Applications of T-spherical fuzzy aczel-alsina power muirhead mean operators in identifying the most effective water purification process for commercial purpose.

One of the key areas of research in ambiguous and inconsistent problems with decision-making is T-spherical multiple-attribute decision making (TP-FMADM). The TP-F number (TP-FN), which is an extension of the fuzzy number, an intuitionistic fuzzy number, and other fuzzy structures, may deal with problems involving a significant amount of incorrect, incomplete, and inconsistent data. Being an extension of various fuzzy structures, the TP-F sets (TP-FSs) provide decision-makers greater freedom to voice their actual opinions and offer a broader range of acceptable membership grades. The Muirhead mean (MM) operator and power aggregation (PA) operators are illustrations of standard aggregation operators. They are better because they can reproduce the relationships amongst input qualities and eliminate the negative influence of obstinate data. Since its publication in 1982, Aczel and Alsina's t-norms have proven to be a highly effective and popular technique for generating aggregation operators of any kind. Furthermore, the parameter belongs to converts the Aczel-Alsina t-norms into a specific instance of the algebraic t-norms. In this article, novel aggregation operators (AGOs) are suggested, considering the advantages of the TP-FN, to handle the multi-criteria decision-making problems. These new AGOs consider how various input data are related to one another and can mitigate the impact of inaccurate data at the same time. To strengthen the adaptability of these new AGOs, this article proposes the T-spherical fuzzy Aczel-Alsina power Muirhead mean (TP-FAAPMM) operator, T-spherical fuzzy Aczel-Alsina power dual Muirhead mean (TP-FAAPDMM) operator which combines the Aczel-Asina operational rules with the power average/geometric operator and the Muirhead/dual Muirhead mean operators. A variety of fundamental characteristics and special cases with respect to the parameters are explored and it is retrieved that various existing AGOs are special cases of these newly initiated AGOs. Further, weighted forms of these AGOs are established. Then, we set up the multiple attribute decision making (MADM) technique using these AGOs that are suggested to solve MADM problems. We then give a numerical example about industrial water purification selection and compare it to other related MADM techniques that are already in use in the TP-FN information to demonstrate the effectiveness and appropriateness of the anticipated technique.