Statistical Quality Control for Sheet Metal Defects

1. (c) Statistical Quality Control Program for Sheet Metal Production

The manufacturing facility specializing in Sheet Metal Production has implemented a Statistical Quality Control Program to monitor and improve process performance. The data provided represents the number of visible surface defects identified on 20 sequential metal sheets, each considered one production unit.

(i) Determine the Center Line (CL), Upper Control Limit (UCL), and Lower Control Limit (LCL)

Given that the data consists of the number of defects per unit (metal sheet) and the sample size (one sheet) is constant, a c-chart is the appropriate control chart to use. The c-chart is designed for monitoring the number of nonconformities per unit when the number of opportunities for nonconformities is large but the probability of any single nonconformity occurring is small.

First, let's list the given data:

Sheet No.	Defect Counts (c_i)
1	5
2	6
3	4
4	4
5	6
6	7
7	0
8	6
9	5
10	3
11	1
12	4
13	5
14	3
15	6
16	4
17	3
18	1
19	3
20	4

Step 1: Calculate the total number of defects ($\Sigma c_i$)

$\Sigma c_i = 5+6+4+4+6+7+0+6+5+3+1+4+5+3+6+4+3+1+3+4 = 80$

Step 2: Determine the number of samples (k)

$k = 20$ (since there are 20 sequential metal sheets)

Step 3: Determine the Center Line (CL)

The Center Line (CL) for a c-chart is the average number of defects per unit, denoted as $\bar{c}$.

$CL = \bar{c} = \frac{\Sigma c_i}{k} = \frac{80}{20} = 4$

Step 4: Calculate the Upper Control Limit (UCL) and Lower Control Limit (LCL)

The formulas for UCL and LCL for a c-chart (assuming a 3-sigma control limit) are:

• $UCL = \bar{c} + 3\sqrt{\bar{c}}$
• $LCL = \bar{c} - 3\sqrt{\bar{c}}$

Substituting the value of $\bar{c} = 4$:

$UCL = 4 + 3\sqrt{4} = 4 + 3 \times 2 = 4 + 6 = 10$

$LCL = 4 - 3\sqrt{4} = 4 - 3 \times 2 = 4 - 6 = -2$

Since the number of defects cannot be negative, if the calculated LCL is less than zero, it is set to 0.

$LCL = \max(0, -2) = 0$

Summary of Control Limits:

• Center Line (CL) = 4
• Upper Control Limit (UCL) = 10
• Lower Control Limit (LCL) = 0

(ii) Plot the appropriate Control Chart and Interpret the Result

The appropriate control chart is a c-chart. Below is a plot of the defect counts for each sheet against the control limits.

Control Chart Plot:

(Note: As an AI, I cannot actually generate images. The description below represents the visual chart and interpretation.)

Visual Representation of the C-Chart:

• The X-axis represents the "Sheet Number" (1 to 20).
• The Y-axis represents the "Number of Defects".
• A horizontal line is drawn at CL = 4.
• A horizontal line is drawn at UCL = 10.
• A horizontal line is drawn at LCL = 0.
• Each observed defect count for sheets 1 to 20 is plotted as a point.

Plotting the Points:

• Sheet 1: 5
• Sheet 2: 6
• Sheet 3: 4
• Sheet 4: 4
• Sheet 5: 6
• Sheet 6: 7
• Sheet 7: 0 (on the LCL)
• Sheet 8: 6
• Sheet 9: 5
• Sheet 10: 3
• Sheet 11: 1
• Sheet 12: 4
• Sheet 13: 5
• Sheet 14: 3
• Sheet 15: 6
• Sheet 16: 4
• Sheet 17: 3
• Sheet 18: 1
• Sheet 19: 3
• Sheet 20: 4

Interpretation of the Result:

Upon plotting the defect counts on the c-chart, we observe the following:

1. All data points lie within the Upper Control Limit (UCL = 10) and the Lower Control Limit (LCL = 0). This indicates that the process is currently in statistical control. There are no points exceeding the control limits, which would signal the presence of assignable (special) causes of variation.
2. No discernible patterns or trends: There are no unusual runs of points (e.g., eight or more points above or below the center line), no consistent upward or downward trends, and no cyclical patterns. This further supports the conclusion that the process variation observed is due to common causes, which are inherent to the process.
3. Point at LCL: Sheet 7 shows 0 defects, which falls exactly on the LCL. While it's on the limit, it does not violate the control limit. This could be a positive sign but does not indicate an out-of-control situation.

Conclusion: Based on the c-chart, the sheet metal production process, with respect to visible surface defects, appears to be in statistical control. The observed variations in defect counts are likely due to common causes inherent in the manufacturing system. Therefore, no immediate intervention is required to address special causes. However, continuous monitoring and efforts towards process improvement (e.g., reducing the average number of defects, thereby shifting the control limits downwards) should be pursued to enhance overall quality and reduce common cause variation.