Question

Beaver Brothers, Inc. is conducting a study to assess the capability of its 150-gram bar soap production line. A critical quality measure is the weight of the soap bars after stamping. The lower and upper specification limits are 162 and 170 grams, respectively. As a part of an initial capability study, 25 samples of size 5 were collected by the quality assurance group and the observations in Table 3.7 were recorded.

Sample	OBS.1	OBS.2	OBS.3	OBS.4	OBS.5
1	167.0	159.6	161.6	164.0	165.3
2	156.2	159.5	161.7	164.0	165.3
3	167.0	162.9	162.9	164.0	165.4
4	167.0	159.6	163.7	164.1	165.4
5	156.3	160.0	162.9	164.1	165.5
6	164.0	164.2	163.0	164.2	163.9
7	161.3	163.0	164.2	157.0	160.6
8	163.1	164.2	156.9	160.1	163.1
9	164.3	157.0	161.2	163.2	164.4
10	156.9	161.0	163.2	164.3	157.3
11	161.0	163.3	164.4	157.6	160.6
12	163.3	164.5	158.4	160.1	163.3
13	158.2	161.3	163.5	164.6	158.7
14	161.5	163.5	164.7	158.6	162.5
15	163.6	164.8	158.0	162.4	163.6
16	164.5	158.5	160.3	163.4	164.6
17	164.9	157.9	162.3	163.7	165.1
18	155.0	162.2	163.7	164.8	159.6
19	162.1	163.9	165.1	159.3	162.0
20	165.2	159.1	161.6	163.9	165.2
21	164.9	165.1	159.9	162.0	163.7
22	167.6	165.6	165.6	156.7	165.7
23	167.7	165.8	165.9	156.9	165.9
24	166.0	166.0	165.6	165.6	165.5
25	163.7	163.7	165.6	165.6	166.2

After analyzing the data by using statistical control charts, the quality assurance group calculated the process capability ratio, Cp, and the process capability index, Cpk. It then decided to improve the stamping process, especially the feeder mechanism. After making all the changes that were deemed necessary, 18 additional samples were collected. The summary data for these samples are

x¯¯=163 grams

R¯¯¯=2.326 grams

σ=1 grams

All sample observations were within the control chart limits. With the new data, the quality assurance group recalculated the process capability measures. It was pleased with the improved Cp but felt that the process should be centered at 166 grams to ensure that everything was in order. Its decision concluded the study.

1. What were the values obtained by the group for CpCp and CpkCpk for the initial capability study? Comment on your findings and explain why further improvements were necessary.

2. What are the CpCp and CpkCpk after the improvements? Comment on your findings, indicating why the group decided to change the centering of the process.

3. What are the CpCp and CpkCpk if the process were centered at 166? Comment on your findings.

Answer 1

a. The control chart as per the data is as follows:

This shows that the process is within statistical control.

b. The values for the control limits are as below:

The Process Capability Index (CPK) and the Process Capability Ratio (CP) Is calculated as given below:

The minimum of:

And

Hence the minimum is CPK = 1.27.

However the Process Control Ratio is:

c. The new process gives the following results for CPK and CP:

The minimum of:

And

Hence the minimum is CPK = 0.33.

However the Process Control Ratio is:

d. The centering process brings the median equidistant from the upper and lower specifications.

The Process Capability Index (CPK) and the Process Capability Ratio (CP) Is calculated as given below:

The minimum of:

And

Hence the minimum is CPK = 1.33.

However the Process Control Ratio is:

This process has improved the Process Capability Index and the Process Capability Ration to 1.33 or 4/3 which is showing four-sigma capability.

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