Iranian journal of fuzzy systems
Volume:13 Issue: 1, Feb 2016

The intuitionistic linguistic set (ILS) is an extension of linguisitc variable. To overcome the drawback of using single real number to represent membership degree and non-membership degree for ILS, the concept of interval-valued intuitionistic linguistic set (IVILS) is introduced through representing the membership degree and non-membership degree with intervals for ILS in this paper. The operation law, score function, accuracy function, and certainty function for interval-valued intuitionistic linguistic varibales (IVILVs) are defined. Hereby a lexicographic method is proposed to rank the IVILVs. Then, three kinds of interval-valued intuitionistic linguistic arithmetic average operators are defined, including the interval-valued intuitionistic linguistic weighted arithmetic average (IVILWAA) operator, interval-valued intuitionistic linguistic ordered weighted arithmetic (IVILOWA) operator, and interval-valued intuitionistic linguistic hybrid arithmetic (IVILHA) operator, and their desirable properties are also discussed. Based on the IVILWAA and IVILHA operators, two methods are proposed for solving multi-attribute group decision making problems with IVILVs. Finally, an investment selection example is illustrated to demonstrate the applicability and validity of the methods proposed in this paper.

In recent times, intuitionistic fuzzy sets introduced by Atanassov has been one of the most powerful and flexible approaches for dealing with complex and uncertain situations of real world. In particular, the concept of divergence between intuitionistic fuzzy sets is important since it has applications in various areas such as image segmentation, decision making, medical diagnosis, pattern recognition and many more. The aim of this paper is to introduce a new divergence measure for Atanassov's intuitionistic fuzzy sets (textit{AIFS)}. The properties of the proposed divergence measure have been studied and the findings are applied in medical diagnosis of some diseases with a common set of symptoms.

Multiple attributes group decision making (MAGDM) is regarded as the process of determining the best feasible solution by a group of experts or decision makers according to the attributes that represent different effects. In assessing the performance of each alternative with respect to each attribute and the relative importance of the selected attributes, quantitative/qualitative evaluations are often required to handle uncertainty, imprecise and inadequate information, which are well suited to represent with fuzzy values. This paper develops a VIKOR method based on intuitionistic fuzzy sets with multi-judges and multi-attributes in real-life situations. Intuitionistic fuzzy weighted averaging (IFWA) operator is used to aggregate individual judgments of experts to rate the importance of attributes and alternatives. Then, an intuitionistic ranking index is introduced to obtain a compromise solution to solve MAGDM problems. For application and validation, this paper presents two application examples and solves the practical portfolio selection and material handling selection problems to verify the proposed method. Finally, the intuitionistic fuzzy VIKOR method is compared with the existing intuitionistic fuzzy MAGDM method for two application examples, and their computational results are discussed.

In this paper fuzzy piecewise cubic interpolation is constructed for fuzzy data based on B-spline basis functions. We add two new additional conditions which guarantee uniqueness of fuzzy B-spline interpolation.Other conditions are imposed on the interpolation data to guarantee that the interpolation function to be a well-defined fuzzy function. Finally some examples are given to illustrate the proposed method.

In this paper, it is shown that the category of $L$-ordered fuzzifying convergence spaces contains the category of pretopological $L$-ordered fuzzifying convergence spaces as a bireflective subcategory and the latter contains the category of topological $L$-ordered fuzzifying convergence spaces as a bireflective subcategory. Also, it is proved that the category of $L$-ordered fuzzifying convergence spaces can be embedded in the category of stratified $L$-ordered convergence spaces as a coreflective subcategory.

This paper focuses on the relationships between stratified $L$-conver-gence spaces, stratified strong $L$-convergence spaces and stratified levelwise $L$-convergence spaces. It has been known that: (1) a stratified $L$-convergence space is precisely a left-continuous stratified levelwise $L$-convergence space; and (2) a stratified strong $L$-convergence space is naturally a stratified $L$-convergence space, but the converse is not true generally. In this paper, a strong left-continuity condition for stratified levelwise $L$-convergence space is given. It is proved that a stratified strong $L$-convergence space is precisely a strongly left-continuous stratified levelwise $L$-convergence space. Then a sufficient and necessary condition for a stratified $L$-convergence space to be a stratified strong $L$-convergence space is presented.

In this paper, the connection between Menger probabilistic norms and H"{o}hle probabilistic norms is discussed. In addition, the correspondence between probabilistic norms and Wu-Fang fuzzy (semi-) norms is established. It is shown that a probabilistic norm (with triangular norm $min$) can generate a Wu-Fang fuzzy semi-norm and conversely, a Wu-Fang fuzzy norm can generate a probabilistic norm.

The $L$-fuzzy approximation operator associated with an $L$-fuzzy approximation space $(X,R)$ turns out to be a saturated $L$-fuzzy closure (interior) operator on a set $X$ precisely when the relation $R$ is reflexive and transitive. We investigate the relations between $L$-fuzzy approximation spaces and $L$-(fuzzy) topological spaces.

The aim of this paper is to introduce the notion of commutative pseudo BE-algebras and investigate their properties. We generalize some results proved by A. Walendziak for the case of commutative BE algebras. We prove that the class of commutative pseudo BE-algebras is equivalent to the class of commutative pseudo BCK-algebras. Based on this result, all results holding for commutative pseudo BCK-algebras also hold for commutative pseudo BE-algebras. For example, any finite commutative pseudo BE-algebra is a BE-algebra, and any commutative pseudo BE-algebra is a join-semilattice. Moreover, if a commutative pseudo BE-algebra is a meet-semilattice, then it is a distributive lattice. We define the pointed pseudo-BE algebras, and introduce and study the relative negations on pointed pseudo BE-algebras. Based on the relative negations we construct two closure operators on a pseudo BE algebra. We also define relative involutive pseudo BE-algebras, we investigate their properties and prove equivalent conditions for a relative involutive pseudo BE-algebra. We define the relative Glivenko property for a relative good pseudo BE-algebra and show that any relative involutive pseudo BE-algebra has the relative Glivenko property.

At present, the filter theory of $BL$textit{-}algebras has been widely studied, and some important results have been published (see for example cite{4}, cite{5}, cite{xi}, cite{6}, cite{7}). In other works such as cite{BP}, cite{vii}, cite{xiii}, cite{xvi} a study of a filter theory in the more general setting of residuated lattices is done, generalizing that for $BL$textit{-}algebras. Note that filters are also characterized by various types of fuzzy sets. Most of such characterizations is trivial but some are nontrivial, for example characterizations obtained in cite{xm}. Both situation have revealed a rich range of classes of filters: Boolean, implicative, Heyting, positive implicative, fantastic (or MV-filter), etc. In this paper we work in the general cases of residuated lattices and put in evidence new types of filters in a residuated lattice (in the spirit of cite {mvl}): semi-G-filterstextit{,}Stonean filters, divisible filters, BL-filters and regular filters.