Estimating process yield based on S-pk for multiple samples

Abstract

Boyles ( 1994) proposed a process measurement called S-pk, which provides an exact measure on the process yield for normal processes. Lee et al. ( 2002) considered an asymptotic distribution for the natural estimator of Spk under a single sample. In this paper, we extend the results for the case of multiple samples. We first compare the yield index S-pk with the most commonly used index, C-pk, and review some results of S-pk under a single sample. Next, we derive the sampling distribution for the estimator (S) over cap'(pk) of S-pk under multiple samples and find that for the same S-pk, the variance of (S) over cap'(pk) would be largest when the process mean is on the centre of specification limits. We calculate the lower bounds for various commonly used quality requirements under the situation with the largest variance of (S) over cap'(pk) for assurance purposes. To assess the normally approximated distribution of (S) over cap'(pk), we simulate with 10 000 replications to generate 10 000 estimates of (S) over cap'(pk), calculate their lower bounds, compare with the real (preset) S-pk and check the actual type I error. We also compute how many sample sizes are required for the normal approximation to converge to S-pk within a designated accuracy. Then, we present a real-world application of the one-cell rechargeable Li-ion battery packs, to illustrate how we apply the lower bounds to actual data collected from factories.