Convex Approximations of a Probabilistic Bicriteria Model with Disruptions

We consider a multiperiod system operation problem with two conflicting objectives, minimizing cost and risk. Risk stems from uncertain disruptions to the system during operation. Whereas a general model would hedge against disruptions in each time period, we study special cases in which only a modest number of disruptions occur. To optimize for risk, we employ a convex approximation based on constraint sampling. We develop a stratified sampling scheme based on distributional information on the time of disruption. We establish that our scheme yields significant savings in sampling costs–up to an order of magnitude in the number of time periods–over naive sampling. Moreover, in the absence of distributional information, we exhibit a sampling strategy that has comparable performance to optimal stratification. We numerically demonstrate that stratification improves cost over naive sampling, improving the solution's proximity to the efficient frontier of the bicriteria problem.