Aggregation of convex intuitionistic fuzzy sets
Introduction
Different kinds of convexity have been introduced in the literature, first to handle non-fuzzy sets and then for fuzzy sets and intuitionistic fuzzy sets. These definitions allow us to deal with any possible kind of non-fuzzy, fuzzy or intuitionistic fuzzy set. Therefore, suitable notions are introduced for cases when the universe is not necessarily a linear space. We work then with abstract convexities that come from [21].
We devote our study to this class of sets because convexity is one of the most important aspects in the study of geometric properties of not only standard sets, but also intuitionistic fuzzy sets, mainly arising –and indeed playing a crucial role– in applications connected to optimization and control (see [1], [27]).
In particular we are interested in the study of the intersection of two convex IF-sets, but we will consider a more general framework, by using aggregation functions instead of the particular case of t-norms. Accordingly, the use of aggregation functions for intuitionistic fuzzy sets immediately carries the question of characterizing those aggregation functions that preserve convexity in some way. In the present paper we address this kind of questions.
The structure of the paper goes as follows: In Section 2, we recall the notions of a fuzzy set and an intuitionistic fuzzy set. In Section 3, we introduce the convex non-fuzzy and fuzzy sets and establish some equivalent definitions. Section 4 is devoted to the study of convexity for intuitionistic fuzzy sets. Finally, in Section 5, we analyze the behaviour of a generalization of the intersection of two convex intuitionistic fuzzy sets. A final section of concluding remarks and some open problems closes the paper.
Section snippets
Basic concepts
In this section we carry out a brief introduction to fuzzy sets and intuitionistic fuzzy sets for a better understanding of the main body of the paper.
Let us start by recalling the standard definition of a fuzzy set. Definition 1 [29]Let X be a nonempty set, usually called the universe. A fuzzy set A in X is defined by means of a map . The map is said to be the membership function (or indicator) of A. The support of A is the non-fuzzy set , whereas the kernel of A is the
Convexity of non-fuzzy and fuzzy sets
As a starting point we will recall here several definitions related to convexity for non-fuzzy and fuzzy sets and we will relate them. The obtained results will also be very important to deal with the convexity of intuitionistic fuzzy sets, because of the strong relationship between them and fuzzy sets. Definition 6 If X is a linear space, a non-fuzzy set is said to be convex if belongs to A for every and every .
Suppose now that we want to drop the restriction of X being a linear
Convex intuitionistic fuzzy sets
Although the theory and application of convex fuzzy sets have been studied intensively, the corresponding research for convex intuitionistic fuzzy sets is rather scarce, which restricts its application a lot. Thus, in [31], Zhang et al. define sixteen kinds of intuitionistic convex fuzzy sets from the point view of cut sets and neighbourhood relations between a fuzzy point and an IF-set. In [28], Xu et al. introduced the concept of quasi-convex IF-set based on convex fuzzy sets and concave
Aggregation functions and convexity
Along this paper we have dealt with the notion of a convex IF-set. One of the properties that make convexity so important is its preservation under intersections. We will study this question for the case of IF-sets. However, instead of the intersection we will consider a wider class of aggregation functions. As the intersection of IF-sets is defined by a triangular norm, which is a particular case of an aggregation operator, our results will be valid also for intersections.
Aggregation functions
Conclusions and further topics
Not only we have introduced a version of convexity for intuitionistic fuzzy sets when the universe is a linear space, but also a more general definition in the general case where the universe is any set. Both concepts have been related and also a cutworthy approach has been taken into account. The particular cases of non-fuzzy and fuzzy sets have also been considered. Moreover the preservation of such convexity under aggregation has also been studied and necessary and sufficient conditions for
Acknowledgements
The second author would express that the research reported on in this contribution has been partially supported by Project MTM2012-37894-C02-02 and Project TIN2013-40765-P, both of the Spanish Ministry of Economy and Competitiveness. Also the third author acknowledges the support of the grant supported by the Campus of International Excellence of the University of Oviedo in the year 2014. We would also like to thank the three anonymous reviewers and the editor for some helpful comments.
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