STATISTICAL QUALITY CONTROL

Submitted to: Sir. HakeemSubmitted by: Ms. Madiha Nabi Roll No. 15 7th Semester

PROCESS CAPABILITY

In process improvement efforts, the process capability index or process capability ratio is a statistical measure of process capability: The ability of a process to produce output within specification limits.[1] The concept of process capability only holds meaning for processes that are in a state of statistical control.

If the upper and lower specifications of the process are USL and LSL, the target process mean is T, the estimated mean of the process is and the estimated variability of the process (expressed as a standard deviation) is , then commonly-accepted process capability indices include:

Index Description

Estimates what the process would be capable of producing if the process could be centered. Assumes process output is approximately normally distributed.

Estimates process capability for specifications that consist of a lower limit only (for example, strength). Assumes process output is approximately normally distributed.

Estimates process capability for specifications that consist of an upper limit only (for example, concentration). Assumes process output is approximately normally distributed.

Estimates what the process is capable of producing if the process target is centered between the specification limits. If the process mean is not centered, overestimates process capability. if the process mean falls outside of the specification limits. Assumes process output is approximately normally distributed.

Estimates process capability around a target, T. is always greater than zero. Assumes process output is approximately normally distributed. is also known as the Taguchi capability index.[2]

Estimates process capability around a target, T, and accounts for an off-center process mean. Assumes process output is approximately normally distributed.

is estimated using the sample standard deviation.

PROCESS CAPABILITY INDICES:

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. 

Process capability indices are constructed to express more desirable capability with increasingly higher values. Values near or below zero indicate processes operating off target ( far from T) or with high variation.

Being the process capability a function of the specification, the Process Capability Index is as good as the specification is. For instance, if the specification came from an engineering guideline without considering the function and criticality of the part, a discussion around process capability is useless, and would have more benefits if focused on what are the real risks of having a part borderline out of specification. The loss function of Taguchi better illustrates this concept.

At least one academic expert recommends[3] the following:

SituationRecommended minimum

process capability for two-sided specifications

Recommended minimum process capability for one-

sided specification

Existing process 1.33 1.25

New process 1.50 1.45

Safety or critical parameter for existing process

1.50 1.45

Safety or critical parameter for new process

1.67 1.60

Six Sigma quality process

2.00 2.00

It should be noted though that where a process produces a characteristic with a capability index greater than 2.5, the unnecessary precision may be expensive[4].

TYPES OF PROCESS CAPBILITY INDICES:

CP

Historically, this is one of the first capability indexes used. The "natural tolerance" of the process is computed as 6s . The index simply makes a direct comparison of the process natural tolerance to the engineering requirements. Assuming the process distribution is normal and the process average is exactly centered between the engineering requirements, a CP index of 1 would give a "capable process." However, to allow a bit of room for process drift, the generally accepted minimum value for CP is 1.33. In general, the larger CP is, the better. The CP index has two major shortcomings. First, it can’t be used unless there are both upper and lower specifications. Second, it does not account for process centering. If the process average is not exactly centered relative to the engineering requirements, the CP index will give misleading results. In recent years, the CP index has largely been replaced by CPK (see below).

CPU AND CPL

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 CPU and Cpl. 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

CR

The CR index is algebraically equivalent to the CP index. The index simply makes a direct comparison of the process to the engineering requirements. Assuming the process distribution is normal and the process average is exactly centered between the engineering requirements, a CR index of 100% would give a "capable process." However, to allow a bit of room for process drift, the generally accepted maximum value for CR is 75%. In general, the smaller CR is, the better. The CR index suffers from the same shortcomings as the CP index.

CM

The CM index is generally used to evaluate machine capability studies, rather than full-blown process capability studies. Since variation will increase when normal sources of process variation are added (e.g., tooling, fixtures, materials, etc.), CM uses a four sigma spread rather than a three sigma spread.

ZU

The ZU index measures the process location (central tendency) relative to its standard deviation and the upper requirement. If the distribution is normal, the value of ZU can be used to determine the percentage above the upper requirement by using Table 4 in the appendix of The Complete Guide to the CQM. The method is the same as described in Chapter III.B using the Z statistic, simply use ZU instead of using Z. In general, the bigger ZU is, the better. A value of at least +3 is required to assure that 0.1% or less defective will be produced. A value of +4 is generally desired to allow some room for process drift.

ZL

The ZL index measures the process location relative to its standard deviation and the lower requirement. If the distribution is normal, the value of ZL can be used to determine the percentage above the upper requirement by using Table 4 in the appendix of The Complete Guide to the CQM. In general, the bigger ZL is, the better. A value of at least +3 is required to assure that 0.1% or less defective will be produced. A value of +4 is generally desired to allow some room for process drift.

ZMIN

The value of ZMIN is simply the smaller of the ZL or the ZU values. It is used in computing CPK.

CPK

The value of CPK is simply ZMIN divided by 3. Since the smallest value represents the nearest specification, the value of CPK tells you if the process is truly capable of meeting requirements. A CPK of at least +1 is required, and +1.33 is preferred. Note that CPK is closely related to CP, and that the difference between CPK and CP represents the potential gain to be had from centering the process.

CPM

A CPM of at least 1 is required, and 1.33 is preferred. CPM is closely related to CP. The difference represents the potential gain to be obtained by moving the process mean closer to the target. Unlike CPK, the target need not be the center of the specification range.

CNPK

Another set of indices, that apply to non-normal distributions is called Cnpk (for non-parametric Cpk). Its estimator is calculated by

Where p(0.995) is the 99.5th percentile of the data and p(.005) is the 0.5th percentile of the data.