The six-sigma advocates claim it is necessary to add a 1.5
σ
shift to the average for most processes, with only personal experiences and three dated empirical studies as justification (see Bender (1975), Evans (1975), Gilson (1951)).In this chapter, we provide Bothe’s statistical rationale regarding this issue. The data in Bothe’s study was assumed to be close to normal distributions. For the process output having non-normal distributions, we also conduct some studies here.
2.1. Process Capability Adjustment for Normal Processes with Mean Shift
Shewhart control charts are very useful in phase I implementation of SPC, where the process is likely to be out of control and experiencing assignable causes that result in large shifts in the monitored parameters. Nevertheless, a major disadvantage of a Shewhart control chart is that it uses only the information about the process contained in the last sample observation and it ignores any information given by the entire sequence of points. This feature makes the Shewhart control chart relatively insensitive to small process shift. This potentially makes Shewhart control charts less useful in phase II monitoring problems, where the process tends to operate in control, reliable estimates of the process parameters (such as the mean and standard deviation) are available, and assignable causes do not typically result in large process upsets or disturbances.This is demonstrated in Table 1, displays the probabilities of detecting changes in μ versus subgroup size for shift 0.5(0.1)3= σ with n =3, 4 and 5 . The probabilities of detecting small shifts in μ are close to zero. As the size of the shift increases, so does the detection power of the X control chart to detect it, with sample subgroup sizes n =3, 4 and 5 eventually close to 100 percent for shifts in excess of 3
σ
.Table 1. Probabilities of detection the mean shift versus subgroup size n.
Subgroup Size Mean Shift Size
3 4 5
In studying the properties of control charts, the emphasis has been on determining the detection power and ARL(Average Run Length) of the chart.
The ARL of a chart is the expected number of samples to be taken before the chart detects a shift in the process characteristic. The ARL should be large when there has been no change in the process, but the ARL should be small when the process having undergone a change. The value of the ARL is depending on the purpose being studied. For any Shewshart control chart, we have noted that the ARL can also be expressed as
for the out-of-control
ARL , where
1α
is the probability of detecting a shift when none has occurred, and β is the probability of failing to detect a real shift in process characteristic. In general, we setARL = in most applications.
1 2 Therefore, the detection power is1
1 1
Detection power 1 0.5,
2 β ARL
= − = = =
that is, the probability of the control chart to detect the small shift immediately within two samples is 50 percent. By this idea, Bothe set the detection power to be 50 percent and computed the several magnitude of adjustments for various sample subgroup sizes. The results are shown in Table 2, which displays shift sizes that have a 50 percent chance of failing to detect the change in μ , which we refer to as
S
50, for various sample subgroup sizes from 1 to 6.Table 2.
S
50 values for normal distribution with various subgroup sizes.S
50Because shifts ranging in size from 0 up to are likely to remain undetected, a conservative approach is to assume that every missed shift is as large as
S
50S
50σ
. And Bothe made the modifications into the Cpk formula, called the dynamic Cpk, defined as follow:Dynamic
Cpkμ σ μ σ
2.2. Process Capability Adjustment for Non-normal Processes with Mean Shift
However, for the majority of cases, normal data seem impossible to be found in real-world situations. Pyzdek (1992) 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 processes.The abundance of outputs from skewed distributions, the censoring effects induced by the finite precision of actual measurements, stratification, etc., makes the normal assumption often unreasonable. Thus, there should be more concern about how the indexes are applied.
In the recent years, several approaches to dealing with 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 et al.(1992)). One approach to dealing with this situation is to transform the data so that in the new, transformed metric the data have normal distribution appearance. There are various graphical and analytical approaches to selecting a transformation, such as Box-Cox power transformation and Johnson’s transformations. And some authors replaced the unknown distribution by a known three or four-parameter distribution. Examples include Clements (1989), Franklin and Wasserman (1992), Shore (1998) and Polansky (1998).
There have also been attempts to modify the usual capability indices so that they are appropriate for both normal and non-normal distributions. The general idea is to use appropriate quantiles of the process distribution, and
, to define a quantile-based PCIs. Good discussions of these approaches are in Kotz and Lovelace (1998).
0.00135
x
0.99865
x
Hsu et al. (2007) examine Bothe’s study and find the detection power was less than 0.5 when data came from Gamma distributions, showing that Bothe’s statistical rationales are inadequate when we had Gamma processes. Then, Hsu et
al. (2007) calculate the magnitude of adjustments which called AS under
50 various sample subgroup sizes and Gamma parametern N , with power fixed
to 0.5. Table 3 displays the magnitude of adjustmentsAS which Hsu et al.
50 provided and data come from with various values ofand .
Gamma( ,1)N N =1(1)10
2(1)6 n =
Table 3.
AS
50 values for various subgroup sizes and various of Gamma(N, 1).n
To consider the undetected process mean shift, Hsu et al. (2007) obtained
Dynamic
Cpk index for non-normal processes by modifying Bothe’s Dynamic Cpk :Dynamic
Cpk 0.5 50 0.5 500.99865 0.5 0.5 0.00135
( ) ( )