Six Sigma and Beyond: Statistical Process Control, Volume IV
The u control chart is made for nonconformities (defects) and offers the same advantages as does the c chart. For u charts , inspection units do not need to be constant ( unequal n ).
STEPS FOR CONSTRUCTING A U CHART
The steps for constructing a u chart are nearly identical to those for the c chart. One extra step is needed to accommodate the unequal sample sizes.
1. Establish a Sampling Plan for the Control Chart
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Determine an appropriate sample size (n). The sample size used to construct a u chart need not be constant. It must be large enough so that a typical sample subgroup contains nonconformities (defects).
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Determine a sampling frequency. Collect sample subgroups often enough to provide diagnostic information. People must identify special causes of variation and search for the cause of the unexpected variation. Problem solving is easier when the subgroups are collected at reasonable intervals. The rate of process change defines an appropriate sampling interval. A very frequent sampling plan is needed for a process that changes often. Stable processes can tolerate less frequent sampling plans. When insufficient information is available for planning a sampling frequency, sample as often as possible. After a stable and in-control process is established, the time interval between sample subgroups may be increased slowly.
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Record the process data on the control chart. Record the number of defects, sample size ( n ), time, shift, and date for each subgroup on an attribute control chart form. Maintain a process log as a record of all major changes to the process (e.g., changes in machinery, material, methods , or personnel).
2. Calculate the Proportion ( u ) of Nonconformities
Use the following formula to calculate the proportion of nonconformities in each inspection unit:
u = c · n
where
| u | = | proportion of nonconformities |
| c | = | number of nonconformities |
| n | = | sample size |
3. Calculate the Process Center Line
The mean number of defects per inspection unit (ubar) is calculated with the following formula:
ubar = & pound ; c · n = ( c 1 + c 2 + ‹ + c k ) · ( n 1 + n 2 ‹ + n k )
where
| ubar | = | mean number of defects |
| C | = | sum of the defects |
| n | = | sum of the sample sizes |
4. Calculate the Standard Deviation ( ƒ u ) for the Number of Defects per Inspection Unit
The standard deviation for the number of defects per inspection unit is calculated as
where
| ƒ u | = | standard deviation for the number of defects per inspection unit |
| ubar | = | mean number of defects |
| nbar | = | mean sample size |
where
| n | = | sum of the sample size |
| k | = | number of samples |
Figure 9.16 shows the sequence of operations needed to calculate the standard deviation for proportions with the Sharp series of calculators .
5. Calculate Control Limits (UCL and LCL)
The upper control limit (UCL) and the lower control limit (LCL) are boundaries of common variation. They are calculated as three standard deviations above and below the process center line.
| UCL u | = | ubar + 3 ƒ u |
| LCL u | = | ubar - 3 ƒ u |
where
| UCL | = | upper control limit |
| LCL | = | lower control limit |
| ubar | = | process center line (mean number of defects per inspection unit) |
| ƒ u | = | standard deviation for number of defects per inspection unit) |
A negative control limit may be calculated for the LCL when the average proportion of defects per inspection unit (ubar) or the mean sample size (nbar) is small. Under these conditions, the lower control limit does not exist.
6. Determine the Vertical Scale for the U Chart
Determine the vertical scale for control charts only after calculating the process center lines and the control limits.
7. Plot the Proportion of Defects on the Control Chart
Plot the u value for each sample on the chart. Place each plotted point directly above the data and time sequence information included in the data table. The plotted points may be connected with a solid line as an aid to the analysis of patterns and trends in the data. Proofread the plotted points before analyzing the control chart.
8. Draw the Process Center Line and Control Limits on the Chart
The ubar line should be drawn as a solid horizontal line. The control limit(s) are usually drawn as a dashed, bold, horizontal line. Date and label all lines.
9. Analyze the U Chart for Process Control
Use the five criteria for process control to evaluate the u chart. Identify and note all out-of-control conditions and factors which contribute to instability. In the process log, note the time, shift, date, material lot number, material vendor, process parameters, and production personnel included in the manufacturing operation during the out-of-control period.
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Are there any point( s ) beyond the control limits?
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Are there runs of seven (7) or more points?
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Are there trends of seven (7) or more points?
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Are there cycles of points?
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Is there unusual variation (the 1/3 or 2/3 rule)?
If all of the plotted data on the u chart are within the control limits and randomly distributed around the center line, the defect rate for the process is stable and predictable. The process is in control. Unless the process is in control, ongoing control limits cannot be calculated, process capability cannot be studied, and experimental studies cannot be generalized to longer and larger production runs.
10. Identify and Eliminate the Special Causes of Variation
How people react to control chart signals is the most critical part of the SPC program. To identify and eliminate special variation, people must analyze the manufacturing operations and the resources used. In addition to the control chart, fishbone diagrams, Pareto charts, process flow charts, and controlled experiments may be needed to identify and resolve factors which create instability in the process. It is very important to react promptly to eliminate and prevent special causes of variation. A production department's reaction to out-of-control conditions indicates its commitment to understanding and controlling the manufacturing process. The control chart provides signals to help production departments maintain good quality.
11. Extend the Control Limits for Ongoing Process Control
When the data from the operating process (at least 25 subgroups) are consistently within the control limits and in control, the control limits may be extended to future samples of data. Production operators and supervisors should closely monitor the charts and act promptly to correct special causes of variation. If the process is improved, or if extreme sources of variation are eliminated, the process center line and control limits may no longer be appropriate. The changes made to the process may make it necessary to recalculate the process center lines and control limits. When a production department continually improves its operations, control charts require frequent recalculation. The improvement activities are recorded on u charts as downward trends and/or runs below ubar. These out-of-control conditions verify that the process has been improved.