# Process Capability and Performance (Cp, Cpk, Pp, Ppk, Cpm)

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## What Is Process Capability Analysis?

Process capability analysis is a statistical tool used to evaluate the ability of a process to produce output within specified limits, or the "potential capability." It is used to determine whether a process is capable of meeting customer requirements and specifications and also to identify opportunities for improvement.

Process capability analysis involves comparing the natural variability of a process with the allowable tolerance range for the output.

Cp and Cpk are commonly referred to as process capability indices. They are statistical measures used in process capability analysis to evaluate the ability of a process to produce output within specified limits.

First, let's understand what Cp is. Cp, or process capability, is the ratio of the spread between the process specification and the six times process standard deviation. In other words, it is a measure of how well a process is capable of meeting customer requirements. A higher value indicates a greater degree of capability.

The formula for Cp (process capability) is:

$$Cp = \frac{USL - LSL}{6\sigma}$$

where USL and LSL are the upper and lower specification limits, respectively.

Cpk is calculated by taking the minimum difference between the process mean and the specification limit (upper or lower specification, whichever is nearest to the process mean) and dividing it by three times the process standard deviation.

The formula for Cpk (process capability index) is:

$$Cpk = \min\left(\frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma}\right)$$

where μ is the mean or average of the process.

## Difference between Cp and Cpk

The main difference between Cp and Cpk is that Cp only considers the spread of the process data relative to the specification limits, while Cpk also considers the location of the process average relative to the specification limits. This means that Cpk provides a more comprehensive view of the process capabilities and limitations and can be a more useful metric for identifying and addressing potential problems.

For example, if a process has a high Cp value but a low Cpk value, it may indicate that it can meet the specification limits in terms of spread, but the process average is not centred. In this case, it may be necessary to adjust the process to ensure that the average is in the middle of specification limits to improve the overall process capability.

## What is Process Performance Analysis?

Process performance analysis is a statistical tool used to evaluate the "actual performance" of a process relative to its desired performance. It is used to determine whether a process is meeting the customer's requirements and specifications and to identify opportunities for improvement.

The process performance analysis uses statistical measures such as the process performance index (Pp and Ppk),.

Pp, is similar to Cp, but it uses the long-term standard deviation instead of the short-term standard deviation. Ppk, is the same as Cpk but uses the long-term standard deviation.

The formula for Pp is:

$$Pp = \frac{USL - LSL}{6\sigma}$$

where USL is the upper specification limit, LSL is the lower specification limit, and σ is the long-term standard deviation.

The formula for Ppk is:

$$Ppk = \min\left(\frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma}\right)$$

where μ is the mean or average of the process. For Ppk, we use long-term standard deviation σ_overall.

## Taguchi Capability Index (Cpm)

Taguchi capability index is a statistical measure used to evaluate the performance of a manufacturing process in relation to its target value and specification limits. It is calculated using the formula:

$$C_{pm}=\frac{USL-LSL}{6\sqrt{\sigma^2+(\mu-Target)^2}}$$

where Cpm is the Taguchi capability index, Target is the target value of the process, USL and LSL are the upper and lower specification limits, and σ is the standard deviation of the process.

This index is named after Japanese engineer Genichi Taguchi, who developed it as part of his "quality loss function" approach to engineering. It is often used in conjunction with other process performance metrics, such as Cp, Cpk, and Ppk, to provide a more comprehensive view of a process's performance.

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