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

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.