The use of Pythagorean fuzzy $N$-soft sets (PFNSs) enables the examination of belongingness and non-belongingness of membership degrees, as well as their combinations with $N$-grading, in the unpredictable nature of individuals. This research aims to enhance our understanding of a popular multi-criteria group decision making (MCGDM) technique, Preference Ranking Organization Method for Enrichment of Evaluations, under the PFNS environment, aiding in making effective decisions for real-life problems, as fuzzy set theory is directly relevant to real-life applications. The PROMETHEE technique's main principle is to calculate the inflow and outflow streams of alternatives based on the deviation of their score degrees, ultimately providing partial and complete rankings of the given options. To capture the uncertainty of human nature, which demands both the association and disassociation of the considered criteria and provision of $N$-grading, the PFNS PROMETHEE technique is introduced in this research article. First, an Analytic Hierarchy Process AHP is used to check the feasibility of the standard weights of the criteria. The article then explains the detailed method of the fuzzy $N$-soft PROMETHEE technique to rank alternatives, with all the steps presented in an extensive flowchart for better understanding of the methodology. Furthermore, the practicality and viability of the proposed technique are demonstrated through an example of selecting the best chemical element in cloud seeding, where the most suitable choice is identified using an outranking directed graph. The credibility of the PFNS PROMETHEE technique is assessed by comparison with an existing method. Finally, the proposed technique's strengths and weaknesses are discussed to demonstrate its efficiency and drawbacks.

In recent years, fossil fuel resources have become increasingly rare and caused a variety of problems, with a global impact on economy, society and environment. To tackle this challenge, we must promote the development and diffusion of alternative fuel technologies. The use of cleaner fuels can reduce not only economic cost but also the emission of gaseous pollutants that deplete the ozone layer and accelerate global warming. To select an optimal alternative fuel, different fuzzy decision analysis methodologies can be utilized. In comparison to other extensions of fuzzy sets, the $T$-spherical fuzzy set is an emerging tool to cope with uncertainty by quantifying acceptance, abstention and rejection jointly. It provides a general framework to unify various fuzzy models including fuzzy sets, picture fuzzy sets, spherical fuzzy sets, intuitionistic fuzzy sets, Pythagorean fuzzy sets and generalized orthopair fuzzy sets. Meanwhile, decision makers prefer to employ linguistic terms when expressing qualitative evaluation in real-life applications. In view of these facts, we develop an extended multi-attributive border approximation area comparison (MABAC) method for solving multiple attribute group decision-making problems in this study. Firstly, the combination of $T$-spherical fuzzy sets with 2-tuple linguistic representation is presented, which provides a general framework for expressing and computing qualitative evaluation. Secondly, we put forward four kinds of 2-tuple linguistic $T$-spherical fuzzy aggregation operators by considering the Heronian mean operator. We investigate some fundamental properties of the proposed 2-tuple linguistic $T$-spherical fuzzy aggregation operators. Lastly, an extended MABAC method based on the 2-tuple linguistic $T$-spherical fuzzy generalized weighted Heronian mean and the 2-tuple linguistic $T$-spherical fuzzy weighted geometric Heronian mean operators is developed. For illustration, a case study on fuel technology selection with 2-tuple linguistic $T$-spherical fuzzy information is also conducted. Moreover, we show the validity and feasibility of our approach by comparing it with several existing approaches.

The extraction of analytical solution of uncertain fractional Langevin differential equations involving two independent fractional-order is frequently complex and difficult. As a result, developing a proper and comprehensive technique for the solution of this problem is very essential. In this article, we determine the explicit and analytical fuzzy solution for various classes of the fuzzy fractional Langevin differential equations (FFLDEs) with two independent fractional-orders both in homogeneous and non-homogeneous cases. The potential solution of FFLDEs is also extracted using the fuzzy Laplace transformation technique. Furthermore, the solution of FFLDEs is defined in terms of bivariate and trivariate Mittag-Leffler functions both in the general and special forms. FFLDEs are a new topic having many applications in science and engineering then to grasp the novelty of this work, we connect FFLDEs with RLC electrical circuit to visualize and support the theoretical results.

Industrial robots enable manufacturers to produce high-quality products at low cost, so they are a key component of advanced production technology. Welding, assembly, disassembly, painting of printed circuit boards, pick-and-place mass production of consumer products, laboratory research, surgery, product inspection and testing are just some of the applications of industrial robots. All functions are done with a high level of endurance, speed and accuracy. Many competing attributes must be evaluated simultaneously in a comprehensive selection method to determine the performance of industrial robots. In this research article, we introduce the 2TL $q$-ROFS as a new advancement in fuzzy set theory to communicate complexities in data and presents a decision algorithm for selecting an arc welding robot utilizing the 2-tuple linguistic $q$-rung orthopair fuzzy (2TL $q$-ROF) set, which can dynamically delineate the space of ambiguous information. We propose the $q$-ROF Hamy mean ( $q$-ROFHM) and the $q$-ROF weighted Hamy mean ( $q$-ROFWHM) operators by combining the $q$-ROFS with Hamy mean operator. We investigate the properties of some of the proposed operators. Then based on the proposed maximization bias, a new optimization model is built, which is able to exploit the DM preference to find the best objective weights among attributes. Next, we extend the COmbinative Distance-Based ASsessment (CODAS) method to 2TL $q$-ROF-CODAS version which not only covers the uncertainty of human cognition but also gives DMs a larger space to represent their decisions. To validate our strategy, we present a case study of arc welding robot selection. Finally, the effectiveness and accuracy of the method are proved by parameter analysis and comparative analysis results. The results show that our method effectively addresses the evaluation and selection of arc welding robots and captures the relationship between an arbitrary number of attributes.