Charting uncertain waters: Insights into objective function formulations under future uncertainty

Optimal management of water resources is challenging due to uncertainty in future conditions. One promising approach is to directly incorporate future uncertainty into objective function formulations of optimization problems, enabling system performance evaluation across multiple potential conditions. However, this creates additional uncertainties as both the choice of objective function formulation and the plausible future conditions included in optimization are subjective. Given the inherent uncertainty in plausible future conditions, it is highly unlikely that future conditions included in optimization can cover all conditions that might occur. Therefore, identifying objective function formulations that perform well regardless of future uncertainties is crucial; however, it has not been formally explored. In this study, the performance of different objective function formulations under both expected (i.e., similar to conditions used in optimization) and unexpected (i.e., vastly different from conditions used in optimization) future conditions is investigated using a real-world case study. Results reveal that percentile and expected-value-based formulations generally perform consistently under both expected and unexpected conditions, whereas extreme-case-based formulations can lead to highly variable results depending on the actual conditions that will be realized in the future. Finally, variance-based formulations offer the greatest consistency across all conditions but may lead to compromised performance under favorable conditions.