Information Technology Reference
In-Depth Information
By using Eqs. (
1.3
) and (
1.4
), Xu and Yager (2006), and Xu (2007) gave the
following method for ranking IFVs:
(1) If
S
(α
i
)>
S
(α
j
)
, then
α
i
is larger than
α
j
.
(2) If
S
(α
i
)
=
S
(α
j
)
, then
(a) If
H
(α
i
)
=
H
(α
j
)
, then
α
i
=
α
j
;
α
i
>α
j
.
Instead of the method above, Li and Rao (2001) gave another ranking technique
by replacing the accuracy function (
1.4
) with the score function (
1.5
).
Then let's redo Example 1.1, we calculate the accuracy values and the score values,
respectively:
(b) If
H
(α
i
)>
H
(α
j
)
, then
H
(α
1
)
=
0
.
4
+
0
.
3
=
0
.
7
,
H
(α
3
)
=
0
.
3
+
0
.
2
=
0
.
5
S
(α
1
)
=
S
(α
3
)
=
1
−
0
.
3
=
0
.
7
,
1
−
0
.
2
=
0
.
8
and
S
(α
1
)<
S
(α
3
)
then
H
.
We find that the two methods have different results. Hong and Choi (2000)'s
method emphasizes the amount of information that an IFV contains, but Li and Rao
(2001)'s method is inclined to choose the IFVwhich has the smaller non-membership
degree.
Although the methods above can be used to rank all IFVs, sometimes it cannot
satisfy our requirements. Let's see an example:
(α
1
)>
H
(α
3
)
Example 1.2
(Zhang and Xu 2012) Suppose that there are two major state-funded
projects
y
1
and
y
2
, and a decision maker wants to select one of them by voting. The
results of the voting are expressed by IFVs, the membership degree represents the
proportion of the voters who agree to a project, the non-membership degree means
the proportion of the voters who against the project, and the hesitation degree denotes
the proportion of the abstainers. The results are listed as below:
(1)
y
1
:
(
0
.
6
,
0
.
15
,
0
.
25
)
—60% in favor, 15% against, and 25% abstain.
(2)
y
2
:
(
0
.
5
,
0
,
0
.
5
)
—50% in favor, 0% against, and 50% abstain.
In real situations, a decision maker may choose the first project because there are
more people who support and believe that the project can be carried out better. But if
we use the ranking methods based on the score function, it will produce the opposite
result.
1.1.2.2 The Method for Ranking IFVs by Using the Positive Ideal
Point
Bustince and Burillo (1995)
introduced the distance between two IFSs
A
i
={
x
,μ
A
i
(
x
),
v
A
i
(
x
)
|
x
∈
X
}
(
i
=
1
,
2
)
as follows:
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