The specifications of the thickness of a lowalloy steel gear

The specifications of the thickness of a low-alloy steel gear bland are 0.300 plusminus 0.015 centimeter. If it is desired that the value of C_pk be at least 1.25, what would be the minimum sigma_x of a centered process? In part (a), if the specifications change to 0.3000 plusminus 0.010 centimeters, what would be the minimum sigma_x for a centered process? If it is desired for the process to have a C_p value of at least 1.25, what is the minimum sigma_x if X = 0.301 Centimeter? What if X = 0.298 centimeter? What does this say about using C_p as a measure of capability?

Solution

(a)

Here USL = 0.300+0.015 and LSL = 0.300-0.015

Cp = (USL – LSL) / 6s where s= standard deviation

Cpk = Min [ (USL-Xbar)/3s , (Xbar-LSS)/3s

Please refere the figure below for understanding the relationship between Cp and Cpk

Here, please note that Cp is dependent on variability within sample produced.

Whereas Cpk is depended on the SPREAD of the samples produced.

For a centred process Cpk and Cp have to be same

Therefore, for minimum Sigma level,

Cp=Cpk=1.25

Which mean

(USL – LSL)/ 6s = 1.25

6s = (USL – LSL )/1.25   ---- Formula 1.

s = (0.015+0.015)/ 6*1.25 = 0.004   (answer)

(b)

If specification changes to 0.3 +/- 0.01,

s = (0.01+0.01) / 6*1.25 = 0.0027

(c)

Equation for Cp is

Cp = (USL – LSL) / 6s where s= standard deviation

Here we see, Cp is not dependent on Xbar, ie. Mean value of the items produced.

Minimum standard deviation for 1.5 Cp will be same as before

(d)

Cp as a measure has some limitation as it does not specify if any process shift happens towards USL or LSL.

Cpk value together with Cp gives a complete indication.