Bipolar Soft Sets Research Articles

A Study on Rough Fuzzy Bipolar Soft Ideals in Semigroups

The theories of rough sets (RSs) and soft sets (SSs) are practical mathematical techniques to accommodate data uncertainty. On the other hand, fuzzy bipolar soft sets (FBSSs) can address uncertainty and bipolarity in various situations. The key objective of this study is to establish the notions of rough fuzzy bipolar soft ideals in semigroup (SG), which is an extension of the idea of rough fuzzy bipolar soft sets in a SG. Also, we have analyzed the roughness in the bipolar fuzzy subsemigroup (BF-SSG) by employing a congruence relation (CR) defined on the SG and investigating several related characteristics. Further, the idea is expanded to the rough fuzzy bipolar soft ideal, rough fuzzy bipolar soft interior ideal, and rough fuzzy bipolar soft bi-ideal in SGs. Moreover, it is observed that CR and complete CR (CCR) are critical in developing rough approximations of fuzzy bipolar soft ideals. Therefore, their related characteristics are studied via CRs and CCRs.

A systematic literature review of soft set theory

AbstractSoft set theory, initially introduced through the seminal article “Soft set theory—First results” in 1999, has gained considerable attention in the field of mathematical modeling and decision-making. Despite its growing prominence, a comprehensive survey of soft set theory, encompassing its foundational concepts, developments, and applications, is notably absent in the existing literature. We aim to bridge this gap. This survey delves into the basic elements of the theory, including the notion of a soft set, the operations on soft sets, and their semantic interpretations. It describes various generalizations and modifications of soft set theory, such as N-soft sets, fuzzy soft sets, and bipolar soft sets, highlighting their specific characteristics. Furthermore, this work outlines the fundamentals of various extensions of mathematical structures from the perspective of soft set theory. Particularly, we present basic results of soft topology and other algebraic structures such as soft algebras and $$\sigma$$ σ -algebras. This article examines a selection of notable applications of soft set theory in different fields, including medicine and economics, underscoring its versatile nature. The survey concludes with a discussion on the challenges and future directions in soft set theory, emphasizing the need for further research to enhance its theoretical foundations and broaden its practical applications. Overall, this survey of soft set theory serves as a valuable resource for practitioners, researchers, and students interested in understanding and utilizing this flexible mathematical framework for tackling uncertainty in decision-making processes.

Application of Bipolar Near Soft Sets

The bipolar soft set is supplied with two soft sets, one positive and the other negative. Whichever feature is stronger can be selected to find the object we want. In this paper, the notion of bipolar near soft set, which near set features are added to a bipolar soft set, and its fundamental properties are introduced. In this new set, its features can be restricted and the basic properties and topology of the set can be examined accordingly. With the soft set close to bipolar, it will be easier for us to decide to find the most suitable object in the set of objects. This new idea is illustrated with real-life examples. With the help of the bipolar near soft set, we make it easy to choose the one closest to the criteria we want in decision making. Among the many given objects, we can find the one with the properties we want by using the ones with similar properties.

Binary Bipolar Soft Points and Topology on Binary Bipolar Soft Sets with Their Symmetric Properties

The aim of this paper is to give an interesting connection between two mathematical approaches to vagueness: binary bipolar soft sets and binary bipolar soft topology. The binary bipolar soft points are defined using binary bipolar soft sets. The binary bipolar soft set will be the binary bipolar soft union of its binary bipolar soft points. Moreover, the notion of binary bipolar soft topological spaces over two universal sets and a parameter set is proposed. Some topological properties of binary bipolar soft sets, such as binary bipolar soft open, binary bipolar soft closed, binary bipolar soft closure, binary bipolar soft interior, and binary bipolar soft boundary, are introduced. Some important properties of these classes of binary bipolar soft sets are investigated. Furthermore, the symmetry relation is compared between binary bipolar soft topology and binary soft topology on a common universe set. Finally, some results and counterexamples are demonstrated to explain this work.

BIPOLAR SOFT CONTINUITY ON BIPOLAR SOFT TOPOLOGICAL SPACES

The striking idea of soft sets was frst claimed that by Molodtsov as a new mathematical tool for dealing with uncertainties which is free from the other theories limitations. After the advent of soft set theory, bipolar soft sets as a generalization of soft sets, a new model of uncertain information, were introduced by Shabir and Naz. The main purpose of this paper is to introduce and investigate the structures of bipolar soft continuity, bipolar soft openness, bipolar soft closedness and bipolar soft homeomorphism.

Novel MCDM Methods and Similarity Measures for Extended Fuzzy Parameterized Possibility Fuzzy Soft Information with Their Applications

Complicated uncertainties arising in the multicriteria decision-making (MCDM) problems that show distinct possible satisfaction of the subjects to favorable and equally unfavorable parameters with varying preferences require reliable decision-making under comprehensive mathematical tools. For such complications, this work aims to develop a novel fuzzy parameterized possibility fuzzy bipolar soft set model as a fuzzy parameterized bipolar soft extension of possibility fuzzy sets. The proposed model efficiently depicts the possibility of fuzzy belongingness of alternatives under fuzzy parameterized bipolar parameters (or attributes). The respective operations and properties such as subset, complement, union, and intersection are presented along with their numerical illustrations. Two logical operations namely “AND” and “OR” operations followed by two corresponding MCDM algorithms have been developed and implemented. Furthermore, similarity measures between fuzzy parameterized possibility fuzzy bipolar soft sets are proposed and applied to an agricultural land selection scenario. Finally, a comparative analysis of current work with existing ones is discussed in detail to show the eminent quality of the proposed work over them.

Optimistic multigranulation roughness of fuzzy bipolar soft sets by soft binary relations and its applications

Two important mathematical methods for addressing uncertainty in data processing are multigranulation rough set (MGRS) and fuzzy bipolar soft set (FBSS). This paper describes a certain kind of multigranulation rough set in the context of multiple soft binary relations. We first define the multigranulation roughness of fuzzy bipolar soft sets in the two universes. Moreover, a detailed study of structural properties has been conducted in order to explore this concept. The key characteristics of the traditional MGRS model are completely preserved in this new approach. Following that, we suggest two decision-making algorithms with respect to aftersets and foresets of the soft binary relations over dual universes. This approach appears to be better suited and more adaptable than other available methods, making it a favorable option for addressing decision-making problems. Finally, we provide a practical application of the suggested approach to a real-world problem.

A Hybridization of Modified Rough Bipolar Soft Sets and TOPSIS for MCGDM

Uncertain data is a challenge to decision-making (DM) problems. Multi-criteria group decision-making (MCGDM) problems are among these problems that have received much attention. MCGDM is difficult because the existing alternatives frequently conflict with each other. In this article, we suggest a novel hybrid model for an MCGDM approach based on modified rough bipolar soft sets (MRBSs) using a well-known method of technique for order of preference by similarity to ideal solution (TOPSIS), which combines MRBSs theory and TOPSIS for the prioritization of alternatives in an uncertain environment. In this technique, we first introduce an aggregated parameter matrix with the help of modified bipolar soft lower and upper matrices to identify the positive and negative ideal solutions. After that, we define the separation measurements of these two solutions and compute relative closeness to choose the best alternative. Next, an application of the proposed technique in the MCGDM problem is introduced. Afterward, an algorithm for this application is developed, which is illustrated by a case study. The application demonstrates the usefulness and efficiency of the proposal. Compared to some existing studies, we additionally present several merits of our proposed technique. Eventually, the paper handles whether additional studies on these topics are needed.

Connectedness, local connectedness, and components on bipolar soft generalized topological spaces

Connectedness represents the most significant and fundamental topological property. It highlights the main characteristics of topological spaces and distinguishes one topology from another. There is a constant study of bipolar soft generalized topological spaces (\( \mathcal{BSGTS}s \)) by presenting \(\mathcal{BS}\) \( \widetilde{\widetilde{\mathfrak{g}}} \)-connected set and \(\mathcal{BS}\) \( \widetilde{\widetilde{\mathfrak{g}}} \)-connected space in \(\mathcal{BSGTS}s\) as well as it is discussing some properties and results for these topics. Additionally, the notion of bipolar soft disjoint sets is put forward, \(\mathcal{BS} \) \(\widetilde{\widetilde{\mathfrak{g}}}\)-separation set, \(\widetilde{\widetilde{\mathfrak{g}}}\)-separated \(\mathcal{BSS}s\) and \(\mathcal{BS}\) \( \widetilde{\widetilde{\mathfrak{g}}} \)-hereditary property. Moreover, there is an extensive study of \(\mathcal{BS}\) \( \widetilde{\widetilde{\mathfrak{g}}} \)-locally connected space and \(\mathcal{BS}\) \( \widetilde{\widetilde{\mathfrak{g}}} \)-component with some related properties and theorems following them, such as the concepts of \( \mathcal{BS} \) \(\widetilde{\widetilde{\mathfrak{g}}}\)-locally connected spaces and \( \mathcal{BS} \) \(\widetilde{\widetilde{\mathfrak{g}}}\)-connected are independent of each other; also determined the conditions under which the \( \mathcal{BS} \) \(\widetilde{\widetilde{\mathfrak{g}}}\)-connected subsets are \( \mathcal{BS} \) \(\widetilde{\widetilde{\mathfrak{g}}}\)-components.

Bipolar soft ideal rough set with applications in COVID-19

Bipolar soft rough set represents an important mathematical model to deal with uncertainty. This theory represents a link between bipolar soft set and rough set theories. This study introduced the concept of topological bipolar soft set by combining a bipolar soft set with topologies. Also, the topological structure of bipolar soft rough set has been discussed by defining the bipolar soft rough topology. The main objective of this paper is to present some solutions to develop and modify the approach of the bipolar soft rough sets. Two kinds of bipolar soft ideal approximation operators which represent extensions of bipolar soft rough approximation operator have been presented. Moreover, a new kind of bipolar approximation space via two ideals, called bipolar soft biideal approximation space, was introduced and studied by two different methods. Their properties are discussed and the relationships between these methods and the previous ones are proposed. The importance of these methods is reducing the vagueness of uncertainty areas by increasing the bipolar lower approximations and decreasing the bipolar upper approximations. Also, the bipolar soft biideal rough sets represent two opinions instead of one opinion. Finally, an application in multicriteria group decision making (MCGDM) in COVID-19 by using bipolar soft ideal rough sets is suggested by using two methods.

Medical decision-making techniques based on bipolar soft information

<abstract><p>Data uncertainty is a barrier in the decision-making (DM) process. The rough set (RS) theory is an effective approach to study the uncertainty in data, while bipolar soft sets (BSSs) can handle the vagueness and uncertainty as well as the bipolarity of the data in a variety of situations. In this article, we introduce the idea of rough bipolar soft sets (RBSSs) and apply them to find the best decision in two different DM problems in medical science. The first problem is about deciding between the risk factors of a disease. Our algorithm facilitates the doctors to investigate which risk factor is becoming the most prominent reason for the increased rate of disease in an area. The second problem is deciding between the different compositions of a medicine for a particular illness having different effects and side effects. We also propose algorithms for both problems.</p></abstract>

Possibility neutrosophic bipolar fuzzy soft sets and their applications

In this work, the possibility neutrosophic bipolar soft sets interact with the possibility bipolar fuzzy soft sets, as well as complementation, union, intersection, AND, and OR. This paper extends the concept of bipolar neutrosophic soft sets to the possibility neutrosophic bipolar soft sets. Our main goal was to demonstrate De Morgan’s law, associate law and distributive law, which are all the laws related to the possibility neutrosophic bipolar soft sets. Also, we present an algorithm that uses a soft set model to solve the decision-making problem primarily in order to simplify the process.

Binary bipolar soft sets

Bu yazıda iki matematiksel yaklaşımı belirsizliğe göre ilginç bir bağlantı veriyoruz: bipolar yumuşak kümeler ve ikili yumuşak kümeler. İkili iki kutuplu yumuşak darı iki evrensel küme üzerine set ve bir parametre seti önerilmiştir. Tamamlayıcı, birleşme, kesişme, sınırlı birleşme, sınırlı kesişme, null ikili bipolar yumuşak seti, mutlak ikili bipolar yumuşak seti, iki ikili bipolar yumuşak seti, “VE”, “VEYA” tarihinde ikili bipolar yumuşak setlerde tanımlanmıştır. İkili bipolar yumuşak kümelerin temel özellikleri de araştırılmış ve tartışılmıştır. Sonunda ikili bipolar yumuşak kümenin karakteristik bir fonksiyonunu veriyoruz.

An Innovative Hybrid Multi-Criteria Decision-Making Approach under Picture Fuzzy Information

These days, multi-criteria decision-making (MCDM) approaches play a vital role in making decisions considering multiple criteria. Among these approaches, the picture fuzzy soft set model is emerging as a powerful mathematical tool for handling various kinds of uncertainties in complex real-life MCDM situations because it is a combination of two efficient mathematical tools, namely, picture fuzzy sets and soft sets. However, the picture fuzzy soft set model is deficient; that is, it fails to tackle information symmetrically in a bipolar soft environment. To overcome this difficulty, in this paper, a model named picture fuzzy bipolar soft sets (PRFBSSs, for short) is proposed, which is a natural hybridization of two models, namely, picture fuzzy sets and bipolar soft sets. An example discussing the selection of students for a scholarship is added to illustrate the initiated model. Some novel properties of PRFBSSs such as sub-set, super-set, equality, complement, relative null and absolute PRFBSSs, extended intersection and union, and restricted intersection and union are investigated. Moreover, two fundamental operations of PRFBSSs, namely, the AND and OR operations, are studied. Thereafter, some new results (De Morgan’s law, commutativity, associativity, and distributivity) related to these proposed notions are investigated and explained through corresponding numerical examples. An algorithm is developed to deal with uncertain information in the PRFBSS environment. To show the efficacy and applicability of the initiated technique, a descriptive numerical example regarding the selection of the best graphic designer is explored under PRFBSSs. In the end, concerning both qualitative and quantitative perspectives, a detailed comparative analysis of the initiated model with certain existing models is provided.

Bipolar Soft Limit Points in Bipolar Soft Generalized Topological Spaces

The concept of soft set theory can be used as a mathematical tool for dealing with problems that contain uncertainty. Then, a new mixed mathematical model called the bipolar soft set is created by merging soft sets and bipolarity, which gave the concept of a binary model of grading. Bipolar soft set is characterized by two soft sets, one of which provides positive information and the other negative. Bipolar soft generalized topology is a generalization of bipolar soft topology. The importance of limit points in all branches of mathematics cannot be ignored. It forms one of the most significant and fundamental concepts in topology. On this basis, the derived set concept is required in the establishment and continuation of some properties. Accordingly, the limit point in bipolar soft generalized theory is defined. In this paper, we present the notion of bipolar soft generalized limit points. We explained the relation between the bipolar soft generalized derived and the bipolar soft generalized closure set. Added to that, we discussed some structures of a bipolar soft generalized topological space such as: <img src=image/13428354_01.gif>-interior point, <img src=image/13428354_01.gif>-exterior point, <img src=image/13428354_01.gif>-boundary point, <img src=image/13428354_01.gif>-neighborhood point and basis on <img src=image/13428354_02.gif>. Finally, we give comparisons among these concepts of bipolar soft generalized topological spaces (<img src=image/13428354_02.gif>) by using bipolar soft point (<img src=image/13428354_03.gif>). Each concept introduced in this paper is explained with clear examples.

Ranking of Downstream Fish Passage Designs for a Hydroelectric Project under Spherical Fuzzy Bipolar Soft Framework

Nowadays, several real-world decision-making problems concerning falling economies, power crises, depleting resources, etc., require efficient decision-making. To solve such problems, researchers proposed several hybrid models by combining the spherical fuzzy sets with other theories, such as spherical fuzzy soft sets, which is an efficient tool to deal with the uncertainties concerning positive, neutral, and negative memberships in the soft environment. However, all the existing hybridizations of spherical fuzzy sets fail to deal with information symmetrically in a bipolar soft environment. Accordingly, this paper presents a novel hybrid model called spherical fuzzy bipolar soft sets (SFBSSs) by fusing spherical fuzzy sets and bipolar soft sets, considering the opposite sets of parameters in symmetry. An example considering the selection of a chief management officer (CMO) for a multi-national company illustrates the proposed model in detail. In addition, some symmetric properties and algebraic operations of the initiated model, including subset, complement, relative null SFBSSs, relative absolute SFBSSs, extended union, extended intersection, restricted union, restricted intersection, AND, and OR operations, are discussed and illustrated via numerical examples. Further, some fundamental results, namely, commutativity, associativity, distribution, and De Morgan’s laws are presented for SFBSSs. Moreover, by considering the massive impact of hydropower in combating the energy crisis and possible dangers to fish migration, a multi-attribute decision-making problem concerning the ranking of downstream fish passage designs for a hydroelectric project is modeled and solved under the developed algorithm based on SFBSSs. Finally, a comparative analysis discusses the supremacy of the initiated model over its building blocks.

Local compactness and paracompactness on bipolar soft topological spaces

A bipolar soft set is given by helping not only a chosen set of “parameters” but also a set of oppositely meaning parameters called “not set of parameters”. It is known that a structure of bipolar soft set is consisted of two mappings such that F : E → P (X) and G :⌉ E → P (X), where F explains positive information and G explains opposite approximation. In this study, we first introduce a new definition of bipolar soft points to overcome the drawbacks of the previous definition of bipolar soft points given in [34]. Then, we explore the structures of bipolar soft locally compact and bipolar soft paracompact spaces. We investigate their main properties and illuminate the relationships between them. Also, we define the concept of a bipolar soft compactification and investigate under what condition a bipolar soft topology forms a bipolar soft compactification for another bipolar soft topology. To elucidate the presented concepts and obtained results, we provide some illustrative examples.

Decision analysis review on the concept of class for bipolar soft set theory

Decision analysis review on the concept of class for bipolar soft set theory

Combination of the Bipolar Soft Set and Soft Expert Set with an Application in Decision Making

In this paper, we propose a novel concept of the bipolar soft expert set by combining the soft expert set and the bipolar soft set. Then, we define its basic operations such as complement, union, intersection, AND and OR for bipolar soft expert sets with illustrative examples. Then, using this set theory, an algorithm is proposed to express an uncertainty problem in the best way. Finally, we exemplify an uncertainty problem on how the proposed algorithm can be applied against uncertain situations that may be encountered in any field and we give its implementation in detail.

A New Optimization Approach Based on Bipolar Type-2 Fuzzy Soft Sets

The main objective of this research study is to amplify the schematic representation of human reasoning by launching the most generalized fantastic theories of bipolar type-2 fuzzy set (BT2FS) and bipolar type-2 fuzzy soft set (BT2FS f S). These incredible models are exclusively developed for the simultaneous capturing of both polarity and abstruseness inherent in equivocal interpretations. The proposed BT2FS f S theory renders an outstanding parameterized framework that skilfully wipes out the high-order uncertainty of imprecise knowledge-based systems. First, we keenly provide the formal structure of both proposed models along with the deep exploration of their elementary properties. Moreover, we explore rudimentary set-theoretical operations of developed frameworks inclusive of equality, subset, complement, union, and intersection with their noticeable results. We brilliantly formulate a highly proficient algorithm using proposed theory to disentangle the real-world multiattribute decision-making conundrums with two-sided ambiguous information. Finally, we holistically scrutinize an empirical analysis for the selection best way to cultivate the dessert city to demonstrate the remarkable accountability of the proposed methodology.