Optimal Process Planning under Uncertainty with Risk Management

We propose a computationally-tractable optimization-based framework for risk management in midterm process planning under uncertainty. We employ stochastic programming to account for the uncertainty in which a scenario-based approach is used to represent the underlying probability distribution of the uncertain parameters. The problem is formulated as a two-stage stochastic program with recourse that is extended to incorporate the statistically-significant risk measure of Conditional Value-at-Risk (CVaR). However, since a large number of scenarios are often required to capture the uncertainty of the problem, the model suffers from the curse of dimensionality. To circumvent this problem, we propose a solution strategy with relatively low computational burden that involves a combination of simulation, scenario-based stochastic programming with recourse appended with risk management, and statistical-based scenario reduction technique. First, we solve an approximation of the risk-inclined multiscenario model for a number of randomly generated scenarios with an objective of minimizing the standard deviation of the Monte Carlo sampling estimator, which results in a convex stochastic quadratic program. The advantages of solving the approximation problem are two-fold: First, it only requires the use of a small number of samples. Second, we may utilize the standard deviation value of the Monte Carlo estimator (i.e., the objective value) within a scenario reduction procedure to determine the minimum number of scenarios that is theoretically required to obtain an optimal solution. Subsequently, the VaR parameters of the model are simulated for incorporation in a mean– CVaR stochastic linear programming approximation of the all-encompassing risk-averse planning model. The proposed strategy is implemented on a petroleum refinery planning case study with satisfactory results that illustrate how solutions with relatively affordable computational expense can be attained in a riskconscious modeling framework in the face of uncertainty.