IEEE Systems, Man and Cybernetics Magazine - April 2021 - 8

if i 1 k
0
w i = * 1/m if k # i 1 k + m
0
if k $ k + m.

Compensative
connectives are
characterized by
a higher degree of
satisfaction of one of
the criteria, which
can compensate for
a lower degree of
satisfaction of another
criterion.

Weighted OWA
There exists a new combination
f u nct ion, t he weig hted OWA
(WOWA), which combines the
advantages of the OWA operator
and the ones of the weighted mean
[18], [23], [24]. The new function
allows the user to weigh the reliability of the information source,
as the weighted mean does, and
the values in relation to their relative position, as the OWA operator.
This aggregation function has two
weighting vectors, one of which
stands for the role of the weighting vector in weighted
means, and the other is for the role of the weighting vector
in the OWA functions.
Let us consider the following situation. A robot has to
aggregate information coming from n different sensors, meaning distances to the obstacles. On one hand, the reliability of
each sensor is known (weights p). On the other hand, independent of their reliability, the distances from the nearest
obstacles are more relevant; therefore, independent from
the reliability of the sensors, the inputs also have to be
weighted according to their numerical value (weighting vector w). Thus, both the numerical value of the inputs as well
as their reliability are to be taken into account. It is exactly
WOWA that provides this type of aggregation.
Generalized OWA
OWA functions have been generalized to generalized OWA
operators (GOWA, also known as ordered weighted quasiarithmetic mean) with the help of generating functions
g := [0, 1] " (- 3, 3), as defined in the next section.
Definition 7
Let g := [0, 1] " (- 3, 3) be a continuous, strictly monotone function; and let w be a weighting vector. Let us
define the function GOWA(x) as
GOWA (x) = g

c | w g (x
n

-1

i

i

) m.

Choosing the Weights for OWA Operators
As we have seen, one important issue in the theory of OWA
operators is the determination of the associated weights to
obtain the appropriate OWA operators in practical usage
[9]. Various methods have been introduced, proving to be
useful for obtaining the weights associated with OWA
operators. Generally, to obtain the value of the weights, we
can use the following two ways.
The first method is the use of a learning mechanism. In
this approach, we have some sample data and associated
aggregated values. The process involves the use of some
kind of a regression model.
A second way is to give some meaning to the weights on
the base, of which we can have the decision maker directly
provide the values for the weights. Let
k

Sk = | wi .
i=0

It is easy to see that S n = 1 and S 0 = 0. If we have an
input vector of criteria satisfaction B such that b j = 1 for j K,
it means that K of the criteria are completely satisfied and
the rest are completely unsatisfied.
Thus,
k

F (B) = | w i = S k .
i=1

(i)

As for OWA, x(i) denotes the ith largest value of x.
Special cases, such as the ordered weighted geometric
function, the power-based generalized OWA, or the
ordered weighted harmonic that function together with
trigonometric, quadratic, and exponential OWA operators,
were studied in [33].
TOWA
In [34], Yager showed how the evaluation of an alternative
involves the determination of the degree to which subsets
8	

of criteria are satisfied by the
alternative, a calculation based
upon an anding of the satisfactions of the individual criteria. He
examined the possibility of using
other t-norms different from min
for the anding operation. Applying
this generalization, he introduced
a further extension of OWA operators, called TOWA, involving the
mixing of the t-norm and the OWA
operator. He produced various
aggregation functions resulting
from this mixing. The concept of
the " power of a t-norm " to provide
an ordering over the t-norm operators was also defined.

IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE Apri l 2021

In other words, S k expresses the degree of satisfaction if
K/N portion of the criteria is satisfied. Moreover, due to the
fact that S k = S k - 1 + W k, we can interpret W as the degree
of additional (marginal) satisfaction we get when we turn
from the satisfaction of K - 1 of the criteria to the satisfaction of K. Note that in this interpretation, the case of
w i = 1/n corresponds to a linear increase for each argument. From a pragmatic point of view, it is more natural
for a decision maker to provide the degree to which he/she
is happy with k criteria being satisfied (the S k function).
From this, we can easily obtain the w j s:



IEEE Systems, Man and Cybernetics Magazine - April 2021

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