Information Technology Reference
In-Depth Information
From Eqs. (
2.208
), (
2.209
) and Xu and Yager (2006)'s ranking method, we can
get an ordering of the alternatives
y
i
and
y
j
. Concretely speaking, (1) if
Z
−
1
j
(
Z
i
)
ij
>
Z
−
1
i
Z
−
1
j
(
Z
j
)
ji
, then the alternative
y
j
is preferred to the alternative
y
i
;(2)if
(
Z
i
)
ij
=
Z
−
1
i
(
Z
j
)
ji
, then there is no difference between the alternatives
y
i
and
y
j
; and (3) if
Z
−
1
j
Z
−
1
i
(
Z
i
)
ij
<(
Z
j
)
ji
, then the alternative
y
i
is preferred to the alternative
y
j
.
Example 2.15
(Wang et al. 2012) We express the evaluation results of the cars
y
i
(
i
=
1
,
2
,
3
,
4
,
5
)
in Table
2.16
as the vectors
Z
i
=
(
z
i
1
,
z
i
2
,...,
z
i
6
)(
i
=
1
,
2
,
3
,
4
,
5
)
, respectively, where
z
ij
=
(μ
ij
,
v
ij
)(
i
=
1
,
2
,
3
,
4
,
5
;
j
=
1
,
2
,
3
,
4
,
5
,
6
)
:
Z
1
=
((
0
.
3
,
0
.
5
),(
0
.
6
,
0
.
1
), (
0
.
4
,
0
.
3
),(
0
.
8
,
0
.
1
),(
0
.
1
,
0
.
6
),(
0
.
5
,
0
.
4
))
Z
2
=
((
0
.
5
,
0
.
3
),(
0
.
5
,
0
.
2
), (
0
.
6
,
0
.
1
),(
0
.
7
,
0
.
1
),(
0
.
3
,
0
.
6
),(
0
.
4
,
0
.
3
))
Z
3
=
((
0
.
4
,
0
.
4
),(
0
.
8
,
0
.
1
), (
0
.
5
,
0
.
1
),(
0
.
6
,
0
.
2
),(
0
.
4
,
0
.
5
),(
0
.
3
,
0
.
2
))
Z
4
=
((
0
.
2
,
0
.
4
),(
0
.
4
,
0
.
1
), (
0
.
9
,
0
.
0
),(
0
.
8
,
0
.
1
),(
0
.
2
,
0
.
5
),(
0
.
7
,
0
.
1
))
Z
5
=
((
0
.
5
,
0
.
2
),(
0
.
3
,
0
.
6
), (
0
.
6
.
0
.
3
),(
0
.
7
,
0
.
1
),(
0
.
6
,
0
.
2
),(
0
.
5
,
0
.
3
))
Then we utilize the intuitionistic fuzzy triangle products Eqs. (
2.208
) and (
2.209
)
to calculate the implication degrees
Z
−
1
j
Z
−
1
i
(
Z
i
)
ij
and
(
Z
j
)
ji
(
i
=
1
,
2
,
3
,
4
,
5
;
j
=
1
,
2
,...,
6
)
respectively:
1
6
k
=
1
μ
z
1
k
→
z
2
k
,
6
6
1
6
Z
−
1
2
(
Z
1
)
12
=
v
z
1
k
→
z
2
k
)
k
=
1
1
6
6
=
min
{
1
,
1
−
μ
1
k
+
μ
2
k
,
1
−
v
2
k
+
v
1
k
}
,
k
=
1
6
1
6
{
,
{
μ
1
k
−
μ
2
k
,
v
2
k
−
v
1
k
}}
max
0
min
k
=
1
=
(
0
.
9500
,
0
.
0167
)
1
6
6
6
1
6
Z
−
1
1
(
Z
2
)
21
=
1
μ
z
2
k
→
z
1
k
,
v
z
2
k
→
z
1
k
)
k
=
k
=
1
1
6
6
=
{
,
−
μ
2
k
+
μ
1
k
,
−
v
1
k
+
v
2
k
}
,
min
1
1
1
k
=
1
6
1
6
max
{
0
,
min
{
μ
2
k
−
μ
1
k
,
v
1
k
−
v
2
k
}}
k
=
1
=
(
0
.
8833
,
0
.
0667
)
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