<p>Score function (SF) is an important measure to rank fuzzy numbers (FNs). In the present study, firstly, a novel SF (NSF) has been proposed under Fermatean Fuzzy Environment (FFE), which can rank Fermatean Fuzzy Numbers (FFNs) regardless of their membership grades (MGs) and non-MGs (NMGs) and satisfies all decisive characteristics of an ideal SF. The authenticity and reliability of the proposed NSF have been validated through numerical examples. Secondly, with the help of the FF weighted power geometric operator (FFWPGO) combined with the proposed NSF, conventional Step-wise Weight Assessment Ratio Analysis (SWARA) and Weighted Aggregated Sum Product Assessment (WASPAS) techniques have been redefined under FFE to make them capable of handling group decision-making (GrDM) situations under uncertainty. Finally, the biomedical waste recycling technology (BMWRT) selection problem has been structured as an MCGrDM problem and has been solved using the suggested NSF-induced integrated technique to validate the applicability of the proposed work. After comparing the findings, it has been seen that the suggested framework is capable of determining the best BMWRT under fuzzy environment. The main advantage of the proposed NSF is that it can overcome the shortcomings of the contemporary Fermatean fuzzy set functions (FFSFs) and can rank any type of FFNs irrespective of their MGs. Here, the proposed integrated approach has identified Plasma Gasification Melting as the best BMWRT based on the assessments received from the decision makers, which can help the stakeholders to provide a pollution-free, clean environment by properly recycling the BMWs. Moreover, the proposed method can handle the uncertainty associated with any real life decision-making problem.</p>