IEEE Systems, Man and Cybernetics Magazine - April 2021 - 10

interval argument OWA operator
constrained-optimization problem.
to the more general case in which
In [12], the authors chose a vector of
The OWAD operator
the argument values have imporweights that maximizes the dispercan be further
tance weights. Using this approach,
sion disp (w) for a given n using
he introduced the idea of an attithe method of Lagrange multipliers.
extended by using
tudina l-ba sed expected va lue
In [13], Fullér and Majlender examother types of
associated with a continuous ranined a minimum-variance method
distances, such
dom variable.
to obtain the minimal variability
OWA operators are also applied
OWA operator weights.
as Euclidean,
in decision making under ignoIn 2009, Yager [36] also considMinkowski, and
rance, an important class of uncerered the possibility of using minimitain decision-making problems. In
zation of dispersion. He discussed
quasi-arithmetic.
this case, there are alternatives
the concerns he had with both the
from which one must be selected,
maximization and minimization of
and there are different states of
dispersion and investigated the
nature. The payoff received depends on the state of nature
possibility of finding an optimal solution intermediate to
and the alternative selected at the same time. Uncertainty
these extremes. He introduced a fundamental requirement
means that the decision maker is unaware of the state of
for a measurement of dispersion called the preference for
nature at the time of choosing the alternative.
equal division.
An extended class of OWA operators, one based on the
relaxation
of requirements on the OWA operators, is introMethods Based on Weight-Generating Functions
duced
in
[31].
This relaxation allows us to consider a new
Weight-generating functions make it possible to obtain
branch of OWA operators, NOMOWA operators, which
weighting vectors of OWA operators and weighted means
have negative weights and exhibit nonmonotonicity. Some
for any number of arguments, which means the acquisition
special cases of these operators are discussed, and then
of extended-aggregation functions. This is very important
we investigate the role of these nonmonotonic operators in
when the number of arguments is not given. It turned out
the formulation of multicriteria decision functions.
to be possible to learn weight-generating functions from
The OWA operator can be also used to provide norms.
empirical data, similar to determining weighting vectors
Several different classes of OWA norms are considered in
of the aggregation functions of a fixed dimension [3]. The
[38]. It is shown that the functional generation of the weights
method relies on representing a weight-generating funcof an OWA norm requires that the weight-generating function
tion with a spline or polynomial and fitting its coefficients
have a nonpositive second derivative. The use of OWA operaby solving a least-squares or least absolute deviation probtors to induce similarity measurements is also discussed.
lem, subject to several linear constraints.
In business and economics, the use of different similarity
measurements such as the Hamming distance [15] is needFurther Applications
ed. These measurements also use aggregation operators.
A wide range of further applications of OWA operators has
The advantage of the Hamming distance in decision makbeen introduced in the literature. Without the desire for
ing is that it permits comparisons of available values with
completeness, a few examples are provided in this section.
some ideal ones. Depending on the particular problem we
There exists an application of fuzzy connectives in stahave, different decision makers may have varying opinions
tistical regression. In it, the standard least squares, least
or interests; therefore, the " best " results are not always the
absolute deviation, and maximum-likelihood criteria can
same for each decision maker. An extreme example of this
all be replaced with an OWA function of the residuals.
would be the concept of dumping, which means that the
Yager and Beliakov [37] presented various approaches
to the numerical solution of regression problems. OWAseller is selling the product at a price that is lower than its
based regression is particularly useful in the presence
production cost. Thus, in the decision process of fixing this
of outliers.
price, the seller is looking for an ideal that it is not the best
In [28], Yager focused on the problem of maximizing the
one. The use of the OWA operator in the Hamming disOWA aggregation of a group of variables that are contance, i.e., the OWA distance (OWAD) operator, has also
strained by a collection of linear inequalities. This procebeen analyzed [19], [20]. Its main advantage is that it produre is extremely useful to provide a solution to fuzzy
vides a parameterized family of distance aggregation operlinear programming problems in which some linguistically
ators between the maximum and minimum distances. The
proscribed number of goals must be satisfied.
OWAD operator can be further extended by using other
An extension of the OWA operator to the case in which
types of distances, such as Euclidean, Minkowski, and quaour argument is a continuous-valued interval rather than
si-arithmetic [19].
a finite set of values is also possible [32]. Moreover,
An extension of OWA operators for n-dimensional fuzzy
Yager considered the extension of the continuous
sets, denoted by MOWA, has recently been proposed by De
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