Assessing the Proportion of Conformance of a Process from a Bayesian Perspective

T he question of assessing the capability of a process to produce according to some specifications connected with a measurable characteristic (X) of the items produced through it is of great importance in statistical process control. These specifications consist of the lower specification limit (L), which is the minimum acceptable process value; the upper specification limit (U), which is the maximum acceptable process value; and the target value (T), which coincides with the most desirable process value. Process capability indices constitute the most commonly used tool for measuring process capability. Every process capability index combines the values of process specifications, with those of the parameters of the process, that is, the mean (μ) and the standard deviation (σ), takes non-negative values, and its large values provide evidence of a capable process. The development and the estimation of a process capability index simplify substantially when the studied process is assumed to be normally distributed and most of the work that has appeared in the literature on this issue has been based on this assumption. For more details on various aspects of process capability indices, such as properties, estimation techniques, and interrelations, the reader is referred to Kotz and Johnson, Kotz and Lovelace, Pearn and Kotz, and Wu et al. Furthermore, Spiring et al. and Yum and Kim provide an extensive coverage of the bibliography on process capability indices. It should be noted that after the publication of the paper by Kane, which undoubtedly motivated many authors to deal with process capability indices, several papers on this issue have been published. The appearance of new papers concerning process capability indices is continuous even in the last 3 years; almost 30 years after the publication of the paper by Kane, many new papers have been published, introducing new process capability indices or new findings on existing ones (see, e.g., Negrin et al., Pearn et al., Perakis and Xekalaki, Jalili et al., Jose and Luke, and Shiau et al.). Another measure that provides information on the capability of a process is the proportion of conformance (yield), which is defined as the probability of producing within the so-called specification area, that is, the interval that consists of all the acceptable process values. Under the assumption that the process is normally distributed, it is defined as