Optimal design under uncertainty via CVaR (conditional value at risk)

Abstract

The aim of this thesis work is to investigate the possibility of controlling and (re)shaping the statistical probability distribution of optimal objective function values in optimization problems related to process synthesis, design, and operation under uncertainty via imposing CVaR (Conditional Value at Risk) constraints. Probability distributions of the process model outputs are obtained by Monte Carlo Sampling/Simulation (MCS). Both the sequential and simultaneous computations of CVaR are studied. In the sequential approach, distribution of the optimal process output is generated first via MCS and then CVaR of this distribution is assessed. In the simultaneous approach, CVaR of the process output’s distribution is obtained in a single stage by augmenting the process/optimization model equations for each and every realization of the input uncertainties and by solving these augmented equations together with the equations of the CVaR. These sequential and simultaneous approaches are applied to simple yet illustrious benzoic acid plant and alkylation plant examples. For the profit and cost distributions of these process models, expected value, skewness/kurtosis, CVaR+, CVaR−, difference between CVaR+ and CVaR−, Rachev Ratio (RR), linearized RR, and some linear combinations of them are considered. CVaR− and CVaR+ are defined for the risk (left) and reward (right) sides of a probability distribution disjointedly. The results show that under the simultaneous scheme, where minimization of the difference between CVaR+ and CVaR− or minimization of the RR is used as the objective, or when they are linearly adjoined to a main objective such as the expected profit it is possible to (re)shape the probability distribution of optimal objective function values. Contrary to applications in economics and finance where CVaR and the RR are exclusively used to make the loss distribution less skewed to the left, in this work, the difference between CVaR+ and CVaR− or the RR are both successfully utilized to compress the distribution of the optimal profit around its mean in order to increase certainty on the mean optimal profit, despite uncertainties in process inputs.