On bipolar fuzzy rough continuous functions

In 1983 the Intuitionistic Fuzzy Sets (IFSs) were introduced as an extension of Zadeh's fuzzy sets. Fifteen years later the Bipolar Fuzzy Sets (BFSs) were introduced as a new extension of fuzzy sets. In the article it is proved that each BFS can be transformed injectively or bijectively to an IFSs.

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 concept of q-rung orthopair hesitant fuzzy set represents an advancement and extension of hesitant fuzzy sets, encompassing both fuzzy sets and q-rung orthopair fuzzy sets. q-rung orthopair hesitant fuzzy set characterizes a set of membership and non-membership grades within the interval [0, 1], which enhances its adaptability compared to existing methods. This flexibility proves invaluable in providing more insightful data about various objects. The primary objective of this research is to introduce a decision-making technique in the context of q-RHF using the theory of set pair analysis (SPA). q-RHFS effectively handles ambiguous data by incorporating membership and non-membership grades, while the connection number (CN) based on SPA theory manages the intricacies of uncertainty and certainty structures by relying on "identity", "discrepancy" and "contrary" grades. Building on the relationship between q-RHFS and the connection number of set pair analysis, a comprehensive framework known as q-rung hesitant fuzzy connection number set (qHCNs) is developed. This model not only addresses uncertainty, but also offers valuable insights. Furthermore, this research introduces similarity measures derived from qHCN and examines their advantages through illustrative examples. Additionally, a novel approach to decision modeling utilizing these measures applied to medical diagnosis is also introduced. The application of this established model contributes to an effective approach and demonstrated its soundness and efficiency. In addition, a detailed comparative study is conducted with the existing models and advantages of proposed model. The research concludes with a summary of the authors' findings, highlighting the consistency and effectiveness of their work.

To address the uncertainty and imprecision in a complex system, the concept of the linear Diophantine fuzzy set (LDFS) was introduced, which is a fuzzy set extension that eliminates the constraints of current methodologies and gives the decision-maker complete freedom to select the grades, producing results that are expressive and flexible. A circular intuitionistic fuzzy set (C-IFS) is depicted by a circle with the membership and non-membership values as its center coordinates, addressing the ambiguity regarding membership and non-membership values. In this paper, we propose the notion of a circular linear Diophantine fuzzy set (C-LDFS) as a hybrid structure of both circular intuitionistic fuzzy sets and linear Diophantine fuzzy sets. In addition, some fundamental set operations on C-LDFSs are presented. Furthermore, a similarity metric for C-LDFS is also introduced in the paper. Additionally, we confirm the axiomatic requirement of the similarity measure and highlight a few of its traits. Finally, the proposed similarity metric is implemented in clinical decision-making to illustrate its efficacy.

In this manuscript we demonstrate interval-valued bipolar complex fuzzy set (IVBCFS) and then interval-valued bipolar complex fuzzy soft set (IVBCFSS), as a generalization of fuzzy set, interval-valued fuzzy set, bipolar fuzzy set, complex fuzzy set and soft set. We also initiate operational laws and basic results and properties for IVBCFS and IVBCFSS. Further explanation is given for the basic algebraic operations like complement, extended union, extended intersection, restricted union, and restricted intersection, AND product and OR product for IVBCFSS. Moreover, we demonstrate some fundamental aggregation operators like IVBCFS average aggregation, IVBCFS geometric and as well as their properties. To emphasize the usefulness and application of the system, we also develop the decision-making method and joint instances of the IVBCFSS (set-ups 1 and 2). In order to describe the effectiveness and influence of the approaching novel work, this study uses a comparative analysis of the new creating concept with prevailing ideas.

In 1994, Zhang (Proceedings of FUZZ-IEEE, 1998) [195], (Proceedings of IEEE conference, 1994) [71] introduced the concept of bipolar fuzzy sets as a generalization of the notion of Zadeh’s fuzzy sets. A bipolar fuzzy subset of a set is a pair of functions one from the set into the interval [0, 1] and the other into the interval \([-1,0].\) In a bipolar fuzzy set, the membership degree 0 of an element can be interpreted that the element is irrelevant to the corresponding property, the membership degree in (0, 1] of an element indicates the intensity that the element satisfies the property, and the membership degree in \([-1,0)\) of an element indicates the element does not satisfy the property. Fuzzy and possibilistic formalisms for bipolar information have been proposed in Dubios et al. Inf Pro Man Unc IPMU’04, 2002, [71] because bipolarity exists when dealing with spatial information in image processing or in spatial reasoning applications.

Understanding bipolar information is crucial as it enables individuals to make informed decisions that consider both extremes of a spectrum, leading to more balanced and effective outcomes. Interval-valued bipolar fuzzy set (IVBFS) has already been introduced in the literature as a great decision-making tool that can capture interval-valued bipolar information to properly address uncertainty. In this article, we introduce a hybrid of Interval-valued bipolar fuzzy set (IVBFS) and bipolar hypersoft sets (BHSS) called interval-valued bipolar fuzzy hypersoft set [Formula: see text], which merges the capabilities of IVBFS and BHSS. The rationale behind the design of the presented data structure is to manipulate and process information in decision-making scenarios when the data is bipolar, has multiple attributes that need to be addressed up to a sub-attributive level to get a proper representation of the data provided, and needs to be presented in the form of intervals. In [Formula: see text], two hyper soft sets (HSSs) are used, one providing positive interval-valued membership information and the other providing negative interval-valued membership information. We outline the essential features and basic operations of [Formula: see text] in this paper, examining its commutative, associative, distributive, and De Morgan laws to ensure a comprehensive analysis. To demonstrate the significance of [Formula: see text], we develop a preferential decision support algorithm for selecting the best alternative in e-learning, such as identifying the most suitable instructional method, which can effectively be formulated as a Multi-Attribute Decision-Making (MADM) problem. This approach allows for the systematic evaluation of various alternatives based on multiple parameters and sub-parameters, enabling a rational and well-informed decision. This algorithm helps select the best alternative from a given set of options, leveraging the versatile nature of [Formula: see text]. The presented study conducts both computation-based and structural comparisons to evaluate the adaptability and reliability of the proposed framework.

In this paper, we introduce the concept of \((\inf, \sup)\)-hesitant fuzzy subalgebras, which is a general concept of interval-valued fuzzy subalgebras, in BCK/BCI-algebras and investigate its properties. We characterize \((\inf, \sup)\)-hesitant fuzzy subalgebras in terms of sets, fuzzy sets, hesitant fuzzy sets, interval-valued fuzzy sets, Pythagorean fuzzy sets, negative fuzzy sets and bipolar fuzzy sets. Furthermore, characterizations of subalgebras, fuzzy subalgebras, anti-fuzzy subalgebras, negative fuzzy subalgebras, Pythagorean fuzzy subalgebras and bipolar fuzzy subalgebras of BCK/BCI-algebras are given in terms of \((\inf, \sup)\)-hesitant fuzzy subalgebras and interval-valued fuzzy subalgebras.

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.

Fuzzy concepts are represented by fuzzy sets, so a fuzzy knowledge is defined to be a collection of fuzzy sets (concepts) which includes two constants, 1 (true) and 0 (false). Abstractly, a logical derivation is a finite series of logical operations acting on a given knowledge. In fuzzy world, logical operations are defined by mathematical operations. So fuzzy logical derivations are mathematical derivations, and vice versa In this paper, the closure of fuzzy reasoning is characterized by fuzzy rough sets, and by topologies (neighborhood systems). A new fuzzy concept F N is derivable from the old knowledge K iff the membership function of F N is continuous in K-topology, or equivalently, iff F N is a rough fuzzy set (concept) of I N D(K),or plainly, iff the membership function of F N is a “step” function in the sense that it takes a constant value in each equivalence class of I N D(K).A fuzzy set L partitions the universe into equivalence classes; each equivalence class consists of those elements which have the same degree of membership. The indiscernibility relation I N D(K)over K is the “intersection” of all the equivalence relations induced by the fuzzy sets in K. The membership function of a fuzzy set is a real valued function, so U can be given the minimal topology such that the membership function of each fuzzy set in K is a continuous function. Such topology is called K-topology.

A novel method for ranking of vague sets for handling the risk analysis of compressor system

On fuzzy quasi continuity and an application of fuzzy set theory

The main purpose of this work is to construct a basic concept of fuzzy topology. In this article, we will try to give definition of fuzzy topology, fuzzy open set, fuzzy closed set, fuzzy interior of a fuzzy set, fuzzy closure of a fuzzy set in fuzzy topology and also, we will try to prove some theorems on fuzzy interior and fuzzy closure of a fuzzy topology with the help of extended definition of fuzzy sets and complement of a fuzzy sets.

In many domains of information processing, both vagueness, or imprecision, and bipolarity, encompassing positive and negative parts of information, are core features of the information to be modeled and processed. This led to the development of the concept of bipolar fuzzy sets, and of associated models and tools. Here we propose to extend these tools by defining algebraic relations between bipolar fuzzy sets, including intersection, inclusion, adjacency and RCC relations widely used in mereotopology, based on bipolar connectives (in a logical sense) and on mathematical morphology operators.

Granular structure plays a very important role in the model construction, theoretical analysis and algorithm design of a granular computing method. The granular structures of classical rough sets and fuzzy rough sets have been proven to be clear. In classical rough set theory, equivalence classes are basic granules, and the lower and upper approximations of a set can be computed by those basic granules. In the theory of fuzzy rough set, granular fuzzy sets can be used to describe the lower and upper approximations of a fuzzy set. This paper discusses the granular structure of type-2 fuzzy rough sets over two universes. Definitions of type-2 fuzzy rough sets over two universes are given based on a wavy-slice representation of type-2 fuzzy sets. Two granular type-2 fuzzy sets are deduced and then proven to be basic granules of type-2 fuzzy rough sets over two universes. Then, the properties of lower and upper approximation operators and these two granular type-2 fuzzy sets are investigated. At last, several examples are given to show the applications of type-2 fuzzy rough sets over two universes.