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Similarly, we have
Z
−
1
3
Z
−
1
1
(
Z
1
)
13
=
(
0
.
9333
,
0
.
0167
), (
Z
3
)
31
=
(
0
.
8333
,
0
.
0500
)
Z
−
1
4
Z
−
1
1
(
Z
1
)
14
=
(
0
.
9500
,
0
.
000
), (
Z
4
)
41
=
(
0
.
8333
,
0
.
1000
)
Z
−
1
5
Z
−
1
1
(
Z
1
)
15
=
(
0
.
9000
,
0
.
0500
), (
Z
5
)
51
=
(
0
.
8167
,
0
.
1000
)
Z
−
1
3
Z
−
1
2
(
Z
2
)
23
=
(
0
.
9333
,
0
.
0333
), (
Z
3
)
32
=
(
0
.
9167
,
0
.
0333
)
Z
−
1
4
Z
−
1
2
(
Z
2
)
24
=
(
0
.
9167
,
0
.
0167
), (
Z
4
)
42
=
(
0
.
8833
,
0
.
0500
)
Z
−
1
5
Z
−
1
2
(
Z
2
)
25
=
(
0
.
9000
,
0
.
0333
), (
Z
5
)
52
=
(
0
.
9000
,
0
.
0500
)
Z
−
1
4
Z
−
1
3
(
Z
3
)
34
=
(
0
.
8667
,
0
.
0000
), (
Z
4
)
43
=
(
0
.
8333
,
0
.
0500
)
Z
−
1
5
Z
−
1
3
(
Z
3
)
35
=
(
0
.
8667
,
0
.
0833
), (
Z
5
)
53
=
(
0
.
8500
,
0
.
0667
)
Z
−
1
5
Z
−
1
4
(
Z
4
)
45
=
(
0
.
8167
,
0
.
1000
), (
Z
5
)
54
=
(
0
.
8833
,
0
.
0833
)
According to Xu and Yager (2006)'s ranking method, we know that
Z
−
1
2
Z
−
1
1
Z
−
1
3
Z
−
1
1
(
Z
1
)
12
>(
Z
2
)
21
,(
Z
1
)
13
>(
Z
3
)
31
Z
−
1
4
Z
−
1
1
Z
−
1
5
Z
−
1
1
(
Z
1
)
14
>(
Z
4
)
41
,(
Z
1
)
15
>(
Z
5
)
51
Z
−
1
3
Z
−
1
2
Z
−
1
4
Z
−
1
2
(
Z
2
)
23
>(
Z
3
)
32
,(
Z
2
)
24
>(
Z
4
)
42
Z
−
1
5
Z
−
1
2
Z
−
1
4
Z
−
1
3
(
Z
2
)
25
>(
Z
5
)
52
,(
Z
3
)
34
>(
Z
4
)
43
Z
−
1
5
Z
−
1
3
Z
−
1
5
Z
−
1
4
(
Z
3
)
35
>(
Z
5
)
53
,(
Z
4
)
45
<(
Z
5
)
54
from which we get
y
4
y
1
.
From the above process, we can see that the intuitionistic fuzzy triangle product
can be used to compare the alternatives in multi-attribute decision making with
intuitionistic fuzzy information, but the computational complexity increases rapidly
as the numbers of the alternatives and attributes increase.
y
5
y
3
y
2
2.9.4 The Application of the Intuitionistic Fuzzy Square Product
From Eq. (
2.204
), we know that the intuitionistic fuzzy square product
Z
2
)
ij
can be interpreted as: it measures the similarity degree of the
i
th row of an intu-
itionistic fuzzy matrix
Z
1
and the
j
th row of an intuitionistic fuzzy matrix
R
2
mathe-
matically. Therefore, considering the problem stated at the beginning of Sect.
2.9.2
,
(
(
Z
1
Z
−
j
)
ij
reflects the similarity of the alternatives
y
i
and
y
j
. We can use the follow-
ing formula to construct an intuitionistic fuzzy similarity matrix for the alternatives
y
i
(
Z
i
i
=
1
,
2
,...,
n
)
:
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