Question

You will consider the process of a roll of plastic material sleeves for clinical use, manufactured through a process called extrusion. The process is continuous and produces a roll of 1000 linear foot plastic tube every 15 minutes. 96 rolls are made in a three-turn run. The manufacturing process for this product is new to the company and they have only produced it using the experience of operators and quality control personnel. The critical parameter for this product is the inner diameter (ID) of the tube (DP), whose specifications are: 10.10 mm + / 1mm. These diameters are measured through a sectional cut. Consider that you have been hired as a quality manager. You have been given the task of establishing a quality control, so you decide to implement control charts. Here is the Data:

Sampling Hour	Shift	Sample 1	Sample 2	Sample 3	Sample 4
6:00 AM	1	10.94	10.74	11.06	10.66
7:00 AM	1	10.66	10.6	10.8	10.68
8:00 AM	1	10.68	10.6	10.54	10.36
9:00 AM	1	10.03	10.72	10.74	10.7
10:00 AM	1	10.7	10.62	11.04	10.58
11:00 AM	1	10.38	10.42	10.66	10.4
12:00 PM	1	10.46	10.6	10.26	10.02
1:00 PM	1	10.66	10.6	10.22	10.68
2:00 PM	2	10.5	10.78	10.68	10.58
3:00 PM	2	10.58	10.48	10.58	10.4
4:00 PM	2	10.8	10.74	10.36	10.6
5:00 PM	2	10.42	10.6	10.64	10.34
6:00 PM	2	10.52	10.6	10.46	10.6
7:00 PM	2	11.04	10.72	10.56	10.5
8:00 PM	2	10.52	10.62	10.68	10.7
9:00 PM	2	10.38	10.42	10.76	10.54
10:00 PM	3	10.56	10.6	10.28	10.68
11:00 PM	3	10.58	10.6	10.58	10.64
12:00 AM	3	10.42	10.78	10.64	10.86
1:00 AM	3	10.01	10.48	10.84	10.7
2:00 AM	3	10.84	10.64	10.4	10.52
3:00 AM	3	10.54	10.36	10.6	11.02
4:00 AM	3	10.68	10.4	10.56	10.32
5:00 AM	3	10.7	10.6	10.74	10.52

1- Which control chart is the most appropriate to evaluate this case? Explain your answer.
2- Construct the process control graph for the critical parameter.
3- Analyze the graph and interpret your results. Determine if the process is in control or not. Explain your answer.
4- Considering the product specifications, determine the process capacity (Cpk).
5- What conclusions can you reach? What would be your next steps?

this is all the information

10.10 mm +/- 1mm (go from 10-11mm)

Answer 1

Answer 1:

The control charts which should be used for the data provided are:

• Mean X̅-chart
• Range R-Chart

As the data provided is samples with 4 observations each (so the accommodate the variability of observations recorded and the spread of observations Mean X̅-chart and Range R-Chart should be used).

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Answer 2:

(X-X) Sample 6:00 AM 7:00 AM 8:00 AM 9:00 AM 10:00 AM 11:00 AM 12:00 PM 1:00 PM 2:00 PM 3:00 PM Mean (X) 10.85 10.69 10.55 10.55 10.74 10.47 10.34 10.54 10.64 10.51 10.63 10.50 10.55 Range (R) 0.40 0.20 0.32 0.71 0.46 0.28 0.58 0.46 0.28 0.18 0.070 0.010 0.002 0.001 0.022 0.015 0.063 0.002 0.002 0.006 0.002 4:00 PM 5:00 PM Sample 1 Sample 2 Sample 3 Sample 4 10.94 10.74 11.06 10.66 10.66 10.60 10.80 10.68 10.68 10.60 10.54 10.36 10.03 10.72 10.74 10.70 10.70 10.62 11.04 10.58 10.38 10.42 10.66 10.40 10.46 10.60 10.26 10.02 10.66 10.60 10.22 10.68 10.50 10.78 10.68 10.58 10.58 10.48 10.58 10.40 10.80 10.74 10.36 10.60 10.42 10.60 10.64 10.34 10.52 10.60 10.46 10.60 11.04 10.72 10.56 10.50 10.52 10.62 10.68 10.70 10.38 10.42 10.76 10.54 10.56 10.60 10.28 10.68 10.58 10.60 10.58 10.64 10.42 10.78 10.64 10.86 10.01 10.48 10.84 10.70 10.84 10.64 10.40 10.52 10.54 10.36 10.60 11.02 10.68 10.40 10.56 10.32 10.70 10.60 10.74 10.52 0.44 0.30 0.007 0.14 0.002 0.54 0.014 6:00 PM 7:00 PM 8:00 PM 9:00 PM 10.71 10.63 0.18 0.002 10.53 0.38 0.004 0.40 0.003 10:00 PM 11:00 PM 12:00 AM 10.53 10.60 10.68 0.06 0.000 0.008 0.44 1:00 AM 10.51 0.83 0.006 0.000 0.002 2:00 AM 3:00 AM 4:00 AM 5:00 AM 10.60 10.63 10.49 10.64 10.59 0.44 0.66 0.36 0.22 0.39 0.009 0.003 0.103 R 0

Based on the sample size, the Control chart constants are:

Sample size	4
A2	0.729
D3	0
D4	2.282

Populate the data for Mean (X̅-chart):

10.69 Sample 6:00 AM 7:00 AM 8:00 AM 9:00 AM 10:00 AM 11:00 AM 12:00 PM 1:00 PM 2:00 PM 3:00 PM 4:00 PM 5:00 PM 6:00 PM 7:00 PM 8:00 PM 9:00 PM 10:00 PM 11:00 PM 12:00 AM 1:00 AM 2:00 AM 3:00 AM 4:00 AM 5:00 AM Mean (X bar) 10.85 10.55 10.55 10.74 10.47 10.34 10.54 10.64 10.51 10.63 10.50 10.55 10.71 10.63 10.53 10.53 10.60 10.68 10.51 10.60 10.63 10.49 10.64 UCL 10.87 10.87 10.87 10.87 10.87 10.87 10.87 10.87 10.87 10.87 10.87 10.87 10.87 10.87 10.87 10.87 10.87 10.87 10.87 10.87 10.87 10.87 10.87 10.87 LCL 10.30 10.30 10.301 10.30 10.30 10.30 10.30 10.30 10.30 10.30 10.30 10.30 10.30 10.30 10.30 10.30 10.30 10.30 10.30 10.301 10.30 10.301 10.30 10.30 CL (X) 10.59 10.591 10.59 10.59 10.59 10.591 10.591 10.59 10.59 10.59 10.59 10.59 10.59 10.591 10.59 10.59 10.59 10.59 10.591 10.59 10.591 10.591 10.59 10.59

Plot the Mean (X̅-chart):

10.00 10.10 10.20 10.30 10.40 10.50 10.60 10.70 10.80 10.90 11.00 6:00 AM 7:00 AM 8:00 AM 9:00 AM 10:00... 11:00... 12:00.. 1:00 PM 2:00 PM 3:00 PM 4:00 PM 5:00 PM 6:00 PM 7:00 PM 8:00 PM 9:00 PM 10:00.. 11:00. 12:00. 1:00 AM 2:00 AM 3:00 AM 4:00 AM 5:00 AM -UCL bar) 101 —Mean

Populate the data for Range (R-chart):

7:00 PM 8:00 PM 9:00 PM 10:00 PM 11:00 PM 12:00 AM 00 390.3 00 0.00 0.00 301 301

Plot the Range (R-chart):

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 6:00 AM 7:00 AM 8:00 AM 9:00 AM 10:00.. 11:00.. 12:00.. 1:00 PM 2:00 PM 3:00 PM 4:00 PM 5:00 PM 6:00 PM 7:00 PM 8:00 PM 9:00 PM 10:00.. 11:00... 12:00... $ 1:00 AM 2:00 AM 3:00 AM 4:00 AM 5:00 AM

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Answer 3:

• All the sample observations for Mean chart are within the control limits (UCL and LCL).
• All the sample observations for Range chart are within the control limits (UCL and LCL).

Hence, the process is under control.

If any sample observation would have breached control limits it could have resulted in process being out of control (which is not the case here).

PROCESS IS UNDER CONTROL.

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Answer 4:

Mean	μ	10.59	minutes
Standard deviation	σ	0.103	minutes
UCL	11.00	minutes
LCL	10.00	minutes

Capability Index	Formula	Value
Lower	Cpl = (μ-LCL)/3σ	1.91
Upper	Cpu = (UCL-μ)/3σ	1.33
Process	Cpk = Min(Cpu,Cpl) =	1.33

Cpk = 1.33

Cpk value of 1.33 indicates that the process is capable and meets specification limits.

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