Renewable energy sources, such as solar, wind, and hydropower, are critical for sustainable development, environmental protection, and long‐term energy security. Selecting the most suitable renewable energy sources involves complex multi‐criteria decision‐making, considering economic, environmental, technological, and reliability aspects. This paper introduces a bipolar q‐fractional fuzzy set model along with corresponding aggregation operators for multi‐criteria decision‐making under uncertainty. The proposed bipolar q‐fractional fuzzy sets combine the flexibility of q‐fractional fuzzy sets with the dual‐perspective capability of bipolar fuzzy sets, enabling simultaneous representation of positive and negative evaluations. To aggregate bipolar q‐fractional fuzzy information across multiple criteria, two operators are defined: the bipolar q‐fractional fuzzy weighted average and the bipolar q‐fractional fuzzy weighted geometric operators. Their key properties, including idempotency, monotonicity, and boundedness, are also established. A systematic decision‐making algorithm is proposed, integrating these operators with score and accuracy functions to rank alternatives effectively. A practical example on renewable energy sources selection demonstrates the model's ability to preserve ranking consistency, maintain smooth score trends, and provide interpretable results. Comparative and sensitivity analyses further illustrate the robustness of the proposed approach.

Healthcare decision-making (DM) often requires evaluating multiple interdependent criteria under uncertainty, balancing desirable and undesirable outcomes. To address the limitations of traditional methods, we propose the q-rung orthopair fuzzy N-bipolar soft set (q-ROFNBSS) framework, a unified model that integrates multilevel evaluation, bipolar information, and higher-order fuzzy membership. The framework defines key operations—including null and whole sets, complement, subset, equality, union, and intersection—along with their algebraic properties, providing a rigorous foundation for systematic analysis. A step-by-step procedure is introduced to compute aggregated positive and negative scores for alternatives. We apply the model to a telemedicine platform selection case study, evaluating five hypothetical platforms across four positive and four negative criteria. The computed net scores, combining positive and negative impacts, are: z_1 = 0.5954, z_2 = 0.6572, z_3 = 0.7745, z_4 = -0.8369, and z_5 = -1.0852, identifying the mobile-first teleconsultation platform (x_3) as the most suitable choice. Comparative and sensitivity analyses illustrate the framework’s superiority relative to existing models and its flexibility to variations in grading intervals and parameter settings. The proposed framework outperforms existing q-rung orthopair fuzzy set (q-ROFS) and N-bipolar soft set (NBSS) models by providing more balanced, trade-off-aware rankings. Overall, this framework offers a versatile and robust tool for complex DM scenarios in healthcare and other domains.

Cybersecurity risk assessment in financial institutions has become more difficult because cyber threats keep changing, many evaluation factors must be considered, and decisions often depend on the opinions of several experts. To deal with these challenges, an intuitionistic fuzzy N-bipolar soft expert set (IFNBSES) model is introduced. This framework integrates intuitionistic fuzzy concepts with N-bipolar soft expert sets (NBSESs). The IFNBSES model considers both degrees of acceptance and rejection, includes positive and negative evaluations, and allows experts to provide multi-level assessments. The model also provides a clear way to combine expert opinions when there is uncertainty. The IFNBSES model is formally defined, and its structure is explained using practical examples. Basic operations of the model are studied, and their algebraic properties are examined to ensure both theoretical soundness and practical usefulness. The model is then applied to multi-criteria group decision-making (MCGDM) problems to show how it supports clear, reliable, and well-informed selection of cybersecurity strategies. The results are discussed along with their implications for researchers and decision-makers, and the robustness of the proposed approach is evaluated. Finally, a comparison with existing methods shows that IFNBSES handles uncertainty more effectively, making it a strong tool for cybersecurity decision-making (DM).

The rapid advancement of smart healthcare systems demands sophisticated mathematical tools to manage uncertainty and conflicting expert opinions in critical decision-making (DM) processes. Multi-criteria group decision making (MCGDM) plays a pivotal role in synthesizing diverse expert evaluations for complex healthcare challenges. However, existing soft set (SS) extensions often struggle to simultaneously capture multinary evaluations, bipolar reasoning, higher-order fuzzy logic, and multi-expert input. To overcome these limitations, we propose the Fermatean fuzzy Nbipolar soft expert set (FFNBSES), which enhances fuzzy representation while integrating multinary, bipolar, and multi-expert evaluations. We formally define the fundamental operations of FFNBSES and demonstrate its algebraic properties. A DM methodology based on FFNBSES is developed and applied to a healthcare case study, showcasing its superior capability to handle expert consensus and disagreement in multi-criteria evaluations. Comparative analysis within the SS theory framework highlights the enhanced flexibility and robustness of FFNBSES for real-world MCGDM problems. This workprovides a powerful and comprehensive approach to support smart healthcare transformation through improved group DM under uncertainty.

Intuitionistic fuzzy set (IFS), introduced by Atanassov, extends classical fuzzy sets (FSs) by incorporating both membership degree (MD) and non-membership degree (NMD), allowing for a more nuanced representation of uncertainty. N-bipolar soft set (NBSS) extends the traditional N-soft set (NSS) by incorporating bipolarity, while also supporting both binary and multinary evaluations. In this paper, we propose an intuitionistic fuzzy N-bipolar soft set (IFNBSS) model, a hybridization of IFS and NBSS that combines intuitionistic fuzzy elements, multinary evaluation, and bipolarity considerations. We define the fundamental operations of IFNBSS, establish their algebraic properties, and illustrate their applicability through examples. In addition, we embed IFNBSS into a multi-criteria decision-making (MCDM) scenario by providing a structured algorithm for determining the best alternative. A numerical example on the assessment of artificial intelligence (AI) in medical diagnosis illustrates the practical efficacy of the model on handling of uncertain and confounding information. We also demonstrate generality and robustness of IFNBSS through a detailed comparison with existing approaches.

Urban transportation planning involves evaluating multiple conflicting criteria such as accessibility, cost-effectiveness, and environmental impact, often under uncertainty and incomplete information. These complex decisions require input from various stakeholders, including planners, policymakers, engineers, and community representatives, whose opinions may differ or contradict. Traditional decision-making approaches struggle to effectively handle such bipolar and multivalued expert evaluations. To address these challenges, we propose a novel decisionmaking framework based on Pythagorean fuzzy N-bipolar soft expert sets. This model allows experts to express both positive and negative opinions on a multinary scale, capturing nuanced judgments with higher accuracy. It introduces algebraic operations and a structured aggregation algorithm to systematically integrate and resolve conflicting expert inputs. Applied to a real-world case study, the framework evaluated five urban transport strategies based on key criteria, producing final scores as follows: improving public transit (− 0.70), optimizing traffic signal timing (1.86), enhancing pedestrian infrastructure (3.10), expanding bike lanes (0.59), and implementing congestion pricing (0.77). The results clearly identify enhancing pedestrian infrastructure as the most suitable option, having obtained the highest final score of 3.10. Comparative analysis demonstrates the framework’s superior capability in modeling expert consensus, managing uncertainty, and supporting transparent multi-criteria group decision-making.

Efficient planning in the healthcare sector requires selecting the most suitable facility location based on multiple, often conflicting criteria and expert evaluations. This study proposes Fuzzy N-Bipolar Soft Expert (FN-BSE) set model, which integrates expert input, bipolar assessments, and multinary evaluation within a fuzzy framework. Theoretical structure of the model is developed through fundamental operations and their algebraic properties. Applying FN-BSE set model to a case study of seven potential healthcare locations, expert judgments were aggregated, and alternatives were ranked. The results indicate that location ω 3 is the optimal choice. Analysis of positive and negative evaluations shows the model effectively balances conflicting opinions, and comparative evaluation demonstrates its superiority over existing approaches in handling uncertainty and supporting robust Multi-Attribute Group Decision-Making (MAGDM).

Sustainability evaluation in manufacturing industries is increasingly vital for promoting responsible growth and long-term competitiveness amid environmental, social, and economic challenges. Effective decision-making (DM) under uncertainty is crucial for managing multiple, often conflicting sustainability objectives. In this paper, we propose a novel hybrid model, termed Pythagorean fuzzy N-bipolar soft sets (PFNBSSs), which integrates Pythagorean fuzzy sets (PFSs), N-soft sets (NSSs), and bipolar soft sets (BSSs) within a unified multi-criteria decision-making (MCDM) framework. For theoretical purposes, we define basic operations and algebraic properties of PFNBSSs, supported by illustrative examples. To demonstrate practical applicability, the PFNBSS model is applied to assess sustainability practices in manufacturing industries through two numerical examples: one focusing on positive and negative sustainability indicators, and another emphasizing comparative sustainability risk assessment across diverse manufacturing sectors. Detailed interpretations of computational results and their relevance in practical DM are provided. This is followed by a comparative analysis confirming the superior discrimination power and expressive capability of the PFNBSS model over existing alternatives. The paper concludes with a critical evaluation of the model and suggestions for future research.

This manuscript introduces N-hypersoft topology (N-HST), an innovative extension of the established hypersoft topology (HST). Unlike its predecessor, the N-hypersoft (N-HS) set, N-HST adopts a multi-opinion approach to decision-making, enhancing robustness and adaptability across various domains. In addressing limitations observed in N-soft topology (N-ST), particularly in handling multi-argument approximate functions, N-HST offers a comprehensive framework. The study delves into various operators within the N-HST framework, including closure, interior, exterior, and boundary operators, elucidating their interrelationships and providing a deeper understanding of the structural components. Furthermore, this manuscript explores the practical implications of N-HST in multi-criteria group decision-making (MCGDM), showcasing its applicability in real-world decision scenarios. In addition to its specific applications in MCGDM, N-HST offers generic aspects that contribute to its versatility and efficacy in decision-making processes. By accommodating multiple opinions and handling complex evaluation factors, N-HST emerges as a superior framework for addressing diverse decision-making challenges. Through a numerical example, this manuscript demonstrates the practical utility and potential impact of N-HST in various domains.

This paper introduces fuzzy N-bipolar soft (FN-BS) sets, a novel mathematical framework designed to enhance multi-criteria decision-making (MCDM) processes under uncertainty. The study addresses a significant limitation in existing models by unifying fuzzy logic, the consideration of bipolarity, and the ability to evaluate attributes on a multinary scale. The specific contributions of the FN-BS framework include: (1) a formal definition and set-theoretic foundation, (2) the development of two innovative algorithms for solving decision-making (DM) problems, and (3) a comparative analysis demonstrating its superiority over established models. The proposed framework is applied to a real-world case study on selecting vaccination programs across multiple countries, showcasing consistent DM outcomes and exceptional adaptability to complex and uncertain scenarios. These results position FN-BS sets as a versatile and powerful tool for addressing dynamic DM challenges.

This paper presents N-bipolar soft expert (N-BSE) sets, a novel framework designed to enhance multi-attribute group decision-making (MAGDM) by incorporating expert input, bipolarity, and non-binary evaluations. Existing MAGDM approaches often lack the ability to simultaneously integrate positive and negative assessments, especially in nuanced, multi-valued evaluation spaces. The proposed N-BSE model addresses this limitation by offering a comprehensive, mathematically rigorous structure for decision-making (DM). Fundamental operations of the N-BSE model are defined and analyzed, ensuring its theoretical consistency and applicability. To demonstrate its practical utility, the N-BSE model is applied to a general case study on sustainable energy solutions, illustrating its effectiveness in handling complex DM scenarios. An algorithm is proposed to streamline the DM process, enabling systematic and transparent identification of optimal alternatives. Additionally, a comparative analysis emphasizes the advantages of the N-BSE model over existing MAGDM frameworks, highlighting its capacity to integrate diverse expert opinions, evaluate both positive and negative attributes, and support multi-valued assessments. By bridging the gap between theoretical development and practical application, this paper contributes to advancing DM methodologies.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.