Process capability index: A better way to assess equipment capability
minimum Cp value should be 1.00, with specifications if it was between 100 mil and tion spread (the numerator) is equal to the
many companies requiring a hurdle of 120 mil (the lower and upper specification process spread, ±3 σ (the denominator).
1.33 or higher. However, Cp is referred to limits, respectively). Before determining The capable process engineer should be
as ‘inherent’ capability because it merely PCI values, assume it is shown that the aware that ±3 σ of a normal distribution
incorporates the spread of the process, but process is normally distributed (e.g., by use accounts for 99.73% of the area under
not where the process mean is located. The of histograms or normal probability plots), the curve. Therefore the area outside of
centering of the process is quite crucial to and that the process is shown to be in con- the curve (and outside specifications of a
yields—one can have very high Cp values yet trol (e.g., by use of control charts)—these centered, Cp=1 process) is 0.27%. When
still have very poor yields if the centering are two important and necessary criteria translated to ppm, 0.27% = 2,700. More
of the process is not near the target value. for process capability analyses. Further let complete tables of PCI values as compared
To accommodate the centering of the us suppose that 30 parts randomly drawn to defect levels can be found in many
process, the Cpk formula was developed and from the process yield a sample standard statistical texts
3
.
is calculated as follows: deviation of 2.35 mil. With this infor- experimental—stencil printing
Because this index incorporates center- mation we can estimate the Cp value as
follows:
Table 1. Process capability index vs. defect level.
While that value appears to be a good
Cp ppm
ing and spread of the process as compared
0.5 133,614
to the specification limits, Cpk has been 1.0 2,700
dubbed as the ‘actual’ process capability. one (e.g., > 1.33), we still recommend plac-
1.3 96
While Cp and Cpk are the most funda- ing a CI around the estimate (the hat over
mental and often used, other PCIs exist. the ‘C’ indicates this is an estimate). Using
1.7 0.34
For example, Cpm and Cpmk incorporate the Cp CI formula, which is a function of
2.0 0.0018
the target of the part’s specification, Cpw is the sample standard deviation, s, sample
similar to Cpm, but incorporates a weighting size, and confidence level
2,3
we obtain the
function, and the Automotive Industry Ac- following 95% confidence interval for Cp: A simple experiment was set up to test
tion Group (AIAG) lobbies for the use of While the 1.42 initial estimate is a the theory in a real application environ-
Pp and Ppk when the process is not known ment on whether a stencil printer with a
to be in a state of statistical control. The stated accuracy can guarantee good results
interested reader is referred to Montgom- (yields). Two printers, Printer 1 and Printer
ery
3
for a review of these indices. good one, it is important to realize that 2, were set up to run 10 boards under
A point we stress time and again is to since the value is based upon a sample, it identical conditions. Both of these printers
make it clear to the practitioner that in all is possible that the Cp value (at 95% con- have a stated accuracy of + 25 µ. The test
likelihood the actual process parameters fidence) may actually be as low as 1.06—a vehicle for this experiment consisted of
of µ (mean) and σ (standard deviation) are borderline value. Thus, when a manufac- a 9” X 14” copper-clad board with pre-
unknown. This is something that is dif- turer reports their Cp, Cpk or other process printed fiducials. The stencil used in this
ficult for some younger engineers to wrap capability index, a knowledgeable process test was a 5 mil thick laser cut, electropol-
their minds around. Seldom, if ever, do we engineer should also ask for the confidence ished stencil with a 3 X 6, 1 mm2 aperture
know the true population parameters of interval and corresponding sample size matrix. Figure 1 shows the schematic of the
mean and sigma; however, we can get good used to estimate the index! stencil design. A commercially available
estimates of them if we obtain large sample type 3 lead-free paste was used for this
sizes from the process. But since they are analysis and use of a pci test. A Kho Young SPI system was used to
estimates, x for µ and s for σ, what we actu- Although PCIs are dimensionless—the characterize the print quality of both of the
ally are obtaining are estimates of Cp, Cpk, dimension of the numerator (e.g., mils, printers.
and other PCIs. And since we are obtain- mil
3
, inches, etc., is the same as the dimen-
ing estimates of these indices, we can and sion of the denominator and thus cancel result and discussion
should place confidence intervals (CIs) out)—their value lies in the ability to almost The distribution plot for Printer 1 and
around them
2
. The confidence interval for instantly equate them to defect levels. Table 2, in respect to the Y offset, is shown in
Cp is exact; Bissel’s approximate CI is often 1 provides a few Cp values and their associ- Figures 2 and 3. Note that assumption of
used for C
2
pk . In order to demonstrate the ated defect counts in terms of parts-per-mil- normality is reasonable due to the shape of
concept of process capability indices and lion (ppm) defects. The table does assume the histograms.
their use, the following section examines that the process is in control, is normally Hence, we can apply the process
a hypothetical process/product example distributed, the process is centered on the capability analysis to understand the actual
followed by an example specific to a stencil target value, and that the specifications are capability of both of the printers. Process
printer. two-sided. capability results from the Kho Young SPI
The interested reader may want to inspection showing the Y offset is shown
hypothetical example know that the defect levels are obtained by in Figures 4 and 5. As it can be seen from
For example purposes, let us suppose we determining the area under a normal distri- these plots, even though both the printers
have a part with the following specifica- bution curve that lies outside the specifi- report to have a Y offset specification of +
tions: 110 mil ± 10 mil. In other words, cation limits. For example, if a process is 25 µ, Printer 1 (see Figure 2) is not capable
the target value is 110 mil, but a part from centered on the target and its Cp value is at all.
the process would be considered to be in 1.00, then that implies that the specifica- This example serves to illustrate that
www.globalsmt.net Global SMT & Packaging – September 2008 – 23
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