A Nonparametric HEWMA-p Control Chart for Variance in Monitoring Processes
Abstract
:1. Introduction
2. Design of the Enhanced HEWMA-p Chart
- Step 1: Select a random sample of size () from the process at time t. Compute using (3) and using (5).
- Step 2: The process is declared to be as out-of-control if or and to be in-control if , here LCL and UCL show the lower control limit and upper control limit.
3. The Average Run Length of Enhanced HEWMA-p Control Chart
- Step 1. Setting specified values of n, pv0, and ARLv0.
- Step 2. Evaluation of proposed control chart coefficients and for in-control process
- 2.1. Generate 10,000 possible values of control chart coefficients and .
- 2.2. When the process is in-control, from a binomial distribution with the in-control parameters a random sample of size 2000 is generated, i.e., at time t.
- 2.3. The enhanced HEWMA-p statistic HEWMA-p is computed for each subgroup of size 2000.
- 2.4. The proposed statistic HEWMA-p is plotted and in-control if go to step 2.5 and the run length for out of control process is noted.
- 2.5. Repeat 10,000 times steps 2.2 through 2.3, to compute run lengths. If the average of these run lengths (ARLs) is equal to the specified ARLv0 note the corresponding values of and , and move to step 3, otherwise select other possible values of and , and repeat the procedure from steps 2.2.
- Step 3 Evaluation of ARLv1 for proposed control chart when the process is shifted
- 3.1. Let the out-of-control proportion, pv1, be a proportion of the in-control proportion, pv0. That is, pv1 = c pv0, c ≠ 1, and 0 < pv1 ≤ 1, where c is the amount of shift in the process proportion, pv0.
- 3.2. From binomial distribution, with the in-control parameters, , a random sample of size 2000 is generated, i.e., at time t.
- 3.3. The Enhanced HEWMA-p Statistic HEWMA-p is Computed for Each Subgroup of Size 2000.
- 3.4. Using the Values of and , the proposed statistic HEWMA-p is plotted and in-control if go to step 3.5 and the run length for out of control process is noted.
- 3.5. Repeat 10,000 times steps 3.2 through 3.3, to compute run lengths. The average of run length (ARLv1) and standard error of run length (SERLv1) for each specified amount of shift is computed.
- 2. The ARLv1 and SERLv1 values decrease when pv1 is far away from pv0.
- 3. The ARLv1 and SERLv1 values decrease more rapidly as c increases rather than it decreases. For example, for 0.5n = 4 and pv1 = 0.05 (c = 0.5) from Table 2 we have ARLv1 = 107.03 and SERLv1 = 0.8830, whereas if pv1 = 0.15 (c = 1.5), we have ARLv1 = 47.74 and SERLv1 = 0.4055. We also observed a similar trend from Table 4.
4. Comparative Study
5. Example
6. Concluding Remarks
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Nomenclature
ARL | Average run length |
HEWMA-p | Hybrid exponentially weighted moving average proportion |
EWMA | Exponentially weighted moving average |
HEWMA | Hybrid exponentially weighted moving average |
EWMA–CUSUM | Exponentially weighted moving average–Cumulative sum |
LCL | Lower control limit |
UCL | Upper control limit |
SERL | Standard error of run length |
Appendix A
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n | 0.5n | UCL | LCL | ||
---|---|---|---|---|---|
8 | 4 | 0.1892 | 0.0126 | 5.3509 | 5.2421 |
10 | 5 | 0.1823 | 0.0252 | 5.5211 | 5.0203 |
12 | 6 | 0.1751 | 0.0314 | 5.5216 | 5.0416 |
14 | 7 | 0.1699 | 0.0364 | 5.5459 | 5.0498 |
16 | 8 | 0.1689 | 0.0430 | 5.8435 | 4.8326 |
18 | 9 | 0.1612 | 0.0431 | 5.5045 | 5.1243 |
20 | 10 | 0.1573 | 0.0448 | 5.4378 | 5.2352 |
22 | 11 | 0.1618 | 0.0525 | 6.1448 | 4.7231 |
24 | 12 | 0.1553 | 0.0524 | 5.7484 | 4.9475 |
26 | 13 | 0.1566 | 0.0561 | 6.1275 | 4.7436 |
28 | 14 | 0.1492 | 0.0541 | 5.5185 | 5.1495 |
30 | 15 | 0.1477 | 0.0558 | 5.5422 | 5.1406 |
n | 0.5n | pv1 = 0.025 | pv1 = 0.050 | pv1 = 0.075 | pv0 = 0.100 | pv1 = 0.125 | pv1 = 0.150 | pv1 = 0.175 | pv1 = 0.200 |
---|---|---|---|---|---|---|---|---|---|
8 | 4 | 35.29 | 107.03 | 227.89 | 370.30 | 108.55 | 47.74 | 27.62 | 18.52 |
0.2125 | 0.8830 | 4.1031 | 3.7450 | 1.0118 | 0.4055 | 0.2032 | 0.1169 | ||
10 | 5 | 22.51 | 55.24 | 204.00 | 370.36 | 107.43 | 45.55 | 25.30 | 16.80 |
0.1067 | 0.4342 | 1.9517 | 3.7060 | 0.9798 | 0.3844 | 0.1749 | 0.0992 | ||
12 | 6 | 19.22 | 44.42 | 168.19 | 370.36 | 101.22 | 40.28 | 22.00 | 14.99 |
0.0794 | 0.3273 | 1.5208 | 3.5468 | 0.9420 | 0.3305 | 0.1508 | 0.0837 | ||
14 | 7 | 16.81 | 37.03 | 141.92 | 370.09 | 98.87 | 36.21 | 19.87 | 13.64 |
0.0630 | 0.2572 | 1.3294 | 3.7582 | 0.9133 | 0.2828 | 0.1276 | 0.0716 | ||
16 | 8 | 14.15 | 28.21 | 100.20 | 370.12 | 106.17 | 37.22 | 18.71 | 12.36 |
0.0473 | 0.1753 | 0.8503 | 3.5983 | 0.9782 | 0.2861 | 0.1260 | 0.0680 | ||
18 | 9 | 14.17 | 29.27 | 116.90 | 370.35 | 82.10 | 29.98 | 16.52 | 11.66 |
0.0443 | 0.1870 | 1.0169 | 3.6899 | 0.7268 | 0.2192 | 0.0986 | 0.0549 | ||
20 | 10 | 13.41 | 27.90 | 114.63 | 370.35 | 74.67 | 26.99 | 15.20 | 9.83 |
0.0411 | 0.1772 | 1.0130 | 3.7271 | 0.6372 | 0.1921 | 0.0860 | 0.0478 | ||
22 | 11 | 11.07 | 20.20 | 65.97 | 370.20 | 101.72 | 31.72 | 15.01 | 9.60 |
0.0285 | 0.1097 | 0.5498 | 3.5711 | 0.9293 | 0.2351 | 0.0942 | 0.0491 | ||
24 | 12 | 10.12 | 19.51 | 71.04 | 370.29 | 78.48 | 24.22 | 12.61 | 8.29 |
0.0270 | 0.1122 | 0.5978 | 3.6796 | 0.7026 | 0.1885 | 0.0756 | 0.0420 | ||
26 | 13 | 9.21 | 17.79 | 57.10 | 370.28 | 90.92 | 24.92 | 11.07 | 7.55 |
0.0219 | 0.0912 | 0.4547 | 3.6574 | 0.7930 | 0.1952 | 0.0787 | 0.0420 | ||
28 | 14 | 9.69 | 19.33 | 70.98 | 370.34 | 62.21 | 21.78 | 11.49 | 7.28 |
0.0231 | 0.1001 | 0.5895 | 3.8081 | 0.5213 | 0.1400 | 0.0600 | 0.0339 | ||
30 | 15 | 9.28 | 18.12 | 65.59 | 370.11 | 60.42 | 20.57 | 11.07 | 7.07 |
0.0213 | 0.0921 | 0.5349 | 3.8897 | 0.5151 | 0.1291 | 0.0566 | 0.0326 |
n | 0.5n | UCL | LCL | ||
---|---|---|---|---|---|
8 | 4 | 0.4404 | 0.1691 | 5.5158 | 5.1435 |
10 | 5 | 0.4342 | 0.1873 | 5.8915 | 4.9485 |
12 | 6 | 0.4137 | 0.1911 | 5.4695 | 5.2405 |
14 | 7 | 0.4049 | 0.1990 | 5.4499 | 5.2481 |
16 | 8 | 0.4040 | 0.2094 | 5.7765 | 5.0350 |
18 | 9 | 0.3931 | 0.2111 | 5.4839 | 5.2395 |
20 | 10 | 0.3924 | 0.2186 | 5.7395 | 5.0565 |
22 | 11 | 0.3883 | 0.2225 | 5.7495 | 5.0485 |
24 | 12 | 0.3803 | 0.2230 | 5.4635 | 5.2414 |
26 | 13 | 0.3815 | 0.2287 | 5.7725 | 5.0515 |
28 | 14 | 0.3925 | 0.2352 | 6.7985 | 4.7655 |
30 | 15 | 0.3887 | 0.2376 | 6.7485 | 4.7465 |
n | 0.5n | pv1 = 0.20 | pv1 = 0.225 | pv1 = 0.250 | pv1 = 0.275 | pv0 = 0.300 | pv1 = 0.325 | pv1 = 0.350 | pv1 = 0.375 | pv1 = 0.400 |
---|---|---|---|---|---|---|---|---|---|---|
8 | 4 | 35.28 | 60.05 | 118.01 | 245.61 | 370.90 | 226.17 | 107.71 | 61.38 | 38.35 |
0.2524 | 0.4932 | 1.0515 | 2.3753 | 3.8363 | 2.2024 | 0.9818 | 0.5119 | 0.3029 | ||
10 | 5 | 26.68 | 43.27 | 84.94 | 188.98 | 370.31 | 280.15 | 124.92 | 63.86 | 38.04 |
0.1754 | 0.3237 | 0.7445 | 1.7778 | 3.5587 | 2.7424 | 1.1877 | 0.5410 | 0.2896 | ||
12 | 6 | 25.94 | 43.63 | 88.21 | 218.61 | 370.36 | 193.66 | 82.10 | 44.11 | 27.30 |
0.1651 | 0.3323 | 0.7546 | 2.1131 | 3.7020 | 1.8944 | 0.7278 | 0.3442 | 0.1879 | ||
14 | 7 | 22.86 | 38.41 | 77.95 | 203.34 | 370.12 | 180.32 | 72.49 | 38.13 | 23.90 |
0.1408 | 0.2824 | 0.6669 | 1.9692 | 3.6434 | 1.7636 | 0.6397 | 0.2950 | 0.1580 | ||
16 | 8 | 19.08 | 30.41 | 59.02 | 155.32 | 370.12 | 213.84 | 80.71 | 40.05 | 23.99 |
0.1092 | 0.2102 | 0.4795 | 1.4364 | 3.6315 | 2.0796 | 0.7144 | 0.3078 | 0.1549 | ||
18 | 9 | 18.64 | 29.35 | 61.47 | 169.83 | 370.12 | 161.54 | 62.54 | 31.54 | 19.87 |
0.0991 | 0.1991 | 0.5082 | 1.6226 | 3.5624 | 1.5162 | 0.5296 | 0.2273 | 0.1222 | ||
20 | 10 | 16.21 | 25.20 | 49.94 | 139.81 | 370.11 | 185.81 | 65.54 | 29.22 | 19.90 |
0.0827 | 0.1632 | 0.3935 | 1.3342 | 3.6760 | 1.8122 | 0.5570 | 0.2378 | 0.1183 | ||
22 | 11 | 15.02 | 23.56 | 45.65 | 128.87 | 370.29 | 176.40 | 61.05 | 27.67 | 16.29 |
0.0716 | 0.1513 | 0.3694 | 1.1736 | 3.5888 | 1.6707 | 0.5079 | 0.2084 | 0.1034 | ||
24 | 12 | 14.60 | 23.50 | 46.43 | 141.41 | 370.11 | 132.92 | 49.45 | 22.94 | 16.06 |
0.0683 | 0.1484 | 0.3717 | 1.2653 | 3.5991 | 1.3040 | 0.4106 | 0.1661 | 0.0836 | ||
26 | 13 | 13.54 | 20.49 | 39.45 | 115.26 | 370.11 | 161.59 | 52.72 | 26.01 | 15.38 |
0.0593 | 0.1236 | 0.3016 | 1.0572 | 3.6564 | 1.5568 | 0.4251 | 0.1730 | 0.0858 | ||
28 | 14 | 12.08 | 17.50 | 32.06 | 88.77 | 370.20 | 356.82 | 88.25 | 29.94 | 13.28 |
0.0489 | 0.0966 | 0.2331 | 0.7514 | 3.6537 | 3.4724 | 0.7457 | 0.2628 | 0.1168 | ||
30 | 15 | 11.49 | 16.59 | 30.72 | 83.96 | 370.08 | 326.41 | 79.90 | 27.51 | 13.22 |
0.0457 | 0.0902 | 0.2163 | 0.7208 | 3.6923 | 3.1849 | 0.6792 | 0.2246 | 0.1001 |
n | 0.5n | pv0 = 0.1 | pv0 = 0.3 | |||||||
---|---|---|---|---|---|---|---|---|---|---|
Yang and Arnold [4] | Yang and Arnold [14] | Enhanced | Yang and Arnold [4] | Yang and Arnold [14] | Enhanced | Yang and Arnold [4] | Yang and Arnold [14] | Enhanced | ||
pv1 = 0.2 | pv1 = 0.2 | pv1 = 0.2 | pv1 = 0.2 | pv1 = 0.2 | pv1 = 0.2 | pv1 = 0.4 | pv1 = 0.4 | pv1 = 0.4 | ||
8 | 4 | 28.2 | 25.50 | 18.52 | 41.90 | 43.59 | 35.28 | 50.50 | 47.55 | 38.35 |
10 | 5 | 22.7 | 19.02 | 16.80 | 33.70 | 34.06 | 26.68 | 40.80 | 40.29 | 38.04 |
12 | 6 | 19 | 17.43 | 14.99 | 28.20 | 31.24 | 25.94 | 34.20 | 31.36 | 27.30 |
14 | 7 | 16.4 | 14.54 | 13.64 | 24.20 | 25.69 | 22.86 | 29.40 | 28.23 | 23.90 |
16 | 8 | 16.4 | 12.55 | 12.36 | 21.30 | 22.42 | 19.08 | 25.80 | 24.58 | 23.99 |
18 | 9 | 13 | 11.85 | 11.66 | 19.00 | 19.96 | 18.64 | 23.00 | 22.13 | 19.87 |
20 | 10 | 11.8 | 10.67 | 9.83 | 17.20 | 17.27 | 16.21 | 20.70 | 20.96 | 19.90 |
22 | 11 | 10.9 | 9.67 | 9.60 | 15.70 | 17.92 | 15.02 | 18.90 | 17.03 | 16.29 |
24 | 12 | 10.1 | 7.47 | 8.29 | 14.50 | 14.77 | 14.90 | 17.40 | 17.25 | 15.96 |
26 | 13 | 9.4 | 7.95 | 7.55 | 13.40 | 14.07 | 13.54 | 16.10 | 15.62 | 15.38 |
28 | 14 | 8.9 | 7.87 | 7.28 | 12.60 | 13.45 | 12.08 | 15.10 | 14.26 | 13.28 |
30 | 15 | 8.4 | 7.51 | 7.07 | 11.80 | 11.98 | 11.59 | 14.10 | 14.23 | 13.22 |
t | X1 | X2 | X3 | X4 | X5 | X6 | X7 | X8 | X9 | X10 | Vt | EWMApt | HEWMApt |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 3.54 | 0.01 | 1.33 | 7.27 | 5.52 | 0.09 | 1.84 | 1.04 | 2.91 | 0.63 | 0 | 0.2480 | 0.2976 |
2 | 0.86 | 1.61 | 1.15 | 0.96 | 0.54 | 3.05 | 4.11 | 0.63 | 2.37 | 0.05 | 0 | 0.1984 | 0.2381 |
3 | 1.45 | 0.19 | 4.18 | 0.18 | 0.02 | 0.70 | 0.80 | 0.97 | 3.60 | 2.94 | 0 | 0.1587 | 0.1905 |
4 | 1.37 | 0.14 | 1.54 | 1.58 | 0.45 | 6.01 | 4.59 | 1.74 | 3.92 | 4.82 | 0 | 0.1270 | 0.1524 |
5 | 3.00 | 2.46 | 0.06 | 1.80 | 3.25 | 2.13 | 2.22 | 1.37 | 2.13 | 0.25 | 0 | 0.1016 | 0.1219 |
6 | 1.59 | 3.88 | 0.39 | 0.54 | 1.58 | 1.70 | 0.68 | 1.25 | 6.83 | 0.31 | 0 | 0.0813 | 0.0975 |
7 | 5.01 | 1.85 | 3.10 | 1.00 | 0.09 | 1.16 | 2.69 | 2.79 | 1.84 | 2.62 | 0 | 0.0650 | 0.0780 |
8 | 4.96 | 0.55 | 1.43 | 4.12 | 4.06 | 1.42 | 1.43 | 0.86 | 0.67 | 0.13 | 0 | 0.0520 | 0.0624 |
9 | 1.08 | 0.65 | 0.91 | 0.88 | 2.02 | 2.88 | 1.76 | 2.87 | 1.97 | 0.62 | 0 | 0.0416 | 0.0499 |
10 | 4.56 | 0.44 | 5.61 | 2.79 | 1.73 | 2.46 | 0.53 | 1.73 | 7.02 | 2.13 | 0 | 0.0333 | 0.0399 |
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Aslam, M.; Rao, G.S.; AL-Marshadi, A.H.; Jun, C.-H. A Nonparametric HEWMA-p Control Chart for Variance in Monitoring Processes. Symmetry 2019, 11, 356. https://doi.org/10.3390/sym11030356
Aslam M, Rao GS, AL-Marshadi AH, Jun C-H. A Nonparametric HEWMA-p Control Chart for Variance in Monitoring Processes. Symmetry. 2019; 11(3):356. https://doi.org/10.3390/sym11030356
Chicago/Turabian StyleAslam, Muhammad, G. Srinivasa Rao, Ali Hussein AL-Marshadi, and Chi-Hyuck Jun. 2019. "A Nonparametric HEWMA-p Control Chart for Variance in Monitoring Processes" Symmetry 11, no. 3: 356. https://doi.org/10.3390/sym11030356
APA StyleAslam, M., Rao, G. S., AL-Marshadi, A. H., & Jun, C.-H. (2019). A Nonparametric HEWMA-p Control Chart for Variance in Monitoring Processes. Symmetry, 11(3), 356. https://doi.org/10.3390/sym11030356