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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
 
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