Interval-valued Fuzzy Set Theory Research Articles

Distributivity of implication operations over t-representable t-norms in interval-valued fuzzy set theory: The case of nilpotent t-norms

In this article we discuss the distributive equation of implications I(x,T1(y,z))=T2(I(x,y),I(x,z)) over t-representable t-norms generated from nilpotent t-norms in interval-valued fuzzy sets theory. As a byproduct result we show all solutions of some functional equation related to this case.

Uninorms which are neither conjunctive nor disjunctive in interval-valued fuzzy set theory

Uninorms are a generalisation of t-norms and t-conorms for which the neutral element is an element of [0,1] which is not necessarily equal to 0 (as for t-norms) or 1 (as for t-conorms). Uninorms on the unit interval are either conjunctive or disjunctive, i.e. they aggregate the pair (0,1) to either 0 or 1. In real-life applications, this kind of aggregation may be counter-intuitive. Interval-valued fuzzy set theory and Atanassov’s intuitionistic fuzzy set theory are extensions of fuzzy set theory which allows to model uncertainty about the membership degrees. In these theories there exist uninorms which are neither conjunctive nor disjunctive. In this paper we study such uninorms more deeply and we investigate the structure of these uninorms. We also give several examples of uninorms which are neither conjunctive nor disjunctive.

Complete solution sets of inf-→ interval-valued fuzzy relation equations

Fuzzy relation equations play an important role in fuzzy set theory. Interval-valued fuzzy set theory is an extension of fuzzy theory in which a closed subinterval of the unit interval is assigned with membership degree. Therefore, it is very significant to study interval-valued fuzzy relation equations from both the theoretical and practical viewpoints. In this paper, the solution sets of interval-valued fuzzy relation equations with inf-→ composition is investigated, where → is interval-valued R-, S- or QL-implication. Necessary and sufficient conditions such that there exist solutions for these equations are first shown. Some sufficient conditions for existence of maximal solutions for these equations are represented, and then it is shown that the complete solution sets of inf-→ interval-valued fuzzy relation equations can be determined by their maximal solutions. Finally, the solution sets of linear interval-valued fuzzy relation equations are described by a method similar to that in linear algebra.

On the representation of cardinalities of interval-valued fuzzy sets: The valuation property

In the previous work, we developed an axiomatic theory of the scalar cardinality of interval-valued fuzzy sets following Wygralak's axiomatic theory of the scalar cardinality of fuzzy sets. Cardinality was defined as a mapping from the set of interval-valued fuzzy sets with finite support to the set of closed subintervals of [0,+∞). We showed that the scalar cardinality of each interval-valued fuzzy set can be characterized using an appropriate mapping called a cardinality pattern. Moreover, we found some basic conditions under which the valuation property, the subadditivity property, the complementarity rule and the Cartesian product rule are satisfied using different cardinality patterns, t-norms, t-conorms and negations on the lattice LI (the underlying lattice of interval-valued fuzzy set theory). This paper is the first in a series that further investigates the proposed theory, providing a description of cardinality patterns, t-norms, t-conorms and negations satisfying the properties mentioned above. This paper focuses on the valuation property.

Triangular norms which are meet-morphisms in interval-valued fuzzy set theory

In this paper we study t-norms on the lattice of closed subintervals of the unit interval. Unlike for t-norms on a product lattice for which there exists a straightforward characterization of t-norms which are join-morphisms, respectively meet-morphisms, the situation is more complicated for t-norms in interval-valued fuzzy set theory. In previous papers several characterizations were given of t-norms in interval-valued fuzzy set theory which are join-morphisms and which satisfy additional properties, but little attention has been paid to meet-morphisms. Therefore, in this paper, we focus on t-norms which are meet-morphisms. We consider a general class of t-norms and investigate under which conditions t-norms belonging to this class are meet-morphisms. We also characterize the t-norms which are both a join- and a meet-morphism and which satisfy an additional border condition.

A genetic tuning to improve the performance of Fuzzy Rule-Based Classification Systems with Interval-Valued Fuzzy Sets: Degree of ignorance and lateral position

Fuzzy Rule-Based Systems are appropriate tools to deal with classification problems due to their good properties. However, they can suffer a lack of system accuracy as a result of the uncertainty inherent in the definition of the membership functions and the limitation of the homogeneous distribution of the linguistic labels. The aim of the paper is to improve the performance of Fuzzy Rule-Based Classification Systems by means of the Theory of Interval-Valued Fuzzy Sets and a post-processing genetic tuning step. In order to build the Interval-Valued Fuzzy Sets we define a new function called weak ignorance for modeling the uncertainty associated with the definition of the membership functions. Next, we adapt the fuzzy partitions to the problem in an optimal way through a cooperative evolutionary tuning in which we handle both the degree of ignorance and the lateral position (based on the 2-tuples fuzzy linguistic representation) of the linguistic labels. The experimental study is carried out over a large collection of data-sets and it is supported by a statistical analysis. Our results show empirically that the use of our methodology outperforms the initial Fuzzy Rule-Based Classification System. The application of our cooperative tuning enhances the results provided by the use of the isolated tuning approaches and also improves the behavior of the genetic tuning based on the 3-tuples fuzzy linguistic representation.

New roughness measures of the interval-valued fuzzy sets

In the paper ( Gong, Sun, & Chen, 2008; Han, Chen, & Chen, 2008), by combining the rough set theory with the interval-valued fuzzy set theory, Gong and Han proposed the concept of the interval-valued rough fuzzy sets and studied a parameters-related roughness measure of the interval-valued fuzzy sets in an approximation space, respectively. Since the measure inherited from Banerjee’s about the fuzzy sets in Banerjee and Pal (1996), it inevitably holds some similar undesirable properties. In this paper, we first remark some defects about the above mentioned measure, and then give a new roughness measure–rough entropy of the interval-valued fuzzy sets in an approximation space by using the information entropy theory, and demonstrate the difference between the two measures by a simple illustration example. Furthermore, in respect that the two measures mentioned above both strongly depend on parameters, we propose a parameter-free roughness measure of the interval-valued fuzzy sets in an approximation space based on the notions of the average mass assignment and average membership function of an interval-valued fuzzy set, adhering to the idea of parameter-free roughness measure of the fuzzy sets proposed in Huynh and Nakamori (2005). Some related properties about the new measure are discussed. Finally, we discuss how use the new measure to evaluate the rough approximation quality of classification in an interval-valued fuzzy decision information system.

Interval-valued intuitionistic fuzzy soft sets and their properties

Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. However, it has been pointed out that classical soft sets are not appropriate to deal with imprecise and fuzzy parameters. In this paper, the notion of the interval-valued intuitionistic fuzzy soft set theory is proposed. Our interval-valued intuitionistic fuzzy soft set theory is a combination of an interval-valued intuitionistic fuzzy set theory and a soft set theory. In other words, our interval-valued intuitionistic fuzzy soft set theory is an interval-valued fuzzy extension of the intuitionistic fuzzy soft set theory or an intuitionistic fuzzy extension of the interval-valued fuzzy soft set theory. The complement, “and”, “or”, union, intersection, necessity and possibility operations are defined on the interval-valued intuitionistic fuzzy soft sets. The basic properties of the interval-valued intuitionistic fuzzy soft sets are also presented and discussed.

Extension of the ELECTRE method based on interval-valued fuzzy sets

Decision-making is the process of finding the best option among the feasible alternatives. In classical multiple criteria decision-making (MCDM) methods, the ratings and the weights of the criteria are known precisely. However, if decision makers cannot reach an agreement on the method of defining linguistic variables based on the fuzzy sets, the interval-valued fuzzy set theory can provide a more accurate modeling. In this paper, the interval-valued fuzzy ELECTRE method is presented aiming at solving MCDM problems in which the weights of criteria are unequal, using interval-valued fuzzy set concepts. For the purpose of proving the validity of the proposed model, we present a numerical example and build a practical maintenance strategy selection problem.

On the Decomposition of Interval-Valued Fuzzy Morphological Operators

Interval-valued fuzzy mathematical morphology is an extension of classical fuzzy mathematical morphology, which is in turn one of the extensions of binary morphology to greyscale morphology. The uncertainty that may exist concerning the grey value of a pixel due to technical limitations or bad recording circumstances, is taken into account by mapping the pixels in the image domain onto an interval to which the pixel's grey value is expected to belong instead of one specific value. Such image representation corresponds to the representation of an interval-valued fuzzy set and thus techniques from interval-valued fuzzy set theory can be applied to extend greyscale mathematical morphology. In this paper, we study the decomposition of the interval-valued fuzzy morphological operators. We investigate in which cases the [? 1,? 2]-cuts of these operators can be written or approximated in terms of the corresponding binary operators. Such conversion into binary operators results in a reduction of the computation time and is further also theoretically interesting since it provides us a link between interval-valued fuzzy and binary morphology.

Extension of VIKOR method based on interval-valued fuzzy sets

Decision making is the process of finding the best option among the feasible alternatives. In classical multiple-criteria decision-making (MCDM) methods, the ratings and the weights of the criteria are known precisely. However, if decision makers cannot reach an agreement on the method of defining linguistic variables based on the fuzzy sets, the interval-valued fuzzy set theory can provide a more accurate modeling. In this paper, the interval-valued fuzzy VIKOR method is presented, aiming at solving MCDM problems in which the weights of criteria are unequal, using interval-valued fuzzy set concepts. For application and verification, this study presents a numerical example and builds a practical maintenance strategy selection problem to verify our proposed method. Moreover, a comparison is made between the interval-valued fuzzy VIKOR and other adapted MCDM interval-valued fuzzy number-based.

On characterization of generalized interval-valued fuzzy rough sets on two universes of discourse

This paper proposes a general study of ( I , T ) -interval-valued fuzzy rough sets on two universes of discourse integrating the rough set theory with the interval-valued fuzzy set theory by constructive and axiomatic approaches. Some primary properties of interval-valued fuzzy logical operators and the construction approaches of interval-valued fuzzy T -similarity relations are first introduced. Determined by an interval-valued fuzzy triangular norm and an interval-valued fuzzy implicator, a pair of lower and upper generalized interval-valued fuzzy rough approximation operators with respect to an arbitrary interval-valued fuzzy relation on two universes of discourse is then defined. Properties of I -lower and T -upper interval-valued fuzzy rough approximation operators are examined based on the properties of interval-valued fuzzy logical operators discussed above. Connections between interval-valued fuzzy relations and interval-valued fuzzy rough approximation operators are also established. Finally, an operator-oriented characterization of interval-valued fuzzy rough sets is proposed, that is, interval-valued fuzzy rough approximation operators are characterized by axioms. Different axiom sets of I -lower and T -upper interval-valued fuzzy set-theoretic operators guarantee the existence of different types of interval-valued fuzzy relations which produce the same operators.

Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory

In this paper we study extensions of the arithmetic operators + , - , · , ÷ to the lattice L I of closed subintervals of the unit interval. Starting from a minimal set of axioms that these operators must fulfill, we investigate which properties they satisfy. We also investigate some classes of t-norms on L I which can be generated using these operators; these classes provide natural extensions of the Łukasiewicz, product, Frank, Schweizer–Sklar and Yager t-norms to L I .

Characterizations of (weakly) Archimedean t-norms in interval-valued fuzzy set theory

In this paper we first give characterizations of the class of continuous t-norms on L I (where L I is the lattice of closed subintervals of the unit interval) which satisfy the residuation principle and which are a natural extension of a t-norm on the unit interval and which satisfy one of the following conditions: the negation generated by their residual implication is involutive; they are (weakly) Archimedean; they are (weakly) nilpotent. We fully characterize the class of continuous t-norms on L I which satisfy the residuation principle, which are a natural extension of a t-norm on the unit interval and which are weakly Archimedean. We construct a separate representation for the t-norms in this class which are weakly nilpotent and for those which are not weakly nilpotent. Finally we give a characterization of the continuous t-norms on L I which satisfy the residuation principle, which are a natural extension of a t-norm on the unit interval and which are strict.

Interval-Valued Fuzzy Soft sets 관한 연구

본 논문은 부정확한 null interval-valued fuzzy soft sets 와 absolute interval-valued fuzzy soft sets 개념의 오류를 지적하였으며 interval-valued fuzzy soft sets를 위해 새롭게 정의된 개념을 소개하며 이를 이용한 기본적인 성질을 조사한다. In this paper, we show that the concepts of null interval-valued fuzzy and absolute interval-valued fuzzy soft sets are not reasonable. Thus we introduce the modified concepts for them and study some properties. Also we introduce an operation for the interval-valued fuzzy soft set theory and study basic properties.

Extension of fuzzy TOPSIS method based on interval-valued fuzzy sets

Decision making is one of the most complex administrative processes in management. In circumstances where the members of the decision making team are uncertain in determining and defining the decision making criteria, fuzzy theory provides a proper tool to encounter with such uncertainties. However, if decision makers cannot reach an agreement on the method of defining linguistic variables based on the fuzzy sets, the interval-valued fuzzy set theory can provide a more accurate modeling. In this paper the interval-valued fuzzy TOPSIS method is presented aiming at solving MCDM problems in which the weights of criteria are unequal, using interval-valued fuzzy sets concepts.

Fuzzy rough set theory for the interval-valued fuzzy information systems

The concept of the rough set was originally proposed by Pawlak as a formal tool for modelling and processing incomplete information in information systems, then in 1990, Dubois and Prade first introduced the rough fuzzy sets and fuzzy rough sets as a fuzzy extension of the rough sets. The aim of this paper is to present a new extension of the rough set theory by means of integrating the classical Pawlak rough set theory with the interval-valued fuzzy set theory, i.e., the interval-valued fuzzy rough set model is presented based on the interval-valued fuzzy information systems which is defined in this paper by a binary interval-valued fuzzy relations R ∈ F ( i ) ( U × U ) on the universe U . Several properties of the rough set model are given, and the relationships of this model and the others rough set models are also examined. Furthermore, we also discuss the knowledge reduction of the classical Pawlak information systems and the interval-valued fuzzy information systems respectively. Finally, the knowledge reduction theorems of the interval-valued fuzzy information systems are built.

Rough set theory for the interval-valued fuzzy information systems

The notion of a rough set was originally proposed by Pawlak [Z. Pawlak, Rough sets, International Journal of Computer and Information Sciences 11 (5) (1982) 341–356]. Later on, Dubois and Prade [D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, International Journal of General System 17 (2–3) (1990) 191–209] introduced rough fuzzy sets and fuzzy rough sets as a generalization of rough sets. This paper deals with an interval-valued fuzzy information system by means of integrating the classical Pawlak rough set theory with the interval-valued fuzzy set theory and discusses the basic rough set theory for the interval-valued fuzzy information systems. In this paper we firstly define the rough approximation of an interval-valued fuzzy set on the universe U in the classical Pawlak approximation space and the generalized approximation space respectively, i.e., the space on which the interval-valued rough fuzzy set model is built. Secondly several interesting properties of the approximation operators are examined, and the interrelationships of the interval-valued rough fuzzy set models in the classical Pawlak approximation space and the generalized approximation space are investigated. Thirdly we discuss the attribute reduction of the interval-valued fuzzy information systems. Finally, the methods of the knowledge discovery for the interval-valued fuzzy information systems are presented with an example.

REPRESENTABILITY IN INTERVAL-VALUED FUZZY SET THEORY

Interval-valued fuzzy set theory is an increasingly popular extension of fuzzy set theory where traditional [0,1]-valued membership degrees are replaced by intervals in [0,1] that approximate the (unknown) membership degrees. To construct suitable graded logical connectives in this extended setting, it is both natural and appropriate to "reuse" ingredients from classical fuzzy set theory. In this paper, we compare different ways of representing operations on interval-valued fuzzy sets by corresponding operations on fuzzy sets, study their intuitive semantics, and relate them to an existing, purely order-theoretical approach. Our approach reveals, amongst others, that subtle differences in the representation method can have a major impact on the properties satisfied by the generated operations, and that contrary to popular perception, interval-valued fuzzy set theory hardly corresponds to a mere twofold application of fuzzy set theory. In this way, by making the mathematical machinery behind the interval-valued fuzzy set model fully transparent, we aim to foster new avenues for its exploitation by offering application developers a much more powerful and elaborate mathematical toolbox than existed before.

Additive and Multiplicative Generators in Interval-Valued Fuzzy Set Theory

In this paper, we introduce the notion of additive and multiplicative generator on LI, which is the underlying lattice of interval-valued fuzzy set theory. A special class of generators on LI is the class of representable generators which can be represented using generators on ([0,1],les). A representation of involutive negators on LI by means of continuous additive generators which is similar to the representation of involutive negators on ([0,1], les) is given. We also investigate additive and multiplicative generators of t-norms on LI