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Diabetes mellitus is one of the most critical diseases, affecting millions of people around the world. This work deals with the fractional optimal control of the dynamics of the population model on diabetes. This framework is based on the fractional order differential problems that describe the population before and after diabetes involving some health problems. We consider the Caputo derivatives for the study of the proposed model. The maximum principle of Pontryagin is utilized to derive the necessary conditions for the optimality of a dynamical system. Using a forward-backward sweep approach with the generalized Euler method accomplishes numerical solutions of formulated issues. [ABSTRACT FROM AUTHOR]

The idea of probabilistic q-rung orthopair linguistic neutrosophic (P-QROLN) is one of the very few reliable tools in computational intelligence. This paper explores a significant breakthrough in nanotechnology, highlighting the introduction of nanoparticles with unique properties and applications that have transformed various industries. However, the complex nature of nanomaterials makes it challenging to select the most suitable nanoparticles for specific industrial needs. In this context, this research facilitate the evaluation of different nanoparticles in industrial applications. The proposed framework harnesses the power of neutrosophic logic to handle uncertainties and imprecise information inherent in nanoparticle selection. By integrating P-QROLN with AO, a comprehensive and flexible methodology is developed for assessing and ranking nanoparticles according to their suitability for specific industrial purposes. This research contributes to the advancement of nanoparticle selection techniques, offering industries a valuable tool for enhancing their product development processes and optimizing performance while minimizing risks. The effectiveness of the proposed framework are demonstrated through a real-world case study, highlighting its potential to revolutionize nanoparticle selection in HVAC (Heating, Ventilation, and Air Conditioning) industry. Finally, this study is crucial to enhance nanoparticle selection in industries, offering a sophisticated framework probabilistic q-rung orthopair linguistic neutrosophic quantification with an aggregation operator to meet the increasing demand for precise and informed decision-making., (© 2024. The Author(s).)