IEEE Systems, Man and Cybernetics Magazine - April 2021 - 6

```w i e [0, 1], 1 # i # n with R in= 1 w i = 1 and F (x 1, x 2, ..., x n) =
OWA w (x) = w 1 x (1) + w 2 x (2) + g + w n x (n), where x (j) is the
jth largest element of the bag 1 x 1, ..., x n 2.
It is important to emphasize that the fundamental
aspect of this operator is the reordering action, i.e., the
weights w i are associated with a particular ordered position rather than with a particular element. It is obvious
that different OWA operators are distinguished by their
weighting functions. Particularly,
◆◆ Max: w ) = (1, 0..., 0) and Fmax (x 1, ..., x n) = max (x 1, ..., x n)
◆◆ Min: w ) = (0, 0..., 1) and Fmin (x 1, ..., x n) = min (x 1, ..., x n)
◆◆ Arithmetic mean: w A = ^ 1/n, 1/nf, 1/n h and FA =
^^ x 1 + g + x n h /n h .
Naturally, OWA operators have the basic properties of
an averaging operator: They are always commutative,
monotonic, and idempotent. The dual of an OWA function
is the so-called reverse OWA, with the vector of weights
w d = (w n, ..., w 1) . OWA functions are continuous, symmetric, homogeneous, and shift invariant. They do not have
neutral or absorbing elements, except for the special
cases of min and max. The OWA functions are special
cases of the Choquet integral with respect to symmetric
fuzzy measures. We can see that the OWA operators provide a parameterized family of aggregation operators,
which include many of the well-known operators, such as
maximum, minimum, k-order statistics, median, and
arithmetic mean.
Obviously, the pure " and, " with its lack of compensation
(anding the criteria means no compensation at all), or the
pure " or, " with its total submission to any good satisfaction
and also with its indifference to the individual criteria (oring
the criteria means full compensation), are not the desired
aggregation functions in most of the cases. A more descriptive name for OWA operators, suggested by Yager [26], would
be " orand " operators because they are, in a sense, acting as a
combination of the " anding " and " oring " operators.
OWA operators allow for a positive compensation
among ratings, which means that a higher degree of satisfying a criterion can compensate, to a certain extent,
for a lower degree of satisfying another one. The level
of compensation can be chosen between the logical
" and " and " or. " Given a decision problem, we can find
an appropriate OWA aggregation operator from some
of the rules and samples, as determined by the decision makers.

STOWA. (OWA functions have recently been mixed with
t-norms and t-conorms to provide new mixed aggregation
functions, known as OWA operators based on a t-norm
(T-OWAs), OWA operators based on an s-conorm (S-OWAs),
and OWA operators based on a t-norm or an s-conorm (STOWAs); these functions have proved to be useful, in particular, in the context of multicriteria decision making.)
Basically, the measurement of orness measures how far a
given averaging function is from the max function (the
weakest disjunctive function). The definition follows in the
next section.

Characterizing Measurements

Obviously, to determine a weighting vector with a
desired orness value, we can use different combinations of w 1, w 2, which all result in different w 3 with the
same orness value. For some special weighting vectors,
the measurement of orness has been precalculated as
well [1].
The measurement of andness can be defined as the
complement of orness,

Measurement of Orness
Yager introduced a measurement associated with an OWA
operator called the measurement of orness [26]. Also called
the degree of orness or attitudinal character, it is an important numerical characteristic of averaging aggregation
functions. It was first defined in 1974 by Dujmovic [7] and
rediscovered several times. It is applicable to any averaging
function and even some other aggregation functions, like
6

IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE Apri l 2021

Definition 2
Let F be an OWA operator with a weighting function w.
1
orness (w) = a n - 1 k \$ | ((n - i) w i) .
n

i=1

We can readily see that, for any w, orness (w) always
lies in the unit interval. Moreover, the nearer w is to " or, "
the closer its measurement is to one, while the nearer it is
to " and, " the closer it is to zero.
◆◆ for w = [1, 0, 0,...] ( " or " ), we get orness (w) = 1
◆◆ for w = [0, 0, 0,...1] ( " and " ), we get orness (w) = 0
◆◆ for w = [1/n, 1/n, ...1/n] (arithmetic mean), we get
orness (w) = 0.5.
Furthermore, the orness is one for only the max function ( " or " ) and zero for only the min function ( " and " ). However, orness can be 0.5 in cases different from the
arithmetic mean as well.
The sum of the orness of an OWA operator and the
orness of its dual is always one, i.e., an OWA function is
self-dual if and only if orness (w) = 0.5.
If the weighting vector is nondecreasing, i.e., w i # w i + 1
for i = 1, ..., n - 1, then orness (w) e 60.5, 1@ . Similarly, if
t h e we i g h t i n g v e c t o r i s n o n i n c r e a s i n g , t h e n
orness (w) e 60, 0.5@ .
If two OWA functions have weighing vectors w 1, w 2
with orness values o 1, o 2, and if
w 3 = a \$ w 1 + (1 - a) \$ w 2,
where ae [0, 1], then, for the OWA function with weighting
vector w 3, the orness value is
orness (w 3) = a \$ o 1 + (1 - a) \$ o 2 .

andness (w) = 1 - orness (w) .

```

IEEE Systems, Man and Cybernetics Magazine - April 2021

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