Abstract
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.
Highlights
1.1 Capability indices Process capability indices (PCIs) are widely used in the manufacturing industry to quantify the capability in different processes [1, 2]
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 method overestimates the proportion of non-conforming parts while the conventional method underestimates it
To handle this, during the estimation of the non-conformity rate the one-sided upper tolerance limit is supposed to be known which is equal to the upper specification limit (USL) (LSL) value and the proportion of the distribution belonging to this tolerance limits are to be calculated
Summary
1.1 Capability indices Process capability indices (PCIs) are widely used in the manufacturing industry to quantify the capability in different processes [1, 2]. Where USL is the upper specification limit, LSL is the lower specification limit, the σ is the square root of the variance of the (according to the conventional interpretation: normal) distribution of the quality characteristic [4]. The expected value of the quality characteristic of interests is equal to the midpoint of the specification interval (μ = T = (USL – LSL)/2). If the quality characteristic is normally distributed and the process is centered in the midpoint of the specifications (USL + LSL)/2, the proportion of non-conforming parts equals 2Φ(−3CP), where Φ(∙) denotes the standard normal distribution function. If the probability model behind the investigated process is different from normal, the proportion of non-conforming parts can still be calculated according to this other model; it will correspond to the sum of the areas of the probability density of x below the LSL and above the USL. The usual standard deviation of the sample is used to estimate σLT
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