This is file Q8IM13C.DOC - The third of 5 files for solutions to this chapter’s problems.
13. For each of the following control charts, assume that the process has been operating in statistical control for some time. What conclusions should the operators reach at this point?
Ans
13. a) The first control chart shows an out-of-control process with a definite downward trend. The last 4 out of 5 points are below one standard error away from the mean. The process needs adjustment upward.
b) The second control chart shows an out of control condition, with the first eight points above the centerline. Then there appears to be a sudden shift in the process average, putting the next six points below the centerline. It is possible that the process is being over-adjusted. It needs to be centered and then watched for out-of-control indications with no unnecessary operator intervention.
c) The third control chart shows the data hugging the centerline, indicating that the process is possibly out of control. If the process has multiple machines or operators, a control chart should be constructed for each machine to avoid "masking" the variation brought on by mixing data from several sources.
d) The fourth control chart shows a process that is stable and in control.
e) The fifth control chart shows an out of control condition, with a point above the upper control limit.
f) The sixth control chart shows that seven out of eight of the most recent points are below the centerline, indicating that the process is out of control.
g) The last control chart shows too many points close to the upper and lower control limits, indicating an out of control condition.
14. Discuss the interpretation of each of the following control charts:
Ans
14. a) Two points outside upper control limit. b) Process is in control c) Mean shift upward in second half of control chart. d) Points hugging upper and lower control limits.
15. PCDrives has a manufacturing process that is normally distributed and has the sample means and ranges for fifteen samples of size 5, found in the worksheet Prob. 13-15. Note that only sample statistics have been given, instead of the raw data from the samples. Determine process capability limits. If specifications are determined to be 70 ± 25, what percentage will be out of specification?
Answer
15. (See spreadsheet Prob.13-15CP-PCD.xls for details.)
For the Center Lines, CL: = 68.567; CLR: = 21.92 Control limits for the - chart are: ± A2 = 68.567 ± 0.577 (21.92) = 55.92 to 81.22
For the R-chart: UCLR = D4 = 2.114 (21.92) = 46.34 LCLR = D3 = 0
The limits above apply to sample groups of 5 items each.
Estimated = / d2 = 21.92 / 2.326 = 9.42
The problem asks that students perform a process capability analysis. This is only justified if the process is in control. The fact that the process is thought to be normally distributed does not establish that it is in control. The - chart shows that the process is, in fact, out of control, because of 4 out of 5 samples within samples 2-6 are more than one standard error away from the center-line. The % outside calculation can be performed as follows. Note the warning given below about the assumption that the process is in control.
Percent outside Specification Limits (45 to 95)
% Below LSL: z = LSL - _
z = 45 - 68.567 = - 2.50 ; P( z < -2.50) = (0.5 - 0.4938) = 0. 0062 that 9.42 items will exceed lower limit
% Above USL: z = USL - _
z = 95 - 68.567 = 2.81 ; P( z > 2.81) = (0.5 - 0.4975) = 0.0025 that 9.42 items will exceed upper limit
Therefore, the percent outside is calculated as: 0.87 %
Although the % outside calculations seem to show that the process has a relatively small % outside specifications, it should be noted that the x-bar chart shows that the process is not in control. Hence, the % outside
