As the production manager for an electronic circuit company you have encountered the following situation.
Part 1
A process for producing electronic circuits has achieved very high yield levels. An average of only 10 defective parts per million is currently produced.
1. What are the upper and lower control limits for a sample size of 100?
2. Recompute the upper and lower control limits for a sample size of 10,000?
3. Which of these two sample sizes would you recommend? Explain.
Part 2
Management has reconsidered the method of quality control and has decided to use process control by variables instead of attributes. For variables control a circuit voltage will be measured based on a sample of only five circuits. The past average voltage for samples of size 5 has been 3.1 volts, and the range has been 1.2 volts.
1. What would the upper and lower control limits be for the resulting control charts (average and range)?
2. Five samples of voltage are taken with the results in the table below. What action should be taken if any?
3. Discuss the pros and cons of using this variables control chart versus the control chart discussed in the first part of the assignment. Which do you prefer?
Sample 1 2 3 4 5
x 3.6 3.3 2.6 3.9 3.4
R 2.0 2.6 0.7 2.1 2.3
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Part 1:
To determine the upper and lower control limits for a sample size of 100, we need the average and standard deviation of the defect rate.
1. For a sample size of 100, the expected number of defective parts would be 10/1,000,000 * 100 = 0.001 (since there are 1 million parts per sample).
To calculate the standard deviation (σ), we can use the formula:
σ = sqrt(p * (1-p)/n)
where p is the defect rate (in this case, 10/1,000,000) and n is the sample size (100).
Using these values, we can calculate the standard deviation for a sample size of 100:
σ = sqrt(0.001 * (1-0.001)/100) = 0.010
The control limits can be determined by adding or subtracting three times the standard deviation from the average defect rate.
Upper control limit = average + (3 * standard deviation)
Lower control limit = average - (3 * standard deviation)
Substituting the values:
Upper control limit = 0.001 + (3 * 0.010) = 0.031
Lower control limit = 0.001 - (3 * 0.010) = -0.029 (but it cannot be negative, so it is zero)
So, the upper control limit is 0.031 and the lower control limit is 0.
2. For a sample size of 10,000, the expected number of defective parts would be 10/1,000,000 * 10,000 = 0.1.
Using the formula mentioned earlier, we can calculate the standard deviation for a sample size of 10,000:
σ = sqrt(0.1 * (1-0.1)/10,000) = 0.003
Upper control limit = average + (3 * standard deviation)
Lower control limit = average - (3 * standard deviation)
Substituting the values:
Upper control limit = 0.1 + (3 * 0.003) = 0.109
Lower control limit = 0.1 - (3 * 0.003) = 0.091
So, the upper control limit is 0.109 and the lower control limit is 0.091.
3. Between the two sample sizes, 100 and 10,000, I would recommend using the sample size of 10,000. This is because a larger sample size provides a more accurate representation of the process and its variability. It helps in detecting smaller variations and gives a better indication of process stability.
Part 2:
1. For variables control, we need to calculate the control limits for the average (x̅) and the range (R) based on the past data.
For the average chart:
The upper control limit (UCL) is given by: x̅ bar + (A2 * R bar)
The lower control limit (LCL) is given by: x̅ bar - (A2 * R bar)
For the range chart:
The upper control limit (UCL) is given by: D4 * R bar
The lower control limit (LCL) is given by: D3 * R bar
In this case, the average (x̅ bar) is 3.1 volts and the range (R bar) is 1.2 volts.
Substituting the values of A2, D3, and D4 (which are constants based on the sample size):
For a sample size of 5, A2 = 0.729, D3 = 0, and D4 = 2.114.
For the average chart:
UCL = 3.1 + (0.729 * 1.2) = 4.155
LCL = 3.1 - (0.729 * 1.2) = 2.045
For the range chart:
UCL = 2.114 * 1.2 = 2.537
LCL = 0 * 1.2 = 0
2. Using the provided data for the five samples:
Sample 1: x = 3.6, R = 2.0
Sample 2: x = 3.3, R = 2.6
Sample 3: x = 2.6, R = 0.7
Sample 4: x = 3.9, R = 2.1
Sample 5: x = 3.4, R = 2.3
To determine if any action should be taken, we compare the values to the control limits.
For the average chart, all the sample averages (x) are within the control limits of 2.045 and 4.155.
For the range chart, all the sample ranges (R) are within the control limits of 0 and 2.537.
Since all the data points are within the control limits, no specific action needs to be taken.
3. The pros of using the variables control chart are:
- It provides more detailed information about process variability through the average and range charts.
- It can help identify specific patterns or trends in the data.
- It allows for more targeted problem-solving and improvement efforts by focusing on the process variables.
The cons of using the variables control chart are:
- It requires collecting and analyzing more data points, which can be time-consuming.
- It requires more knowledge and expertise to calculate the control limits and interpret the charts accurately.
- It may not be suitable for all types of processes where attributes control charts might be more appropriate.
As an AI bot, I don't have personal preferences. However, the choice between using variables control charts and attributes control charts depends on the specific requirements and characteristics of the process being monitored. Variables control charts are generally preferred for continuous measurements where the focus is on process variability and improvement, while attributes control charts are useful for discrete measurements where the focus is on defect rates and quality levels. The decision ultimately depends on the context and goals of the quality control process.