On Designing of Bayesian Shewhart-Type Control Charts for Maxwell Distributed Processes with Application of Boring Machine
Abstract
:1. Introduction
2. Preliminaries
2.1. Maxwell Distribution
2.2. Distribution of Statistic V
2.3. V Control Chart
2.4. CUSUMV Control Chart
2.5. EWMAV Control Chart
3. Proposed Bayesian Shewhart-Type Control Charts
3.1. Proposed Bayesian-I Shewhart-Type Control Chart
3.2. Proposed Bayesian-II Shewhart-Type Control Chart
4. Proposed Bayesian Shewhart-Type Control Charts
4.1. Simulation Study
4.2. Average Run-Length
4.3. Overall Performance Measure
4.3.1. Extra Quadratic Loss
4.3.2. Relative Average Run Length
4.3.3. Performance Comparison Index
4.4. Sensitivity Analysis of Hyperparameters
- The Bayesian-I and Bayesian-II Shewhart-type control charts are very sensitive to hyperparameter values. A slight change in hyperparameters significantly affects the ARL performance. For example, for n = 2 and δ = 1.5, the ARL for the proposed Bayesian-I Shewhart-type control chart is 211.94, if (a, b) = (0, 0) and when (a, b) = (1, 0) then ARL is 183.01 (see Table 3). Similarly, for the same n = 2, δ = 1.5 the ARL for the proposed Bayesian-II Shewhart-type control chart is 221.54, if (a, b) = (0, 0) and when (a, b) = (1, 0) then ARL is 187.48 (see Table 4);
- The detection ability of the proposed Bayesian-I and Bayesian-II Shewhart-type control chart improves when a gets larger and b becomes smaller at the same time. For example, for n = 2 and δ = 1.5, the ARL for the proposed Bayesian-I Shewhart-type control chart is 34.00, if (a, b) = (25.5, 1.5), whereas when (a, b) = (40.9, 0.005) then ARL is reduced to 18.35 (see Table 3). Similarly, the ARL for the proposed Bayesian-II Shewhart-type control chart with n = 2, δ = 1.5, is 51.064 when (a, b) = (25.5, 1.5), while for (a, b) = (40.9, 0.005) the ARL for the proposed Bayesian-II Shewhart-type control chart is 1.01 (see Table 4);
- The constants A1, A2 and A3 reduce as a increases. For instance, when n = 2 and α = 0.0027, then A1 = 0.04105, A2 = 0.08955, and A3 = 0.25601 if a = 8.5 and a = 25.2 then A1 = 0.02114, A2 = 0.03539, and A3 = 0.06592 (see Table 1);
- Similarly, the values of B1, B2, and B3 decrease as a gets larger. For example, if n = 2, α = 0.0027, then B1 = 0.00127, B2 = 0.10600, and B3 = 0.94696. Likewise, when a = 8.5 and 25.5 then B1 = 0.00052, B2 = 0.04188, and B3 = 0.31137 (see Table 2).
5. Performance Comparison and Illustration of Results
5.1. Proposed versus CUSUMV Control Chart
5.2. Proposed versus EWMAV Control Chart
5.3. Proposed versus V Control Chart
5.4. Main Finding of the Study
- The Bayesian-I and Bayesian-II Shewhart-type control charts are very sensitive to hyperparameter values. A slight change in hyperparameter values significantly affects the performance of the proposed Bayesian-I and Bayesian-II Shewhart-type control charts in terms of the ARL measure (see, Table 3 and Table 4);
- The proposed Bayesian-I and Bayesian-II Shewhart-type control charts have improved ARL performance than the CUSUMV, EWMAV and V control charts, particularly when hyperparameters a and b increase (see, Table 5);
6. Real Data Analysis
7. Summary, Conclusions, and Recommendation
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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a | n | False Alarm Rate α | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
0.005 | 0.0027 | 0.002 | ||||||||
A1 | A2 | A3 | A1 | A2 | A3 | A1 | A2 | A3 | ||
1 | 0.04725 | 0.10345 | 0.29759 | 0.04510 | 0.10345 | 0.32434 | 0.04414 | 0.10345 | 0.33790 | |
2 | 0.04290 | 0.08955 | 0.23675 | 0.04105 | 0.08955 | 0.25601 | 0.04022 | 0.08955 | 0.26571 | |
8.5 | 3 | 0.03935 | 0.07895 | 0.19505 | 0.03773 | 0.07895 | 0.20963 | 0.03701 | 0.07895 | 0.21692 |
4 | 0.03639 | 0.07059 | 0.16494 | 0.03495 | 0.07059 | 0.17639 | 0.03431 | 0.07059 | 0.18209 | |
5 | 0.03387 | 0.06383 | 0.14232 | 0.03259 | 0.06383 | 0.15156 | 0.03201 | 0.06383 | 0.15615 | |
1 | 0.01025 | 0.01405 | 0.01998 | 0.01005 | 0.01405 | 0.02050 | 0.00995 | 0.01405 | 0.02075 | |
2 | 0.01007 | 0.01376 | 0.01949 | 0.00987 | 0.01376 | 0.02000 | 0.00978 | 0.01376 | 0.02024 | |
70 | 3 | 0.00990 | 0.01348 | 0.01903 | 0.00970 | 0.01348 | 0.01951 | 0.00961 | 0.01348 | 0.01974 |
4 | 0.00973 | 0.01322 | 0.01858 | 0.00954 | 0.01322 | 0.01905 | 0.00945 | 0.01322 | 0.01928 | |
5 | 0.00957 | 0.01296 | 0.01816 | 0.00938 | 0.01296 | 0.01861 | 0.00930 | 0.01296 | 0.01883 |
a | n | False Alarm Rate α | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
0.005 | 0.0027 | 0.0027 | ||||||||
B1 | B2 | B3 | B1 | B2 | B3 | B1 | B2 | B3 | ||
1 | 0.00082 | 0.04423 | 0.29951 | 0.00054 | 0.04423 | 0.33103 | 0.00045 | 0.04423 | 0.34661 | |
2 | 0.00078 | 0.04188 | 0.28191 | 0.00052 | 0.04188 | 0.31137 | 0.00042 | 0.04188 | 0.32592 | |
25.5 | 3 | 0.00074 | 0.03977 | 0.26626 | 0.00049 | 0.03977 | 0.29391 | 0.00040 | 0.03977 | 0.30754 |
4 | 0.00071 | 0.03786 | 0.25225 | 0.00047 | 0.03786 | 0.27829 | 0.00038 | 0.03786 | 0.29112 | |
5 | 0.00067 | 0.03612 | 0.23963 | 0.00045 | 0.03612 | 0.26424 | 0.00036 | 0.03612 | 0.27636 | |
1 | 0.00058 | 0.03102 | 0.20310 | 0.00038 | 0.03102 | 0.22362 | 0.00031 | 0.03102 | 0.23371 | |
2 | 0.00056 | 0.02984 | 0.19483 | 0.00037 | 0.02984 | 0.21445 | 0.00030 | 0.02984 | 0.22408 | |
36.9 | 3 | 0.00054 | 0.02876 | 0.18721 | 0.00036 | 0.02876 | 0.20599 | 0.00029 | 0.02876 | 0.21522 |
4 | 0.00052 | 0.02774 | 0.18016 | 0.00034 | 0.02774 | 0.19818 | 0.00028 | 0.02774 | 0.20702 | |
5 | 0.00050 | 0.02680 | 0.17362 | 0.00033 | 0.02680 | 0.19093 | 0.00027 | 0.02680 | 0.19943 |
δ | |||||||||
---|---|---|---|---|---|---|---|---|---|
1 | 1.25 | 1.5 | 1.75 | 2 | 2.25 | 2.5 | 3 | 6 | |
n | a = 0, b = 0 | ||||||||
2 | 369.75 | 265.15 | 211.94 | 159.73 | 134.96 | 116.21 | 94.10 | 82.09 | 33.00 |
5 | 369.83 | 203.85 | 116.15 | 79.44 | 57.77 | 47.03 | 34.10 | 23.62 | 6.02 |
9 | 371.53 | 129.99 | 86.30 | 42.92 | 27.24 | 16.46 | 12.50 | 8.19 | 1.85 |
a = 1, b = 0 | |||||||||
2 | 369.66 | 242.51 | 183.01 | 128.38 | 104.10 | 86.52 | 66.71 | 56.43 | 18.95 |
5 | 371.08 | 193.86 | 105.37 | 69.96 | 49.72 | 39.90 | 28.35 | 19.23 | 4.71 |
9 | 369.83 | 123.53 | 80.64 | 39.11 | 24.48 | 14.61 | 11.04 | 7.19 | 1.68 |
a = 8.5, b = 0.005 | |||||||||
2 | 371.52 | 148.09 | 83.00 | 40.85 | 27.46 | 19.67 | 12.51 | 9.49 | 2.23 |
5 | 371.50 | 138.00 | 56.49 | 31.72 | 19.98 | 14.92 | 9.69 | 6.08 | 1.59 |
9 | 370.10 | 89.93 | 52.45 | 21.74 | 12.57 | 7.06 | 5.24 | 3.39 | 1.12 |
a = 25.5, b = 0.005 | |||||||||
2 | 371.67 | 81.95 | 27.57 | 10.80 | 6.40 | 4.30 | 2.65 | 2.06 | 1.02 |
5 | 370.91 | 73.92 | 23.10 | 10.77 | 6.20 | 4.15 | 2.86 | 1.89 | 1.02 |
9 | 371.99 | 53.46 | 22.33 | 9.00 | 4.89 | 2.75 | 2.11 | 1.51 | 1.00 |
a = 25.5, b = 1.5 | |||||||||
2 | 370.02 | 95.26 | 40.00 | 15.24 | 9.22 | 6.21 | 3.77 | 2.85 | 1.09 |
5 | 371.57 | 85.36 | 29.71 | 14.53 | 8.45 | 6.08 | 3.86 | 2.47 | 1.06 |
9 | 371.41 | 62.31 | 28.18 | 11.53 | 6.33 | 3.51 | 2.64 | 1.82 | 1.01 |
a = 40.9, b = 0.005 | |||||||||
2 | 370.74 | 60.70 | 18.35 | 5.86 | 3.62 | 2.39 | 1.60 | 1.34 | 1.00 |
5 | 371.11 | 52.85 | 14.37 | 6.36 | 3.46 | 2.64 | 1.80 | 1.32 | 1.00 |
9 | 369.03 | 38.98 | 13.61 | 5.62 | 3.08 | 1.83 | 1.48 | 1.18 | 0.99 |
a = 70, b = 0.2 | |||||||||
2 | 369.21 | 29.36 | 8.58 | 2.65 | 1.70 | 1.32 | 1.09 | 1.03 | 1.00 |
5 | 370.48 | 37.24 | 7.01 | 3.05 | 1.86 | 1.47 | 1.17 | 1.04 | 1.00 |
9 | 370.33 | 23.42 | 6.90 | 2.91 | 1.73 | 1.22 | 1.10 | 1.02 | 1.00 |
a = 110, b = 0.005 | |||||||||
2 | 369.44 | 23.78 | 3.78 | 1.86 | 1.25 | 1.11 | 1.05 | 1.00 | 1.00 |
5 | 373.26 | 16.07 | 3.47 | 1.79 | 1.18 | 1.06 | 1.02 | 1.00 | 1.00 |
9 | 369.37 | 14.12 | 3.25 | 1.76 | 1.22 | 1.04 | 1.01 | 1.00 | 1.00 |
δ | |||||||||
---|---|---|---|---|---|---|---|---|---|
1 | 1.25 | 1.5 | 1.75 | 2 | 2.25 | 2.5 | 3 | 6 | |
n | a = 0, b = 0 | ||||||||
2 | 371.04 | 274.85 | 221.54 | 181.87 | 144.91 | 136.23 | 95.18 | 68.31 | 28.53 |
5 | 371.26 | 146.33 | 98.83 | 47.75 | 36.48 | 24.90 | 12.97 | 8.69 | 1.13 |
9 | 369.99 | 50.36 | 29.14 | 9.99 | 5.95 | 2.11 | 1.50 | 1.00 | 1.00 |
a = 1, b = 0 | |||||||||
2 | 370.76 | 249.31 | 187.48 | 144.49 | 107.06 | 98.71 | 61.61 | 39.84 | 12.78 |
5 | 370.90 | 129.66 | 83.37 | 36.70 | 27.13 | 17.70 | 8.53 | 15.48 | 1.00 |
9 | 369.45 | 43.67 | 24.32 | 7.78 | 4.49 | 1.56 | 1.15 | 1.00 | 1.00 |
a = 8.5, b = 0.005 | |||||||||
2 | 371.35 | 123.94 | 56.46 | 27.82 | 12.38 | 10.02 | 3.03 | 1.16 | 1.00 |
5 | 371.14 | 54.01 | 24.50 | 5.58 | 3.35 | 1.73 | 1.00 | 1.00 | 1.00 |
9 | 369.50 | 15.65 | 6.65 | 1.45 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
a = 25.5, b = 0.005 | |||||||||
2 | 369.37 | 25.72 | 4.04 | 1.12 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
5 | 370.41 | 7.94 | 1.97 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
9 | 370.91 | 1.90 | 1.01 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
a = 25.5, b = 1.5 | |||||||||
2 | 369.39 | 119.39 | 51.06 | 23.12 | 9.05 | 7.06 | 1.72 | 1.00 | 1.00 |
5 | 369.09 | 45.86 | 18.92 | 3.44 | 1.93 | 1.11 | 1.00 | 1.00 | 1.00 |
9 | 369.81 | 11.45 | 4.37 | 1.06 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
a = 40.9, b = 0.005 | |||||||||
2 | 369.58 | 6.77 | 1.01 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
5 | 371.59 | 1.82 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
9 | 371.50 | 1.01 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
a = 70, b = 0.2 | |||||||||
2 | 371.36 | 1.21 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
5 | 370.55 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
9 | 370.31 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
a = 2.5, b = 0.005 | |||||||||
2 | 371.87 | 222.32 | 153.44 | 109.32 | 73.91 | 66.50 | 35.97 | 20.36 | 4.70 |
5 | 370.23 | 109.67 | 65.89 | 25.48 | 18.00 | 11.05 | 4.81 | 2.93 | 1.00 |
9 | 369.50 | 35.76 | 18.87 | 5.47 | 3.04 | 1.13 | 1.00 | 1.00 | 1.00 |
CUSUMV | EWMAV | V | ||
---|---|---|---|---|
n | δ | ARL | ARL | ARL |
1 | 369.8 | 370.19 | 372.02 | |
1.25 | 33.93 | 63.02 | 116.26 | |
1.5 | 16.1 | 20.63 | 39.64 | |
2 | 1.75 | 10.88 | 9.52 | 18.67 |
2 | 8.19 | 5.71 | 10.84 | |
2.25 | 6.39 | 3.95 | 7.08 | |
3 | 4.36 | 2.13 | 3.33 | |
6 | 2.13 | 1.14 | 1.37 | |
1 | 373.98 | 371.90 | 374.69 | |
1.25 | 18.25 | 32.03 | 69.04 | |
1.5 | 9.15 | 8.22 | 17.55 | |
5 | 1.75 | 6.06 | 3.65 | 7.36 |
2 | 4.58 | 2.17 | 4.19 | |
2.25 | 3.82 | 1.62 | 2.75 | |
3 | 2.51 | 1.21 | 1.51 | |
6 | 1.32 | 1.01 | 1.02 | |
1 | 371.61 | 371.40 | 369.29 | |
1.25 | 12.65 | 18.28 | 42.4 | |
1.5 | 5.98 | 4.13 | 8.97 | |
9 | 1.75 | 4.11 | 1.94 | 3.76 |
2 | 3.09 | 1.35 | 2.21 | |
2.25 | 2.64 | 1.13 | 1.60 | |
3 | 1.82 | 1.01 | 1.11 | |
6 | 1.04 | 1.00 | 1.00 |
n | CUSUMV | EWMAV (λ = 0.75) | V | Bayesian-I (a = 110, b = 0.005) | Bayesian-II (a = 25.5, b = 0.005) | |
---|---|---|---|---|---|---|
EQL | 57.97 | 40.60 | 57.49 | 26.79 | 26.75 | |
2 | PCI | 2.17 | 1.52 | 2.15 | 1.00 | 1.00 |
RRL | 4.10 | 2.65 | 4.49 | 1.05 | 1.00 | |
EQL | 38.31 | 29.53 | 36.20 | 26.20 | 25.14 | |
5 | PCI | 1.52 | 1.17 | 1.44 | 1.04 | 1.00 |
RARl | 2.62 | 1.65 | 2.63 | 1.14 | 1.00 | |
EQL | 30.91 | 26.53 | 29.93 | 25.92 | 24.57 | |
9 | PCI | 1.26 | 1.08 | 1.22 | 1.05 | 1.00 |
RARL | 2.27 | 1.67 | 2.76 | 1.49 | 1.00 |
Sample Number | Vi | Ci+ | Zi | Sample Number | Vi | Ci+ | Zi |
---|---|---|---|---|---|---|---|
1 | 3,336,713 | 22,832 | 3,204,768 | 16 | 4,354,789 | 8,273,484 | 3,746,652 |
2 | 2,859,270 | 0 | 3,118,393 | 17 | 2,805,902 | 7,765,506 | 3,511,464 |
3 | 3,666,550 | 352,669 | 3,255,433 | 18 | 4,588,520 | 9,040,146 | 3,780,728 |
4 | 3,132,794 | 171,582 | 3,224,773 | 19 | 4,829,929 | 10,556,194 | 4,043,029 |
5 | 3,781,886 | 639,588 | 3,364,051 | 20 | 3,910,538 | 11,152,851 | 4,009,906 |
6 | 2,378,780 | 0 | 3,117,733 | 21 | 2,841,708 | 10,680,678 | 3,717,856 |
7 | 1,759,270 | 0 | 2,778,118 | 22 | 2,566,451 | 9,933,248 | 3,430,005 |
8 | 4,370,996 | 1,057,115 | 3,176,337 | 23 | 4,267,153 | 10,886,521 | 3,639,292 |
9 | 4,503,843 | 2,247,078 | 3,508,214 | 24 | 4,202,461 | 11,775,101 | 3,780,084 |
10 | 4,761,577 | 6,504,774 | 3,821,554 | 25 | 3,406,360 | 11,867,580 | 3,686,653 |
11 | 2,893,367 | 6,084,261 | 3,589,508 | 26 | 3,459,810 | 12,013,509 | 3,629,942 |
12 | 2,931,706 | 5,702,087 | 3,425,057 | 27 | 3,723,778 | 12,423,407 | 3,653,401 |
13 | 3,068,791 | 5,456,997 | 3,335,991 | 28 | 3,430,499 | 12,540,025 | 3,597,676 |
14 | 2,934,246 | 5,077,362 | 3,235,554 | 29 | 4,419,754 | 13,645,899 | 3,803,195 |
15 | 4,469,095 | 6,232,576 | 3,543,940 | 30 | 3,378,578 | 13,710,596 | 3,697,041 |
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Alshahrani, F.; Almanjahie, I.M.; Khan, M.; Anwar, S.M.; Rasheed, Z.; Cheema, A.N. On Designing of Bayesian Shewhart-Type Control Charts for Maxwell Distributed Processes with Application of Boring Machine. Mathematics 2023, 11, 1126. https://doi.org/10.3390/math11051126
Alshahrani F, Almanjahie IM, Khan M, Anwar SM, Rasheed Z, Cheema AN. On Designing of Bayesian Shewhart-Type Control Charts for Maxwell Distributed Processes with Application of Boring Machine. Mathematics. 2023; 11(5):1126. https://doi.org/10.3390/math11051126
Chicago/Turabian StyleAlshahrani, Fatimah, Ibrahim M. Almanjahie, Majid Khan, Syed M. Anwar, Zahid Rasheed, and Ammara N. Cheema. 2023. "On Designing of Bayesian Shewhart-Type Control Charts for Maxwell Distributed Processes with Application of Boring Machine" Mathematics 11, no. 5: 1126. https://doi.org/10.3390/math11051126
APA StyleAlshahrani, F., Almanjahie, I. M., Khan, M., Anwar, S. M., Rasheed, Z., & Cheema, A. N. (2023). On Designing of Bayesian Shewhart-Type Control Charts for Maxwell Distributed Processes with Application of Boring Machine. Mathematics, 11(5), 1126. https://doi.org/10.3390/math11051126