Yield-related process capability indices for processes of multiple quality characteristics

Process capability indices (PCIs) have been widely used in industries for assessing the capability of manufacturing processes. Castagliola and Castellanos (Quality Technology and Quantitative Management 2005, 2(2):201-220), viewing that there were no clear links between the definition of the existing multivariate PCIs and theoretical proportion of nonconforming product items, defined a bivariate C_pk and C_p (denoted by BC_pk and BC_p, respectively) based on the proportions of nonconforming product items over four convex polygons for bivariate normal processes with a rectangular specification region. In this paper, we extend their definitions to MC_pk and MC_p for multivariate normal processes with flexible specification regions. To link the index to the yield, we establish a 'reachable' lower bound for the process yield as a function of MC_pk. An algorithm suitable for such processes is developed to compute the natural estimate of MC_pk from process data. Furthermore, we construct via the bootstrap approach the lower confidence bound of MC_pk, a measure often used by producers for quality assurance to consumers. As for BC _p, we first modify the original definition with a simple preprocessing step to make BC_p scale-invariant. A very efficient algorithm is developed for computing a natural estimator BC^p of BC_p. This new approach of BC_p can be easily extended to MC_p for multivariate processes. For BC_p, we further derive an approximate normal distribution for BC^p, which enables us to construct procedures for making statistical inferences about process capability based on data, including the hypothesis testing, confidence interval, and lower confidence bound. Finally, the proposed procedures are demonstrated with three real data sets.