In this research paper, the compatible maps and maps of type \(\alpha\) and \(\beta\) within the context of neutrosophic metric spaces (NMS) has been established. Additionally, the interconnections between \(\alpha\) and \(\beta\) types maps has been elucidated. Furthermore, the concept of R-weakly commuting mappings and weakly commuting in NMS has been introduced. The foundational work on Pant’s theorem, for intuitionistic fuzzy metric spaces (IFMS) is extended, by adapting it for NMS. While IFMS only consider membership and non-membership degrees, our extension incorporates the crucial element of uncertainty inherent in NMS. This expansion addresses a significant gap in previous research and provides a more robust framework for analyzing scenarios where ambiguity plays a key role. An illustrative example is presented to validate the result. To demonstrate the practical significance and applicability, VIKOR method is applied within a Neutrosophic Multi-criteria Decision-Making (MCDM) framework for sustainable water resource management.
