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Using Method of Moments in Schedule Risk Analysis

Risk Track




A program schedule is a critical tool of program management. Program schedules based on discrete estimates of time lack the necessary information to provide a robust determination of schedule uncertainty and therefore the risk that the proposed schedule will be completed on time. To determine the risk that a proposed, discrete schedule will meet a particular schedule and to find the probable critical paths (i.e., the criticality index), a probabilistic schedule risk analysis (SRA) is performed.
SRA is a process by which probability density functions (PDFs), usually triangular, are defined at the task level in an effort to quantify the uncertainty of each task element. The network structure provides the relationships of successor and predecessor tasks that define the mathematical problem to be solved. Typically, a statistical sampling technique (e.g., Monte Carlo sampling) is used to find a final, probabilistic project end date and criticality index.

The primary method of performing SRA is through a statistical simulation (i.e., Monte Carlo approach), however there are major drawbacks: Correlated random variables representing probabilistic schedule durations can lead to unwieldy correlation matrices, and the amount of time required to perform a meaningful SRA can be very time consuming. Another method is an analytic approach using the method of moments, whereby the moments (e.g., the mean and variance) of the input distributions defined at the task level are used to determine the moments of the probability distribution representing the task finish date. This method has been and continues to be used in the cost risk analysis community to nearly instantaneously and precisely determine the primary moments of WBS summations. Shortcomings of MoM when applied to SRA include problems with calculating the PDFs where tasks merge (i.e., merge bias) and calculating a criticality index.

The author of this paper has found a method of analytically determining the schedule PDF using MoM that solves these two problems. Journal literature from the mathematical and electrical engineering community (specifically the IEEE transactions on VLSI) demonstrates how these problems can be overcome, and provides formulae for computing the distributions formed by merging parallel schedule tasks. Additional assumptions pertaining to the correlation of merged parallel paths allow the use of MoM in SRAs that provide instantaneous solutions with minimal definition of correlation matrices.


Raymond P. Covert
Raymond P. Covert, Technical Director and Chief Practitioner for Cost and Schedule Analysis at MCR, is responsible for ensuring technical excellence of MCR products, services, and processes by encouraging process improvement, maintaining quality control, and training employees and government and industrial customers in cost and schedule analysis and associated program-control disciplines. He has given numerous technical and tutorial presentations on cost-risk analysis, cost-estimating relationship development, and other statistical aspects of cost and economics to DoD, NASA, and EACE (European Aerospace Working Group on Cost Engineering) Cost Symposia, the AF/NASA/ESA Space Systems Cost Analysis Group (SSCAG), and professional societies such as the International Society of Parametric Analysts (ISPA), Society for Cost Estimating and Analysis (SCEA), Military Operations Research Society (MORS), U.K. Association of Cost Engineers (ACostE), and the American Institute of Aeronautics and Astronautics (AIAA).