Conditional Probability and Conditional Expectation of a Multivariate Normal Random Variable Over a Rectangle

The idea of orthonormal variates, designed by Deak to find conditional probabilities of a multivariate normal random variable, is extended to compute the conditional expectation given that the observation falls inside an n-dimensional rectangle. Another possible technique is presented in the form of a transformation to independence. Numerical results are provided to illustrate the performance of the methods

A Standard Input Format for Multiperiod Stochastic Linear Programs

Data conventions for the automatic input of multiperiod stochastic linear programs are described. The input format is based on the MPSX standard and is designed to promote the efficient conversion of originally deterministic problems by introducing stochastic variants in separate files. A flexible "header" syntax generates a useful variety of stochastic dependencies. An extension using the NETGEN format is proposed for stochastic network programs

Models and model value in stochastic programming

Finding optimal decisions often involves the consideration of certain random or unknown parameters. A standard approach is to replace the random parameters by the expectations and to solve a deterministic mathematical program. A second approach is to consider possible future scenarios and the decision that would be best under each of these scenarios. The question then becomes how to choose among these alternatives. Both approaches may produce solutions that are far from optimal in the stochastic programming model that explicitly includes the random parameters. In this paper, we illustrate this advantage of a stochastic program model through two examples that are representative of the range of problems considered in stochastic programming. The paper focuses on the relative value of the stochastic program solution over a deterministic problem solution.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44253/1/10479_2005_Article_BF02031741.pd

On solving stochastic production planning problems via scenario modelling

production planning, scenario tree, compact representation, splitting variable representation, nonanticipativity,