The processes capability adjustment for normal and non-normal distributions had been researched. In this section, we will review these papers about adjustments for normal processes, Gamma processes and Weibull processes.
2.1. Process Capability Adjustment for Normal Processes
Bothe (2002) provided a statistical reason why to add a 1.5
σ
shift to the average. Assuming the processes approximately normal distribution, control charts can’t reliably detect small movement in average. Table 1 displays the probabilities of detecting changes in μ versus subgroup size for shift=0.5(0.5)3σ
with n=3, 4 and 5. When μ had a small movement (ex: 0.5σ
, 1σ
) and the detection power of Shewhart X control chart is too small to discover. Then, small mean movement affects the PCIs accuracy. However, the probability of nonconformance will increase obviously. For example, when is 1.33, the probability of nonconformance is 64 ppm. If average occur pkC 1
σ
shift that be difficultly detected by control chart, the probability of nonconformance becomes 1350 ppm. The probability of nonconformance will increase twenty-fold.Bothe considered that adjustments should accord with the same detection standard.
Table 1. Probabilities of detection changes in μ versus subgroup size.
Subgroup Size
When subgroup size is 4 and mean shift is 1.5
σ
, the detecting power will be 0.5. Bothe (2002) considered providing the same detecting power in order to define the several adjustments with different subgroup size and called the adjustmentsS
50. By this idea, he set the detecting power to 50 percent and computed the several adjustments for different subgroup size. The reason which Bothe set the power to 50 percent was we want detect the processes out of control immediately if the process mean shifts and theARL (average run length)=1 is
1 the perfect condition. But in fact, theARL = is impossible. For this reason we
1 1 can just only set theARL =
1 2, and the detection power is 1 ARL , so we can 1 know ifARL =
1 2 the detecting power is 0.5. The results showed in Table 2.Table 2 displays shift sizes that have 50 percent chance of remaining undetected for subgroup sizes 1 through 6. Because shifts ranging in size from 0 up to
S
50σ
are the ones likely to remain undetected, a conservative approach is to assume that every missed shift is as large asS
50σ
. And Bothe invented dynamic Cpk beTable 2. Adjustment values for normal distribution with several subgroup size.
Subgroup Size
S
502.2. Process Capability Adjustment for Gamma Processes
When using the index , one of the most essential is that the process monitored is supposed to be stable and the output is approximately normally distributed. When the distribution of a process characteristic is non-normal, PCIs calculated using conventional methods could often lead to erroneous and misleading interpretation of the process’s capability. In the recent years, several approaches to the problems of PCIs for the non-normal populations have been suggested (see e.g. Pal (2005), Ding (2004), Pearn and Chen (1997), Kotz and Lovelace (1998), Somerville and Montgomery (1996), Kocherlakota and Kirmani (1992)). Several authors used data transformation techniques such as the Box-Cox power transformation, Johnson’s transformations and quantile transform techniques to solve this problem. And some authors replaced the unknown distribution by a known three or four-parameter distribution. Examples include Clments (1989), Franklin and Wasserman (1992), Shore (1998) and Polansky (1998).
Cpk
Hsu et al. (2007) provided the process capability adjustment for gamma process. For small process mean shifts, it is beyond the control chart detection power when process assumed gamma distribution and the process capability will be overestimated. They examine Bothe’s approach and find the detection power was less than 0.5 when data came from gamma distribution, showing that Bothe’s
adjustments are inadequate when we had gamma processes. Then, they calculate adjustments which called
AS under various sample sizes and gamma
50 parametern
N , with power fixed to 0.5. Table 3 displays the magnitude of
adjustmentsAS which they provided and data comes from Gamma (
50 ) with various values of andHsu et al. (2007) used the most common method for modifying PCIs in the non-normal case is the technique of quantile estimation. Analogous to the normal case, where the “natural” process width is between the 0.135th percentile and the 99.865th percentile, PCIs can be redefined in terms of their quantiles for possible modification in the non-normal case. The quantile definition for Cpk are defined as:
so that the normality assumption can be verified simultaneously. To consider the undetected process mean shift, they obtained Dynamic Cpk index for non-normal process by modifying Bothe’s Dynamic Cpk:
0.5 50 0.5 50
By considering an adjustment
AS
50σ
in this assessment for undetected shifts in process median, the estimate of dynamic index Cpk will decrease and the expected total number of nonconforming parts will increase. This nonconforming level assumes that undetected shifts are happening almost constantly and that every one is equal toAS
50σ .
2.3. Process Capability Adjustment for Weibull Processes
Li (2007) provided the process capability adjustment for Weibull process.
Weibull distribution doesn’t have reproductive, and the parameter of the X distribution can’t be found easily. They used a reference which Lu (2003) provided to approximate the cumulative density function of Xn of Weibull processes. The
and was 99.865
UCL LCL
th and 0.135th percentile of Xn distribution. We call the control chart they used is percentile Weibull control chart. Then they used the control limits to calculate the detection power for Weibull processes under the subgroup size and shape parametern
γ .Table 4.
AS
50 values for several n and various γ values whenk >
0.AS
50 Weibull distribution(1,γ ) for right shiftn 1 2 3 4 5 6 7 8 9 10
AS
50 Weibull distribution(1, γ ) for left shiftn 1 2 3 4 5 6 7 8 9 10
Since the shape of the Weibull distribution changing from positive skewness to negative skewness with increasing the shape parameter, they discussed two different cases. Process mean had right and left shifts. They used this cumulative
density function to compute the relationship between the mean shift and Type Ⅱ error and calculate the mean shift adjustment which means that the processes mean shift sigma when the detection power of control chart is 0.5. Table 4 and Table 5 display the magnitude of mean shift adjustments based on the detection power is 0.5 and data from Weibull (1,
AS
50AS
50AS
50γ ) distribution for various value of γ =1(1)10 and n=2(1)10 with right shift ( ) and left shift ( ). They also used the most common method for modifying PCIs in the non-normal case is the technique of quantile estimation, and the dynamic
k >
0 0k <
Cpk
was as the same as gamma processes which Hsu et al. (2007) provided.
The adjustments of Weibull processes are related that which control chart you choose to control the process. The Shwehart X control chart assumed that the process data come from a normal or near-normal distribution. When the data come from Weibull distribution, we should choose control charts for non-normal processes or for Weibull processes to control production process. Padgett and Supurrier (1990) use Monte Carlo simulation to construct Shewhart-type control charts for percentiles of strength distributions. Chan and Cui (2003) provided a skewness correction X and R charts for skewed distribution. This control chart proposed a skewness correction method for constructing the X and R control charts for skewed process distributions. Their asymmetric control limits are based on the degree of skewness estimated from the subgroups. Nichols and Padgett (2006) provided a bootstrap Weibull control chart. This control chart is use bootstrap method to simulate the UC and for monitoring Weibull percentiles. Erto (2006) provided a Weibull control chart which was used Bayes theorem to calculate the sampling distribution of Weibull percentile.