2025-01

Mathematics and Computational Applications (Issue : 4) (Volume : 29)

M-parametrized N-soft set (MPNSS), an extension of N-soft set (N-SS) theory, is instrumental in addressing the challenges of assigning non-binary evaluations to both alternatives and attributes. Recognizing the inherent duality in human decisionmaking, where choices are influenced by both positive and negative aspects, we enhance the MPNSS framework by incorporating bipolarity. This addition, aimed at capturing the dual nature of decision processes, results in the development of bipolar Mparametrized N-soft set (BMPNSS) model. In the context of BMPNSS, we present some related definitions such as incomplete, negatively efficient, positively efficient, and totally efficient. Additionally, for the complement of MPNSS, we introduce four distinct definitions: complement, weak complement, top weak complement, and bottom weak complement. Set-theoretic operations, including extended and restricted union and intersection, are explored accompanied by a discussion of their properties, providing a comprehensive understanding of the behavior of these operations within the BMPNSS framework. To facilitate understanding, we include an illustrative example. The decision-making procedure introduces alternative ranking based on extended choice and extended weight choice values, demonstrated through a numerical example. In our comparative analysis, BMPNSS is positioned against existing models, emphasizing its distinctive features and advantages in diverse decision-making scenarios.

2024-07

Mathematics and Computational Applications (Issue : 4) (Volume : 29)

This article presents a pioneering mathematical model, fuzzy bipolar hypersoft (FBHS) sets, which combines the bipolarity of parameters with the fuzziness of data. Motivated by the need for a comprehensive framework capable of addressing uncertainty and variability in complex phenomena, our approach introduces a novel method for representing both the presence and absence of parameters through FBHS sets. By employing two mappings to estimate positive and negative fuzziness levels, we bridge the gap between bipolarity, fuzziness, and parameterization, allowing for more realistic simulations of multifaceted scenarios. Compared to existing models like bipolar fuzzy hypersoft (BFHS) sets, FBHS sets offer a more intuitive and user-friendly approach to modeling phenomena involving bipolarity, fuzziness, and parameterization. This advantage is underscored by a detailed comparison and a practical example illustrating FBHS sets’ superiority in modeling such phenomena. Additionally, this paper provides an in-depth exploration of fundamental FBHS set operations, highlighting their robustness and applicability in various contexts. Finally, we demonstrate the practical utility of FBHS sets in problem-solving and introduce an algorithm for optimal object selection based on available information sets, further emphasizing the advantages of our proposed framework.

2024-05

PLoS ONE (Issue : 19) (Volume : 5)

This paper investigates N-bipolar hypersoft topology (N-BHST), a novel extension of both the well-established N-hypersoft topology (N-HST) and hypersoft topology (HST). Deviating significantly from its precursor, the N-bipolar hypersoft (N-BHS) set, N-BHST introduces a multi-opinion approach to decision-making, augmenting robustness and adaptability. This innovative framework addresses identified limitations in N-bipolar soft topology (N-BST), especially in managing multi-argument approximate functions. The study analyzes various operators (closure, interior, exterior, and boundary) within the N-BHST framework, elucidating their interrelationships. Additionally, an examination is carried out on the enhancement of multi-criteria group decision-making (MCGDM) using N-BHST, setting it apart from existing models. A numerical example is presented to illustrate its application in real-world decision scenarios.

2024-01

PLoS ONE (Issue : 1) (Volume : 19)

This paper introduces N-bipolar hypersoft (N-BHS) sets, a versatile extension of bipolar hypersoft (BHS) sets designed to effectively manage evaluations encompassing both binary and non-binary data, thereby exhibiting heightened versatility. The major contributions of this framework are twofold: Firstly, the N-BHS set introduces a parameterized representation of the universe, providing a nuanced and finite granularity in perceiving attributes, thereby distinguishing itself from conventional binary BHS sets and continuous fuzzy BHS sets. Secondly, this model signifies a new area of research aimed at overcoming limitations inherent in the N-bipolar soft set when handling multi-argument approximate functions. Through the strategic partitioning of attributes into distinct subattribute values using disjoint sets, the N-BHS set emerges as a powerful tool for effectively addressing uncertainty-related problems. In pursuit of these objectives, the paper outlines various algebraic definitions, including incomplete N-BHS sets, efficient N-BHS sets, normalized N-BHS sets, equivalence under normalization, N-BHS complements, and BHS sets derived from a threshold, exemplified through illustrative examples. Additionally, the article explores set-theoretic operations within the N-BHS sets framework, such as relative null/whole N-BHS sets, N-BHS subsets, and two distinct approaches to N-BHS extended/restricted union and intersection. Finally, it proposes and compares decision-making methodologies regarding N-BHS sets, including a comprehensive comparison with relevant existing models.

2023-09

Symmetry (Issue : 9) (Volume : 15)

This paper introduces N-hypersoft (N-HS) sets—an enriched and versatile extension of hypersoft (HS) sets—designed to handle evaluations involving both binary and non-binary data while embodying an inherent sense of structural symmetry. The paper presents several algebraic definitions, including incomplete N-HS sets, efficient N-HS sets, normalized N-HS sets, equivalence under normalization, N-HS complements, and HS sets derived from a threshold. These definitions are accompanied by illustrative examples. Additionally, the paper delves into various set-theoretic operations within the framework of N-HS sets, such as relative null/whole N-HS sets, N-HS subsets, and N-HSextended/restricted union and intersection, presented in two different ways. Finally, the paper presents and compares decision-making methodologies regarding N-HS sets.

2023-04

Neutrosophic Sets and Systems (Volume : 55)

For a mathematical model to describe vague (uncertain) problems effectively, it must have the ability to explain the links between the objects and parameters in the problem in the most precise way. There is no suitable model that can handle such scenarios in the literature. This deficiency serves as motivation for this study. In this article, the bipolar hypersoft set (abbreviated, BHSS) is considered since the parameters and their opposite play a symmetrical role. We present a novel theoretical technique for solving decision-making problems using BHSS and investigate parameter reductions for these sets. Algorithms for parameter reduction are provided and explained with examples. The findings demonstrate that our suggested parameter reduction strategies remove unnecessary parameters and still retain the same decision-making options.

2023-01

International Journal of Neutrosophic Science (Issue : 1) (Volume : 20)

According to its definition, a topological space could be a highly unexpected object. There are spaces (indiscrete space) which have only two open sets: the empty set and the entire space. In a discrete space, on the other hand, each set is open. These two artificial extremes are very rarely seen in actual practice. Most spaces in geometry and analysis fall somewhere between these two types of spaces. Accordingly, the separation axioms allow us to say with confidence whether a topological space contains a sufficient number of open sets to meet our needs. To this end, we use bipolar hypersoft (BHS) sets (one of the efficient tools to deal with ambiguity and vagueness) to define a new kind of separation axioms called BHS Ti-space (i = 0,1,2,3,4). We show that BHS Ti-space (i = 1,2) implies BHS Ti−1-space; however, the converse is false, as shown by an example. For i = 0,1,2,3,4, we prove that BHS Ti-space is hypersoft (HS) Ti-space and we present a condition so that HS Ti-space is BHS Ti-space. Moreover, we study that a BHS subspace of a BHS Ti-space is a BHS Ti-space for i = 0,1,2,3.

2023-01

Filomat (Issue : 19) (Volume : 36)

In this manuscript, we continue to study the hypersoft topological space (for short, HSTS) by presenting hypersoft (HS) separation axioms, called HS Ti-spaces for i = 0,1,2,3,4. The notions of HS regular and HS normal spaces are explained in detail. We discuss the connections between them and present numerous examplestohelpclarify the interconnections between the different types of these spaces. Wepoint out that HS Ti-axioms imply HS Ti−1 for i = 1,2,3, and with the help of an example we show that HST4 space need no t be HS T3-space. We also clarify that the property that an HS space being HSTi-spaces (i = 0,1,2,3) is HS hereditary. Finally, we provide a diagram to illustrate the relationships between our proposed axioms.

2023-01

Neutrosophic Sets and Systems (Volume : 53)

Mappings are significant mathematical tools with many applications in our daily lives. The bipolar hypersoft set is one of the effective tools for dealing with ambiguity and vagueness. The purpose of this article is to define mappings between the classes of bipolar hypersoft sets. The notions of bipolar hypersoft image and bipolar hypersoft inverse image of bipolar hypersoft sets are then defined, and some of their properties are studied. Moreover, we discuss the relations between the bipolar hypersoft image and the bipolar hypersoft inverse image of the bipolar hypersoft sets. This proposed work can be extended to IndetermSoft Set, IndetermHyperSoft Set and TreeSoft Set and their corresponding Fuzzy, Intuitionistic Fuzzy, Neutrosophic forms and other Fuzzy-extension.

2023-01

Demonstratio Mathematica (Issue : 1) (Volume : 56)

The main objective of this study is to define a new class of bipolar soft (BS) separation axioms known as BS Ti-space (i = 0,1,2,3,4). This type is defined in terms of ordinary points. We prove that BS Ti -space implies BS Ti_1-space for i =1,2 ; however, the opposite is incorrect, as demonstrated by an example. For i=0,1,2,3,4, we investigate that every BS Ti-space is soft Ti-space; and we set up a condition in which the reverse is true. Moreover, we point out that a BS subspace of a BS Ti-space is a BS Ti-space for i=0,1,2,3.

2022-11

International Journal of Neutrosophic Science (Issue : 2) (Volume : 19)

Hypersoft topology (HST) is the study of a structure based on all hypersoft (HS) sets on a given set of alternatives. In continuation of this concern, in this article, we introduce new maps namely HS continuous, HS open, HS closed, and HS homomorphism. We examine the main characteristics of each of these maps. Furthermore, we study HS compact space and discuss some of its properties. We point out that HS compactness preserved under HS continuous map.

2022-11

International Journal of Neutrosophic Science (Issue : 2) (Volume : 19)

Herein, we further contribute and promote topological structures via bipolar hypersoft (BHS) setting by introducing new types of maps called BHS continuous, BHS open, BHS closed, and BHS homeomorphism maps. We investigate their characterizations and establish their main properties. By providing a thorough picture of the proposed maps, we investigate the concept of BHS compact space and obtain several results relating to this concept. We point out that BH compactness preserved under BH continuous map. The relationships among these concepts with their counterparts in hypersoft (HS) structures are discussed.

2022-10

Neutrosophic Sets and Systems (Volume : 51)

Connectedness (resp. disconnectedness), which reflects the key characteristic of topological spaces and helps in the differentiation of two topologies, is one of the most significant and fundamental concept in topological spaces. In light of this, we introduce hypersoft connectedness (resp. hypersoft disconnectedness) in hypersoft topological spaces and investigate its properties in details. Furthermore, we present the concepts of disjoint hypersoft sets, separated hypersoft sets, and hypersoft hereditary property. Also, some examples are provided for the better understanding of these ideas.

2022-08

Journal of Intelligent & Fuzzy Systems (Issue : 4) (Volume : 43)

The most significant and fundamental topological property is connectedness (resp. disconnectedness). This property highlights the most important characteristics of topological spaces and helps to distinguish one topology from another. Taking this into consideration, we investigate bipolar hypersoft connectedness (resp. bipolar hypersoft disconnectedness) for bipolar hypersoft topological spaces. With the help of an example, we show that if there exist a non-null, non-whole bipolar hypersoft sets which is both bipolar hypersoft open and bipolar hypersoft closed over U, then the bipolar hypersoft space need not be a bipolar hypersoft disconnected. Furthermore, we present the concepts of separated bipolar hypersoft sets and bipolar hypersoft hereditary property.

2022-04

Neutrosophic Sets and Systems (Volume : 49)

Smarandache [48] introduced the concept of hypersoft set which is a generalization of soft set. This notion is more adaptable than soft set and more suited to challenges involving decision-making. Consequently, topology defined on the collection of hypersoft sets will be in great importance. In this paper, we introduce hypersoft topological spaces which are defined over an initial universe with a fixed set of parameters. The notions of hypersoft open sets, hypersoft closed sets, hypersoft neighborhood, hypersoft limit point, and hypersoft subspace are introduced and their basic properties are investigated. Finally, we introduce the concepts of hypersoft closure, hypersoft interior, hypersoft exterior, and hypersoft boundary and the relationship between them are discussed.

2022-02

Journal of Mathematics (Volume : 2022)

In this article, we introduce bipolar hypersoft topological spaces over the collection of bipolar hypersoft sets. It is proven that a bipolar hypersoft topological space gives a parametrized family of hypersoft topological spaces, but the converse does not hold in general, and this is shown with the help of an example. Furthermore, we give a condition on a given parametrized family of hypersoft topologies, which assure that there is a bipolar hypersoft topology whose induced family of hypersoft topologies is the given family. The notions of bipolar hypersoft neighborhood, bipolar hypersoft subspace, and bipolar hypersoft limit points are introduced. Finally, we define bipolar hypersoft interior, bipolar hypersoft closure, bipolar hypersoft exterior, and bipolar hypersoft boundary, and the relations between them, differing from the relations on hypersoft topology, are investigated.

2021-08

Mathematics (Issue : 15) (Volume : 9)

Hypersoft set theory is an extension of soft set theory and is a new mathematical tool for dealing with fuzzy problems; however, it still suffers from the parametric tools’ inadequacies. In order to boost decision-making accuracy even more, a new mixed mathematical model called the bipolar hypersoft set is created by merging hypersoft sets and bipolarity. It is characterized by two hypersoft sets, one of which provides positive information and the other provides negative information. Moreover, some fundamental properties relative to it such as subset, superset, equal set, complement, difference, relative (absolute) null set and relative (absolute) whole set are defined. Furthermore, some set-theoretic operations such as the extended intersection, the restricted union, intersection, union, AND-operation and OR-operation of two bipolar hypersoft sets with their properties are discussed and supported by examples. Finally, tabular representations for the purposes of storing bipolar hypersoft sets in computer memory are used.