In: Operations Management
Question 1: A manager wants to investigate a bottling process by using a sample mean chart. He knows from his experience that the process standard deviation is 4 mL (milliliter). Each day last week, he randomly selected 9 bottles and measured each. The data (in mL) from that activity appear below.
Weight |
|||||||||
Day |
Bottle 1 |
Bottle 2 |
Bottle 3 |
Bottle 4 |
Bottle 5 |
Bottle 6 |
Bottle 7 |
Bottle 8 |
Bottle 9 |
Monday |
323 |
322 |
323 |
324 |
323 |
322 |
323 |
324 |
322 |
Tuesday |
323 |
321 |
319 |
321 |
323 |
321 |
319 |
321 |
323 |
Wednesday |
320 |
319 |
320 |
321 |
320 |
319 |
320 |
321 |
320 |
Thursday |
318 |
319 |
320 |
319 |
318 |
319 |
320 |
319 |
328 |
Friday |
318 |
320 |
322 |
320 |
318 |
320 |
322 |
320 |
320 |
c. Based on the x-bar chart, is this process in control? Create a X-bar chart in Excel
Use the data of question 1 to (a) create Range chart in Excel. (b) Based on the R chart, is this process in control?
The information given in the question is as under:
Bottles |
|||||||||
Days |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
Monday |
323 |
322 |
323 |
324 |
323 |
322 |
323 |
324 |
322 |
Tuesday |
323 |
321 |
319 |
321 |
323 |
321 |
319 |
321 |
323 |
Wednesday |
320 |
319 |
320 |
321 |
320 |
319 |
320 |
321 |
320 |
Thursday |
318 |
319 |
320 |
319 |
318 |
319 |
320 |
319 |
328 |
Friday |
318 |
320 |
322 |
320 |
318 |
320 |
322 |
320 |
320 |
Answer 1. (a):
Calculation of all the sample means (X Bar) and the mean of all the sample means (X double bar).
Sample mean (X Bar) for each days 9 sample = Average of all 9 sample on a particular day.
Like,
Sample mean (X Bar) for Monday = (323+322+323+324+323+322+323+324+322)/9 = 322.889
Mean of all the sample means (X double bar) = Average of means of all 5 days = (322.889+321.222+320.000+320.000+320.000)/5 = 320.822
Bottles |
||||||||||
Days |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
Mean (X bar) |
Monday |
323 |
322 |
323 |
324 |
323 |
322 |
323 |
324 |
322 |
322.889 |
Tuesday |
323 |
321 |
319 |
321 |
323 |
321 |
319 |
321 |
323 |
321.222 |
Wednesday |
320 |
319 |
320 |
321 |
320 |
319 |
320 |
321 |
320 |
320.000 |
Thursday |
318 |
319 |
320 |
319 |
318 |
319 |
320 |
319 |
328 |
320.000 |
Friday |
318 |
320 |
322 |
320 |
318 |
320 |
322 |
320 |
320 |
320.000 |
X double bar |
320.822 |
Answer 1. (b):
For the sample size of 9, the Control chart constant A2 is as under:
Sample size |
9 |
A2 |
0.337 |
For the calculation of UCL and LCL of X bar chart, value of R bar needs to be calculated.
Range for a particular day = Max of value – Min of value of the particular day
Like,
Range for Monday = (Max of Monday)-(Min of Monday) = 324-322 = 2.000
R bar = Average of all Range from Monday through Friday = (2+4+2+10+4)/5 = 4.400
Bottles |
||||||||||
Days |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
Range (R ) |
Monday |
323 |
322 |
323 |
324 |
323 |
322 |
323 |
324 |
322 |
2.000 |
Tuesday |
323 |
321 |
319 |
321 |
323 |
321 |
319 |
321 |
323 |
4.000 |
Wednesday |
320 |
319 |
320 |
321 |
320 |
319 |
320 |
321 |
320 |
2.000 |
Thursday |
318 |
319 |
320 |
319 |
318 |
319 |
320 |
319 |
328 |
10.000 |
Friday |
318 |
320 |
322 |
320 |
318 |
320 |
322 |
320 |
320 |
4.000 |
R bar |
4.400 |
Upper x-bar chart control limits = UCLx bar = X double bar + A2*R bar = 320.822+(0.337*4.400) = 322.305
Lower x-bar chart control limits = LCLx bar = X double bar - A2*R bar = 320.822-(0.337*4.400) = 319.339
Answer 1. (c):
X bar chart:
Populate the values in the table:
Sample |
1 |
2 |
3 |
4 |
5 |
Average (X bar) |
322.889 |
321.222 |
320.000 |
320.000 |
320.000 |
UCL |
322.305 |
322.305 |
322.305 |
322.305 |
322.305 |
LCL |
319.339 |
319.339 |
319.339 |
319.339 |
319.339 |
CL |
320.822 |
320.822 |
320.822 |
320.822 |
320.822 |
Create the X bar chart in Excel by using above data and selecting “Line Chart”:
Based on the X bar chart, the process in not in process control as the X bar value is observed beyond UCL.
R chart:
Populate the values in the table:
Sample |
1 |
2 |
3 |
4 |
5 |
Range (R ) |
2.000 |
4.000 |
2.000 |
10.000 |
4.000 |
UCL |
7.990 |
7.990 |
7.990 |
7.990 |
7.990 |
LCL |
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