Question

In-Class Exercise - Cpk Problem Design specifications require that a key dimension on a product measure 110 + 13 units. A process being considered for producing this product has with a mean of 110 units and a standard deviation of four units. a. What can you say (quantitatively) regarding the process capability? b. Suppose the process average shifts to 98 units. Calculate the new process capability. c. What can you say about the process after the shift? Approximately what percentage of the items produced will be defective?

Answer 1

We have upper specification USL = 110+13 = 123 and lower specification LSL = 110-113 =97. The means = 110 and std dev = 4

a.

The process capability Cp is calculated as Cp = (USL-LSL)/6std dev = (123-97)/(6*4) = 1.08

Since the process average is centered, the value of process capability Cp and process capability index Cpk will be the same.

b.

If the process average becomes 98 then the process capability Cp will remain the same. However, the value of Cpk will be affected. We calculate the value of Cpk with the formula Cpk = min (CpU, CpL) where

CpU = (USL-mean)/3std dev = (123-98)/(3*4) = 2.08

CpL = (mean-LSL)/3std dev = (98-97)/(3*12) = 0.08

The new process capability index will be Cpk = 0.08

c.

The defect percentage is given by the probability of occurrence beyond the USL or LSL. In this case, the area that will be below the LSL will have a z value of (97-98)/4 = -0.25. This makes the corresponding probability as 0.4012

The area below the USL will be (123-98)/4 = 6.25. This means area above USL will nearly 0 %

Thus the percentage of defect is 0.4012 or 40.12%

The USL = 4.003 and the LSL = 3.997. The mean = 4.001 and std dev = 0.002

a.

CpU = (4.003-4.001)/(3*0.002) = 0.33

CpL = (4.001-3.997)/(3*0.002) = 0.66

Cpk = min (0.33, 0.66) = 0.33

b.

No. The value of Cpk 0.33 means that the process will have a very high rate of defect and will not be advisable to use the machine.

c.

Below the LSL z = (3.997-4.001)/0.002 = -2

Probability = 0.0227

Below the USL = (4.003-4.001)/0.002 = 1

Probability = 0.8413

Area under the specification = 0.8413 – 0.0227 = 0.8186

Defect percentage = 1 – 0.8186 = 0.1814 or 18.14%