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, C and p C discussed in Kane (1986), and more-advanced indices pk 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:
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 first capability index C considers the overall process variability relative p to the manufacturing tolerance, reflecting product quality consistency. Due to the design is simplicity, Cp can not reflect the tendency of process centering.
The index C was created in Japan to offset some of the weaknesses in pk C , primarily because the fact that p C measures capability in terms of process p variation only and does not take process location into consideration. However, C considers process variation and the location of process mean. It has been pk
regarded as a yield-based index since it provides lower bounds on process yield, and is always used to measure the quality of the process. For example, when the
pk = 1
C means that the product’s fractions of defectives is not more than 2700 parts per million (ppm) fall outside the specification limits. At Cpk = 1.33, the defect rate drops to 66 ppm. To achieve that defect rate less than 0.544 ppm, a C level of 1.67 is required. At a pk C level of 2.0, the defective rate reduced to pk 0.002 ppm. The exact number of nonconformities with fixed C is very pk
detect this movement obviously. So that the C will be underestimated the true pk number of nonconformities. At the present time, the C index is used more pk than any other index for measuring process capability. It is the reason why we study C more than other indices here. pk
Since Motorola, Inc. introduced its Six Sigma quality initiative, followers of this philosophy notion should add 1.5σ when estimating process capability.
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 C be overestimated, and pk he 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. 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 β
, ), with various values of scale parameter α and shape parameterβ
, 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
For some non-normal cases, Hsu et al. (2007) provided the process capability adjustment for gamma processes, and Li (2007) provided the process capability adjustment for Weibull processes, but they only investigate the change of the process mean shift. In real world, the process is dynamic, the mean and variance could change with small movement for momentary. In this thesis, we focus on the process variance change for non-normal cases.
We investigate Weibull distribution to calculate the ARL(average run length) by simulation. We also show the detection power performance of S2chart under variance change. In the cases, we show that the detection power in this control chart is very sensitive. When the data are from Weibull distribution, we provide the statistical derived variance change adjustment based on the chart subgroup size and distribution parameter to calculate the estimate of dynamic C when pk the data is non-normal distribution. It can make sure our process capability do not overestimate.
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 for mesn shift are included, and adjustment for Weibull process is also in Chapter 2. In Chapter 3, we introduce the characteristic of Weibull distribution, and introduce some properties for S of Weibull process. Then, we compare the difference 2 between normal process and Weibull process on variance distribution. In chapter 4, we use the MATLAB program to create a Monte-Carlo simulation to find upper and lower control limits for detecting variance change. We provide the simulation derived adjustments based on the chart’s subgroup size (for Weibull distribution) to calculate the estimator of dynamic C when the data is Weibull pk distribution. For illustrative purpose, application is presented in Chapter 5. Finally, we give some conclusion in Chapter 6.