4. The Joint SDDS EWMA-AM and SDDS EWMA-AV Charts
4.3. Detection Performance of the Joint EWMA-AM and EWMA-AV
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4.3. Detection Performance of the Joint EWMA-AM and EWMA-AV Charts
For the single sampling approach, using both the EWMA-AM chart and EWMA-AV chart (joint EWMA-AM and EWMA-AV charts) in Yang and Arnold (2014), may monitor the process mean and variance simultaneously. We measure the detection performance of the joint EWMA-AM and EWMA-AV charts in this Section.
To measure the detection performance, we calculated the ARL and the 25th, 50th, 75th percentiles of run length using the computer simulation of the process with 10,000 times. The of the joint EWMA-AM and EWMA-AV charts depends on the corresponding coefficients of their control limits ( , , , , , ,
, , ). Since EWMA-AM and EWMA-AV statistics are dependent, to determine the coefficients of the control limits with acceptably fixed , we use direct search approach to adjust the combinations of the coefficients of the control limits by computer simulation.
Table 68 shows the combinations of the coefficients of the control limits of the joint EWMA-AM and EWMA-AV charts with and for various combinations of ( , , ). Table 69 shows the combinations of the coefficients of the control limits of the joint EWMA-AM and EWMA-AV charts with and for various distributions and .
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Table 68. Coefficients of the control limits of the joint EWMA-AM and EWMA-AV charts with and under various
( , , ).
combination 1 2 3 4 5 6 7
0.1 0.1 0.3 0.3 0.5 0.5 0.4
0.1 0.4 0.1 0.4 0.1 0.4 0.3
10 10 10 10 10 10 4
2.58 2.58 2.47 2.47 2.44 2.42 2.47
2.34 2.34 2.47 2.47 2.54 2.51 2.46
2.66 2.47 2.66 2.47 2.69 2.47 2.56
2.24 2.46 2.24 2.46 2.26 2.46 2.37
182.39 181.84 181.19 181.55 187.60 181.4 185.18
Table 69. Coefficients of the control limits of the joint EWMA-AM and EWMA-AV charts with and for various distributions and .
distribution N(0, 1) DE(0, 1)
DE(0, ) Unif( , ) Exp(1)
0.5000 0.5000 0.5000 0.3679
0.3173 0.2706 0.3466 0.2432
2.42 2.42 2.42 2.47
2.51 2.51 2.52 2.46
2.50 2.54 2.60 2.54
2.43 2.37 2.35 2.39
184.97 181.95 183.32 188.87
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For the out-of-control process, it is assumed that both the IC process mean and the variance have shifted, that is and .
Table 70–76 show the out-of-control ARL (the first row of each cell), 25th, 50th, 75th percentiles of run length (the second row of each cell) of the joint EWMA-AM and EWMA-AV charts with and under 0.1–0.9, 0.1–0.9 and various combinations of ( , , ), respectively.
Their corresponding coefficients of the control limits are listed in Table 66. It can be seen that the run length has right-skewed distributions and the out-of-control ARLs decrease when is far away from and/or is far away from .
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Table 70. The out-of-control ARL, 25th, 50th, 75th percentiles of run length of the joint EWMA-AM and EWMA-AV charts with , and under , .
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 (1) 182.39 16.51 7.50 4.89 3.63 2.91 2.40 2.13 2.01 (2) 61-129-246 10-14-21 5-7-9 4-5-6 3-3-4 2-3-3 2-2-3 2-2-2 2-2-2 0.2 (1) 10.61 9.26 6.59 4.73 3.64 2.88 2.41 2.12 2.01
(2) 7-10-13 6-9-11 5-6-8 4-5-5 3-4-4 2-3-3 2-2-3 2-2-2 2-2-2 0.3 (1) 5.04 4.97 4.64 4.05 3.38 2.83 2.38 2.12 2.01
(2) 4-5-6 4-5-6 4-5-5 3-4-5 3-3-4 2-3-3 2-2-3 2-2-2 2-2-2 0.4 (1) 3.43 3.43 3.39 3.23 2.95 2.64 2.32 2.11 2.01
(2) 3-3-4 3-3-4 3-3-4 3-3-4 2-3-3 2-3-3 2-2-3 2-2-2 2-2-2 0.5 (1) 2.69 2.67 2.66 2.62 2.52 2.37 2.20 2.06 1.99
(2) 2-3-3 2-3-3 2-3-3 2-3-3 2-2-3 2-2-3 2-2-2 2-2-2 2-2-2 0.6 (1) 2.20 2.20 2.19 2.18 2.16 2.09 2.03 1.98 1.96
(2) 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 0.7 (1) 1.90 1.89 1.89 1.90 1.89 1.88 1.85 1.86 1.85
(2) 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 0.8 (1) 1.62 1.63 1.62 1.62 1.62 1.63 1.62 1.63 1.63
(2) 1-2-2 1-2-2 1-2-2 1-2-2 1-2-2 1-2-2 1-2-2 1-2-2 1-2-2 0.9 (1) 1.25 1.27 1.26 1.27 1.26 1.27 1.26 1.26 1.27
(2) 1-1-2 1-1-2 1-1-2 1-1-2 1-1-2 1-1-2 1-1-2 1-1-2 1-1-2 (1) is the ARL, (2) is the 25th, 50th and 75th percentiles of run length.
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Table 71. The out-of-control ARL, 25th, 50th, 75th percentiles of run length of the joint EWMA-AM and EWMA-AV charts with , and under , .
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 (1) 7.13 11.63 28.63 181.84 28.36 11.84 7.33 5.38 4.36 (2) 6-7-8 9-11-14 17-25-36 62-129-247 16-24-36 8-11-14 6-7-9 5-5-6 4-4-5 0.2 (1) 6.64 8.71 10.28 10.55 10.20 8.66 6.70 5.23 4.33 (2) 6-6-7 7-8-10 7-9-13 7-10-13 7-9-12 6-8-10 5-6-8 5-5-6 4-4-5 0.3 (1) 4.80 5.03 5.06 5.05 5.06 5.01 4.78 4.41 4.02
(2) 4-5-6 4-5-6 4-5-6 4-5-6 4-5-6 4-5-6 4-5-6 4-4-5 4-4-4 0.4 (1) 3.45 3.48 3.46 3.44 3.45 3.44 3.46 3.39 3.34 (2) 3-3-4 3-3-4 3-3-4 3-3-4 3-3-4 3-3-4 3-3-4 3-3-4 3-3-4 0.5 (1) 2.67 2.67 2.66 2.67 2.67 2.68 2.67 2.66 2.67 (2) 2-3-3 2-3-3 2-3-3 2-3-3 2-3-3 2-3-3 2-3-3 2-3-3 2-3-3 0.6 (1) 2.20 2.20 2.20 2.20 2.20 2.19 2.20 2.20 2.19 (2) 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 0.7 (1) 1.90 1.90 1.90 1.90 1.90 1.89 1.89 1.90 1.89 (2) 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 2-2-2 0.8 (1) 1.62 1.62 1.63 1.62 1.63 1.63 1.63 1.64 1.63 (2) 1-2-2 1-2-2 1-2-2 1-2-2 1-2-2 1-2-2 1-2-2 1-2-2 1-2-2 0.9 (1) 1.26 1.27 1.26 1.26 1.26 1.26 1.27 1.26 1.27 (2) 1-1-2 1-1-2 1-1-2 1-1-2 1-1-2 1-1-2 1-1-2 1-1-2 1-1-2 (1) is the ARL, (2) is the 25th, 50th and 75th percentiles of run length.
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Table 72. The out-of-control ARL, 25th, 50th, 75th percentiles of run length of the joint EWMA-AM and EWMA-AV charts with , and under , .
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 (1) 7.17 6.97 6.02 4.71 3.63 2.88 2.41 2.12 2.01 (2) 6-7-8 6-7-8 5-6-7 4-5-6 3-4-4 2-3-3 2-2-3 2-2-2 2-2-2 0.2 (1) 16.76 12.30 7.29 4.86 3.65 2.89 2.40 2.13 2.01 (2) 12-15-20 9-11-15 5-7-9 4-5-6 3-4-4 2-3-3 2-2-3 2-2-2 2-2-2 0.3 (1) 181.19 16.65 7.49 4.89 3.65 2.89 2.40 2.13 2.01 (2) 64-132-245 10-15-21 5-7-9 4-5-6 3-4-4 2-3-3 2-2-3 2-2-2 2-2-2 0.4 (1) 16.68 11.97 7.21 4.90 3.63 2.88 2.41 2.13 2.01 (2) 11-15-21 8-11-15 5-7-9 4-5-6 3-3-4 2-3-3 2-2-3 2-2-2 2-2-2 0.5 (1) 7.31 7.04 5.90 4.60 3.59 2.87 2.41 2.12 2.01 (2) 6-7-9 5-7-8 5-6-7 4-5-5 3-3-4 2-3-3 2-2-3 2-2-2 2-2-2 0.6 (1) 4.78 4.75 4.51 4.03 3.40 2.83 2.39 2.12 2.01 (2) 4-5-5 4-5-5 4-4-5 3-4-5 3-3-4 2-3-3 2-2-3 2-2-2 2-2-2 0.7 (1) 3.61 3.59 3.55 3.39 3.08 2.70 2.37 2.12 2.01 (2) 3-4-4 3-4-4 3-3-4 3-3-4 3-3-4 2-3-3 2-2-3 2-2-2 2-2-2 0.8 (1) 2.90 2.89 2.89 2.83 2.71 2.52 2.28 2.09 2.01 (2) 3-3-3 3-3-3 3-3-3 3-3-3 2-3-3 2-2-3 2-2-3 2-2-2 2-2-2 0.9 (1) 2.32 2.32 2.32 2.31 2.26 2.20 2.11 2.04 2.00 (2) 2-2-3 2-2-3 2-2-3 2-2-3 2-2-3 2-2-2 2-2-2 2-2-2 2-2-2 (1) is the ARL, (2) is the 25th, 50th and 75th percentiles of run length.
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Table 73. The out-of-control ARL, 25th, 50th, 75th percentiles of run length of the joint EWMA-AM and EWMA-AV charts with , and under , .
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 (1) 6.34 6.99 7.17 7.17 7.15 6.93 6.25 5.24 4.35 (2) 6-6-7 6-7-8 6-7-8 6-7-8 6-7-8 6-7-8 5-6-7 5-5-6 4-4-5 0.2 (1) 7.10 10.66 15.14 16.80 14.87 10.58 7.23 5.38 4.36
(2) 6-7-8 8-10-12 11-14-18 12-15-20 11-14-18 8-10-13 6-7-8 5-5-6 4-4-5 0.3 (1) 7.10 11.60 28.77 181.55 28.48 11.72 7.31 5.38 4.35
(2) 6-7-8 9-11-14 17-25-36 61-131-248 16-24-36 8-11-14 6-7-9 5-5-6 4-4-5 0.4 (1) 7.04 10.38 14.86 16.83 14.62 10.27 7.15 5.38 4.37
(2) 6-7-8 8-10-12 10-14-18 11-15-21 10-13-18 8-10-12 6-7-8 5-5-6 4-4-5 0.5 (1) 6.14 6.99 7.34 7.34 7.28 6.93 6.10 5.12 4.31
(2) 5-6-7 6-7-8 6-7-9 6-7-9 6-7-9 5-7-8 5-6-7 4-5-6 4-4-5 0.6 (1) 4.70 4.77 4.78 4.79 4.80 4.76 4.66 4.42 4.05
(2) 4-5-5 4-5-5 4-5-5 4-5-5 4-5-5 4-5-5 4-5-5 4-4-5 4-4-4 0.7 (1) 3.59 3.60 3.59 3.59 3.59 3.59 3.59 3.58 3.51
(2) 3-4-4 3-4-4 3-4-4 3-3-4 3-3-4 3-3.5-4 3-4-4 3-4-4 3-4-4 0.8 (1) 2.90 2.90 2.90 2.89 2.90 2.90 2.91 2.89 2.91
(2) 3-3-3 3-3-3 3-3-3 3-3-3 3-3-3 3-3-3 3-3-3 3-3-3 3-3-3 0.9 (1) 2.33 2.33 2.33 2.31 2.32 2.32 2.32 2.32 2.32
(2) 2-2-3 2-2-3 2-2-3 2-2-3 2-2-3 2-2-3 2-2-3 2-2-3 2-2-3 (1) is the ARL, (2) is the 25th, 50th and 75th percentiles of run length.
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Table 74. The out-of-control ARL, 25th, 50th, 75th percentiles of run length of the joint EWMA-AM and EWMA-AV charts with , and under , .
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 (1) 3.95 3.93 3.87 3.66 3.28 2.82 2.41 2.13 2.02 (2) 4-4-4 4-4-4 4-4-4 3-4-4 3-3-4 2-3-3 2-2-3 2-2-2 2-2-2 0.2 (1) 5.29 5.25 4.96 4.29 3.53 2.89 2.43 2.15 2.02
(2) 4-5-6 4-5-6 4-5-6 4-4-5 3-3-4 2-3-3 2-2-3 2-2-2 2-2-2 0.3 (1) 8.28 7.88 6.35 4.78 3.65 2.92 2.43 2.14 2.02
(2) 6-8-10 6-8-9 5-6-8 4-5-6 3-3-4 2-3-3 2-2-3 2-2-2 2-2-2 0.4 (1) 19.58 13.04 7.38 4.92 3.68 2.90 2.43 2.14 2.02
(2) 13-18-24 9-12-16 5-7-9 4-5-6 3-4-4 2-3-3 2-2-3 2-2-2 2-2-2 0.5 (1) 187.60 16.80 7.59 4.94 3.67 2.89 2.44 2.14 2.02
(2) 64-133-255 10-15-21 5-7-9 4-5-6 3-3-4 2-3-3 2-2-3 2-2-2 2-2-2 0.6 (1) 18.37 12.87 7.32 4.91 3.65 2.91 2.42 2.14 2.02
(2) 12-16-23 9-12-16 5-7-9 4-5-6 3-3-4 2-3-3 2-2-3 2-2-2 2-2-2 0.7 (1) 7.93 7.57 6.19 4.77 3.65 2.90 2.43 2.14 2.02
(2) 6-8-9 6-7-9 5-6-7 4-5-6 3-3-4 2-3-3 2-2-3 2-2-2 2-2-2 0.8 (1) 5.10 5.07 4.81 4.23 3.52 2.89 2.42 2.14 2.02
(2) 4-5-6 4-5-6 4-5-5 4-4-5 3-3-4 2-3-3 2-2-3 2-2-2 2-2-2 0.9 (1) 3.76 3.77 3.71 3.53 3.20 2.78 2.42 2.14 2.02
(2) 3-4-4 3-4-4 3-4-4 3-4-4 3-3-4 2-3-3 2-2-3 2-2-2 2-2-2 (1) is the ARL, (2) is the 25th, 50th and 75th percentiles of run length.
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Table 75. The out-of-control ARL, 25th, 50th, 75th percentiles of run length of the joint EWMA-AM and EWMA-AV charts with , and under , .
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 (1) 3.93 3.93 3.92 3.92 3.92 3.92 3.92 3.90 3.85 (2) 4-4-4 4-4-4 4-4-4 4-4-4 4-4-4 4-4-4 4-4-4 4-4-4 4-4-4 0.2 (1) 5.13 5.21 5.23 5.25 5.24 5.21 5.07 4.71 4.23
(2) 4-5-6 4-5-6 4-5-6 4-5-6 4-5-6 4-5-6 4-5-6 4-5-5 4-4-4 0.3 (1) 6.54 7.71 8.11 8.20 8.11 7.62 6.46 5.24 4.34
(2) 6-6-7 6-7-9 6-8-9 6-8-10 6-8-9 6-7-9 5-6-7 5-5-6 4-4-5 0.4 (1) 7.10 10.79 16.51 19.19 16.29 10.71 7.22 5.38 4.35
(2) 6-7-8 9-10-13 12-15-20 12-17-24 11-15-20 8-10-13 6-7-8 5-5-6 4-4-5 0.5 (1) 7.15 11.62 28.64 181.40 28.43 11.75 7.31 5.39 4.37
(2) 6-7-8 9-11-14 17-25-36 61-130-248 16-24-36 8-11-14 6-7-8 5-5-6 4-4-5 0.6 (1) 7.08 10.63 15.81 18.14 15.65 10.57 7.18 5.37 4.36
(2) 6-7-8 8-10-12 11-14-19 12-16-22 11-14-19 8-10-13 6-7-8 5-5-6 4-4-5 0.7 (1) 6.43 7.45 7.81 7.83 7.81 7.34 6.34 5.22 4.35
(2) 6-6-7 6-7-9 6-7-9 6-7-9 6-7-9 6-7-8 5-6-7 5-5-6 4-4-5 0.8 (1) 4.97 5.06 5.07 5.07 5.06 5.03 4.91 4.65 4.18
(2) 4-5-6 4-5-6 4-5-6 4-5-6 4-5-6 4-5-6 4-5-5 4-5-5 4-4-4 0.9 (1) 3.68 3.69 3.68 3.68 3.68 3.69 3.69 3.67 3.62
(2) 3-4-4 3-4-4 3-4-4 3-4-4 3-4-4 3-4-4 3-4-4 3-4-4 3-4-4 (1) is the ARL, (2) is the 25th, 50th and 75th percentiles of run length.
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Table 76. The out-of-control ARL, 25th, 50th, 75th percentiles of run length of the joint EWMA-AM and EWMA-AV charts with , and under , .
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 (1) 8.21 8.26 8.22 8.20 8.08 7.80 7.14 6.26 5.39
(2) 7-8-9 7-8-9 7-8-9 7-8-9 7-8-9 7-7-9 6-7-8 5-6-7 5-5-6 0.2 (1) 12.38 13.29 13.52 13.15 12.01 10.07 8.09 6.51 5.41
(2) 10-12-14 10-13-16 10-13-16 10-12-16 9-11-14 8-10-12 6-8-9 5-6-8 5-5-6 0.3 (1) 17.01 27.67 33.63 27.19 17.43 11.66 8.52 6.58 5.43
(2) 13-16-20 18-24-34 19-28-42 16-24-34 12-16-21 8-11-14 6-8-10 5-6-8 5-5-6 0.4 (1) 18.25 48.41 185.18 49.45 20.12 12.10 8.58 6.59 5.40
(2) 13-17-21 26-40-62 63-132-250 24-39-64 12-18-25 9-11-15 6-8-10 5-6-8 5-5-6 0.5 (1) 16.82 27.00 33.41 26.60 17.22 11.54 8.53 6.60 5.42
(2) 13-16-20 17-24-33 18-28-42 16-23-33 11-15-21 8-11-14 6-8-10 5-6-8 5-5-6 0.6 (1) 12.21 13.40 13.62 13.28 11.88 9.91 8.04 6.48 5.42
(2) 9-12-15 9-12-16 9-12-17 9-12-16 9-11-14 8-9-12 6-8-9 5-6-8 5-5-6
0.7 (1) 8.33 8.37 8.36 8.41 8.18 7.73 6.99 6.11 5.32
(2) 7-8-10 7-8-10 7-8-10 7-8-10 7-8-9 6-8-9 6-7-8 5-6-7 5-5-5
0.8 (1) 6.03 6.03 6.00 6.03 6.01 5.87 5.72 5.42 5.05
(2) 5-6-7 5-6-7 5-6-7 5-6-7 5-6-7 5-6-7 5-5-6 5-5-6 5-5-5
0.9 (1) 4.67 4.65 4.65 4.66 4.65 4.64 4.62 4.57 4.53
(2) 4-4-5 4-4-5 4-4-5 4-4-5 4-4-5 4-4-5 4-4-5 4-4-5 4-4-5 (1) is the ARL, (2) is the 25th, 50th and 75th percentiles of run length.
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4.4. Performance Comparison with Existing Control Charts
We compare the performance of the joint SDDS EWMA-AM and SDDS EWMA-AV charts with other existing mean and variance control charts under different distributions of quality variable. For performance comparison, values of the competing charts are fixed at (or very close to) an acceptably value, such as 185 or 500, with the compared sample size, , and then compare their out-of-control ARLs.
If the in-control distributions are unknown, we consider different values of and , which depends on the distributions. Tables 77–78 show the detection performance of the joint charts using Method 1 (the first row of each cell), and Method 2 (the second row of each cell), and the joint EWMA-AM and EWMA-AV charts (the third row of each cell) with and and for 0.1–0.9 and 0.1–0.9 under two combinations of ( , ), respectively. Tables 79–82 show the detection performance of the joint charts using Method 1 (the first row of each cell) and the joint EWMA-AM and EWMA-AV charts (the second row of each cell) with , and for
0.1–0.9 and 0.1–0.9 under four combinations of ( , ), respectively.
It can be seen that the detection performance of the joint SDDS charts using Method 2 perform better than Method 1, and the joint SDDS charts have superior out-of-control detection performance than the joint EWMA-AM and EWMA-AV charts for and .
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Table 77. Detection performance (ARL) of the joint SDDS charts using Method 1 and 2, and the joint EWMA-AM and EWMA-AV charts with ,
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Table 78. Detection performance (ARL) of the joint SDDS charts using Method 1 and 2, and the joint EWMA-AM and EWMA-AV charts with ,
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Table 79. Detection performance (ARL) of the joint SDDS charts using Method 1 and the joint EWMA-AM and EWMA-AV charts with , and under , .
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 (1) 3.42 7.30 23.23 182.00 24.48 7.79 3.81 2.27 1.49 (2) 7.13 11.63 28.63 181.84 28.36 11.84 7.33 5.38 4.36 0.2 (1) 2.82 4.42 6.05 6.69 5.96 4.31 2.89 1.99 1.41 (2) 6.64 8.71 10.28 10.55 10.20 8.66 6.70 5.23 4.33 0.3 (1) 1.95 2.24 2.38 2.39 2.34 2.16 1.85 1.55 1.25 (2) 4.80 5.03 5.06 5.05 5.06 5.01 4.78 4.41 4.02 0.4 (1) 1.44 1.49 1.49 1.50 1.46 1.45 1.36 1.25 1.13 (2) 3.45 3.48 3.46 3.44 3.45 3.44 3.46 3.39 3.34 0.5 (1) 1.17 1.17 1.17 1.17 1.17 1.15 1.14 1.10 1.05 (2) 2.67 2.67 2.66 2.67 2.67 2.68 2.67 2.66 2.67 0.6 (1) 1.05 1.05 1.05 1.06 1.05 1.05 1.04 1.03 1.02 (2) 2.20 2.20 2.20 2.20 2.20 2.19 2.20 2.20 2.19 0.7 (1) 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.00 (2) 1.90 1.90 1.90 1.90 1.90 1.89 1.89 1.90 1.89 0.8 (1) 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 (2) 1.62 1.62 1.63 1.62 1.63 1.63 1.63 1.64 1.63 0.9 (1) 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 (2) 1.26 1.27 1.26 1.26 1.26 1.26 1.27 1.26 1.27 (1) is the ARL1 of the joint SDDS charts using Method 1,
(2) is the ARL1 of the joint EWMA-AM and EWMA-AV charts.
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Table 80. Detection performance (ARL) of the joint SDDS charts using Method 1 and the joint EWMA-AM and EWMA-AV charts with , and under , .
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 (1) 3.45 3.04 2.38 1.81 1.44 1.22 1.10 1.03 1.00 (2) 7.17 6.97 6.02 4.71 3.63 2.88 2.41 2.12 2.01 0.2 (1) 11.94 7.16 3.59 2.15 1.52 1.23 1.10 1.03 1.00 (2) 16.76 12.30 7.29 4.86 3.65 2.89 2.40 2.13 2.01 0.3 (1) 186.17 11.63 3.97 2.17 1.52 1.24 1.09 1.03 1.00 (2) 181.19 16.65 7.49 4.89 3.65 2.89 2.40 2.13 2.01 0.4 (1) 12.33 7.12 3.40 2.06 1.49 1.23 1.09 1.03 1.00 (2) 16.68 11.97 7.21 4.90 3.63 2.88 2.41 2.13 2.01 0.5 (1) 3.94 3.33 2.40 1.78 1.40 1.19 1.08 1.03 1.00 (2) 7.31 7.04 5.90 4.60 3.59 2.87 2.41 2.12 2.01 0.6 (1) 2.18 2.01 1.74 1.47 1.27 1.14 1.06 1.02 1.00 (2) 4.78 4.75 4.51 4.03 3.40 2.83 2.39 2.12 2.01 0.7 (1) 1.53 1.46 1.37 1.26 1.15 1.08 1.04 1.01 1.00 (2) 3.61 3.59 3.55 3.39 3.08 2.70 2.37 2.12 2.01 0.8 (1) 1.21 1.19 1.16 1.10 1.07 1.03 1.02 1.01 1.00 (2) 2.90 2.89 2.89 2.83 2.71 2.52 2.28 2.09 2.01 0.9 (1) 1.04 1.04 1.03 1.02 1.01 1.01 1.00 1.00 1.00 (2) 2.32 2.32 2.32 2.31 2.26 2.20 2.11 2.04 2.00 (1) is the ARL1 of the joint SDDS charts using Method 1,
(2) is the ARL1 of the joint EWMA-AM and EWMA-AV charts.
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Table 81. Detection performance (ARL) of the joint SDDS charts using Method 1 and the joint EWMA-AM and EWMA-AV charts with , and under , .
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 (1) 2.63 3.17 3.44 3.47 3.32 3.03 2.52 1.93 1.44 (2) 6.34 6.99 7.17 7.17 7.15 6.93 6.25 5.24 4.35 0.2 (1) 3.31 6.09 10.18 12.02 9.95 6.06 3.56 2.25 1.48 (2) 7.10 10.66 15.14 16.80 14.87 10.58 7.23 5.38 4.36 0.3 (1) 3.43 7.25 23.99 182.01 24.33 7.73 3.85 2.23 1.49 (2) 7.10 11.60 28.77 181.55 28.48 11.72 7.31 5.38 4.35 0.4 (1) 3.23 5.80 10.18 12.34 9.89 5.85 3.42 2.15 1.45 (2) 7.04 10.38 14.86 16.83 14.62 10.27 7.15 5.38 4.37 0.5 (1) 2.57 3.37 3.87 3.99 3.80 3.29 2.56 1.89 1.40 (2) 6.14 6.99 7.34 7.34 7.28 6.93 6.10 5.12 4.31 0.6 (1) 1.94 2.12 2.16 2.17 2.15 2.03 1.83 1.57 1.28 (2) 4.70 4.77 4.78 4.79 4.80 4.76 4.66 4.42 4.05 0.7 (1) 1.50 1.52 1.53 1.54 1.52 1.49 1.41 1.31 1.16 (2) 3.59 3.60 3.59 3.59 3.59 3.59 3.59 3.58 3.51 0.8 (1) 1.21 1.21 1.21 1.21 1.21 1.19 1.18 1.13 1.07 (2) 2.90 2.90 2.90 2.89 2.90 2.90 2.91 2.89 2.91 0.9 (1) 1.04 1.04 1.04 1.04 1.04 1.03 1.03 1.03 1.01 (2) 2.33 2.33 2.33 2.31 2.32 2.32 2.32 2.32 2.32 (1) is the ARL1 of the joint SDDS charts using Method 1,
(2) is the ARL1 of the joint EWMA-AM and EWMA-AV charts.
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Table 82. Detection performance (ARL) of the joint SDDS charts using Method 1 and the joint EWMA-AM and EWMA-AV charts with , and under , .
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.1 (1) 1.22 1.24 1.23 1.23 1.22 1.20 1.18 1.13 1.07 (2) 3.93 3.93 3.92 3.92 3.92 3.92 3.92 3.90 3.85 0.2 (1) 1.81 1.93 1.96 1.94 1.91 1.84 1.67 1.45 1.21 (2) 5.13 5.21 5.23 5.25 5.24 5.21 5.07 4.71 4.23 0.3 (1) 2.77 3.57 4.02 4.15 3.85 3.28 2.56 1.89 1.37 (2) 6.54 7.71 8.11 8.20 8.11 7.62 6.46 5.24 4.34 0.4 (1) 3.58 6.50 11.29 13.84 10.73 5.98 3.47 2.18 1.45 (2) 7.10 10.79 16.51 19.19 16.29 10.71 7.22 5.38 4.35 0.5 (1) 3.84 7.86 24.59 183.77 23.66 7.61 3.81 2.24 1.51 (2) 7.15 11.62 28.64 181.40 28.43 11.75 7.31 5.39 4.37 0.6 (1) 3.63 6.45 11.34 13.74 10.55 6.05 3.49 2.17 1.47 (2) 7.08 10.63 15.81 18.14 15.65 10.57 7.18 5.37 4.36 0.7 (1) 2.77 3.58 4.01 4.11 3.88 3.28 2.54 1.88 1.38 (2) 6.43 7.45 7.81 7.83 7.81 7.34 6.34 5.22 4.35 0.8 (1) 1.81 1.93 1.94 1.96 1.94 1.83 1.67 1.46 1.22 (2) 4.97 5.06 5.07 5.07 5.06 5.03 4.91 4.65 4.18 0.9 (1) 1.22 1.22 1.23 1.22 1.23 1.21 1.17 1.13 1.06 (2) 3.68 3.69 3.68 3.68 3.68 3.69 3.69 3.67 3.62 (1) is the ARL1 of the joint SDDS charts using Method 1,
(2) is the ARL1 of the joint EWMA-AM and EWMA-AV charts.
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If the in-control distributions are known, we compare their out-of-control ARLs for various mean and variance shifts scale and (for IC process the and ). For the standard normal distribution N(0, 1), Table 83 shows the detection performance of the joint SDDS charts using Method 1 and 2 ( ), the joint EWMA-AM and EWMA-AV charts ( ) and the Shewhart and charts with and , Table 84 shows the detection performance of the joint SDDS charts using Method 1 and 2 ( ), the Max chart, the GLR chart and the Fisher chart (Deng (2009)) with and , Table 85 shows the detection performance of the joint SDDS charts using Method 1 ( ), the MEW, the NCS, the WLC, the ELR (Zhang et al. (2010)) and the CEW (2010) charts with and , Table 86 shows the detection performance of the joint SDDS charts using Method 1 ( ), the SC and the SL charts (Chowdhury et al. (2014)) with and , Table 87 shows the detection performance of the joint SDDS charts using Method 1 ( ), the joint DS and S charts, the standard, the TSS and the VSS joint and R charts (He and Grigoryan (2006)) with and .
It can be seen that under the standard normal distribution, (1) the joint SDDS charts using Method 1 have superior out-of-control detection performance than the joint EWMA-AM and EWMA-AV charts for and , superior than the and charts for and (see Table 83); (2) the joint SDDS charts using Method 1 have superior out-of-control detection performance than the Max chart for and , superior than the GLR chart for and , superior than the Fisher chart for and (see Table 84); (3) the joint SDDS charts
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using Method 1 have superior out-of-control detection performance than the MEW chart for and , superior than the NCS chart for and , superior than the WLC chart for and , superior than the ELR chart for and , superior than the CEW chart for , (see Table 85); (4) the joint SDDS charts using Method 1 have superior out-of-control detection performance than the SL charts for and , superior than the SC charts for and (see Table 86); (5) the joint SDDS charts using Method 1 have superior out-of-control detection performance than the joint DS and S charts for and , superior than the standard joint and R charts for and , superior than the TSS joint and R charts for and , superior than the VSS joint and R charts for and (see Table 87).
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Table 83. Detection performance (ARL1) of the joint SDDS charts using Method 1 and 2, joint EWMA-AM and EWMA-AV charts, and Shewhart charts
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Table 83. (Continued)
1 1.1 1.2 1.3 1.4 1.5 1.75
2.00 joint SDDS charts using Method 1 1.01 1.03 1.05 1.08 1.12 1.16 1.28 joint SDDS charts using Method 2 1.01 1.03 1.05 1.08 1.12 1.16 1.29 joint EWMA-AM and EWMA-AV charts 3.03 3.09 3.18 3.30 3.43 3.59 3.99
and charts 1.00 1.00 1.00 1.00 1.01 1.01 1.02
2.50 joint SDDS charts using Method 1 1.00 1.00 1.01 1.01 1.03 1.04 1.10 joint SDDS charts using Method 2 1.00 1.00 1.01 1.02 1.03 1.04 1.10 joint EWMA-AM and EWMA-AV charts 3.00 3.00 3.02 3.05 3.10 3.18 3.43
and charts 1.00 1.00 1.00 1.00 1.00 1.00 1.00
3.00 joint SDDS charts using Method 1 1.00 1.00 1.00 1.00 1.00 1.01 1.04 joint SDDS charts using Method 2 1.00 1.00 1.00 1.00 1.01 1.01 1.03 joint EWMA-AM and EWMA-AV charts 3.00 3.00 3.00 3.00 3.01 3.03 3.13
and charts 1.00 1.00 1.00 1.00 1.00 1.00 1.00
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Table 84. Detection performance (ARL1) of the joint SDDS charts using Method 1 and 2, the Max chart, the GLR chart and the Fisher chart with
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Table 84. (Continued)
0.66 0.76 0.87 1.00 1.15 1.32 1.5
1.25 joint SDDS charts using Method 1 1.0 1.1 1.1 1.3 1.4 1.6 1.8
joint SDDS charts using Method 2 1.0 1.1 1.1 1.2 1.4 1.6 1.8
Max chart 11.3 10.1 8.1 6.4 5.0 3.6 2.5
GLR chart 4.3 6.5 8.3 8.3 6.4 4.2 2.8
Fisher chart 5.1 7.2 8.4 7.2 4.9 3.1 2.1
1.50 joint SDDS charts using Method 1 1.0 1.0 1.0 1.1 1.2 1.3 1.5
joint SDDS charts using Method 2 1.0 1.0 1.0 1.1 1.2 1.3 1.5
Max chart 5.1 4.7 4.2 3.7 3.3 2.7 2.1
GLR chart 2.7 3.8 4.6 4.7 4.1 3.1 2.3
Fisher chart 3.0 4.0 4.5 4.2 3.3 2.4 1.8
2.00 joint SDDS charts using Method 1 1.0 1.0 1.0 1.0 1.0 1.1 1.2
joint SDDS charts using Method 2 1.0 1.0 1.0 1.0 1.0 1.1 1.2
Max chart 1.6 1.7 1.7 1.8 1.8 1.7 1.6
GLR chart 1.4 1.7 1.9 2.1 2.0 1.9 1.6
Fisher chart 1.4 1.7 1.9 1.9 1.8 1.6 1.4
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Table 85. Detection performance (ARL1) of the joint SDDS charts using Method 1, the MEW, the NCS, the WLC, the ELR and the CEW charts with
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Table 85. (Continued)
1.0 1.2 1.4 1.6 1.8 2.0
1.50 joint SDDS charts using Method 1 2.6 3.0 3.5 4.0 4.3 4.6
MEW chart 2.2 2.2 2.2 2.1 2.1 2.0
NCS chart 2.2 2.0 1.9 1.7 1.6 1.5
WLC chart 1.7 1.6 1.5 1.5 1.4 1.4
ELR chart 2.1 2.0 1.9 1.8 1.7 1.6
CEW chart 6.3 - - - - -
2.00 joint SDDS charts using Method 1 2.2 2.4 2.7 3.0 3.2 3.5
MEW chart 1.7 1.7 1.7 1.7 1.7 1.6
NCS chart 1.2 1.3 1.3 1.3 1.3 1.2
WLC chart 1.2 1.2 1.2 1.2 1.2 1.2
ELR chart 1.5 1.5 1.4 1.4 1.4 1.3
CEW chart 4.0 - - - - -
3.00 joint SDDS charts using Method 1 2.0 2.1 2.1 2.3 2.4 2.6
MEW chart 1.2 1.2 1.2 1.2 1.2 1.2
NCS chart 1.0 1.0 1.0 1.0 1.0 1.0
WLC chart 1.0 1.0 1.0 1.0 1.0 1.0
ELR chart 1.0 1.0 1.0 1.0 1.0 1.0
CEW chart 2.1 - - - - -
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Table 86. Detection performance (ARL1) of the joint SDDS charts using Method 1, the SC and the SL charts with and under N( ).
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Table 87. Detection performance (ARL1) of the joint SDDS charts using Method 1, the joint DS and S charts, the standard, the TSS and the VSS joint and R
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Table 87. (Continued)
1 1.1 1.2 1.3 1.4 1.5 2
1.00 joint SDDS charts using Method 1 4.09 4.57 5.13 5.57 6.02 6.39 6.56
Joint DS-2 chart 2.34 2.33 2.29 2.20 2.09 1.97 1.55
Joint DS-3 chart 2.10 2.13 2.13 2.08 1.99 1.88 1.42
chart 6.39 5.49 4.79 4.20 3.69 3.25 1.92
chart 2.44 2.43 2.40 2.35 2.26 2.16 1.68
chart 2.57 2.51 2.45 2.38 2.29 2.19 1.77
chart 2.79 2.73 2.66 2.58 2.48 2.36 1.85
chart 2.95 2.83 2.72 2.61 2.50 2.39 1.91
1.25 joint SDDS charts using Method 1 3.17 3.50 3.86 4.23 4.52 4.84 5.56
Joint DS-2 chart 1.71 1.73 1.73 1.71 1.67 1.62 1.41
Joint DS-3 chart 1.46 1.50 1.53 1.54 1.53 1.50 1.30
chart 3.07 2.92 2.78 2.63 2.48 2.33 1.69
chart 1.76 1.78 1.80 1.80 1.78 1.76 1.57
chart 1.99 1.97 1.95 1.93 1.90 1.87 1.67
chart 2.17 2.15 2.12 2.08 2.04 2.00 1.73
chart 2.38 2.32 2.26 2.20 2.14 2.08 1.80
1.50 joint SDDS charts using Method 1 2.70 2.91 3.16 3.44 3.67 3.95 4.82
Joint DS-2 chart 1.39 1.42 1.45 1.45 1.44 1.42 1.31
Joint DS-3 chart 1.21 1.24 1.27 1.29 1.30 1.29 1.21
chart 1.84 1.85 1.85 1.84 1.81 1.77 1.50
chart 1.49 1.52 1.53 1.55 1.55 1.55 1.48
chart 1.67 1.68 1.69 1.69 1.69 1.68 1.58
chart 1.89 1.87 1.86 1.83 1.81 1.79 1.63
chart 2.05 2.02 1.99 1.96 1.92 1.88 1.70
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For the double exponential distribution DE(0, ) with in-control mean 0 and variance 1 and DE(0, 1) with in-control mean 0 and variance 2, Table 88 shows the detection performance of the joint SDDS charts using Method 1 and 2, the joint EWMA-AM and EWMA-AV charts with , and under DE( , ), Table 89 shows the detection performance of the joint SDDS charts using Method 1 ( ), the SC and the SL charts (Chowdhury et al. (2014)) with and under DE( , ). It can be seen that (1) under DE( , ), the joint SDDS charts using Method 1 have superior out-of-control detection performance than the joint EWMA-AM and EWMA-AV charts for and (see Table 88); (2) under DE( , ), the joint SDDS charts using Method 1 have superior out-of-control detection performance than the SL charts for and , superior than the SC charts for and (see Table 89).
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Table 88. Detection performance (ARL1) of the joint SDDS charts using Method 1 and 2, and the joint EWMA-AM and EWMA-AV charts with ,
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Table 89. Detection performance (ARL1) of the joint SDDS charts using Method 1, the SC and the SL charts with and under DE( , ).
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For the uniform distribution, Table 90 shows the detection performance of the joint SDDS charts using Method 1 and 2, the joint EWMA-AM and EWMA-AV charts with , and under Unif . It can be seen that the joint SDDS charts using Method 1 have superior out-of-control detection performance than the joint EWMA-AM and EWMA-AV charts for and .
For the exponential distribution, Table 91 shows the detection performance of joint SDDS charts using Method 1 and 2, the joint EWMA-AM and EWMA-AV charts with , and under Exp( ) . It can be seen that the joint SDDS charts using Method 1 have superior out-of-control detection performance than the joint EWMA-AM and EWMA-AV charts for and .
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Table 90. Detection performance (ARL1) of the joint SDDS charts using Method 1 and 2, and the joint EWMA-AM and EWMA-AV charts with ,
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Table 91. Detection performance (ARL1) of the joint SDDS charts using Method 1 and 2, and the joint EWMA-AM and EWMA-AV charts with ,
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For the Gamma distribution G(a, b) with parameters (a, b), in-control mean
and in-control variance , the average time to signal (ATS) are used to measure the detection performance. For measuring out-of-control ATS, we only change the shift in parameter b. Table 92 shows the detection performance of the joint SDDS charts using Method 1 and Lower-sided Gamma chart (Zhang et al. (2007)) with , and under G(1, 0.01), G(1, 0.001), G(4, 0.01) and G(4, 0.001). It can be seen that under the Gamma distribution, the joint SDDS charts using Method 1 have superior out-of-control detection performance than those of Lower-sided Gamma chart.
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Table 92. Detection performance (ATS) of the joint SDDS charts using Method 1 and the Lower-sided Gamma chart with and under the Gamma distribution.
In-control distribution b joint SDDS charts
using Method 1 Lower-sided Gamma chart
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, The SDDS EWMA-AV chart:
, , ,
, (38) ,
.
The control limits of the joint SDDS charts using Method 2 with and are the same as Method 1(see Table 52).
Table 93 and Figure 10 show the plotting statistics of the joint SDDS charts using Method 1 for t=1, 2, , 15. It can be seen that all plotted statistics and , So no need to take the second sample and the process is in-control for t=1, 2, , 15.
Table 94 and Figure 11 show the plotting statistics of the joint SDDS charts using Method 2 for t=1, 2, , 15. It can be seen that all plotted statistics and , So no need to take the second sample and the process is in-control for t=1, 2, , 15.
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Table 93.The plotting statistics of the joint SDDS charts with using Method 1 for t=1, 2, , 15.
t Stage 1
detect result 1 -1.633 -0.926 CR, CR 2 -1.569 -0.190 CR, CR
3 -0.999 0.223 CR, CR
4 -0.073 0.520 CR, CR
5 -0.366 0.757 CR, CR
6 -0.136 0.956 CR, CR
7 0.501 0.457 CR, CR
8 0.625 1.316 CR, CR
9 0.737 1.453 CR, CR
10 0.838 0.978 CR, CR
11 0.542 0.553 CR, CR
12 0.655 0.170 CR, CR
13 0.016 -0.179 CR, CR 14 -0.204 0.054 CR, CR 15 -0.047 0.268 CR, CR
Figure 10. The joint SDDS charts using Method 1 for t=1, 2, , 15.
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Table 94. The plotting statistics of the joint SDDS charts with using Method 2 for t=1, 2, , 15.
t Stage 1
detect result 1 -1.633 -0.926 CR, CR 2 -1.569 -0.190 CR, CR
3 -0.999 0.223 CR, CR
4 -0.073 0.520 CR, CR
5 -0.366 0.757 CR, CR
6 -0.136 0.956 CR, CR
7 0.501 0.457 CR, CR
8 0.625 1.316 CR, CR
9 0.737 1.453 CR, CR
10 0.838 0.978 CR, CR
11 0.542 0.553 CR, CR
12 0.655 0.170 CR, CR
13 0.016 -0.179 CR, CR 14 -0.204 0.054 CR, CR 15 -0.047 0.268 CR, CR
Figure 11. The joint SDDS charts using Method 2 for t=1, 2, , 15.
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For future 10 days (t=16, 17, , 25), the new automatic service system of the bank branch is used. Table 95 and Figure 12 show the plotting statistics of the joint SDDS charts using Method 1 for t=1, 2, , 25. It can be seen that at one days (t=20) we need to take the second sample for the SDDS EWMA-AM chart, at two days (t=24, 25) we need to take the second sample for the SDDS EWMA-AV chart, and there are 6 out-of-control mean signals but no out-of-control variance signal.
Table 96 and Figure 13 show the plotting statistics of the joint SDDS charts using Method 2 for t=1, 2, , 25. It can be seen that there are 3 days (t=20, 24, 25) we take double sample, and there are 6 out-of-control mean signals after t=20 and 1 out-of-control variance signals at t=25.
For comparing out-of-control detection performance, we constructed the corresponding joint EWMA-AM and EWMA-AV charts, and transformed Shewhart and S charts (see Section 2.4 and 3.4). The joint EWMA-AM and EWMA-AV charts with , , , and are constructed as follows (see Table 66):
EWMA-AM , EWMA-AM ,
EWMA-AV , (39) EWMA-AV .
The transformed Shewhart and S charts with are constructed as equation (18) and (34).
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Figure 12. The joint SDDS charts using Method 1 for t=1, 2, , 25.
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Figure 13. The joint SDDS charts using Method 2 for t=1, 2, , 25.
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The joint EWMA-AM and EWMA-AV charts show 5 out-of-control mean signals after t=21 and no out-of-control variance signal (see Figure 14), the transformed Shewhart and S charts show one out-of-control mean signal at t=18 and no out-of-control variance signal (see Figure 15)
In this case, it can be seen that (1) the joint SDDS charts using Method 2 have superior out-of-control variance detection than the joint SDDS charts using Method 1, the joint EWMA-AM and EWMA-AV charts, the transformed Shewhart and S charts, and the transformed EWMA- and EWMA-S charts; (2) The joint SDDS charts have superior out-of-control mean detection than the joint EWMA-AM and EWMA-AV charts, transformed Shewhart and S charts; (3) The joint SDDS charts using Method 1 and Method 2 have the same out-of-control mean detection.
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立 政 治 大 學
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Figure 14. The joint EWMA-AM and EWMA-AV charts.
Figure 15. The transformed Shewhart and S charts.
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立 政 治 大 學
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N a tio na
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