Health-care waste (HCW) contains numerous hazardous, infectious, and radioactive contents that may cause serious health issues for society and drastic environmental problems. Therefore, it is crucial to dispose of HCW safely and appropriately. Four commonly used HCW disposal methods are incineration, steam sterilization, microwave, and landfill disposal. To select an appropriate HCW disposal method, there are several conflicting criteria. Furthermore, when information is lacking due to uncertainty and criteria weighing conditions are also unknown, the decision to choose an appropriate HCW disposal method becomes more challenging. So, the objective of this paper is to introduce a novel integrated multi-criteria group decision-making (MCGDM) approach to evaluate HCW disposal methods in which incomplete information is quantified using intuitionistic fuzzy sets (IFSs), unknown criteria weights are evaluated using the maximizing deviation method, and the evaluation of HCW disposal methods is done by applying the Archimedean t-norm and t-conorm-based interactive intuitionistic fuzzy weighted aggregated sum product assessment (WASPAS) method. Herein, the Archimedean t-norm and t-conorm cover a wide range of t-norms and t-conorms, including the Algebraic, Einstein, Hamacher, and Frank, etc., by adopting prescribed combinations of additive generators. The proposed MCGDM approach has been applied to evaluate the above-listed four HCW disposal methods for a case study conducted by Mishra et al. (2020) in the context of India. A sensitivity analysis is performed to examine the impact of changes in the parameters involved in the proposed MCGDM approach. A comparative study is also performed with some well-known multi-criteria decision making(MCDM) and MCGDM methods to demonstrate the consistency and stability of the proposed approach. Findings suggest that the newly developed MCGDM approach is more general, flexible, and provides realistic results to evaluate HCW disposal methods under uncertainty and unknown criteria weight information.