Technical Note—Multivariate Partial-Expectation Results for Exact Solutions of Two-Stage Problems

Saurabh Bansal () and James S. Dyer ()
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Saurabh Bansal: Pennsylvania State University, University Park, Pennsylvania 16802
James S. Dyer: University of Texas at Austin, Austin, Texas 78712

Operations Research, 2017, vol. 65, issue 6, 1526-1534

Abstract: 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 payoff is observed. We focus on problems where the payoff 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.

Keywords: two-stage problems; optimization; decision making under uncertainty (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (2)

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