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.

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-HS extended/restricted union and intersection, presented in two different ways. Finally, the paper presents and compares decision-making methodologies regarding N-HS sets.

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.

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.

\(M\)-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 decision-making‎, ‎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 \(M\)-parametrized \(N\)-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.

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.