Tolerance Limit-based Estimation of the Proportion of Non-conforming Parts in a Multiple Stream Process

The conventional way to characterize the proportion of non-conforming parts in a process is to calculate process capability indices and transform them into a ratio. These widely used indices are able to give digestible information about the ratio of non-conforming parts if some assumptions are fulfilled. A correct estimation method should be based on the output distribution of the process, and the uncertainty of the parameter estimates should be considered, as well. In this article, a special case of the output distribution is examined: a mixture of normal distributions is considered. In practice, this output distribution appears if a multiple stream process is investigated. The novelty of this study is to apply the tolerance interval-based estimation method for the proportion of non-conforming parts in a case study of a multiple stream process and to qualify the limitations of the proposed estimation method. A simulation study is performed to investigate the bias, mean square error, and root mean square error of the estimates from the two estimation methods (process performance index-based and tolerance interval-based) for different sample sizes for each stream (N ). It was found that, if it may be assumed that the speed of the streams is equal in the case of the sample sizes investigated (N = 25, 50, 100 per head), the proposed (tolerance interval-based) method overestimates the proportion of non-conforming parts while the conventional (process performance index-based) method underestimates it. The tolerance-limit based estimation method has asymptotically better properties than the process performance index-based estimation method.