fuzzy similarity matrix.
Proof (1) Since Z 1 is an intuitionistic fuzzy similarity matrix, by Corollary 2.1, the
composition matrix Z of Z 1 and itself is an intuitionistic fuzzy matrix.
(2) Since
n
k = 1 (
z ( 1 )
ik
z ( 1 )
ki
z ii =
) = (
ma k {
min
{ μ z ( 1 )
ik z ( 1 )
ki }} ,
mi k {
max
{
v z ( 1 )
ik ,
v z ( 1 )
ki }} )
(2.28)
then when k
=
i ,wehave
z ( 1 )
ii
z ( 1 )
ii
) = (
1
,
0
) (
1
,
0
) }= (
1
,
0
)
(2.29)
So
n
k = 1 (
z ( 1 )
ik
z ( 1 )
ki
z ii =
) = (
1
,
0
)
(2.30)
 
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