Least Squares Approximation to the Distribution of Project Completion Times with Gaussian Uncertainty

This paper is motivated by the following question: How to construct good approximation for the distribution of the solution value to linear optimization problem, when the random objective coefficients follow a multivariate normal distribution? Using Stein’s Identity, we show that the least squares normal approximation of the random optimal value can be computed by solving the persistency problem, first introduced by Bertsimas et al. (2006). We further extend our method to construct a least squares quadratic estimator to improve the accuracy of the approximation, in particular, to capture the skewness of the objective. Computational studies show that the new approach provides more accurate estimates of the distributions of project completion times compared to existing methods.