A class of aggregation functions encompassing two-dimensional OWA operators
Introduction
In 1983 Atanassov introduced a new operator [2] allowing to associate a fuzzy set with each Atanassov intuitionistic fuzzy set or interval-valued fuzzy set (IVFS) [17], [20]. In fact, this operator, which we denote by , takes a value from the interval representing the membership to the IVFS and defines that value to be the membership degree to a fuzzy set [26], [27]. In this way, it is possible, for instance, to recover all the usual fuzzy set theoretic results when dealing with IVFS. In 1988 Yager presented the definition of an OWA operator [22].
Comparison of the results of Atanassov and Yager reveals that in two dimensions the numerical results provided by Atanassov operators and OWA operators are the same. This numerical coincidence prompted us to introduce and define new operators by suitably modifying the domain for the definition of Atanassov’s operators. Analysis of the properties required for Atanassov’s operators has allowed us to consider a class of aggregation functions that are a generalization of Atanassov’s operators [6], [7], [8]. In particular, it would be interesting to determine whether some of the properties that are usually required for aggregation functions, such as bisymmetry, strictness, monotonicity, etc., also hold for this class of generalized Atanassov operators.
As already stated, the original aim of Atanassov was to build fuzzy sets from IVFS. We have readdressed this aim for our generalized Atanassov operators. This enables us to use these new operators in all the fields in which Atanassov operators have worked well. For instance, because there is quite a simple way of associating each image with an IVFS in such a way that the membership interval represents to some extent the properties of a piece of the image, we can use our generalized Atanassov operators and the results linking them to OWA operators for image processing.
The remainder of the paper is organized as follows. The concepts of operators, aggregation functions, OWA operators and IVFS are described in Section 2. Section 3 presents the relation between OWA and operators. In Section 4 we present a generalization of the operator properties and two construction theorems. In the same section, we define a new family of operators acting on pairs of real numbers and investigate their main properties. In Section 5 we propose two methods to obtain fuzzy sets from IVFS by means of generalized operators. Section 6 concludes the paper.
Section snippets
Preliminary definitions
In fuzzy set theory, a strictly decreasing and continuous function such that is called a strict negation. If, in addition, N is involutive, then we say that it is a strong negation. We call automorphism of the unit interval every function that is continuous, strictly increasing and such that and .
In 1979 Trillas [21] presented the following theorem of characterization of strong negations. Theorem 1 A function is a strong negation if and
OWA operators and operators
As stated in the introduction, Atanassov proposed a family of operators to associate a fuzzy set to each IVFS [2], [3]. Definition 7 The operator is given by , with each operator defined as a convex combination of its boundary arguments bywhere for any we write .
Clearly the following properties hold.
- (i)
for all .
- (ii)
for all .
- (iii)
for
Generalized operators
Observe that, if we denote by K the system of operators , then K can be regarded as an operator on with values in [0, 1]. To generalize this operator, the following definition was proposed by Bustince et al. [6], [8], [7]. Definition 8 A GK operator is a mapping such that, if we denote , the following properties hold: If , then . for all . If , with , then . Let . If ,
Construction of a FS from a IVFS and an operator
In this section we present two methods to associate with each IVFS over the referential U a fuzzy set over the same referential. In both methods we use the operators .
Conclusions and future research
We established a link between operators and OWA operators of dimension 2. This relation led to the definition of a class of aggregation functions, the operators, in terms of operators in such a way that the resulting class encompasses OWA operators of dimension 2.
This generalization retains most of the important features of Atanassov’s operators. We presented two construction theorems for our functions and studied under which conditions they are bisymmetric.
Regarding future lines of
Acknowledgments
This research was partially supported by the Grants TIN2009-07901, TIN2007-65981, APVV-0012-07 and VEGA 1/0080/10. The authors would like to acknowledge both referees and editors for their useful comments and suggestions. Some of their comments have been reproduced in the text.
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