The T-Spherical Fuzzy set (T-SPHFS) is one of the core simplifications of quite a lot of fuzzy concepts such as fuzzy set (FS), intuitionistic fuzzy set (ITFS), picture fuzzy set (PIFS), Q-rung orthopair fuzzy set (Q-RUOFS), etc. T-SPHFS reveals fuzzy judgment by the degree of positive membership, degree of abstinence, degree of negative membership, and degree of refusal with relaxed conditions, and this is a more powerful mathematical tool to pair with inconsistent, indecisive, and indistinguishable information. In this article, several novel operational laws for T-SPFNs based on the Schweizer–Sklar t-norm (SSTN) and the Schweizer–Sklar t-conorm (SSTCN) are initiated, and some desirable characteristics of these operational laws are investigated. Further, maintaining the dominance of the power aggregation (POA) operators that confiscate the ramifications of the inappropriate data and Heronian mean (HEM) operators that consider the interrelationship among the input information being aggregated, we intend to focus on the T-Spherical fuzzy Schweizer–Sklar power Heronian mean (T-SPHFSSPHEM) operator, the T-Spherical fuzzy Schweizer–Sklar power geometric Heronian mean (T-SPHFSSPGHEM) operator, the T-Spherical fuzzy Schweizer–Sklar power weighted Heronian mean (T-SPHFSSPWHEM) operator, the T-Spherical fuzzy Schweizer–Sklar power weighted geometric Heronian mean (T-SPHFSSPWGHEM) operator, and their core properties and exceptional cases in connection with the parameters. Additionally, deployed on these newly initiated aggregation operators (AOs), a novel multiple attribute decision making (MADM) model is proposed. Then, the initiated model is applied to the City of Penticton (British Columbia, Canada) to select the best choice among the accessible seven water reuse choices to manifest the practicality and potency of the preferred model and a comparison with the proffered models is also particularized.

Understanding the complexity of reaction mechanisms is pivotal for advancing scientific and engineering solutions. This study addresses the challenge of distinguishing between slow and fast species in high-dimensional reaction mechanism, where direct analytical solutions are intractable. Through a hybrid approach combining analytical derivations and computational techniques including numerical simulations via MATLAB toolboxes we elucidate the kinetic behavior of species, quantify parameter sensitivities and calculating Surface Bifurcation. This paper provides a comprehensive examination of complex reaction mechanisms. The findings based on the analysis of the elementary steps involved in product formation show that, the hydrogen-assisted propagation step offers the dominant net contribution, while the direct radical recombination step stays close to equilibrium. The relative availability of molecular hydrogen, reactive radicals, and product-driven reverse reactions control step dominance. The integration of computational modeling with analytical insights not only clarifies reaction dynamics but also facilitates the optimization of reaction conditions and the design of efficient analytical frameworks. This work highlights the critical role of model reduction strategies in demystifying complex systems, offering a robust foundation for future innovations in sustainable technology and industrial applications.