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.
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?
Ex. 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
Answer
1. To calculate the upper and lower control limits for a sample size of 100, we can use the formula for control limits in a binomial distribution:
Upper Control Limit (UCL) = p + 3 * sqrt(p * (1 - p) / n)
Lower Control Limit (LCL) = p - 3 * sqrt(p * (1 - p) / n)
Given that the average defective parts per million (DPPM) is 10 (0.001%), this implies that the defect rate (p) is 0.00001. Substituting these values into the formula:
UCL = 0.00001 + 3 * sqrt(0.00001 * (1 - 0.00001) / 100)
LCL = 0.00001 - 3 * sqrt(0.00001 * (1 - 0.00001) / 100)
2. To compute the upper and lower control limits for a sample size of 10,000, we can use the same formula:
UCL = 0.00001 + 3 * sqrt(0.00001 * (1 - 0.00001) / 10000)
LCL = 0.00001 - 3 * sqrt(0.00001 * (1 - 0.00001) / 10000)
3. Comparing the two sample sizes, we can see that for a larger sample size of 10,000, the upper and lower control limits become narrower due to the square root of the sample size in the denominator. This means that there is less variation allowed within the control limits for a larger sample size, resulting in a stricter control.
Therefore, it is recommended to use a sample size of 10,000 for better process control and to minimize the chances of producing defective parts, as it provides tighter control limits.
Moving on to the second part of the question:
1. For variables control, the control limits for the average chart can be calculated as:
Upper Control Limit (UCL) = X-bar + 3 * (R-bar / d2)
Lower Control Limit (LCL) = X-bar - 3 * (R-bar / d2)
Where X-bar is the sample average voltage, R-bar is the average range (the average of the ranges from all the samples collected), and d2 is a constant from statistical tables based on sample size (5 in this case).
The control limits for the range chart can be calculated as:
Upper Control Limit (UCL) = D4 * R-bar
Lower Control Limit (LCL) = D3 * R-bar
Where D4 and D3 are constants from statistical tables based on sample size (5 in this case).
2. To determine what action should be taken based on the sample results provided, we can plot the x-bar (average) and range values on the control charts and check if any points fall outside the control limits. If any point falls outside the control limits or shows a non-random pattern (e.g., a trend or too many consecutive points), it indicates a process is out of control, and corrective action should be taken.
3. The pros of using variables control charts for circuit voltage measurement are:
- It allows for more precise monitoring of variations in process performance, as it uses continuous measurement data, providing more insights into the process.
- It can detect small shifts or trends in process performance, making it more sensitive to process changes compared to attribute control charts.
- Variables control charts provide statistical measures (average and range) that can be used to estimate process capability.
The cons of using variables control charts are:
- It requires more effort and resources compared to attribute control charts, as continuous measurement data needs to be collected and analyzed.
- Variables control charts are more sensitive to measurement errors or variability, so the accuracy and reliability of the measurement system become critical.
- Variables control charts might be unnecessary for processes with stable and predictable performance, where attribute control charts are often sufficient.
Considering the given context of monitoring circuit voltage, which requires continuous measurement data and a smaller sample size, variables control charts would be more suitable. However, the preference for control charts ultimately depends on the specific process, its characteristics, and the information required for effective process control and improvement.