The Fermatean fuzzy set, in contrast to other generalizations of fuzzy sets like PFS and IFS, has a wide range of acceptance for both MF and NMF. In light of this, the Fermatean fuzzy set performs as an efficient, flexible, and comprehensive representation in situations that lack certainty. Here, the weaker forms of Fermatean fuzzy sets are introduced, and their traits are analyzed. Decomposition and continuity of the Fermatean fuzzy α-open set are also accustomed. With the goal of safeguarding our green environment, hiring the best supplier is of the utmost significance in the construction industry. Using outranking techniques, Visual PROMETHEE Academic Edition 1.4 is a live multi-criteria decision aid software program. It runs virtual analysis through GAIA and applies selected criteria to contrast parameters. It also saves them for possible export and editing. In this article, the PROMETHEE II method is applied for Fermatean fuzzy numbers with FF(α,β)-level for selecting the optimal green supplier for a construction company. Because of its ability to handle vagueness, the FF PROMETHEE II method emerges as a valuable tool in Multi-criteria decision making. Furthermore, this study assesses the efficacy of the proposed technique by comparing its results with those obtained through other established methods.