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 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 7

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