Elsevier

Information Sciences

Volume 181, Issue 6, 15 March 2011, Pages 1116-1124
Information Sciences

On averaging operators for Atanassov’s intuitionistic fuzzy sets

https://doi.org/10.1016/j.ins.2010.11.024Get rights and content

Abstract

Atanassov’s intuitionistic fuzzy set (AIFS) is a generalization of a fuzzy set. There are various averaging operators defined for AIFSs. These operators are not consistent with the limiting case of ordinary fuzzy sets, which is undesirable. We show how such averaging operators can be represented by using additive generators of the product triangular norm, which simplifies and extends the existing constructions. We provide two generalizations of the existing methods for other averaging operators. We relate operations on AIFS with operations on interval-valued fuzzy sets. Finally, we propose a new construction method based on the Łukasiewicz triangular norm, which is consistent with operations on ordinary fuzzy sets, and therefore is a true generalization of such operations.

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 A on X is defined as A={x,μA(x),νA(x)|xX}, where μA(x) and νA(x) are the degrees of membership and nonmembership of x in A, which satisfy μA(x),νA(x)[0,1] and 0μA(x)+νA(x)1.

Definition 2

An IVFS A on X is defined as A={x,[lA(x),rA(x)]|xX}, where lA(x) and rA(x) are the lower and upper ends of the membership interval, and satisfy 0lA(x)rA(x)1.

Obviously an ordinary fuzzy set can be

Aggregation functions on the set of AIFV

The following operations are defined for AIFV [4].A+B=μA+μB-μAμB,νAνB,A·B=μAμB,νA+νB-νAνB.

From these formulas one obtains the following equations [16]nA=A++A=1-(1-μA)n,νAn,An=A··A=μAn,1-(1-νA)n,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].A<B

Alternative definitions of aggregation functions on AIFV

Let us write the operation A + B on AIFV as followsA+B=S(μA,μB),T(νA,νB),where 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.

References (43)

  • M. Grabisch et al.

    Aggregation functions: Means

    Information Sciences

    (2011)
  • J. Montero et al.

    On the relevance of some families of fuzzy sets

    Fuzzy Sets and Systems

    (2007)
  • C. Tan et al.

    Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making

    Expert Systems with Applications

    (2010)
  • G. Wei

    Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making

    Applied Soft Computing

    (2010)
  • Z. Xu

    Choquet integrals of weighted intuitionistic fuzzy information

    Information Sciences

    (2010)
  • L.A. Zadeh

    Fuzzy sets

    Information Control

    (1965)
  • L.A. Zadeh

    Is there a need for fuzzy logic?

    Information Sciences

    (2008)
  • L.A. Zadeh

    Toward extended fuzzy logic – A first step

    Fuzzy Sets and Systems

    (2009)
  • J. Aczél

    Lectures on Functional Equations and their Applications

    (1966)
  • K. Atanassov

    More on intuitionistic fuzzy sets

    Fuzzy Sets and Systems

    (1989)
  • G. Beliakov et al.

    Appropriate choice of aggregation operators in fuzzy decision support system

    IEEE Transactions on Fuzzy Systems

    (2001)
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