Question

There are two machines for sale that you are considering purchasing for your sawmill to produce hardwood flooring. You want to find the one that has a higher process capability index, or Cpk. The goal is to produce flooring that is between 46 and 50 millimeters thick. The first machine is more accurate on average, producing to a mean of 48 millimeters...but unfortunately it has more variation with a standard deviation of 7 millimeters. The second machine is not as accurate, with a mean of 47mm, but does deliver a more consistent output, with standard deviation of 3mm.

[ Select] What is the Cpk of machine 1?
[Select] What is the Cpk of machine 2?
[ Select] If your goal is to be capable', what would you do?
[ Select] If (somehow) you could combine the best of both machines (the centering or average of machine 1 coupled with the constancy or standard deviation of machine 2, what would the Cpk be?

Answer 1

Machine 1 = 0.092

Machine 2 = 0.111

Combined = 0.222

Step-by-step explanation:

Given the following :

Lower specification limit (LSL) = 46 mm

Upper specification limit (USL) = 50 mm

MACHINE 1:

Mean 1 (m1) = 48

Standard deviation 1 (σ1) = 0.7

MACHINE 2:

Mean 2 (m2) = 47

Standard deviation 2 (σ2) = 0.3

Cpk formula:

Min(USLcpk, LSLcpk)

USLcpk = (USL - m) / 3σ

LSLcpk = (m - LSL) / 3σ

FOR MACHINE 1:

USLcpk = (50 - 48) / 3(7) = 0.0952

LSLcpk = (48 - 46) / 3(7) = 0.0952

Cpk = Min(0.952, 0.952) = 0.952

FOR MACHINE 2:

USLcpk = (50 - 47) / 3(3) = 0.333

LSLcpk = (47 - 46) / 3(3) = 0.111

Min(USLcpk, LSLcpk)

Cpk = Min(0.333, 0.111) = 0.111

When combined :

Mean = 48

σ = 3

USLcpk = (50 - 48) / 3(3) = 0.222

LSLcpk = (48 - 46) / 3(3) = 0.222

Min(USLcpk, LSLcpk)

Cpk = Min(0.222, 0.222) = 0.222