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A Probabilistic Approach to Determining the Number of Widgets to Build in a Yield-Constrained Process

Models and Methods Track




Many cost estimating problems involve determining the number of Widgets to build when it takes, on average, M attempts to produce N successes. Examples include computer chips, focal plane arrays, circuit boards, FPGAs, etc. The simplistic approach to this problem is to multiply the number of Widgets needed, N, by the average number of attempts needed to produce a success, M. For example, if a contractor reports that it takes, on average, 10 attempts to build one working focal plane array (FPA), and if four such FPAs are needed for a space-borne application, then the simplistic approach would be to plan for 4 x 10 = 40 units, and estimate the cost accordingly. However, if the cost analyst uses the simplistic approach, he/she is likely to be disappointed, as the probability that 40 attempts will actually produce 4 working FPAs is only about 57%. Consequently, there is a 43% probability that 40 attempts will be insufficient. In fact, if the analyst wants to have, say, 80% confidence that four working FPAs will be available, then he/she should plan for 54 attempts! Obviously, this could have a huge impact on the cost estimate.

The purpose of this presentation is to describe the nature of the problem, to model the problem as a Negative Binomial distribution, and to develop the necessary thought process that one must go through in order to adequately determine the number of widgets to build given a desired level of confidence. This will be of great benefit to cost analysts who are in the position of estimating costs when certain hardware elements behave as described previously. The technique will also be very useful in cost uncertainty analysis, enabling the cost analyst to determine the appropriate probability distribution for the number of widgets needed to achieve success in their programs.


Timothy P. Anderson
The Aerospace Corporation
Mr. Timothy P. Anderson is a Senior Engineering Specialist for The Aerospace Corporation, and a professional cost analyst and operations research analyst with over sixteen years experience, primarily in the context of Department of Defense (DoD) weapon systems and national security space acquisition. Prior to returning to The Aerospace Corporation, Tim served for two years as a Technical Manager with MCR, LLC. His areas of interest are cost analysis, cost uncertainty analysis, operations research, and decision analysis. Mr. Anderson served for 20 years in the U.S. Navy and began working in the cost estimating field in 1994 while assigned to the Naval Center for Cost Analysis. Following that he served as a military professor at the Naval Postgraduate School, teaching cost estimation, operations research, and other technical courses. He retired from the Navy in June 2001. Mr. Anderson has a B.S. in Industrial and Operations Engineering from the University of Michigan, and an M.S. in Operations Research from the Naval Postgraduate School. He is a Society of Cost Estimating and Analysis (SCEA) certified cost estimator/analyst; a board member of the Washington D.C. Area chapter of SCEA; an adjunct professor in the Systems Engineering/Operations Research Department at George Mason University; an adjunct professor in the Systems Engineering Department (Distributed Learning Programs) of the Naval Postgraduate School; recognized as the 2010 SCEA National Estimator/Analyst of the Year for Technical Achievement; and a frequent presenter of topics related to cost estimating and cost uncertainty analysis at forums including SCEA, the Military Operations Research Society (MORS), the DoD Cost Analysis Symposium (DoDCAS) and the Space Systems Cost Analysis Group (SSCAG).