Distribution of the Optimal Value of a Stochastic Mixed Zero-One Linear Optimization Problem under Objective Uncertainty

This paper is motivated by the question to approximate the distribution of the completion time of a project network with random activity durations. In general, we consider the mixed zero-one linear optimization problem under objective uncertainty, and develop an approach to approximate the distribution of its optimal value when the random objective coefficients follow a multivariate normal distribution. Linking our model to the classical Stein’s Identity, we show that the best normal approximation of the random optimal value, under the L-norm, can be computed by solving the persistency problem, first introduced by Bertsimas et al. (2006). We further extend our method to the minimum quadratic regret problem, and show that for any general mixed zero-one linear optimization problem, the minimum quadratic regret solution can be computed by solving a related persistency problem. Extensive computational studies on are presented to demonstrate the advantages of the new method.