IEEE Systems, Man and Cybernetics Magazine - April 2021 - 7

Generally, an OWA operator with much of its nonzero
weights near the top will be an orlike operator, with
orness (w) $ 0.5, and when much of its weights are nonzero near the bottom, the OWA operator will be and like, with
andness (w) $ 0.5.
We can also see that, as we move the weight up the vector we increase the orness, while moving the weight down
causes us to decrease it [27]; that is, if we have two weighting vectors, w 1 and w 2, such that w 1 = a 1, a 2, ..., a n and
w 2 = a 1, ..., a j + e, ..., a k - e, ..., a n, where e 2 0, j 1 k, then
orness (w 1) 2 orness (w 2) .
The measurement of the attitudinal character can be
directly associated with the weight-generating function f
(see the " Quantifier-Guided Aggregation " section). In
particular,
orness (f) =

#

0

1

f (x) dx.

An important class of weighting vectors that generate
orness (w) = 0.5 are the symmetric weighting vectors.
This measurement of 0.5, of course, does not mean that
preference is given to central scores. For example, take
w = [0.5, 0...0, 0.5] . If we want to prefer the argument values
lying in the middle, we can use the centered OWA (COWA)
operators defined in the " COWA Operators " section [35].
Considering the family of regular increasing monotone
(RIM) quantifiers (see the " Quantifier-Guided Aggregation " section)

It is easy to see that the entropy measures the degree to
which we use all of the aggregates equally.
Special Types of OWA Operators
COWA Operators
Definition 4
An OWA operator is said to be centered if its associated
weighting vector w satisfies the following conditions:
◆◆ symmetric: w i = w i + n - 1
◆◆ strongly decaying: If i 1 j # (n + 1) /2, then w i < w j,
and, if i 2 j $ (n + 1) /2, then w i 1 w j
◆◆ inclusive: w i 2 0.
One can see that these types of aggregation operators
give the most weight to the central scores in the argument
tuples and less weight to the extreme values. Among other
applications, this type of operator can be useful for some
kinds of smoothing (e.g., for group preference aggregation).
The median, for instance, is a prototypical example of this
class. In this case, we want to aggregate so that we can eliminate the extreme values. In the case of COWA operators,
with the help of dispersion, a measurement called strength of
centering can be provided as defined in the next section.
Definition 5
The strength of centering of a centered weighting vector w
is defined as

Q a (r) = r a, a $ 0,

disp (w)
Cent (w) = 1 - ln (n) .

we obtain
1
orness (Q a) = 1 + a ,
which means that orness (Q a) 1 0.5 for a 2 1, orness
(Q a) = 0.5 for a = 1, and orness (Q a) 2 0.5 for a 1 1.
Entropy
Another important measurement is dispersion (a measurement of entropy) [26], which reflects how uniformly the
weights are distributed. It has as its extremes the case
when all the weights are the same and the case when all
the weights are zero except one. It is defined in the following section.
Definition 3
Let w be a weighting vector [w i] . The measurement of the
dispersion of w is
disp (w) = - | w i ln (w i) .
i

A useful normalization of this measurement is
n
ln (w i)
D (W) = - | w i ln (n) .
i=1

Here, D (W) e [0, 1] .
	

A useful way to specify basic unit interval monotonic
(BUM) functions (see the " Quantifier-Guided Aggregation "
section) for COWA vectors in terms of their derivative is
discussed in [35]. The concept of a centering function is
introduced as defined in the following.
Definition 6
A function g: [0, 1] " R is called a centering function if
◆◆ g (x) 2 0
◆◆ g is symmetric, roughly 0.5, i.e., g (0.5 + z) = g (0.5 - z)
for ze [0, 0.5]
◆◆ g is unimodal:
** g (x) 1 g (y) for x 1 y # 0.5
** g (x) 1 g (y) for x 2 y $ 0.5.
Especially, g can be a piecewise-linear, compressed piecewise-linear, step-like, extreme step-like, or parabolic-type
centering function.
A deeply examined class of COWA operators is based
on using Gaussian-type weights [35] or the Olympic type
introduced by Yager in 1993 [27]. The window type of OWA
operators also form an important class. An operator is of
window type if it takes the average of the m arguments
about the center, omitting the best and worst scores. This
means that we have
Ap ri l 2021

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IEEE Systems, Man and Cybernetics Magazine - April 2021

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