Process capability indices (PCIs) are widely-used statistical quality control tools for measuring process performance. In manufacturing industries, PCIs can be used to help companies stay competitive by providing understanding of process capability and guidance for quality improvement. However, the normality assumption for traditional PCIs, such as C_p and C_pk, might be violated in real-world manufacturing processes, and this can lead to erroneous conclusions of process capability and eventually financial losses for companies. Hence, many researchers modify existing PCIs or construct new PCIs to evaluate the performance of non-normal processes. Given various non-normal PCIs that have been proposed, there is an interesting in knowing how well these methods are, and there are demands for a proper comparative analysis.

In this study, we investigate four non-normal PCIs, C_Npk, C_s , C_y and C_py. We first compare these PCIs in terms of their capability to reflect true process yields under different parameter combinations of Weibull distribution, then examining their magnitude of estimation biases when implemented in different sample sizes. The result shows that C_y is the most exceptional among the four PCIs in the above two aspects, and has a recommended sampling size of 50 or more. Moreover, we employ a well-known inference approach, Bayesian inference with Markov-chain Monte Carlo (MCMC), to review the sampling variation derived from parametric estimation of the Weibull distribution. A series of simulations for establishing MCMC credibility intervals is conducted on C_y with the adaptive rejection Metropolis sampling (ARMS) algorithm. The coverage rates and average widths of credibility intervals are adopted as two criteria to assess the accuracy and precision of the MCMC method, respectively. It turns out that the MCMC credibility intervals perform satisfactorily in different sample sizes and under various parameter combinations of Weibull distribution.