Process or Product Monitoring and Control
Process capability compares the output of an in-control process
to the specification limits by using capability indices. The
comparison is made by forming the ratio of the spread between the
process specifications (the specification "width") to the spread of
the process values, as measured by 6 process standard deviation units
(the process "width").
Process Capability Indices
|A process capability index uses both the process variability and the process specifications to determine whether the process is "capable"||
We are often required to compare the output of a stable process with
the process specifications and make a statement about how well the process
meets specification. To do this we compare the natural variability
of a stable process with the process specification limits.
A process where almost all the measurements fall inside the specification limits is a capable process. This can be represented pictorially by the plot below:
There are several statistics that can be used to measure the capability of a process: Cp, Cpk, Cpm.
Most capability indices estimates are valid only if the sample size used is 'large enough'. Large enough is generally thought to be about 50 independent data values.
The Cp, Cpk, and Cpm statistics assume that the population of data values is normally distributed. Assuming a two-sided specification, if and are the mean and standard deviation, respectively, of the normal data and USL, LSL, and T are the upper and lower specification limits and the target value, respectively, then the population capability indices are defined as follows:
|Definitions of various process capability indices||
|Sample estimates of capability indices||
Sample estimators for these indices are given below.
(Estimators are indicated with a "hat" over them).
The estimator for Cpk can also be expressed as Cpk = Cp(1-k), where k is a scaled distance between the midpoint of the specification range, m, and the process mean, .
Denote the midpoint of the specification range by m = (USL+LSL)/2. The distance between the process mean, , and the optimum, which is m, is - m, where . The scaled distance is
(the absolute sign takes care of the case when ). To determine the estimated value, , we estimate by . Note that .
The estimator for the Cp index, adjusted by the k factor, is
Since , it follows that .
|Plot showing Cp for varying process widths||
To get an idea of the value of the Cp statistic for
varying process widths, consider the following plot
This can be expressed numerically by the table below:
|Translating capability into "rejects"||
where ppm = parts per million and ppb = parts per billion. Note that the reject figures are based on the assumption that the distribution is centered at .
We have discussed the situation with two spec. limits, the USL and LSL. This is known as the bilateral or two-sided case. There are many cases where only the lower or upper specifications are used. Using one spec limit is called unilateral or one-sided. The corresponding capability indices are
|One-sided specifications and the corresponding capability indices||
Estimators of Cpu and Cpl are obtained by replacing and by and s, respectively. The following relationship holds
Cp = (Cpu + Cpl) /2.This can be represented pictorially by
Note that we also can write:
|Confidence Limits For Capability Indices|
|Confidence intervals for indices||
Assuming normally distributed process data, the distribution of the
follows from a Chi-square distribution and
have distributions related to the non-central t distribution.
Fortunately, approximate confidence limits related to the normal
distribution have been derived. Various approximations to the
have been proposed, including those given by
Bissell (1990), and we will use a normal approximation.
The resulting formulas for confidence limits are given below:
100(1-)% Confidence Limits for Cp
ν = degrees of freedom.
|Confidence Intervals for Cpu and Cpl||
confidence limits for Cpu with sample size n are:
Limits for Cpl are obtained by replacing by .
|Confidence Interval for Cpk||
Zhang et al. (1990)
derived the exact variance for the estimator of
Cpk as well as an approximation for
large n. The reference paper is Zhang, Stenback and Wardrop (1990),
"Interval Estimation of the process capability index", Communications
in Statistics: Theory and Methods, 19(21), 4455-4470.
The variance is obtained as follows:
|Capability Index Example|
For a certain process the USL = 20 and the LSL = 8. The observed
= 16, and the
standard deviation, s = 2. From this we obtain
But it doesn't, since = 16. The factor is found by
|What happens if the process is not approximately normally distributed?|
|What you can do with non-normal data||
The indices that we considered thus far are based on normality of the
process distribution. This poses a problem when the process distribution
is not normal. Without going into the specifics, we can list some
There is, of course, much more that can be said about the case of nonnormal data. However, if a Box-Cox transformation can be successfully performed, one is encouraged to use it.