Solution to a Class of Two-stage Stochastic Programs with Multivariate Normal Uncertainty

We provide results for an efficient analytical valuation of partial moments of the multivariate Gaussian distribution over convex polyhedrons to aid the solution, sensitivity analysis and structural analysis of a large number of two-stage resource acquisition and allocation problems. These results decompose a partial multivariate moment into a function of multivariate probabilities that can be easily determined using the existing efficient numerical routines. The results are most useful in practice when (i) the structure of the Stage 2 allocation problem is known a-priori, for example when a greedy allocation algorithm is known to be optimal as is commonly the case for many resource sharing problems, or (ii) when the performances of various a-priori policies need to be evaluated, or (iii) when the number of uncertainties is small. We illustrate the use and benefit of the results over traditional simulation based approaches using a multi-resource newsvendor problem of practical size