This is file Q8Intl-IM13C.doc - The third of 5 files for solutions to this chapter.

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1 This is file Q8Intl-IM13C.doc - The third of 5 files for solutions to this chapter. 11. 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? 1

2 Statistical Process Control 2 Answer 11. 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.

3 Statistical Process Control 3 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. 12. Discuss the interpretation of each of the following control charts:

4 Statistical Process Control 4 Answer 12. 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. 13. Consider the data for the time required to begin loading the homepage in an Internet Answer browser for Slowbay s Web site. Fifteen samples of size 5 are shown in the worksheet Prob Specifications are ± a. Compute control limits for an x-chart (chart for individuals) using the statistic R / d 2 as an estimate of the standard deviation and using the actual standard deviation for the data. Why might they be different? b. Construct the x - and R charts and an x - chart for individuals using the data. Interpret the results. c. Estimate the process capability by using the estimated sample standard deviation, available on the x - and R chart spreadsheet templates on this CD. 13. See spreadsheet Prob13-13IV.xls for details. a) Preparing a chart for individuals: Center Lines, CL x : x = 0.076; CL R : R = Estimated = R / d 2 = / = 0.002; actual = x -± 3 est = ± 3 (0.002) = to 0.082; x -± 3 actual = ± 3 (0.002) = to These limits apply to individual items only. In this case, a perfect estimate of the standard deviation was based on the moving range, using 2. This will not happen in every case, because = R / d 2 is only an estimator for the population standard deviation. Individual items can only be plotted on x-charts, as shown below.

5 Statistical Process Control Prob Individuals (X) Chart Individuals Upper control limit Center line Lower control limit e a lu V Observation number b) Although extraneous to the question of how individual values should be plotted on x- charts, students need to understand how x - chart and R-chart results differ in comparison with the above. See spreadsheet Prob13-13CP.xls for details. For the x - chart: x ± A 2 R = ± (0.005) = 0.07 to 0.08 For the R-chart: UCL R = D 4 R = (0.005) = LCL R = D 3 R = 0 Of course, the limits above apply to sample groups of 5 items each. Problem x - Chart for Sample Groups

6 Statistical Process Control 6 Prob X-bar Chart for Sample Groups Ave rages Lower control limit Up per control limit Center line s e g ra v e A Problem 14-13: R chart for Sample Groups Sample number Problem R-Chart for Sample Groups Ranges Lower control limit Upper control limit Center line s e0.008 g0.006 n a0.004 R Sample number c) The comparisons of process capability using the estimated value is shown on the following chart. (See spreadsheet Prob13-13CP.xls for details.). Note that C p, C pl, C pu, and C pk from the spreadsheet are based on estimated values using = R / d 2 = / = 0.002

7 Statistical Process Control 7 Nominal specification Average Cp Upper tolerance limit Standard deviation Cpl Lower tolerance limit Cpu Cpk Actual = 0.002, as calculated previously, would make a slight difference in process capability calculations 14. Fujiyama Electronics, Inc. has been having difficulties with circuit boards purchased from an outside supplier. Unacceptable variability occurs between two drilled holes that are supposed to be 5 cm apart on the circuit boards. Thirty samples of 4 boards each were taken from shipments sent by the supplier as shown in the data listed in the worksheet for Prob a. Construct x - and R charts for these data. b. If the supplier s plant quality manager admitted that they were experiencing quality problems for shipments 18-21, how would that affect your control chart? Show this adjustment on revised x - and R charts for these data. c. Ten more observations were taken, as shown in the second table in the worksheet. Using the revised x - and R charts from part (b), comment on what the chart shows after extending it with the new data. Answer 14. See data and control charts below. See spreadsheets Prob13-14AXR.xls, Prob13-14BXR.xls, and Prob13-14CXR.xls for details. a) Center Lines, CL x : x = 5.100; CL R : R = Control limits for the x - chart are: x ± A 2 R = ± (1.083) = 4.31 to 5.89

8 Statistical Process Control 8 For the R-chart: UCL R = D 4 R = (1.083) = 2.47 LCL R = D 3 R = 0

9 Statistical Process Control 9 b) We can see from the above x - chart that points 19 and 21 are out of control and the R- chart shows point 18 is out of control on the range. We obtain the following control limits and related charts after dropping these 3 points: New Center Lines: Center Lines, CL x : x = 5.037; CL R : R = Control limits for the x - chart are: x ± A 2 R = ± (1.057) = to For the R-chart: UCL R = D 4 R = (1.057) = LCL R = D 3 R = 0 The x and R-charts, below, now show that the process is in control.

10 Statistical Process Control 10 c) The additional data shows that the process is still operating within control limits. See the "composite" control charts below.

11 Statistical Process Control Squawk Boxes, Inc., a speaker manufacturer, has a process that is normally distributed and has the sample means and ranges for 15 samples of size 5 found in the worksheet

12 Statistical Process Control 12 Prob Determine process capability limits. If specifications are determined to be 50 ± 25, what percentage would be out of specifications? Answer 15. (See spreadsheet Prob13-15CP.xls for details. Note that the formulas have been adjusted to permit use of means and ranges, rather than detailed sample data.) For the Center Lines, CL x : x = ; CL R : R = Control limits for the x - chart are: x ± A 2 R = ± (17.667) = to For the R-chart: UCL R = D 4 R = (17.667) = LCL R = D 3 R = 0 These limits apply to sample groups of 5 items each. Estimated = R / d 2 = / = 7.60 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 x-bar chart, however, shows that the process is, in control. Spreadsheets showing process control conditions

13 Statistical Process Control 13 Nominal specification 50 Average Cp Upper tolerance limit 75 Standard deviation Cpu Lower tolerance limit 25 Cpl Cpk 1.047

14 Statistical Process Control 14 Percent outside Specification Limits (25 to 75) % Below LSL: z = LSL - x _ z = = ; P( z < -3.44) = ( *) = that 7.60 items will exceed lower limit *Note: This was taken from an outside table since Appendix A extends only to z = % Above USL: z = USL - x _ z = = 3.14 ; P( z > 3.14) = ( ) = that 7.60 items will exceed upper limit *Note: This was taken from an outside table since Appendix A extends only to z = 3.09 Therefore, the percent outside is calculated as: 0.11 % The percent outside calculations shows that the process has a very low percent outside of specifications. This verifies what the capability indexes show. 16. Ricardo s Widgets makes a critical part for a popular brand of cell phones. Consider the data for 15 samples of size 4 of a key dimension for the part, shown in the worksheet Prob Specifications are ± a. Compute control limits for an x-chart using the statistic R /d 2 as an estimate of the standard deviation and using the actual standard deviation for the data. Why are they different? b. Construct the x - and R charts and an x- chart for individuals using the data. Interpret the results. c. Estimate the process capability by using the actual sample standard deviation. Answer 16. a) See spreadsheet Prob13-16IV.xls (X and MR chart template) for details.

15 Statistical Process Control 15 CL x : x = ; CL R : R = (with a 4-period moving average) Estimated = R / d 2 = / = ; actual = , close to the estimate. x ± 3 est = ± 3 (0.0060) = to ; for x ± 3 = ± 3 (0.0055) = to The limits above apply to individual items only. Individual items can only be plotted on x-charts. See the chart on individuals, below, and the previous problem, for a more thorough discussion Prob Individuals (X) Chart Individuals Upper control limit Center line Lower control limit e a lu V Observation number

16 Statistical Process Control Prob Moving Range Chart Mo ving ran ges Lo wer control limit C enter line U pper control limit s e g n ra g in v o M Observation number b) The detailed comparisons of process capability using estimated are shown in a table on the spreadsheets Prob13-16CP.xls Although individual values must be plotted on x-charts, as shown above, students need to understand their relationship to x - chart and R-chart results for comparison with the charts for individuals. For the x - chart: x ± A 2 R = ± (0.0092) = to For the R-chart: UCL R = D 4 R = (0.0092) = LCL R = D 3 R = 0 These limits apply to sample groups of 4 items each.

17 Statistical Process Control 17 Prob X-bar Chart for Sample Groups Ave rages Lower control limit Up per control limit Center line 0.12 s e g ra v e A Sample number Problem R-Chart for Sample Groups Ranges Lower control limit Upper control limit Center line s e0.015 g n 0.01 a R Sample number c) The calculations of process capability using the estimated value is shown on the following table. (See spreadsheet Prob13-16CP.xls for details.) Nominal specification Average Cp Upper tolerance limit Standard deviation Cpu Lower tolerance limit Cpl Cpk

18 Statistical Process Control 18 Estimated = R / d 2 = / = ; actual = , as shown in part a, above. 17. Suppose that in Problem 8, after it was revised, the upper specification limit was set at USL = 475, and the lower specification limit became LSL = 325. Compute the process capability and the modified control limits. Answer 17. See previous spreadsheet s (Prob13-8XR.xls), Tab B, Process Capability Analysis section and the calculated data, below. With data from Problem 8 and USL = 475, LSL = 325, note that spreadsheet Prob13-08XR.xls has been modified to show an estimated standard deviation of: = R / d 2 = / = As shown on the spreadsheet: c p = c pl = c pu = c pk = In the spreadsheet Prob13-17CP.xls the actual standard deviation of: been calculated. = has From this, we obtain modified C p, C pl, C pu, and C pk of: Nominal specification 400 Average Cp Upper tolerance limit 475 Standard deviation Cpl Lower tolerance limit 325 Cpu % outside indicates that the process is well within specification limits. The modified control limits based on R and special A m factors are: URL x = US - A m R = (0.749) (30.957) = LRL x = LS + A m R =325 + (0.749) (30.957) = Cpk 1.324

19 Statistical Process Control 19 ` Note that the factor of comes from the ASQ control chart table in Figure

20 Statistical Process Control 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 Determine process capability limits. If specifications are determined to be 70 ± 25, what percentage will be out of specification? Answer 18. (See spreadsheet Prob13-18CP.xls for details.) For the Center Lines, CL x : x = ; CL R : R = Control limits for the x - chart are: x ± A 2 R = ± (21.920) = to For the R-chart: UCL R = D 4 R = (21.920) = LCL R = D 3 R = 0 The limits above apply to sample groups of 5 items each. Estimated = R / d 2 = / = 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 x - chart shows that the process is, in fact, out of control, because of 4 out of 5 samples within samples 6-10 being on one side of the center-line. The % outside calculation can be performed as follows. Note the warning given below, however. Percent outside Specification Limits (45 to 95) % Below LSL: z = LSL - x _ z = = ; P( z < -2.56) = ( ) = that items will exceed lower limit

21 Statistical Process Control 21 % Above USL: z = USL - x _ z = = 2.74 ; P( z > 2.74) = ( ) = that items will exceed upper limit Therefore, the percent outside is calculated as: 0.83 % 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 calculation is going to generate questionable results. Charts showing out-of-control conditions

22 Statistical Process Control The Bell Vader Company, which produces heavy-duty electrical motors, machines a part called an end cap. To meet competitive pressures, the company began to apply statistical quality control to its processes. Because each motor produced by the company uses two end caps that could cost as much as $200 each, the company sees the importance of bringing the process under control. The table in the worksheet Prob shows data collected to construct a control chart. a. Compute control limits and construct and analyze the x and R charts for this process. What conclusions can you reach about the state of statistical control? b. Using the control chart, estimate the process capability. The specification limits for the end cap are to Note that the data are coded, so 75 = , 77 = , and so on. Determine what percentage of end caps would be expected to fall outside specifications. What conclusions and recommendations can you make? Answer 19. See spreadsheet Prob13-19XR.xls for details.

23 Statistical Process Control 23 a) The control chart shown below was constructed. Note that coded data has been used on the control chart. For example, 75 represents Parameters for the charts will be calculated in coded form and translated at the end of the problem: For the center lines, CL x : x = (actual = ); CL R : R = (actual = ) Control limits for the x - chart are: x ± A 2 R = ± (3.300) = to For the R-chart: UCL R = D 4 R = (3.300) = LCL R = D 3 R = 0 Although the tendency is for the data to hug the centerline, the control charts establish that the process is in control. b. The limits above apply to sample groups of 5 items each. % outside calculations are based on specification limits for individual items. Estimated = R / d 2 = 3.300/2.326 = C p = UTL - LTL = = ; very poor capability 6 6 (1.419) The % outside calculation is performed as follows. Percent outside Specification Limits (actual = to ) % Below LSL: z = LSL - x z = = ; P( z < -1.69) = ( ) = that items will exceed lower limit % Above USL: z = USL - x

24 Statistical Process Control 24 z = = ; P( z > 1.831) = ( ) = that items will exceed upper limit Therefore, the total % outside is calculated as 7.9 % Obviously, the problem lies in the fact that the process is not capable of producing good end caps that consistently fall within specification limits. Even though the process is under control, the Bell Vader Company needs to analyze the process to determine what may be done to make it capable. This investigation should look at current materials, equipment, methods, and other pertinent areas. The process needs to be improved, or new equipment purchased in order to improve capability and reduce costs. x -R Chart for Bell Vader Problem Prob X-bar Chart Averages Lower control limit Upper control limit Center line e s e r a g v A Sample number

25 Statistical Process Control 25 Prob R-Chart Ranges Lower control limit Upper control limit Center line s 5 e g n 4 a R Sample num ber 20. Suppose that in Problem 13-14, after it was revised, the upper specification limit was set at USL = 6.75, and the lower specification limit was set at LSL = Compute the process capability and the modified control limits. Answer 20. With data from Problem and using USL = 6.75, LSL = 3.25, from the Prob13-20CP.xls spreadsheet we see: Upper specification 6.75 Cp Lower specification 3.25 Cpl Nominal 5.00 Cpu specification Cpk Note that the spreadsheet uses an estimated standard deviation of: Estimated = R / d 2 = / =

26 Statistical Process Control 26 From this, we obtain: Percent outside Specification Limits (3.25 to 6.75) % Below LSL: z = LSL - x z = = ; P( z < -3.48) = ( *) = that items will exceed lower limit *Note: This was taken from an outside table since Appendix A extends only to z = % Above USL: z = USL - x z = = 3.33 ; P( z > 3.33) = ( ) = that items will exceed upper limit Therefore, the percent outside is calculated as: 0.07 % These calculations show that the process has a relatively small % outside specifications. In problem 14b, points that showed assignable causes were eliminated, so the process should be in control. The process still needs some fine tuning in order to become more capable (ideally the Cp should be 2.0, or better) as shown by the % outside calculation and the capability indexes. The control charts show the following patterns:

27 Statistical Process Control 27 The modified control limits (not shown on the chart) are: URL x = US - A m R = (0.728) (1.057) = 5.98 LRL x = LS + A m R = (0.728) (1.057) = 4.02 See file Q8Intl-IM13D.doc - The next file for more solutions to end-of-chapter problems for Chapter 13.