Comparison of new conditional value‐at‐risk‐based management models for optimal allocation of uncertain water supplies

Abstract

The paper studies the effect of incorporating the conditional value‐at‐risk (CVaRα) in analyzing a water allocation problem versus using the frequently used expected value, two‐stage modeling, scenario analysis, and linear optimization tools. Five models are developed to examine water resource allocation when available supplies are uncertain: (1) a deterministic expected value model, (2) a scenario analysis model, (3) a two‐stage stochastic model with recourse, (4) a CVaRα objective function model, and (5) a CVaRα constraint model. The models are applied over a region of east central Florida. Results show the deterministic expected value model underestimates system costs and water shortage. Furthermore, the expected value model produces identical cost estimates for different standard deviations distributions of water supplies with identical mean. From the scenario analysis model it is again demonstrated that the expected value of results taken from many scenarios underestimates costs and water shortages. Using a two‐stage stochastic mixed integer formulation with recourse permits an improved representation of uncertainties and real‐life decision making which in turn predicts higher costs. The inclusion of CVaRα objective function in the latter provides for the optimization and control of high‐risk events. Minimizing CVaRα does not, however, permit control of lower‐risk events. Constraining CVaRα while minimizing cost, on the other hand, allows for the control of high‐risk events while minimizing the costs of all events. Results show CVaRα exhibits continuous and consistent behavior with respect to the confidence level α, when compared to value‐at‐risk (VaRα).