1.1. Research Background and Motivation
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). 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:
μ μ
USL LSL USL LSL USL LSL
C C C
where
USL
is the upper specification limit,LSL
is the lower specification limit, μ is the process mean,σ
is the process standard deviation, 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, and is always used to measure the quality of the process.When data come from normal distribution, for a Cpk level of 1, statistically one would expect that the product’s fractions of defectives, is no more than 2700 parts per million (ppm) fall outside the specification limits. At Cpk=1.33, the defect rate drops to 66 ppm. To attain less than 0.544 ppm defect rate, a Cpk level of 1.67 is required. At a Cpk level of 2.0, the defective rate reduced to 0.002 ppm.
The exact number of nonconformities with fixed Cpk is very depending upon the location of the process mean and the magnitude of the process variation. Cpk is calculated under assumptions that the process is stable (the process mean and variation are not change), but in practice, the process is dynamic and the mean and variation always change with small movement for momentary, and the some control charts can’t detect obviously so that the Cpk will underestimated the true number of nonconformities.
The changes of various magnitudes not only happen on normal distribution, but also on non-normal distribution. Pyzdek (1995) has mentioned the distributions of certain chemical processes such as zinc plating thickness of a hot-dip galvanizing process are very quite often skewed. Choi (1996) presents an example of a skewed distribution in the ‘active area’ shaping stage of the wafer’s production process. Cygan et al. (1989) have mentioned that the lifetimes of
polypropylene films under high ac and dc field stresses be shown as a two-parameter Weibull distribution. The Weibull distribution, denoted as Weibull (α γ ), with various values of scale parameter ,
α
and shape parameter γ , covers a wide class of non-normal applications, including product life, product reliability and tensile strength of brittle materials, such as carbon and boron. The abundance of outputs from skewed distribution, the censoring, etc, makes the normality assumption often being illegitimate. Specifically, we assure the product lifetime which be from skewed distribution by statistic test and historical data. It will lead to underrate the probability of nonconformance that using the adjustment for normal case to adjust the non-normal cases.1.2. Research Purpose and Objectives
Ever since Motorola, Inc. introduced its Six Sigma quality initiative, followers of this philosophy notion should add 1.5
σ
when estimating process capability. By this idea we will find that 6-sigma actually translates to about 2 defects per billion opportunities, and 3.4 defects per million opportunities, which we normally define as Six Sigma, really corresponds to a sigma value of 4.5.When asked the reason for such an adjustment, six-sigma user claim it is necessary, but offer only personal experiences and three dated empirical literature.
Bothe (2002) provided a statistical reason to adjust the overestimated Cpk. Bothe set the adjustment of shift in average that was dependent on the same detection power of the control chart, and the data of Bothe’s study was assumed to be approximately normality distribution. However effectively non-normal process occurs frequently in practice. If the process capability indices based on the normal assumption concerning the data are used with non-normal observations, the value of the process capability indices may, in a majority of situation, be incorrect and quite likely misrepresent the actual product quality.
The control charts are commonly used in many industries for providing early warning for the shift in process mean. If the control chart detects a process mean shift, then the process is not under control. The well-known and usual Shewhart
X control charts assume that the observed process data come from a
near-normal distribution. However, when the process distribution is unknown or non-normal, the parameter estimators of sampling distribution may not be available theoretically. We can use approximation or simulation to estimate the parameters, such as percentile Weibull control chart which uses simulation to get the (upper control limits) and (lower control limits). But for Weibull processes, Erto (2007) used Bayes theorem to provide a Weibull control chart. If data come from Weibull distribution, we can control the process more exactly than non-normal control chart.UCL LCL
In this research, we show that the detection power performance of three Weibull control charts under the same mean shift adjustment which Bothe provided when the processes in control is very sensitive to the assumption of normality. Then, we compare with the detection power performances of the three control charts. Using the most powerful control chart to provide the statistical derived mean shift adjustment based on the chart subgroup size and distribution
parameter to calculate the estimator of when the data is non-normal distribution for Weibull distribution. pk
C
1.3. Thesis Organization
First, we introduce the research motivation and purpose in Chapter 1.
Secondly, a brief introduction of Bothe’s study and adjustment reason are included and adjustment for Gamma processes and Weibull processes in Chapter 2. In Chapter 3, we introduce the characteristic of Weibull distribution, and introduce some Weibull control charts for Weibull processes, and calculate the detection powers of control charts under the same shift for Weibull processes. We compare the detection powers to choose the best one. After that we calculate the adjustment for Weibull processes by using the best powerful Weibull control chart.
In Chapter 4, we introduce the calculation of dynamic non-normal index , and show the dynamic for Weibull processes. For illustrative purpose, an application is presented in Chapter 5. Finally, we give some conclusions in Chapter 6.
Cpk
Cpk