On Univariate Process Capability Indices: Some Issues and Applications

Abstract

Process capability indices (PCIs) are proposed to assess the performance of manufacturing processes. PCIs evaluate the process performance by comparing the manufacturing tolerance with the natural variability of the process. Since process parameters are unknown, therefore, PCIs are estimated based on a random sample. Assessment of performance of a process based on estimated PCI values may be unreliable due to sampling fluctuations. Hence, it is essential to study the statistical properties of the estimators of PCI. Thus far, most of the studies on PCIs have been done under the assumption that the quality characteristic under investigation follows an independent and identically normal distribution and that measurements are free from error. However, it is not uncommon for consecutive observations to be autocorrelated for some processes, especially in the chemical and process industries. Furthermore, even with the use of highly sophisticated measuring instruments, it is very difficult to avoid measurement errors. Some quality characteristics like taper, ovality, concentricity etc. are non-normal and skewed. In our study, we have considered some of these issues of autocorrelation, measurement error and non-normality on PCIs. We derive the expectation, the variance and the mean square error (MSE) of the estimator of a few PCIs. We consider the index Cp, the Taguchi index Cpm and the incapability index C′′pp when the quality characteristic is autocorrelated and measurement error is present. In particular, we have discussed a special type of autocorrelation process, namely an AR(1) process, which often occurs. Some properties of bias and MSE are discussed. Numerical values of bias and MSE are simulated for different combinations of sample size, autocorrelation and measurement error. Our analysis shows that the estimator behaves differently based on the level of autocorrelation and measurement errors. Due to sampling error, interval estimators are considered more reliable than point estimators. Estimation of an exact confidence interval is very difficult for PCIs as their distributions are very complicated. We propose a reliable confidence interval of the PCIs CNpk and Cpy using fiducial inference for general location-scale distributions. The performance of the proposed interval is compared with some bootstrap confidence intervals (BCI) in terms of coverage probability and average width. Simulation performance shows that our proposed confidence interval performs better than BCIs. The application of the proposed method for the selection of suppliers with superiorquality products is illustrated. We also derive the lower confidence bound for the lifetime performance index CL using the fiducial approach when the quality characteristic is type-II censored and follows location-scale distributions. Directions for further work are indicated at the end.