IEEE Systems, Man and Cybernetics Magazine - April 2021 - 11

Table 1. A literature overview.
Operator

Main Attributes

OWA [26]

Weights are associated directly with an ordered position

COWA [27], [35]

Gives preference to argument values that lie in the middle; the most weight is given
to the central scores

WOWA [18], [23], [24]

Allows for weighing the reliability of the information source and the values in relation to their
relative position; it has two weighting vectors

GOWA [3], [33]

Ordered weighted quasi-arithmetic mean; the special cases are, e.g., the ordered weighted geometric function, the power-based generalized OWA, and the ordered weighted harmonic function

TOWA [34]

Using other t-norms different from min for the anding operation; mixing of the t-norm
and the OWA operator

OWA-based regression [37]

The standard least squares, least absolute deviation, and maximum-likelihood criteria
are replaced by an OWA operator; particularly useful in the presence of outliers

OWA in fuzzy linear
programming [28]

Interrelated and constrained variables maximizing OWA aggregation from a
collection of linear inequalities

Continuous interval argument [32]

An extension of the OWA operator to a continuous-valued interval as an argument

Nonmonotonic OWA [31]

Negative weights, nonmonotonicity

Inducing norms [38]

A weight-generating function with a nonpositive second derivative

OWAD [15], [19], [20]

A parameterized family of distance aggregation operators between the maximum
and the minimum distance

MOWA [6], [22]

An extension of OWA operators for n-dimensional fuzzy sets

OWA operators that are robust
against outliers [4]

A penalty-based method comprising both outlier detection and the reallocation
of weights of the OWA

Miguel et al. [6] by making use of a linear extension of the
product order. Unfortunately, the consideration of a linear
extension of the product order extends OWA operators to
the multidimensional setting at the cost of losing continuity and robustness in the presence of outliers. In [22], it is
shown that this focus ultimately results in a forfeiture of
the properties of continuity and robustness in the presence
of outliers. A further interesting question is the robustness
of the aggregation functions in terms of the ranking of the
alternatives they produce. In [4], the authors propose a version of OWA operators that are, in contrast to standard
OWA operators, robust against inputs with outliers.
Summary and Outlook
The aim of this work was to provide a short review of OWA
operators, to synthesize some of the most significant
developments, and to identify trends in the literature. As
we have seen, the concept of the OWA operator, a symmetric aggregation function that allocates weights according
to the input value and unifies in one operator the conjunctive and disjunctive behavior, was introduced by Yager in
1988. Since then, the family of these functions has been
axiomatized and extended in various ways. For an overview of the operators and problems considered in this
article, see Table 1. In addition, OWA operators provide a
	

parameterized family of aggregation functions, including
many of the well-known operators. The development of an
appropriate methodology for obtaining the weights is still
an issue of considerable interest. For more details about
recent developments in OWA operators, the reader is
referred to [39].
Future studies should concentrate on a systematic comparison of outlier-detection methods, that is, to the best of
the author's knowledge, still missing in the literature. With
respect to multidimensional settings, an interesting open
problem is how the use of geometric quantiles, instead of
linear extensions of the product order in the construction
of OWA operators, could contribute to solving the problem
of robustness.
About the Author
Orsolya Csiszár (ocsiszar@hs-esslingen.de) earned her
Ph.D. degree (summa cum laude) in mathematics and computer science from Óbuda University, Budapest, Hungary, in
2016. Currently, she is an assistant professor with the Physiological Controls Research Center, the Institute of Applied
Mathematics, Óbuda University, Budapest, 1034, Hungary
and an academic fellow at the University of Applied Sciences, Esslingen, 73728, Germany, where she develops artificial
intelligence-based systems in collaboration with international
Ap ri l 2021

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