Intuitionistic fuzzy integrals based on Archimedean t-conorms and t-norms
Introduction
Since the fuzzy set theory was proposed by Zadeh [32] in 1965, the concept of “fuzzy” and some traditional disciplines have been combined into many new research fields, some of which have received great attention [5], [17], [19], [30]. Because of the complexity of data and the ambiguity of the human mind, it is difficult to use a membership function to characterize the accurate membership degrees in the classical fuzzy set theory. After analyzing this shortcoming of the fuzzy set, Atanassov [1] extended it through introducing two functions, i.e., the membership function and the non-membership function, and defined intuitionistic fuzzy set (IFS). In the past decades, the IFS theory has penetrated into many fields of modern life, such as clustering analysis [27], decision making [2], [8], [9], [15], [18], [29], medical diagnosis [10], [20], pattern recognition [10], [16], [23], and so on. In addition, Yu and Shi [31] researched the development of the IFS theory.
Moreover, lots of literature [3], [4], [26] has investigated the aggregation techniques in intuitionistic fuzzy environments, which can fuse multiple data in various practical situations. It shows an advantage of aggregation techniques and a reason why people have studied much about them. The majority of previous works only deals with a limited number of IFNs and discrete intuitionistic fuzzy information. Based on Archimedean t-conorm and t-norm, Xia et al. extended [25] some common aggregation techniques for discrete IFNs into more general forms. However, in some practical applications, we not only need to deal with the discrete intuitionistic fuzzy data, but also solve plenty of problems related to the continuous intuitionistic fuzzy information, such as the question proposed in Ref. [14] about how to aggregate all IFNs that the Beijing citizens give as assessments for Beijing. Hence, the reason to study these integrals on IFNs is similar to the ones in the probability theory and the mathematical statistics where we study not only the discrete random variables, but also the continuous random variables. Recently, Lei and Xu proposed [14] a special kind of intuitionistic fuzzy integral in order to aggregate continuous intuitionistic fuzzy information, which means that each point in a two-dimensional plane is an IFN we want to aggregate. It is only a preliminary attempt of the investigation for intuitionistic fuzzy integrals. But there is still a lot of work to be done for establishing intuitionistic fuzzy integral theory and using it in practical applications. In this paper, we investigate the intuitionistic fuzzy integral techniques in much more general forms by utilizing Archimedean t-conorms and t-norms.
The remainder of this paper is set out as follows: Section 2 introduces some basic knowledge on the IFS theory and Archimedean t-conorms and t-norms. Section 3 defines the concepts of additive definite integral and multiplicative definite integral based on Archimedean t-conorm and t-norm, and investigates their specific mathematical expressions. Section 4 analyzes the properties of these definite integrals for intuitionistic fuzzy information. In Section 5, we apply them to derive the general aggregation techniques dealing with continuous intuitionistic fuzzy data, and also make a detailed comparison with traditional aggregation operators for the discrete IFNs. Finally, the paper ends up with some concluding remarks in Section 6.
Section snippets
Preliminaries
First, we introduce some basic notions related to IFSs which were defined by Atanassov [1] as follows:
Definition 2.1 [1]. Let X be a fixed set. An IFS is expressed as:
where each element is characterized by a membership function μA(x) and a non-membership function νA(x), with two conditions 0 ≤ μA(x), νA(x) ≤ 1 and , for any x ∈ X. Moreover, is called a hesitancy function. When (∀x ∈ X), the IFS reduces to a traditional fuzzy set [32].
Definite integrals of IFNs based on Archimedean t-conorm and t-norm
In this section, we derive the expressions for definite integrals of IFNs based on Archimedean t-conorm and t-norm:
For convenience, we first denote the two-dimensional plane as IFNS which consists of all IFNs (shown as the triangular area in Fig. 1), that is
Definition 3.1 [14]. A region D of IFNs is a subset included in IFNS.
A region D of IFNs as defined in Definition 3.1 is shown in the following Fig. 2:
The properties of integrals based on Archimedean t-conorm and t-norm
In this section, we investigate in detail the properties of the additive definite integral (1) and the multiplicative definite integral (2).
Integral aggregation operators based on Archimedean t-conorm and t-norm
Before defining the integral aggregation operators, we first give the concept of weight density function [14]:
Definition 5.1 [14]. Let D be a region of IFNs. A weight density function (WDF) over D is any non-negative function satisfying , where δ denotes the Lebesgue measure.
If f(μ, ν) in the integral is a WDF, then we call the additive definite integral in Theorem 3.1 an Archimedean t-conorm and t-norm based intuitionistic fuzzy additive integral (ATS–IFAI) operator.
Theorem 5.1 Let
Concluding remarks
In this paper, we have defined the additive definite integral and the multiplicative definite integral of IFNs based on Archimedean t-conorm and t-norm. After introducing their specific mathematical expressions, we have discussed their basic properties, such as finite additivity and linearity. In addition, we have proved the relationships among our integrals of IFNs and those in Ref. [14], and shown that the results in Ref. [14] are only special cases of the results in this paper. Moreover, we
Acknowledgements
The work is supported by the National Natural Science Foundation of China (no. 61273209), the Central University Basic Scientific Research Business Expenses Project (no. skgt201501), and the Spanish Government under Project TIN2013-40765-P.
References (33)
Intuitionistic fuzzy set
Fuzzy Sets Syst.
(1986)- et al.
On averaging operators for Atanassov's intuitionistic fuzzy sets
Inf. Sci.
(2011) - et al.
Generation of linear orders for intervals by means of aggregation functions
Fuzzy Sets Syst.
(2013) - et al.
Hesitant fuzzy ELECTRE II approach: a new way to handle multi-criteria decision making problems
Inf. Sci.
(2015) - et al.
Group decision making based on incomplete intuitionistic multiplicative preference relations
Inf. Sci.
(2015) - et al.
Intuitionistic fuzzy set vs. fuzzy set application in medical pattern recognition
Artif. Intell. Med.
(2009) - et al.
An overview of fuzzy research with bibliometric indicators
Appl. Soft Comput.
(2015) - et al.
An alternative to fuzzy methods in decision-making problems
Expert Syst. Appl.
(2012) - et al.
Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm
Knowl. Based Syst.
(2012) - et al.
Researching the development of Atanassov intuitionistic fuzzy set: using a citation network analysis
Appl. Soft Comput.
(2015)