1 Introduction
Process capability indices (PCIs) which provide numerical measure of production characteristic to reflect the quality of product have been used in the manufacturing industry. Those indices have become popular as unit-less measures on process potential and performance. The most commonly used ones, Cp and Cpk discussed in Kane (1986), and more advanced indices Cpm and Cpmk developed by Chan et al. (1988) and Pearn et al. (1992). Many authors have promoted the use of various PCIs for evaluating a supplier’s process capability. Based on analyzing the PCIs, a production department can trace and improve a poor process so that the quality level can be enhanced and the requirements of the customers can be satisfied. These PCIs have been defined explicitly as:
where USL is the upper specification limit, LSL is the lower specification limit, is the process mean, is the process standard deviation (overall process variation), and T is the target value. The index Cp considers the overall process variability relative to the manufacturing tolerance, reflecting product quality consistency. The index Cpk takes the magnitude of process variance as well as process departure from target value, and has been regarded as a yield-based index since it providing lower bounds on process yield. The index Cpm emphasizes on measuring the ability of the process to cluster around the target, which therefore reflects the degrees of process targeting (centering).
Since the design of Cpm is based on the average process loss relative to the manufacturing tolerance, the index Cpm provides an upper bound on the average process loss, which has been alternatively called the Taguchi index. The index Cpmk is constructed by appropriately combining the yield-based
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index Cpk and the loss-based index Cpm, accounting for the process yield as well as the process loss.
Since Motorola, Inc. introduced its Six Sigma quality initiative in the 1980s, quality practitioners have questioned why the followers of this initiative have added a 1.5σ shift to the process mean when estimating process capability. The advocates of Six Sigma have claimed that such an adjustment is necessary, but they have offered only personal experiences and three dated empirical studies as justification for this claim (see Bender (1975); Evans (1975); Gilson (1951)).
By examining the sensitivity of control charts to detect changes of various magnitudes, Bothe (2002) provided a statistically based reason for this claim. In his study, Bothe assumed that the process data is approximately normally distributed. However, non-normal processes occur frequently, in particular, in the semiconductor industry. Pyzdek (1992) mentioned that the distributions of certain chemical processes, such as zinc plating in a hot-dip galvanizing process, are very often skewed.
Choi et al. (1996) presented an example of a skewed distribution in the ‘‘active area’’ shaping stage of the wafer’s production processes. The abundance of outputs from skewed distributions, the stratification, tec., makes the normality assumption often unreasonable. The non-central chi-square distribution plays an important role in communications, for example in the analysis of mobile and wireless communication systems. It not only includes the important cases of a squared Rayleigh distribution and a squared Rice distribution, but also the generalizations to a sum of independent squared Gaussian random variables of identical variance with or without mean, i.e., a "squared MIMO Rayleigh" and "squared MIMO Rice" distribution. Therefore, a non-central chi-square process for data analysis has been chosen for this study. Moreover, if the capability indices based on the normal assumption concerning the data are used to deal with non-normal observations, the values of the capability indices may, in a majority of situations, be incorrect and quite likely misrepresent the actual product quality.
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The control charts are commonly used in many industries for providing early warning for the shift in the process mean. If the control chart detects a process mean shift, then the process is not under control. However, for momentary process mean shifts, it may be beyond the control chart detection power. Consequently, the undetected shifts may result in overestimating process capability.
If the process mean shifts are not detected, then unadjusted Cpk would overestimate the actual process yield. Bothe (2002) provided a statistical reason for considering such a shift in the process mean for normal processes. However, if the capability indices are based on the assumption of a normal distribution of data but are used to deal with non-normal observations, the values of the capability indices may, in the majority of situations, misrepresent actual product quality.
This paper is organized as follows. We first introduce the characteristic of non-central chi-squared distribution in Section 2. In Section 3, we examine Bothe’s approach and finds that the detection power of the control chart is less than 0.5 when data comes from non-central chi-squared distribution. This shows that Bothe’s adjustments are inadequate when we have non-central chi-squared processes. Therefore, we calculate the adjustments under various subgroup sizes (n) and non-central chi-square parameters λ with a fixed detection power of 0.5. Further, we provide the adjusted process capability formula to accommodate the undetected shifts when data is non-central chi-squared distribution. As a result, our adjustments provide significantly more accurate calculations of the capability of non-central chi-squared processes. In Section 4, we apply our method to asset of real data to illustrate the applicability of the process capability index. Finally, we conclude the paper with a brief summary in Section 5.
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