Developing a Sampling Plan Based on Cpk

Abstract

ABSTRACT The common sampling plans for variables rely solely on process performance and do not take into account the engineering specifications while determining the sample size needed. When the process capability index Cpk is good, the limits of the distribution of the lot (the γ/2th percentile and the 1−γ/2th percentile) lie inside the engineering specifications. In this case the area of the tail of the normal distribution that lies outside the engineering specifications is very small, and hence the effect of the sampling error (which is the result of estimating the mean) on this area is negligible. However, when the limits of the distribution lie close to the engineering specifications (that is, when the Cpk is marginal), the area of the tail that lies outside the engineering specifications can be meaningful and therefore the sampling error may result in inaccuracies in estimating the number of defective items in the lot. The objective of this research is to develop a multistage sampling plan, based on Cpk and adjusted to acceptance sampling for variables, taken as a random sample from a lot of size N (given) having (approximately) a normal distribution with a known variance. The sampling plan was compared to MIL-STD-414 (1957) and it was found (via simulations) that the Cpk sampling plan has a smaller probability to reject good lots (0–0.03% vs. 0–0.27% for the MIL-STD-414) and a smaller probability to accept defective lots (0–0.01% vs. 5.2–8.37% for the MIL-STD-414). In addition, the required sample size is smaller for processes with high capability index.