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

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

Journal of Cost Analysis and Parametrics

#### Abstract:

Many cost estimating problems involve determining the number of units to build in a yield-constrained manufacturing process, when it takes, on average, n attempts to produce m successes (m ≤ n). Examples include computer chips, focal plane arrays, circuit boards, ﬁeld programmable gate arrays, etc. The simplistic approach to this problem is to multiply the number of units needed, m, by the expected number of attempts needed to produce a single success, n. For example, if a contractor reports that it takes, on average, 10 attempts to build one working unit, and if four such units are needed for a space-borne application, then the simplistic approach would be to plan for 4 × 10 = 40 units, and estimate the cost accordingly. However, if the cost analyst uses the simplistic approach, he or she is likely to be disappointed, as the probability that 40 attempts will actually produce four working units 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 units 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 research is to describe the nature of the problem, to justify modeling the problem in terms of a negative binomial random variable, and to develop the necessary thought process that one must go through in order to adequately determine the number of units to build given a desired level of conﬁdence. This understanding 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 units needed to achieve success in their programs.

Authors:

Timothy P. Anderson is Systems Director for NASA Program Assessments with The Aerospace Corporation, Arlington VA, and a professional cost analyst and operations research analyst with over 17 years experience, primarily in the context of Department of Defense (DoD) weapon systems and national security space acquisition. After retiring from the U.S. Navy in 2001 after 20 years service, he joined The Aerospace Corporation, serv-ing from 2001 to 2008 and again beginning in 2010, with a stint as a Technical Manager at MCR, LLC in the intervening period. Among his areas of expertise are cost analysis, cost uncertainty analysis, operations research, and decision analysis. His introduction to the estimating ﬁeld occurred in 1994 when the Navy assigned him to the Naval Center for Cost Analysis. His next, and last, Navy assignment was teaching cost estimation, opera-tions research, and other technical courses as a military professor at the Naval Postgraduate School, Monterey, CA. He is a SCEA Certiﬁed Cost Estimator/Analyst (CCEA®), a board member of the Washington DC Area chapter of SCEA, a former adjunct professor in the Systems Engineering/Operations Research Department of George Mason University, and an adjunct professor in the Systems Engineering and Operations Research departments of the Naval Postgraduate School. Mr. Anderson was recognized as the 2010 SCEA National Estimator/Analyst of the Year for Technical Achievement and was awarded the 2010 Wayne E. Meyer Award for Teaching Excellence by the Naval Postgraduate School. He is a frequent presenter of topics related to cost estimating and cost uncertainty analy-sis at forums, including SCEA, the Military Operations Research Society (MORS), the DoD Cost Analysis Symposium (DoDCAS), the Integrated Program Management (IPM) Conference, and the Space Systems Cost Analysis Group (SSCAG). 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.