An Updating Procedure for the Efficient Estimation of Probabilistic Constraints

Stochastic optimization techniques allow the decision making process to incorporate variability and uncertainty leading to decisions which perform satisfactorily under a range of conditions. When considering probabilistic constraints, there are both robust and reliable formulations. Robust formulations may not be reliable when simulating a different representation of the variable and uncertain parameters. Utilizing the Bayesian Binomial-Beta conjugate pair to represent the distribution of sample violations, a reliable constraint formulation is developed. Applying the reliable constraint formulation to particulate removal through conventional drinking water treatment plants increased the design cost by 5.1% relative to the robust formulation. Unlike the robust solution, the reliable solution satisfied the probability constraint under multiple simulation sets. The reliable formulation was then modified to terminate the simulation loop within the stochastic optimization routine when the proportion of violations was determined to be statistically greater or less than the desired value (under-or over-designed, respectively); thus removing wasted computational effort. The over-designed option did not reduce the computational efficiency. The under-designed option required one more iteration in the optimization routine, however, the total number of process model calls was reduced by 19.4%.