Information Technology Reference
In-Depth Information
1
10
Q
=
0
.
7
×
0
.
5
+
0
.
6
×
0
.
3
+
0
.
2
×
0
.
5
=
0
.
63
that implies
Q
=
6
.
3. Provided the three referees have the same weight, it is
W
=
p
1
+
p
2
+
p
3
1
1
1
1
1
1
3
7
+
6
+
5
(
3
,
3
,
3
)
, and then,
Q
=
p
1
·
3
+
p
2
·
3
+
p
3
·
=
=
=
6, is
3
3
just the arithmetic
mean
of the three qualifications.
Another way of obtaining the final qualification, this time by ignoring the referee's
character, is by the
geometric mean
√
7
√
p
1
3
Q
=
3
·
p
2
·
p
3
=
×
6
×
5
=
5
.
94
,
=
=
,
=
=
sh
owing that
in a problem with
p
1
p
2
10
p
3
0, it results
Q
√
10
20
3
3
×
10
×
0
=
0
,
when the arithmetic mean is
=
6
.
67.
2.3.1 Aggregation Functions
Most of these problems are “represented” by the so-called
Aggregation Functions
,
that is, functions
n
A
:[
0
,
1
]
ₒ[
0
,
1
]
,
such that
1.
A
is continuous in all variables
2.
A
(
0
,...,
0
)
=
0, and
A
(
1
,...,
1
)
=
1
3. If
x
1
y
1
,...,
x
n
y
n
, then
A
(
x
1
,...,
x
n
)
A
(
y
1
,...,
y
n
)
.
Sometimes it is said that
A
is an
n-dimensional aggregation function
. Continuous
t-norms and continuous t-conorms are 2-dimensional aggregation functions.
Of the many types of aggregation functions, a particular and important type are
the
quasi-linear means
,
f
−
1
n
M
(
x
1
,...,
x
n
)
=
p
i
.
f
(
x
i
)
i
=
1
i
=
1
n
with
(
p
1
,...,
p
n
)
in
[
0
,
1
]
, verifying
p
i
=
1, and
f
:[
0
,
1
]ₒR
, continuous,
one-to-one, and monotonic. Function
f
is called the
generator
of
M
.
Notice that if
f
is the identity
f
(
x
)
=
x
, we get the
weighted means
:
n
M
(
x
1
,...,
x
n
)
=
p
i
·
x
i
,
i
=
1
1
n
that with
p
1
=
(
1
i
n
)
is the arithmetic mean
Search WWH ::
Custom Search