Healthcare workers, including doctors and medical staff, must consistently make informed decisions that significantly affect patients on individual, community, national, and global scales. Healthcare practitioners must occasionally make judgments with constrained information, resources, and knowledge, yet it is anticipated that these choices are meticulously measured and precise. Familiarizing oneself with precise concepts pertaining to medical decision-making is essential. The complexity of decision-making (DM) arises from the environment’s ambiguous, imprecise, and uncertain characteristics, especially when several attributes are present and further categorized. We have used the hypersoft set concept to address such complex challenges. This article defines interval-valued Fermatean neutrosophic super hyper-soft sets. Its fundamental operations are subsets, equality, null sets, complements, unions, and intersections. Using these algebraic procedures, we establish a numerical example for the prioritization of patients awaiting organ transplantation. This hybrid environment may effectively manage uncertainty and yield distinctive results.

Industry 5.0 has transformed the logistics value chain with human-centricity, sustainability, and resilience at its core, with circular supply chain strategies playing a pivotal role through reuse, remanufacturing, and recycling. Circular logistics, however, brings complex uncertainties such as quality issues with returns, volume fluctuations, conflicting information, and multi-attribute decision-making, which cannot be captured by traditional models. The paper proposes a new model based on Fermatean Neutrosophic Hypersoft Sets (FNHSS) and machine learning to optimize circular supply chain management in the era of Industry 5.0. The framework makes four contributions: first, it provides a mathematical framework for FNHSS that accounts for three-dimensional uncertainty (truth, indeterminacy, and falsity) under Fermatean constraints (0 ≤ φ³ + υ³ ≤ 1), along with hypersoft multi-attribute parameterization; second, it introduces dual solution paths, including numerical defuzzification optimized using the Modified Distribution method and a deep reinforcement learning architecture based on a transformer encoder and a graph neural network (GNN); third, it proposes a circular economy reward function that balances logistics costs, returns, and circularity; fourth, it validates the framework through an automotive electric vehicle battery recycling case study. Empirical results show that the uncertainty capture index reaches 0.87, significantly above fuzzy machine learning baselines, with cost improvements of 6.16% and 1.99% compared to Type-1 fuzzy and intuitionistic fuzzy approaches, respectively. The numerical and ML solutions show no significant difference (p > 0.05), and the trained ML model enables constant-time inference essential for dynamic Industry 5.0 contexts. The FNHSS-ML architecture offers a robust approach for resilient, adaptable, and sustainable intelligent logistics systems.

This paper introduces a comprehensive mathematical framework that integrates Pythagorean fuzzy sets (PyFS), soft sets (SS), and hypersoft set theory to establish robust methodologies for addressing complex uncertainties in multi-attribute decision-making (MADM). We present two formal approaches: Pythagorean soft sets (PySS) and Pythagorean hypersoft sets (PyHSS), which enhance the conventional soft set and hypersoft set frameworks by incorporating the expressiveness of Pythagorean fuzzy sets (PyFS). The theoretical framework demonstrates that these structures preserve all fundamental set-theoretic operations and facilitate the representation of the membership degree (MD) and non-membership degree (n-MD) with sums not exceeding 1, contingent upon satisfying the Pythagorean condition ψ² + φ² ≤ 1. We illustrate the applicability of our approach through extensive digital implementations in technology selection, cloud-based configuration, and educational recommendation systems. The mathematical functionalities exhibited, including operation closure and generalisation hierarchies, render PySS and PyHSS formidable decision support system instruments for managing imprecise and ambiguous information across several criteria.