This paper compares a number of approximations used to estimate means and variances of continuous random variables and/or to serve as substitutes for the probability distributions of such variables, with particular emphasis on three-point approximations. Numerical results from estimating means and variances of a set of beta distributions indicate surprisingly large differences in accuracy among approximations in current use, with some of the most popular ones such as the PERT and triangular-density-function approximations faring poorly. A simple new three-point approximation, which is a straightforward extension of earlier work by Pearson and Tukey, outperforms the others significantly in these tests, and also performs well in related multivariate tests involving the Dirichlet family of distributions. It offers an attractive alternative to currently used approximations in a variety of applications.probability modeling, approximation, decision analysis

This paper builds upon earlier work from the decision/risk analysis area in presenting simple, easy-to-use approximations for the mean and variance of PERT activity times. These approximations offer significant advantages over the PERT formulas currently being taught and used, as well as over recently proposed modifications. For instance, they are several orders of magnitude more accurate than their PERT counterparts in estimating means and variances of beta distributions if the data required for all methods are obtained accurately. Moreover, they utilize probability data that can be assessed more reliably than those required by the PERT formulas, while still requiring just three points from each activity time probability distribution. Using the proposed approximations can significantly improve the accuracy of probability statements about project completion time, and their use complements ongoing efforts to improve PERT analyses of networks involving multiple critical paths.PERT, three-point approximations, activity time estimation, project management

No abstract

Three-point discrete-distribution approximations are often used in decision and risk analyses to represent probability distributions for continuous random variables---e.g., as probability nodes in decision or probability trees. Performance evaluations of such approximations have generally been based on their accuracies in estimating moments for the underlying distributions. However, moment-based comparisons for recently proposed approximations have been limited. Moreover, very little research has addressed the accuracies of any such approximations in representing expected utilities or certainty equivalents of the underlying distributions. This is of potential concern since recent research shows three-point discrete-distribution approximations exist that match the first several moments of an underlying distribution exactly while, counterintuitively, approximating its certainty equivalents poorly. This paper compares the best two approximations for estimating means and variances identified in an earlier study with promising approximations proposed more recently. Specifically, it examines how accurately six simple general-purpose three-point approximations represent certainty equivalents for continuous random variables as the level of risk aversion is varied, as well as how accurately they estimate means and variances. The results show that several of these are quite accurate over a variety of test distributions when the level of risk aversion and the characteristics of the distributions are within reasonable bounds. Their robust performance is significant for decision analysis practice.decision analysis, approximation of expected utilities, certainty equivalents, probability distributions

We developed an optimization model for scheduling work, workers, and machines at the check-processing facility of Signet Bank, a $12 billion financial institution based in Richmond, Virginia.