Assessing multivariate process/product yield via discrete point approximation

The assessment of multivariate yield is central to the robust design of products/processes. Currently, yield is evaluated via Monte Carlo simulation. However, it requires thousands of replications per simulation to achieve an acceptably precise estimate of yield, this is often tedious and time consuming, thereby rendering it unattractive as an evaluation tool. We propose a discrete point approximation on each design variable, using general Beta distributions, for assessing reasonably precise multivariate yield estimates, which require only a minute fraction of the Monte Carlo replications/simulations required to estimate yield (e.g., 3 and 5 design variables would require only 33=27 and 35=243 replications, respectively). The Beta distribution has the desirable property of being able to characterize a wide variety of processes that may or may not be symmetric and which may or may not have a finite operating range. Using an approach that computes the roots of a polynomial, whose degree is determined by the number of discrete points, discrete three point approximations are obtained and tabulated for twenty-five different Beta distributions. Based on several test examples, where design parameters are modeled as independent Beta random variates, our approach appears to be highly accurate, achieving virtually the same multivariate yield estimate as that obtained via Monte Carlo simulation. The substantial reduction in the number of replications and associated computational time required to assess yield makes the iterative adjustment of design parameters a more practical design strategy.