## 3.3.5 Estimating Process Capability Indices for Autocorrelated Processes

Nien Fan Zhang

Statistical Engineering Division, ITL

Process capability indices (CPI) have been widely used in manufacturing industries to measure a process' performance in meeting preset specification limits. They are also used by supplier companies to demonstrate the quality of their products. Among all the capability indices, Cp and Cpk are the most widely used. In recent years there have been a lot of discussions and debates about the use of process capability indices. Interval estimation of the process capability indices was proposed. In practice, there is also a concern about the assumption of the mutual independence of the process observations. It is well known that in practice process data are often autocorrelated. This is especially true for continuous manufacturing processes such as chemical processes. When the sampling frequency is not too low, the observations are often autocorrelated. In process industries, it is common for quality personnel and process operators to use the capability indices to monitor the process performance. In this case, the variances of the sample CPI's when the data are autocorrelated are needed to construct the interval estimates of CPI.

We assume that the process is a discrete weakly stationary process. Cp and Cpk are defined in the same way as when the process observations are independent. Under the above assumption, the expectation and variance of the sample process variance were derived. It also has been shown that the covariance between the sample process mean and sample process variance is zero when the process is weakly stationary.

Approximate variances of Cp, one-sided Cpk, and Cpk have been derived in similar forms when the process observations are independent. These variances can be easily calculated based on the corresponding CPI, sample size, the process variance and autocorrelations. Thus, interval estimators of capability indices can be constructed when the process is stationary. In particular, when the process is a first order autoregressive (AR(1)) process, the approximate variances are expressed in terms of the process parameter phi, sample size, process variance and CPI. For a fixed process parameter, the attached figure shows how the variances of Cp and Cpk decrease as the sample size increases. In the figure, the curves with markers of o", *", and +" correspond to the AR(1) processes with phi=0.25, 0.50 and 0.75 respectively.

Simulations have been done to find the coverage probability of k-sigma intervals of Cp and Cpk. The results show that the true Cp and Cpk lie within the interval roughly 99% of the times when k=3 and about 93% of the times when k=2.

This work was presented at the 1996 Joint Statistical Meetings.

Figure 25: This figure shows how the variances of Cp and Cpk decrease as the sample size increases.

Date created: 7/20/2001
Last updated: 7/20/2001