Technical note-Multivariate partial-expectation results for exact solutions of two-stage problems

We consider two-stage sequential decision-making problems where in Stage 1 an initial decision is made under a multivariate uncertainty, and in Stage 2 the uncertainty is resolved, a further decision is made based on the uncertainty realization, and the payo is observed. We focus on problems where the payo is a linear function of the multivariate uncertainty realization. Such problems can be written as single-stage nonlinear optimization problems composed of partial polyhedral expectations of the multivariate uncertainty. We identify the structural characteristics of multivariate probability density functions under which the integral expressions for the partial expectations can be directly evaluated for an exact value. We then focus on elliptical distributions, which are frequently used in operations management and do have these characteristics. We develop a sequence of three results to determine partial polyhedral expectations of elliptical probability distributions, with a special emphasis on the normal distribution. These results are useful for solving several commonly faced two-stage problems in operations management for an exact solution, performing a comparative static analysis, and rank ordering the alternatives.