6: II 7: III
4.5 Process Capability Analysis
At the completion of a Phase I study, where control charts have shown that the process is in control at what appears to be an acceptable level, then cal- culating a process capability ratio (PCR) is an appropriate way to document the current state of the process.
For example, X−R charts were made using the data in Table 4.3taken from Christensen, Betz and Stein[21]
TABLE 4.3: Groove Inside Diameter Lathe Operation
Subgroup No. Measurement 1 7.124 7.122 7.125 7.125 2 7.123 7.125 7.125 7.128 3 7.126 7.128 7.128 7.125 4 7.127 7.123 7.123 7.126 5 7.125 7.126 7.121 7.122 6 7.123 7.129 7.129 7.124 7 7.122 7.122 7.124 7.124 8 7.128 7.125 7.126 7.123 9 7.125 7.125 7.121 7.122 10 7.126 7.123 7.123 7.125 11 7.126 7.126 7.127 7.128 12 7.127 7.129 7.128 7.129 13 7.128 7.123 7.122 7.124 14 7.124 7.125 7.129 7.127 15 7.127 7.127 7.123 7.125 16 7.128 7.122 7.124 7.126 17 7.123 7.124 7.125 7.122 18 7.122 7.121 7.126 7.123 19 7.128 7.129 7.122 7.129 20 7.125 7.125 7.124 7.122 21 7.125 7.121 7.125 7.128 22 7.121 7.126 7.120 7.123 23 7.123 7.123 7.123 7.123 24 7.128 7.121 7.126 7.127 25 7.129 7.127 7.127 7.124
When variables control charts were used in Phase I, then the PCR’s Cp
andCpkare appropriate. These indices can be calculated with the formulas:
Cp= U SL−LSL
6σ (4.5)
Cpk= M in(CP u, CP l)
3σ , (4.6)
where
CP u =U SL−X σ
CP l=X−LSL
σ (4.7)
The measure of the standard deviation σ used in these formulas is estimated from the subgroup data in Phase I control charts. If X −R charts were used, then σ = R/d2, where d2 is given in the vignette SixSigma::ShewhartConstants in the R package SixSigma. They can also be found in Section 6.2.3.1 of the online NIST Engineering Statistics Hand- book. IfX−scharts were used, thenσ=s/c4, wherec4is found in the same tables for the appropriate subgroup sizen.
Assuming the data in Table 4.3 shows the process is in control, then the processCapability()function in the R packageqcc, thecp()function in the R package qualityTools, or thess.ca.study() function in the R package SixSigmacan compute the capability indices and display the histogram of the data with the specification limits superimposed as inFigure 4.21. The code to do this is shown using theprocessCapability() function in the R package qcc. The output is below the code.
R>Lathe<-read.table("Lathe.csv",header=TRUE,sep=","
,na.strings="NA",dec=".",strip.white=TRUE) R>library(qcc)
R>qcc(Lathe,type="R",plot=FALSE)
R>pc<-qcc(Lathe,type="xbar",plot=FALSE)
R>processCapability(pc,spec.limits=c(7.115,7.135)) -- Process Capability Analysis --- Number of obs = 100 Target = 7.125
Center = 7.1249 LSL = 7.115
StdDev = 0.002098106 USL = 7.135 Capability indices Value 2.5% 97.5%
Cp 1.59 1.37 1.81 Cp_l 1.57 1.38 1.76 Cp_u 1.60 1.41 1.80 Cp_k 1.57 1.34 1.80 Cpm 1.59 1.37 1.81
FIGURE 4.21: Capability Analysis of Lathe Data inTable 4.3
Exp<LSL 0% Obs<LSL 0%
Exp>USL 0% Obs>USL 0%
If the process is exactly centered between the lower and upper specification limits, thenCp =Cpk. In the example below, the center lineX = 7.1249 is very close to the center of the specification range and thereforeCp and Cpk are nearly equal. When the process is not centered,Cpkis a more appropriate measure than Cp. The indices Cpl = 1.573 and Cpu = 1.605 are called ZL andZU in some textbooks, and they would be appropriate if there was only a lower or upper specification limit. Otherwise Cpk = min(Cpl, Cpu). Since the capability indices are estimated from data, the report also gives 95%
confidence intervals for them.
To better understand the meaning of the capability index, a portion of the table presented by Montgomery[74] is shown inTable 4.4. The second column of the table shows the process fallout in ppm when the mean is such thatCpl
orCpu is equal to the PCR shown in the first column. The third column shows the process fallout in ppm when the process is centered between the two spec limits so thatCP is equal to the PCR. The fourth and fifth columns show the process fallout in ppm if the process mean shifted left or right by 1.5σafter the PCR had been established.
In this table it can be seen that a Cpl = 1.50 orCpu = 1.50 (when there is only a lower or upper specification limit) would result in only 4 ppm (or a proportion of 0.000004) out of specifications. IfCp = 1.50 and the process is
TABLE 4.4: Capability Indices and Process Fallout in PPM
Process Fallout (in ppm) Process Fallout with 1.5σshift PCR Single Spec Double Specs Single Spec Double Specs
0.50 66,807 133,614 500,000 501,350
1.00 1,350 2700 66,807 66,811
1.50 4 7 1,350 1,350
2.00 0.0009 0.0018 4 7
centered between upper and lower specification limits, then only 7 ppm (or a proportion of 0.000007) would be out of the specification limits. As shown in the right two columns of the table, even if the process mean shifted 1.5σ to the left or right after Cp was shown to be 1.50, there would still be only 1,350 ppm (or 0.00134 proportion) nonconforming. This would normally be an acceptable quality level (AQL).
Table 4.4 makes it clear why Ford motor company mandated that their suppliers demonstrate their processes were in a state of statistical control and had a capability index of 1.5 or greater. That way Ford could be guaranteed an acceptable quality level (AQL) for incoming components from their suppliers, without the need for acceptance sampling.
The validity of the PCRCp is dependent on the following.
1. The quality characteristic measured has a normal distribution.
2. The process is in a state of control.
3. For two sided specification limits, the process is centered.
The control chart and output of theprocess.capability()function serve to check these. The histogram of the Lathe data inFigure 4.21appears to jus- tify the normality assumption. Normal probability plots would also be useful for this purpose.
When Attribute control charts are used in a Phase I study, Cp and Cpk
are unnecessary. When using a por np chart, the final estimatepis a direct estimate of the process fallout. Forcanducharts, the average count represents the number of defects that can be expected for an inspection unit.