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Hi-Sport Manufacturing Co. Inc.
Hi-Sport is a small company that manufactures logo sporting jackets. A key goal of the company has always been excellent quality. This has been achieved largely through rigorous inspection, a process that has come to be known as “inspecting the quality in.” As a result, the firm has always had a high reject rate at final inspection. This has necessarily resulted in too many jackets being scrapped or sold below cost as “seconds.” It has also resulted in a bothersome percentage of imperfect jackets “slipping through” inspection and ending up in the hands of customers. The impact has been a so-so reputation with customers, and prices that are too high to be competitive with the imports or major U.S. manufacturers. Management tried and tried to get the production workers to do better, but it seemed that every effort to reduce defects came to nothing. Sometimes it appeared that good ideas and the best of intentions only made matters worse. A few weeks ago the managers retained a manufacturing consultant with statistics and process control credentials. He told them their first priority should be to get their processes under control. With the consultant’s help, they started their program by identifying and eliminating several special causes of variation. These special causes had included machines that needed maintenance and calibration, some employees with insufficient training, and the absence of written work instructions for certain procedures. By the 6-week mark, Hi-Sport’s quality had noticeably improved. Management decided it was time to attempt the development of a control chart. Because rejects were based on pass/fail criteria for various characteristics, the managers needed a control chart that could respond to nonmeasurable attributes. Three common charts meet that requirement: the p -chart, np -chart, and c -chart. The p -chart could help control the percentage of defective jackets. The np -chart could help control the number of defective jackets. The c -chart could help control the number of defects in a jacket. They decided the c -chart would give them what they wanted—namely, using one jacket as the sample and tracking the defects found in the sample. For the initial chart-development data, one jacket was inspected each hour for 30 consecutive working hours. The data are listed in the accompanying table. The data were recorded in the five most common defect categories, with a sixth column collecting all other types of defects encountered. At the end of the 30th hour, 46 total defects had been recorded from the 30 jacket samples.
SPC Exercise: HI-Sport’s Control Chart
1. You are the consultant, and because this is Hi-Sport’s introduction to control charts, you will have to help it develop the chart. From the data table, compute c , UCL c , and LCL t .
2. Next, construct a control chart. Be sure to include some “headroom” above the upper control limit to allow for any out-of-limit events that may be encountered. Also make the chart long enough horizontally to include all the data points in the table, plus a few more days of real-time data points that will need to be plotted as SPC tracking begins.
3. Now plot the appropriate data from the table on the chart. Is the process in control, or is one or more special causes still lingering? Where do you go from here? Complete your task by (a) providing Hi-Sport with a control chart that reflects an in-control process or (b) abandoning the current data and starting over again by seeking out the special causes that prevented success the first time.
To see how the authors resolved this issue, see the next section.
Authors’ Solution to HI-Sport’s Control Chart Exercise
In practice, it is not uncommon to encounter a special cause while developing a control chart. This may happen because the special cause is so subtle or infrequently recurring that it has gone unnoticed or simply because there were other special causes, now gone, that had masked this one. Suppose you have collected your data over 5 working days, made the necessary calculations, and plotted the new control chart. To your great consternation, you find that one of your data points is beyond the upper control limit. Have you lost a week’s work because there was still a special cause of variation lurking in your process? No. You can salvage it as we do here:
The control chart identified as Hi-Sport v1.0 was constructed using these , UCL, and LCL values. Note that sample 25 is above the upper control limit, which indicates that it is the result of a special cause. Because the control chart must be developed from natural (common-cause) variation data only, sample 25 would seem to rule out the entire 30 hours of data. However, if sample 25 is eliminated from the data and if UCL, and LCL are recalculated on the 29 remaining samples, the control chart will remain valid.
Recalculating, , UCL, and LCL without sample 25 yields the following:
The control chart constructed from these values is shown as Hi-Sport v1.1. Note that with the removal of sample 25 data, (the average number of defects per sample) decreased from 1.533333 to 1.37931. At the same time, the UCL decreased from over 5 to under 5. This is a narrowing of limits, which is good. However, sample 7 is now beyond the upper control limit. The same procedure, deleting sample 7 data, can be repeated. Recalculating the remaining 28 samples yields the following:
The corresponding control chart is shown as Hi-Sport v1.2. Note that all sample data points are now inside the control limits and there are no protracted runs below or above the line. With that, Hi-Sport’s consultant pronounced the chart ready for use. This was a close call. If you encounter an out-of-limits data point in your initial chart development data, it is OK to simply eliminate the offending sample data from the calculations and proceed. Doing the same again (as was done here) for a second out-of-limits sample is usually acceptable, but if a third one crops up in a set of 30 to 50 data point samples, the process is too unstable for a control chart. Go back to eliminating special causes before collecting all-new data from which to construct your control chart.
You might ask, “If the control chart showed an out-oflimits condition for data sample 7, why not eliminate that special cause before you go on?” If the special cause was known, then certainly it should be eliminated forthwith. (We would still have to eliminate the data sample and recalculate on the remaining data.) The problem is that it occurred several hours (or maybe days) ago and may no longer be apparent. The trail may have become obscured. That is why, when a control chart is in place and an out-of-limits data point is taken, it is important to stop the process immediately to find the cause. The more time that passes between the event and the search, the less likely the cause will be discovered.
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