# Suppose that a process has upper and lower specifications at USL = 62 and LSL = 38, and a Target...

## Question:

Suppose that a process has upper and lower specifications at USL = 62 and LSL = 38, and a Target = 50. A sample from this process reveals that the process mean |x= 52 and that the standard deviation S = 2. The product cost is \$100. Calculate {eq}C_p, C_{pk} {/eq}, and {eq}C_{pm} {/eq}? Is the process capable?

## Process Capability

A process is capable if a target fraction of its output satisfies customer requirements, i.e., its parameters are within the specification limits. Common ways to measure process capability include the process capability ratio, process capability index, and Taguchi capability index.

The process capability ratio can be found as

{eq}C_p=\frac{USL-LSL}{6\times std.deviation}\\ \\ =\frac{62-38}{6\times 2}\\ \\ =2 {/eq}

Next, we find the process capability index:

{eq}C_{pk}=\min\left\{ \left ( \frac{USL-mean}{3\times std.deviation} \right ), \left ( \frac{mean-LSL}{3\times std.deviation} \right )\right\}\\ \\ =\min\left\{ \frac{62-52}{3\times 2}, \frac{52-38}{3\times 2} \right\}\\ \\ =\min\left\{1.67, 2.33 \right\}\\ \\ =1.67 {/eq}

Finally, we find the Taguchi capability index:

{eq}C_{pm}= \frac{USL-LSL}{6\sqrt{\sigma ^2+(\mu -\text{target})^2}}\\ \\ =\frac{62-38}{6\sqrt{2^2+(52-50)^2}}\\ \\ =\frac{24}{6\sqrt{4+4}}\\ \\ =\frac{4}{\sqrt{8}}\\ \\ =1.41 {/eq}

As all these metrics are greater than 1, we conclude that the process is capable.