OWA operators with functional weights
Introduction
Ordered weighted averaging (OWA) aggregation operators were introduced by Yager in the eighties [23] as a solution to the concerned problem of aggregating multi-criteria to provide an overall decision making. They have been studied from a theoretical [2], [5], [13], [18], [26], [28] and applicational point of view. For example, they have been defined on complete lattices [9], [16], [19], in order to allow the consideration of incomparable elements to be aggregated [5], [13]. OWA operators have been applied to many fields, such as in economy, linguistic, clustering, energy, urban wastewater, etc. For example, a multiattribute decision-making method was introduced in [10] to deal with linguistic membership and nonmembership degrees. They have also been applied to linguistic decision judgments [4], noise removal in computer vision [11], in fuzzy clustering to handle noise and outliers [21], in order to neutralize the negative effects of possible outlier fuzzy data in the clustering process [8], to provide a decomposition for all the rank-dependent poverty measures in terms of incidence, intensity and inequality [1], to allow experts to express multiple self-confidence levels when providing their preferences [12], etc.
Specifically, an n-ary ordered weighted averaging (OWA) operator is a function associated with a list of weights , such that and , and defined as where is a permutation on the index set satisfying that . Hence, the greatest value of the tuple is multiplied by the first weight , the second to the second weight and so on. Since two values and , with , can be equals: , at least other different permutation σ can exist satisfying the ordering among the values: . However, the function is well defined because, if this happens, the equality holds. Since provides a trivial case, usually as we will consider in this paper.
The mathematical formulation of OWA operators allows the user to consider a multicriteria, from the least case given by the conjunction “and” and represented by the minimum operator, and the greatest one given by the disjunction “or” and represented by the maximum. Therefore, for all . Moreover, two other very interesting properties of the OWA operators are:
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Idempotent: , for all .
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Symmetric: , for all permutation and .
In this paper OWA operators will be extended to provide a multi-criteria overall decision making more flexible. Usually, the OWA operator is applied on the whole cartesian product , obtaining a constant character through all elements in . However, in different problems it is interesting to have a criterion with a subset of elements, other criterion to other ones and so on. For example, depending on the values, if we are considering low values, the user can prefer to consider the minimum, if they are medium, the user use an average and if they are high, we can consider the maximum. This versatility will be offered by the use of functional weights instead of constant weights. The definition of ordered functional weighted averaging (OFWA) operator will be introduced and its main properties. We will show that it is not monotonic in general, and necessary and sufficient conditions will be presented, as well. Particular useful OFWA operators will be studied based on an aggressive or conservative character. Moreover, an alternative sufficient condition to check the monotonicity of continuous OFWA operators is also studied and different examples will be presented.
Section snippets
OWA operator with functional weights
Summarization of data is a very useful methodology for understanding the information given in large datasets, which has been widely studied in different papers [6], [15], [24]. The use and definition of special operators with specific properties is important in this methodology. One interesting operator to be considered in this framework is the following one: , defined as: for all . This operator considers the minimum of
OFWA operators on a partition
This section will consider a proper partition of the domain , in order to define monotonic OFWA operators on regions. Given a family of weights , with n elements each one, we can consider the associated OWA operators: , and a list of values , from which we define the OFWA operator for all as follows: that is
Sufficient condition to the OFWA monotonicity
Since Proposition 9 is mainly based on the definition of OFWA operator, we should study alternative conditions to ensure that this operator based on the functional weights is an n-ary aggregator operator. The condition we are considered is based on the notion introduced below.
Definition 19 We say that a list of functions , with , denoted as , is ratio increasing (r-increasing, for short) if it satisfies that: for all
Conclusions and future work
This paper has introduced a flexible extension of OWA operators allowing functions as weights. This enrichment allows the consideration of different multicriteria on the same dataset, depending on the character of the values to be aggregated. The main properties of these operators called OFWA operators have been presented, highlighting that they can be non-monotonic and introducing a mechanism to check when they verify this important property. The OFWA operators defined on a partition of the
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The author would like to thank the anonymous referees for their careful reading of the paper and useful suggestions to clarify this work.
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Partially supported by the 2014-2020 ERDF Operational Programme in collaboration with the State Research Agency (AEI) in projects TIN2016-76653-P and PID2019-108991GB-I00, and with the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia in project FEDER-UCA18-108612, and by the European Cooperation in Science & Technology (COST) Action CA17124.