This study proposes novel operational laws that extend the Frank t-norm and t-conorm to develop a new class of aggregation operators (AOs), namely the \(p,q-\) quasirung orthopair fuzzy Frank weighted average, weighted geometric, ordered weighted average, and ordered weighted geometric operators. These operators are specifically designed to manage uncertain and imprecise information within multi-attribute group decision-making (MGADM) environments. The proposed operators exhibit desirable mathematical properties such as flexibility, robustness, and compatibility, making them highly suitable for complex fuzzy decision contexts. Flexibility is notably enhanced through the independent tuning of the parameters \(p\), \(q\), and \(\tau\), allowing for more refined control over membership (MD), non-membership (NMD), and interaction behaviors. An entropy-based approach is employed to objectively determine unknown attribute weights, minimizing subjective bias. A real-world case study on the selection of an optimal investment location demonstrates the practical applicability of the proposed method. The results show an improvement in decision-making accuracy by approximately 7.5% compared to traditional approaches. Sensitivity analysis confirms the stability and reliability of the proposed operators under varying conditions. Comparative results further highlight the method’s superiority in terms of accuracy, interpretability, and adaptability to input variations. The paper concludes by outlining special cases and acknowledging certain limitations, offering directions for future research.

This paper introduces the concept of Fermatean Double Valued Neutrosophic Soft Set (FDVNSS), an advanced mathematical framework for modeling uncertainty, vagueness, and indeterminacy in complex systems. A FDVNSS extends traditional neutrosophic and soft set theories by incorporating two layers of truth and falsity degrees: absolute and relative. This multi-dimensional representation allows for more nuanced decision-making, particularly in domains characterized by conflicting or imprecise information, such as medical diagnostics. The formal definition of FDVNSS is established, including the conditions under which membership degrees satisfy the Fermatean constraint, i.e., the sum of the cubes of the degrees lies within a defined range. Key operations such as subset, null and absolute FDVNSS, complement, union, and intersection are rigorously defined and exemplified. Furthermore, the algebraic properties of FDVNSSs under these operations are explored, and foundational propositions-analogous to classical set theory- are proven to hold. Examples drawn from health care contexts, such as evaluating diseases, symptoms, or medical responses, are provided to illustrate the applicability of FDVNSS in realworld problems. Overall, this framework provides a more expressive and flexible tool for reasoning under high levels of uncertainty and offers significant potential for application in intelligent decision support systems. The notion of a Fermatean Double Valued Neutrosophic Soft Topological Space (FDVNSTS) is introduced. The concepts of semi-open (s-open), pre-open (p-open), and b-open sets are defined within the context of FDVNSTS. Among these generalized open sets, the pre-open set is selected for further exploration, and several fundamental topological notions are developed based on this definition. These include the closure, exterior, boundary, and interior in FDVNSTS.

In this paper, we introduced the concept of the heptapartitioned neutrosophic soft set (HP-NSS), a novel extension and generalization of the neutrosophic soft set theory. To improve accuracy, the indeterminacy is divided into five additional possibilities, which are as follows: absolute true, relative true, contradiction, unknown, and ignorance, relative false and absolute false. We extend the concept of neutrosophic soft topological spaces by introducing heptapartitioned neutrosophic soft topological spaces (HPNSTS), a novel generalization that incorporates seven distinct partitions to model uncertainty, vagueness, and indeterminacy in topological structures. Three new definitions are introduced and these definitions are p-open, pre-open and semi-open sets. Special attention is focused on p-open sets, and a number of results related to p-open sets are addressed. Machine learning and graphical algorithms, such as K-Means clustering, Heat maps, Elbow method, Feature correlation, 2D-normalized t-SNE, and parallel coordinates of 3D T-SNE, were used and visualized for a real-world application involving the romantic feelings of young boys and girls across various dimensions.

This research presents a new theoretical framework and shows how to use it e ectively to data analysis. First, we introduce a novel method to complex neutrosophic soft sets (CNSS) and ensure the generalize ability of their use by de ning basic operations such as union, intersection, and complement. We go on to construct a complete one-value neutrosophic soft topology, de ning the most important topological properties such as interior, closure, and other related theoretical aspects, to give a strong mathematical foundation. An example is then provided to illustrate this frameworks analytical capabilities. Using a comprehensive visualizing package that includes spline-smoothed functional curves, 3D surface plots, and 2D Heatmaps, we apply our method to address a signal-template alignment problem. The Cotangent Similarity Measure (Cot SM) is used to systematically examine three input signals (S1 S3) and three templates (T1 T3). The ndings are striking and unambiguous: a worldwide maximum indicates a high degree of condence, and the correlation between S2 T1 (CotSM = 06653) is widely recognized as the strongest and most conclusive connection. Subsequent analysis shows that while signal S2 is the most important overall, signals S1 and S3 are helpful for particular targets.