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 Cp and Cpk 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:
Targetheight = 9 inches
USL height = 10 inches
LSL height = 8 inches
Targetwidth = 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 Cp & Cpk for height diameter separately
Use the formula for process capability as follows:
Cp_height = (USLheight - LSLheight)/(6*σheight)
Cpu_height = (USLheight - XBar_height)/(3*σheight)
Cpl_height = (XBar_height-LSLheight )/(3*σheight)
Cpk_height = min (Cpu_height , Cpl_height )
Measure Cp & Cpk for width diameter separately
Cp_width = (USLwidth - LSLwidth)/(6*σwidth)
Cpu_width = (USLwidth - XBar_width)/(3*σwidth)
Cpl_width = (XBar_height-LSLheight )/(3*σwidth)
Cpk_width = min (Cpu_width , Cpl_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 Cp and Cpk are calculated - Cp is calculated to get an estimate of % out of spec, assuming the process is centered; and Cpk 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 Cp and Cpk values to equivalent DPMO or % out of spec, and then add them up, then reverse lookup for equivalent Cp and Cpk values.