Waste management has gained global importance, aligning with the escalating impact of the COVID-19 pandemic and the associated concerns regarding medical waste, which poses threats to public health and environmental sustainability. In Istanbul, medical waste is considered a significant concern due to the rising volume of this waste, along with challenges in collection, incineration and storage. At this juncture, precise estimation of the waste volume is crucial for resource planning and allocation. This study, thus, aims to estimate the volume of medical waste in Istanbul using the nonlinear grey Bernoulli model (NGBM(1,1)) and the firefly algorithm (FA). In other words, this study introduces a novel hybrid model, termed as FA-NGBM(1,1), for predicting waste amount in Istanbul. Within this model, prediction accuracy is enhanced through a rolling mechanism and parameter optimization. The effectiveness of this model is compared with the classical GM(1,1) model, the GM(1,1) model optimized with the FA (FA-GM(1,1)), the fractional grey model optimized with the FA (FA-FGM(1,1)) and linear regression. Numerical results indicate that the proposed FA-NGBM(1,1) hybrid model yields lower prediction error with a mean absolute percentage error value 3.47% and 2.57%, respectively, for both testing and validation data compared to other prediction algorithms. The uniqueness of this study is rooted in the process of initially optimizing the parameters for the NGBM(1,1) algorithm using the FA for medical waste estimation in Istanbul. This study also forecasts the amount of medical waste in Istanbul for the next 3 years, indicating a dramatic increase. This suggests that new policies should be promptly considered by decision-makers and practitioners.