On averaging operators for Atanassov’s intuitionistic fuzzy sets
Introduction
Since the introduction of fuzzy sets by Zadeh [39] many attempts have been made to generalize the notion of fuzzy sets. Among others, Zadeh introduced the idea of type-2 fuzzy sets and interval valued fuzzy sets [40], see also [18], [41], [42]. Later in [2], Atanassov introduced the idea of intuitionistic fuzzy sets (AIFS). Recently, several authors [23], [25], [26], [27], [28], [29], [30] have used AIFS in different applications. In [3], [8], [16] authors advanced the theory of operators and relations for intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets. Bustince and Burillo [8] and Deschrijver and Kerre [11] made theoretical development relating to composition of intuitionistic fuzzy relations.
In many decision making applications, it becomes necessary to aggregate several fuzzy sets, particularly when preference is expressed by fuzzy sets [6], [7], [22], [37]. Weighted means and the Ordered Weighted Averaging functions (OWA) [37] have been applied to this problem, along with triangular norms, conorms and uninorms. Various aggregation functions are discussed in detail in [7], [19], [20], [32].
In a recent series of papers [31], [33], [34], [35], [36], [38], [43] the authors defined some averaging aggregation functions for Atanassov’s intuitionistic fuzzy sets, including weighted means, OWA and Choquet integrals. Their definitions are based on the operations of addition and multiplication for AIFS [3], [4], which involve the product t-norm and its dual t-conorm. A problem with their definitions is that they do not lead to the standard aggregation operations on fuzzy sets in the limiting case (see Example 2).
On the other hand, as noticed in [5] and later developed in [8], [10], [14], the AIFS are mathematically equivalent to the interval-valued fuzzy sets (IVFS). There exist several papers that relate other extensions of fuzzy set theory to AIFS [9], [17], [24]. It is straightforward to define aggregation functions for IVFS, see, e.g. [38], by applying a fixed aggregation function to the ends of the membership interval, the membership and (transformed) non-membership degrees. Such functions are called representable in [15]. This approach is fully consistent with the limiting case of the ordinary fuzzy sets, but there are various non-representable AIFS aggregation functions, such as those presented in [12], [13], [15], and those discussed in this paper.
In this paper we develop an approach to extending aggregation operators to AIFS and IVFS, by using additive generators of the t-norm and t-conorm in the arithmetical operations for AIFS. We provide simple tools to construct both representable and non-representable extensions. We establish several interesting properties of the operators constructed by using Łukasiewicz t-norm and t-conorm in the operations for AIFS. Our approach will eliminate the need for complex and explicit constructions in [34], [35], [36], [43].
The structure of this paper is as follows. We review operations on AIFS and IVFS in Section 2. In Section 3 we present several types of averaging aggregation functions on AIFS based on arithmetical operations on AIFS. In Section 4 we relate the mentioned aggregation functions to those defined for IVFS, and present several more general approaches to aggregation of AIFS. We will show that the use of Łukasiewicz t-norm and t-conorm in the definition of addition of AIFS guarantees consistency of aggregation of AIFS with aggregation of ordinary fuzzy sets. This section is then followed by conclusions.
Section snippets
Operations on AIFS and IVFS
We review several relevant concepts and highlight the correspondence between the notions in AIFS and IVFS [2]. Definition 1 An AIFS on X is defined as , where and are the degrees of membership and nonmembership of x in , which satisfy and . Definition 2 An IVFS on X is defined as , where and are the lower and upper ends of the membership interval, and satisfy .
Obviously an ordinary fuzzy set can be
Aggregation functions on the set of AIFV
The following operations are defined for AIFV [4].
From these formulas one obtains the following equations [16]for any n = 1, 2, … , which can then be extended for positive real n.
When an aggregation function requires the sort operation (in the OWA function and the discrete Choquet integral), one needs to define a total order on AIFVs. The following total order on AIFVs was used in [31], [35], [34], [36].
Alternative definitions of aggregation functions on AIFV
Let us write the operation A + B on AIFV as followswhere T = TP is the product t-norm and S = SP is its dual t-conorm (called probabilistic sum), defined by SP(x, y) = 1 − TP(1 − x, 1 − y). Here we can use any pair of dual t-norm and t-conorm. We first concentrate on continuous Archimedean t-norms and t-conorms, and in particular on the product t-norm, because it was used in many existing definitions. An Archimedean t-norm is strict if it is continuous and strictly increasing.
It is well
Conclusion
We have examined various definitions of aggregation operators for AIFS, which have appeared recently in the literature and are based on the operation of addition of AIFV. We have found that in all such cases, the respective expressions can be written with the help of an additive generator of the t-norm used in the operation of addition. As a consequence, most properties of aggregation operators for AIFS defined in this way follow automatically.
We looked at generalizations of averaging
Acknowledgement
H. Bustince wishes to acknowledge partial support through the grant TIN2010-15055 from the Government of Spain.
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