A class of aggregation functions encompassing two-dimensional OWA operators

doi:10.1016/j.ins.2010.01.022

Information Sciences

Volume 180, Issue 10, 15 May 2010, Pages 1977-1989

https://doi.org/10.1016/j.ins.2010.01.022 Get rights and content

Abstract

In this paper we prove that, under suitable conditions, Atanassov’s $K_{α}$ operators, which act on intervals, provide the same numerical results as OWA operators of dimension two. On one hand, this allows us to recover OWA operators from $K_{α}$ operators. On the other hand, by analyzing the properties of Atanassov’s operators, we can generalize them. In this way, we introduce a class of aggregation functions – the generalized Atanassov operators – that, in particular, include two-dimensional OWA operators. We investigate under which conditions these generalized Atanassov operators satisfy some properties usually required for aggregation functions, such as bisymmetry, strictness, monotonicity, etc. We also show that if we apply these aggregation functions to interval-valued fuzzy sets, we obtain an ordered family of fuzzy sets.

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 $K_{α}$ , 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 $K_{α}$ operators, aggregation functions, OWA operators and IVFS are described in Section 2. Section 3 presents the relation between OWA and $K_{α}$ operators. In Section 4 we present a generalization of the $K_{α}$ 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 $K_{α}$ operators. Section 6 concludes the paper.

Section snippets

Preliminary definitions

In fuzzy set theory, a strictly decreasing and continuous function $N : [0, 1] \to [0, 1]$ such that $N (0) = 1, N (1) = 0$ 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 $φ : [0, 1] \to [0, 1]$ that is continuous, strictly increasing and such that $φ (0) = 0$ and $φ (1) = 1$ .

In 1979 Trillas [21] presented the following theorem of characterization of strong negations.

Theorem 1

A function $N : [0, 1] \to [0, 1]$ is a strong negation if and

OWA operators and $K_{α}$ 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 $K : [0, 1] \times L ([0, 1]) \to [0, 1]$ is given by $K = (K_{α})_{α \in [0, 1]}$ , with each operator $K_{α} : L ([0, 1]) \to [0, 1]$ defined as a convex combination of its boundary arguments by $K_{α} (x) = α \cdot \bar{x} + (1 - α) \cdot \underset{̲}{x},$ where for any $x \in L ([0, 1])$ we write $x = [\underset{̲}{x}, \bar{x}]$ .

Clearly the following properties hold.

(i)
$K_{0} (x) = \underset{̲}{x}$ for all $x \in L ([0, 1])$ .
(ii)
$K_{1} (x) = \bar{x}$ for all $x \in L ([0, 1])$ .
(iii)
$K_{α} (x) = K_{α} ([K_{0} (x), K_{1} (x)]) = K_{0} (x) + α (K_{1} (x) - K_{0} (x)) = \underset{̲}{x} + α (\bar{x} - \underset{̲}{x})$ for

Generalized $K_{α}$ operators

Observe that, if we denote by K the system of operators $(K_{α})_{α \in [0, 1]}$ , then K can be regarded as an operator on $[0, 1] \times L ([0, 1])$ 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 $GK : [0, 1] \times L ([0, 1]) \to [0, 1]$ such that, if we denote ${GK}_{α} (x) = GK (α, x)$ , the following properties hold:

(i)
If $\underset{̲}{x} = \bar{x}$ , then ${GK}_{α} (x) = \underset{̲}{x}$ .
(ii)
${GK}_{0} (x) = \underset{̲}{x}, {GK}_{1} (x) = \bar{x}$ for all $x \in L ([0, 1])$ .
(iii)
If $x ⩽_{L} y$ , with $x, y \in L ([0, 1])$ , then ${GK}_{α} (x) ⩽ {GK}_{α} (y)$ .
(iv)
Let $β \in [0, 1]$ . If $α ⩽ β$ ,

Construction of a FS from a IVFS and an operator ${GK}_{α}$

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 ${GK}_{α}$ .

Conclusions and future research

We established a link between $K_{α}$ operators and OWA operators of dimension 2. This relation led to the definition of a class of aggregation functions, the $K_{α}$ operators, in terms of $K_{α}$ 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.

References (27)

K. Atanassov
Intuitionistic fuzzy sets
Fuzzy Sets and Systems
(1986)
P. Burillo et al.
Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets
Fuzzy Sets and Systems
(1996)
H. Bustince
Construction of intuitionistic fuzzy relations with predetermined properties
Fuzzy Sets and Systems
(2000)
T. Calvo et al.
The functional equations of Frank and Alsina for uninorms and nullnorms
Fuzzy Sets and Systems
(2001)
C. Cornelis et al.
Advances and challenges in interval-valued fuzzy logic
Fuzzy Sets and Systems
(2006)
M.B. Gorzalczany
A method of inference in approximate reasoning based on interval-valued fuzzy sets
Fuzzy Sets and Systems
(1987)
R.R. Yager
Families of OWA operators
Fuzzy Sets and Systems
(1993)
L.A. Zadeh
Fuzzy sets
Information Control
(1965)
L.A. Zadeh
The concept of a linguistic variable and its application to approximate reasoning – I
Information Sciences
(1975)
L.A. Zadeh
Is there a need for fuzzy logic?
Information Sciences
(2008)

L.A. Zadeh

Toward a generalized theory of uncertainty (GTU) - an outline

Information Sciences

(2005)

J. Aczél

On mean values

Bulletin of the American Mathematical Society

(1948)

K. Atanassov, Intuitionistic fuzzy sets, VIIth ITKR Session, Deposited in the Central Science and Technology Library of...

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A class of aggregation functions encompassing two-dimensional OWA operators

Abstract

Introduction

Section snippets

Preliminary definitions

OWA operators and Kα operators

Generalized Kα operators

Construction of a FS from a IVFS and an operator GKα

Conclusions and future research

Acknowledgments

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Information Control

Information Sciences

Information Sciences

Information Sciences

On mean values

Bulletin of the American Mathematical Society

OWA operators and $K_{α}$ operators

Generalized $K_{α}$ operators

Construction of a FS from a IVFS and an operator ${GK}_{α}$