Uncertainty analysis in river quality management considering failure probability: controllable and uncontrollable input pollutants

River quality management involves complex challenges due to inherent uncertainties in various parameters, especially when dealing with controllable and uncontrollable pollutants. This study integrates a finite volume approach, called SEF (symmetric exponential function), with Monte Carlo simulations in MATLAB to solve the advection-dispersion equation, focusing on evaluating river quality protection tools by considering failure probability (Pf). Critical specifications for maintaining reliable river ecosystem performance are identified. We simulate assimilation capacity for managing river water quality against controllable pollutants to satisfy allowable pollution concentration at the high-reliability index. Using the Genetic Programming (GP) algorithm, a new accurate equation for assimilation capacity calculation is presented considering Pf for the first time. Results indicate that flow velocity significantly affects river assimilation capacity: increasing velocity can shift the river to a hazardous state while decreasing it allows for greater pollutant assimilation. Sustainable protection tools, including dilution flow and detention time, are considered to manage uncontrollable pollutants within a specific time (Tc) and river length constraints (Lc), safeguarding river water quality for both human and animal populations. Dilution flow is practical for specific base velocities but ineffective at high base flow rates. Conversely, detention time consistently protects water quality across all base flow velocities within the Lc constraint. Moreover, this study introduces the ratio of detention time to initial pollution contact duration as a vital water quality index to protect the rivers' environment. Combining numerical methods with reliability analysis and soft computing techniques, this research provides valuable insights into river system dynamics and protecting river water quality.