Process Analysis - Process (Machine) Capability Analysis - Process Capability Indices
- Process range
- First, it is customary to establish the ± 3 Sigma limits around the nominal specifications. Actually, the Sigma limits should be the same as the ones used to bring the process under control using Shewhart control charts (see Quality Control). These limits denote the range of the process (i.e., process range). If we use the ± 3 Sigma limits, then, based on the normal distribution, we can estimate that approximately 99% of all piston rings fall within these limits.
- Specification limits LSL, USL
- Usually, engineering requirements dictate a range of acceptable values. In our example, it may have been determined that acceptable values for the piston ring diameters would be 74.0 ± .02 millimeters. Thus, the lower specification limit (LSL) for our process is 74.0 - 0.02 = 73.98; the upper specification limit (USL) is 74.0 + 0.02 = 74.02. The difference between USL and LSL is called the specification range.
Potential capability (Cp). This is the simplest and most straightforward indicator of process capability. It is defined as the ratio of the specification range to the process range; using ± 3 Sigma limits we can express this index as:
Cp = (USL-LSL)/(6*Sigma)
Put into words, this ratio expresses the proportion of the range of the normal curve that falls within the engineering specification limits (provided that the mean is on target, that is, that the process is centered, see below).
Bhote (1988) reports that prior to the widespread use of statistical quality control techniques (prior to 1980), the normal quality of US manufacturing processes was approximately Cp=.67. This means that the two 33/2 percent tail areas of the normal curve fall outside specification limits. As of 1988, only about 30% of US processes are at or below this level of quality (see Bhote, 1988, p. 51). Ideally, of course, we would like this index to be greater than 1, that is, we would like to achieve a process capability so that no (or almost no) items fall outside specification limits. Interestingly, in the early 1980's the Japanese manufacturing industry adopted as their standard Cp=1.33! The process capability required to manufacture high-tech products is usually even higher than this; Minolta has established a Cp index of 2.0 as their minimum standard (Bhote, 1988, p. 53), and as the standard for its suppliers. Note that high process capability usually implies lower, not higher costs, taking into account the costs due to poor quality. We will return to this point shortly.
Capability ratio (Cr). This index is equivalent to Cp; specifically, it is computed as 1/Cp (the inverse of Cp).
Lower/upper potential capability: Cpl, Cpu. A major shortcoming of the Cp (and Cr) index is that it may yield erroneous information if the process is not on target, that is, if it is not centered. We can express non-centering via the following quantities. First, upper and lower potential capability indices can be computed to reflect the deviation of the observed process mean from the LSL and USL. Assuming ± 3 Sigma limits as the process range, we compute:
Cpl = (Mean - LSL)/3*Sigma and Cpu = (USL - Mean)/3*Sigma
Obviously, if these values are not identical to each other, then the process is not centered.
- Non-centering correction (k)
- We can correct Cp for the effects of non-centering. Specifically, we can compute:
k=abs(D - Mean)/(1/2*(USL - LSL))
Where D = ( USL+LSL)/2. This correction factor expresses the non-centering (target specification minus mean) relative to the specification range.
Demonstrated excellence (Cpk). Finally, we can adjust Cp for the effect of non-centering by computing:
Cpk = (1-k)*Cp
If the process is perfectly centered, then k is equal to zero, and Cpk is equal to Cp. However, as the process drifts from the target specification, k increases and Cpk becomes smaller than Cp.
Potential Capability II: Cpm. A recent modification ( Chan, Cheng, & Spiring, 1988) to Cp is directed at adjusting the estimate of Sigma for the effect of (random) non-centering. Specifically, we may compute the alternative Sigma (Sigma2) as:
Sigma2 = { Σ (xi - TS)2/(n-1)} ½
where
Sigma2 is the alternative estimate of sigma xi is the value of the i'th observation in the sample TS is the target or nominal specification n is the number of observations in the sample We can then use this alternative estimate of Sigma to compute Cp as before; however, we will refer to the resultant index as Cpm.