## Use Case of process capability estimation for a part with ovality

This post is an attempt to answer a practical issue faced when we try to estimate the process capability of a process for parts that "can be" oval in nature, but not necessarily expected to be oval.

Let's take an (exaggerated) example, for the sake of visualization of the concept. The customer has specified that the inner diameter of a ring should be 9 inches, with a tolerance of +/- 1 inches. However, let's say e measured a manufactured part and found it to be oval, with a minimum internal diameter of 8 inches and a maximum internal diameter of 10 inches.

## A different approach - multiple process capability

In a typical process capability study C_{p} and C_{pk} are estimated based on measurements taken in a single direction, on multiple samples manufactured from the same process.

However, in this case, we have measurements to be made in a two directions. Let's call them height(the longest diameter) and width (the shortest diameter). Now let's re-read customer specification as follows:

Target_{height} = 9 inches

USL _{height} = 10 inches

LSL _{height} = 8 inches

Similarly,

Target_{width} = 9 inches

USL _{width} = 10 inches

LSL _{width} = 8 inches

Now, follow the following steps:

- Draw multiple samples from the process.
- Measure the height and width inner diameter of each ring.
- Important: Do-not take average of the height and width.
- Test for stability of measured height and width diameters separately - If you are using Minitab, this step is built in in the Minitab assistant feature.
- If the height and width diameters measured are found to be a from a stable process, use the standard deviation of height and width diameter measured separately as follows:
_{ }

## Measure C_{p} & C_{pk} for height diameter separately

Use the formula for process capability as follows:

C_{p_height} = (USL_{height} - LSL_{height})/(6*σ_{height})

C_{pu_height} = (USL_{height} - X_{Bar_height})/(3*σ_{height})

C_{pl_height} = (X_{Bar_height}-LSL_{height} )/(3*σ_{height})

C_{pk_height} = min (C_{pu_height} , C_{pl_height} )

## Measure C_{p} & C_{pk} for width diameter separately

C_{p_width} = (USL_{width} - LSL_{width})/(6*σ_{width})

C_{pu_width} = (USL_{width} - X_{Bar_width})/(3*σ_{width})

C_{pl_width} = (X_{Bar_height}-LSL_{height} )/(3*σ_{width})

C_{pk_width} = min (C_{pu_width} , C_{pl_width} )

## Consolidate the process capability measure of height and width

Now, the above calculations are good for an internal quality department interest. However, the customer may be interested in only one value of process capability, and not separately for height and width dimensions we defined above. Going by the understanding of why C_{p }and C_{pk} are calculated - C_{p }is calculated to get an estimate of % out of spec, assuming the process is centered; and C_{pk} is measured to get an estimate of % out of spec irrespective of whether the process is centered or not.

One approach could be to convert the above height and width C_{p }and C_{pk} values to equivalent DPMO or % out of spec, and then add them up, then reverse lookup for equivalent C_{p }and C_{pk} values.