Bipolar Fuzzy Sets Are Intuitionistic Fuzzy Sets

The extension of fuzzy sets in which membership degrees are intervals in [0, 1], not only real numbers, is called interval-valued fuzzy sets (IVFSs). They include traditional fuzzy sets as particular cases, if looking at real membership degrees as at degenerated intervals. It is useful when real membership degrees are hard or even impossible to be determined, e.g. when experts differ in their opinions on membership degrees: the answer is correct to a degree of 0.5 up to 0.7. The paper presents applications of IVFSs and their cylindric extensions to representation of linguistic data. We introduce cylindric extensions of IVFSs and we formalize unions and intersections of IVFSs in different universes of discourse via cylindric extensions, to represent properties of objects described by attributes having their values in different domains, e.g. tall in cm and rich in €. We apply the described methods to data linguistic summarization in the sense of Yager (1982) but based on IVFSs. In particular, we propose to re-express some algorithms using cylindric extensions to make computations simplified and more human-oriented.

The extensions of ordinary fuzzy sets such as intuitionistic fuzzy sets (IFS), Pythagorean fuzzy sets (PFS), and neutrosophic sets (NS), whose membership functions are based on three dimensions, aim at collecting experts’ judgments more informatively and explicitly. In the literature, generalized three-dimensional spherical fuzzy sets have been developed by Kutlu Gündoğdu and Kahraman [1], including their arithmetic operations, aggregation operators, and defuzzfication operations. Spherical Fuzzy Sets (SFS) are a new extension of Intuitionistic, Pythagorean and Neutrosophic Fuzzy sets. In this paper, our aim is to employ SF-VIKOR method to waste management problems. We handle a waste disposal site selection problem with five alternatives and four criteria in order to demonstrate the performance of the proposed SF-VIKOR method.

<abstract><p>The study on N-soft sets (NSSs) has been significantly developed recently. Hybrid models such as fuzzy N-soft sets, Intuitionistic fuzzy N-soft sets, and hesitant fuzzy N-soft sets were introduced to combine fuzzy sets, intuitionistic fuzzy sets and hesitant fuzzy sets with NSSs. Related to the hybrid models, it was also constructed some complements, operations and related properties. This article aims to construct a new hybrid model called hesitant intuitionistic fuzzy N-soft sets (HIFNSSs) to combine intuitionistic fuzzy N-soft sets and hesitant fuzzy N-soft sets. Moreover, we generalise HIFNSSs to generalized hesitant intuitionistic fuzzy N-soft sets (GHIFNSSs) as a hybrid model between generalized hesitant intuitionistic fuzzy sets and N-soft sets. It was also defined some complements of GHIFNSSs, intersection and union operations between GHIFNSSs, and proved that the operations between some particular complements hold De Morgan Law. In applying a GHIFNSS, we provide an algorithm for decision-making problems and its numerical illustration.</p></abstract>

The concept of bipolar fuzzy sets is an extension of fuzzy sets. Our aim is to define bipolar fuzzy rough topology by means of the lower and upper approximations on bipolar fuzzy sets, bipolar fuzzy rough image, bipolar fuzzy rough inverse image, bipolar fuzzy rough subspace and bipolar fuzzy rough continuous functions between these topologies. We construct some functions which are bipolar fuzzy rough continuous and further prove pasting lemma for bipolar fuzzy rough continuous mappings.

As the extension of fuzzy set, intuitionistic fuzzy set, Pythagorean fuzzy set, and picture fuzzy set, the spherical fuzzy set is characterized by three functions expressing the positive-membership degree, the neutral-membership degree and the negative-membership degree which the sum squares of them is equal or less than 1. In this work, we shall present some novel Dice similarity measures of spherical fuzzy sets and the generalized Dice similarity measures of spherical fuzzy sets and indicates that the Dice similarity measures and asymmetric measures (projection measures) are the special cases of the generalized Dice similarity measures in some parameter values. Then, we propose the generalized Dice similarity measures-based multiple attribute group decision making models with spherical fuzzy information. Then, we apply the generalized Dice similarity measures between spherical fuzzy sets to multiple attribute group decision making. Finally, an illustrative example is given to demonstrate the efficiency of the similarity measures for selecting the desirable ERP system.

A. q-Rung orthopair fuzzy sets (qROFS) constitute a robust extension of Intuitionistic Fuzzy Sets (IFS) and Pythagorean Fuzzy Sets (PyFS).They provide an advanced framework for handling complex information in an uncertain environment.The REGIME method is an MCDM approach used to provide a definitive ranking of alternatives.In the literature, a few extensions have integrated REGIME with IFS and PyFS.However, not only do very few of these extensions address group decision-making problems, but they are also limited when dealing with q-ROFS type data.In this article, we propose an extension of the REGIME method to group decision-making in a qROFS environment.Our methodology proceeds in three phases: aggregation of fuzzy decision matrices, formulation of a non-linear programming model based on qROFS entropy for the objective weighting of criteria, and adaptation of REGIME to evaluate the options.This new methodology extends, for the first time, the REGIME method to a qROFS environment.The proposed method is applied to a numerical example on supplier selection to demonstrate its effectiveness.

The notion of fuzzy sets initiated to overcome the uncertainty of an object. Fuzzy topological space, in- tuitionistic fuzzy sets in topological structure space, vagueness in topological structure space, rough sets in topological space, theory of hesitancy and neutrosophic topological space, etc. are the extension of fuzzy sets. Soft set is a family of parameters which is also a set. Fuzzy soft topological space, intuitionistic fuzzy soft and neutrosophic soft topological space are obtained by incorporating soft sets with various topological structures. This motivates to write a review and study on various soft set concepts. This paper shows the detailed review of soft topological spaces in various sets like fuzzy, Intuitionistic fuzzy set and neutrosophy. Eventually, we compared some of the existing tools in the literature for easy understanding and exhibited their advantages and limitations.

Atanassov (Intuitionistic fuzzy set, Fuzzy Sets and Systems 20 (1986) 87–96) proposed the intuitionistic fuzzy set (A-IFS) that is a generalization of Zadeh's fuzzy set. The essential elements of an A-IFS are intuitionistic fuzzy values (IFVs), each of which is depicted by a membership degree and a non-membership degree. The existing literature only gives the explanation of the meaning and the aggregation operators of IFVs, which is very hard and non-visualized for us to comprehend. To overcome the above shortcomings, this paper puts forward the diagram method to analyze the structures of the basic aggregation operations of IFVs. In addition, this paper utilizes the diagram method which is similar to joint probability distribution to describe the components of aggregation operations of IFVs, and each IFV depicts a 3-dimensional vector which is regarded as the marginal distributions of the joint probability distribution.

<abstract><p>Soft set has limitation for the consideration of disjoint attribute-valued sets corresponding to distinct attributes whereas hypersoft set, an extension of soft set, fully addresses this scarcity by replacing the approximate function of soft sets with multi-argument approximate function. Some structures (i.e., possibility fuzzy soft set, possibility intuitionistic fuzzy soft set) exist in literature in which a possibility of each element in the universe is attached with the parameterization of fuzzy sets and intuitionistic fuzzy sets while defining fuzzy soft set and intuitionistic fuzzy soft set respectively. This study aims to generalize the existing structure (i.e., possibility intuitionistic fuzzy soft set) and to make it adequate for multi-argument approximate function. Therefore, firstly, the elementary notion of possibility intuitionistic fuzzy hypersoft set is developed and some of its elementary properties i.e., subset, null set, absolute set and complement, are discussed with numerical examples. Secondly, its set-theoretic operations i.e., union, intersection, AND, OR and relevant laws are investigated with the help of numerical examples, matrix and graphical representations. Moreover, algorithms based on AND/OR operations are proposed and are elaborated with illustrative examples. Lastly, similarity measure between two possibility intuitionistic fuzzy hypersoft sets is characterized with the help of example. This concept of similarity measure is successfully applied in decision making to judge the eligibility of a candidate for an appropriate job. The proposed similarity formulation is compared with the relevant existing models and validity of the generalization of the proposed structure is discussed.</p></abstract>

The aim of this article is to study the cut sets of interval valued intuitionistic fuzzy sets. By considering the cut sets of an interval-valued intuitionistic fuzzy set as five valued fuzzy sets, the definitions of four cut sets on interval-valued intuitionistic fuzzy sets are given, which are the generalizations of cuts sets on Zadeh fuzzy sets and the intuitionistic fuzzy sets. It is pointed that the cuts sets of interval-valued intuitionistic fuzzy sets have the same properties as that of the intuitionistic fuzzy sets and Zadeh fuzzy sets. These works provide a powerful tool to study interval-valued intuitionistic fuzzy sets.

In this paper we introduce the concentration and dilation operators on interval valued intuitionistic fuzzy soft sets of root type which is a generalization of fuzzy set, intuitionistic fuzzy set, interval valued intuitionistic fuzzy set and intuitionistic fuzzy set of root type and establish some properties of these operators. We define Hamming distance between two interval valued intuitionistic fuzzy soft sets of root type and it is shown that it is a metric. A similarity measure based on this Hamming distance is defined and some properties are established. We also develop a decision making method based on the similarity measure of Hamming distance between interval valued intuitionistic fuzzy soft sets of root type. We develop an algorithm for the decision making problem and illustrate its working by means of examples.

To deal with vagueness, falsity, attributive values and inconsistency, we introduce the cubic intuitionistic fuzzy soft set (CIFS-set) which is the extension of the cubic intuitionistic fuzzy set and propose distance measure, Hamming distance, Euclidean distance and separation measures of CIFS-set. Moreover, we present aggregate operator (P-union, R-intersection) of CIFS-sets. The proposed CIFS-set is more reliable, efficient and accurate. In future MCDM and MCGDM techniques can be propoed to deal with real life issues, and also this CIFS-set can be extended for its hybrids.

Design for six sigma (DFSS) is a structured approach for designing new products, processes and/or services or redesigning the existing ones. It builds quality into the product during the development and ensures the customer expectations are understood, met, or exceed. In DFSS, variation is aimed to be minimized by applying the techniques of the six sigma approach. Some of these techniques are process control and process capability analysis. Vagueness and impreciseness in these analyses such as the statistical distribution of the considered process can be handled by the fuzzy set theory. Recently, ordinary fuzzy sets have been extended to several new types of fuzzy sets such as Pythagorean fuzzy sets, spherical fuzzy sets, picture fuzzy sets, and intuitionistic fuzzy sets. In this paper, we propose a new extension of intuitionistic fuzzy sets, which is called Penthagorean fuzzy sets and employ it in the process capability analysis of a production process. Penthagorean fuzzy process capability indices are developed and applied to an Autonomous Underwater Vehicle (AUV) design example.

This paper proposes a novel approach to analyze the relationship between interval valued fuzzy sets (IVFS) and Atanassov’s intuitionistic fuzzy sets (AIFS), based on the recently introduced notion of paired structure. It is suggested that the different semantics of IVFS and AIFS, with respect to their particular purposes as extensions of fuzzy sets, can be formally recovered and separated in the framework of paired structures, pointing out that both models actually provide a quite different representational performance.

Intuitionistic fuzzy integrals based on Archimedean t-conorms and t-norms