Control Chart for Failure-Censored Reliability Tests under Uncertainty Environment
Abstract
:1. Introduction
2. State of the Art
- Choose a random sample of the size and begin the test. Continue with the test until are reached and note the ith failure time, say (i = 1, …, ).
- Compute the following statistic under NSIM:
- Declare the process in the control state if where and denote the neutrosophic lower control limit (NLCL) and neutrosophic upper control limit (NUCL), respectively.
- Specify , and .
- Specify and determine the neutrosophic control limits such that .
- Several combinations exist that satisfy the condition . However, choose the combination of the neutrosophic parameters where is very close to .
- Use neutrosophic control limits to find .
3. Advantages of the Proposed Chart
4. Case Study
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Neutrosophic Control Limits | |||
---|---|---|---|
[0.0595, 0.778] | [0.0484, 0.708] | [0.0401, 0.621] | |
[6.1131, 11.498] | [6.6267, 11.801] | [6.0133, 11.331] | |
0.1 | [10.28, 1.45] | [13.43, 1.49] | [9.81, 1.43] |
0.2 | [28.02, 2.91] | [40.55, 3.11] | [26.63, 2.81] |
0.3 | [57.00, 6.20] | [88.45, 6.86] | [55.42, 5.87] |
0.4 | [95.77, 13.22] | [155.28, 15.12] | [98.58, 12.32] |
0.5 | [137.44, 27.61] | [226.82, 32.64] | [155.05, 25.52] |
0.6 | [172.70, 55.32] | [283.97, 67.72] | [218.65, 51.63] |
0.7 | [195.74, 102.02] | [316.96, 130.22] | [279.38, 100.34] |
0.75 | [202.48, 131.52] | [324.90, 172.01] | [305.64, 136.51] |
0.8 | [206.48, 161.66] | [328.31, 216.90] | [328.06, 181.27] |
0.85 | [208.20, 188.06] | [328.22, 258.76] | [346.29, 233.11] |
0.9 | [208.11, 206.64] | [325.53, 290.79] | [360.32, 287.91] |
0.92 | [207.68, 211.35] | [323.91, 299.73] | [364.80, 309.24] |
0.95 | [206.66, 215.43] | [321.01, 308.62] | [370.39, 339.15] |
0.98 | [205.29, 216.16] | [317.68, 312.24] | [374.70, 365.18] |
0.99 | [204.76, 215.74] | [316.49, 312.36] | [375.88, 372.78] |
1 | [204.20, 215.021] | [315.26, 311.99] | [376.92, 379.77] |
1.1 | [197.40, 196.27] | [301.78, 288.42] | [381.40, 413.47] |
1.2 | [189.38, 168.95] | [287.42, 248.89] | [377.72, 392.50] |
1.3 | [181.06, 142.89] | [273.34, 210.15] | [369.13, 347.01] |
1.4 | [172.94, 120.90] | [260.06, 177.26] | [357.92, 298.46] |
1.5 | [165.25, 103.03] | [247.76, 150.53] | [345.55, 255.18] |
1.6 | [158.07, 88.61] | [236.47, 129.01] | [332.90, 218.90] |
1.7 | [151.42, 76.92] | [226.14, 111.61] | [320.47, 189.04] |
1.8 | [145.30, 67.36] | [216.70, 97.42] | [308.54, 164.53] |
1.9 | [139.65, 59.47] | [208.06, 85.73] | [297.23, 144.29] |
2 | [134.44, 52.89] | [200.13, 76.01] | [286.60, 127.46] |
2.5 | [113.69, 32.17] | [168.76, 45.55] | [242.88, 75.01] |
3 | [99.01, 21.79] | [146.74, 30.45] | [211.32, 49.30] |
4 | [79.63, 12.21] | [117.73, 16.65] | [169.38, 26.17] |
5 | [67.30, 8.05] | [99.31, 10.76] | [142.69, 16.46] |
6 | [58.70, 5.87] | [86.47, 7.69] | [124.09, 11.50] |
Neutrosophic Control Limits | |||
---|---|---|---|
[0.0865, 1.08] | [0.0693, 0.961] | [0.0621, 0.933] | |
[8.7698, 16.05] | [9.1713, 16.073] | [9.2462, 17.103] | |
0.1 | [1.47, 1.01] | [1.51, 1.00] | [1.52, 1.00] |
0.2 | [2.73, 1.14] | [2.92, 1.14] | [2.96, 1.12] |
0.3 | [5.42, 1.64] | [6.05, 1.64] | [6.17, 1.55] |
0.4 | [11.04, 2.96] | [12.83, 2.97] | [13.21, 2.66] |
0.5 | [22.54, 6.42] | [27.35, 6.46] | [28.43, 5.44] |
0.6 | [45.01, 15.99] | [57.26, 16.15] | [60.31, 12.76] |
0.7 | [84.15, 43.51] | [112.87, 44.57] | [121.43, 33.20] |
0.75 | [109.92, 71.90] | [151.64, 75.13] | [165.63, 54.94] |
0.8 | [137.49, 114.27] | [195.01, 124.35] | [216.93, 91.1923033] |
0.85 | [163.49, 165.47] | [237.85, 193.61] | [270.02, 148.57] |
0.9 | [184.36, 206.18] | [273.92, 266.89] | [317.40, 228.62] |
0.92 | [190.70, 214.58] | [285.30, 289.78] | [333.15, 264.14] |
0.95 | [197.82, 217.32] | [298.53, 310.99] | [352.37, 314.84] |
0.98 | [202.13, 210.10] | [307.11, 314.91] | [365.94, 354.61] |
0.99 | [202.98, 206.08] | [308.97, 312.77] | [369.22, 364.14] |
1 | [203.55, 201.49] | [310.37, 309.19] | [371.89, 371.54] |
1.1 | [197.60, 145.94] | [304.06, 235.50] | [371.27, 343.99] |
1.2 | [180.35, 101.35] | [277.91, 163.84] | [342.17, 250.67] |
1.3 | [161.00, 71.96] | [247.86, 115.38] | [306.11, 177.06] |
1.4 | [143.14, 52.65] | [220.06, 83.59] | [272.03, 127.50] |
1.5 | [127.65, 39.62] | [195.98, 62.26] | [242.29, 94.23] |
1.6 | [114.45, 30.55] | [175.50, 47.53] | [216.92, 71.35] |
1.7 | [103.22, 24.08] | [158.11, 37.08] | [195.36, 55.21] |
1.8 | [93.64, 19.35] | [143.28, 29.49] | [176.96, 43.55] |
1.9 | [85.40, 15.82] | [130.53, 23.86] | [161.16, 34.95] |
2 | [78.26, 13.13] | [119.51, 19.60] | [147.48, 28.49] |
2.5 | [53.68, 6.21] | [81.58, 8.83] | [100.47, 12.32] |
3 | [39.61, 3.67] | [59.90, 4.97] | [73.63, 6.67] |
4 | [24.73, 1.92] | [37.06, 2.38] | [45.38, 2.97] |
5 | [17.31, 1.37] | [25.72, 1.58] | [31.38, 1.85] |
6 | [13.03, 1.15] | [19.20, 1.26] | [23.33, 1.40] |
Neutrosophic Control Limits | |||
---|---|---|---|
[0.119, 1.27] | [0.106, 1.07] | [0.0882, 0.955] | |
[12.029, 18.76] | [13.87, 18.84] | [13.089, 18.984] | |
0.1 | [1.00, 1.00] | [1.00, 1.00] | [1.00, 1.00] |
0.2 | [1.0, 1.00] | [1.02, 1.00] | [1.01, 1.00] |
0.3 | [1.09, 1.00] | [1.12, 1.00] | [1.11, 1.00] |
0.4 | [1.35, 1.01] | [1.47, 1.01] | [1.42, 1.01] |
0.5 | [2.03, 1.09] | [2.41, 1.09] | [2.24, 1.09] |
0.6 | [3.85, 1.44] | [5.19, 1.44] | [4.57, 1.45] |
0.7 | [9.45, 2.74] | [15.11, 2.77] | [12.40, 2.81] |
0.75 | [16.32, 4.52] | [28.77, 4.59] | [22.75, 4.70] |
0.8 | [29.86, 8.52] | [58.00, 8.7.00] | [44.56, 8.99] |
0.85 | [56.52, 18.41] | [117.97, 18.96] | [90.98, 19.80] |
0.9 | [104.00, 45] | [214.7, 47.4] | [180.79, 50.26] |
0.92 | [128.33, 65.61] | [254.10, 70.55] | [229.02, 75.59] |
0.95 | [164.80, 113.08] | [295.89, 129.5] | [302.41, 142.57] |
0.98 | [191.91, 173.85] | [307.79, 226.65] | [355.84, 263.97] |
0.99 | [197.40, 191.41] | [305.72, 264.90] | [366.03, 317.43] |
1 | [200.93, 204.62] | [301.36, 302.15] | [372.12, 374.36] |
1.1 | [166.99, 127.68] | [214.78, 262.21] | [300.84, 411.25] |
1.2 | [114.56, 53.35] | [144.52, 106.83] | [204.15, 166.01] |
1.3 | [79.65, 24.85] | [100.14, 47.67] | [141.14, 72.39] |
1.4 | [57.02, 12.92] | [71.51, 23.63] | [100.48, 34.96] |
1.5 | [41.92, 7.41] | [52.44, 12.87] | [73.43, 18.52] |
1.6 | [31.55, 4.65] | [39.36, 7.64] | [54.91, 10.67] |
1.7 | [24.26, 3.16] | [30.16, 4.91] | [41.91, 6.63] |
1.8 | [19.00, 2.31] | [23.55, 3.38] | [32.59, 4.42] |
1.9 | [15.15, 1.81] | [18.70, 2.49] | [25.76, 3.14] |
2 | [12.26, 1.49] | [15.08, 1.94] | [20.67, 2.37] |
2.5 | [5.18, 1.03] | [6.23, 1.08] | [8.29, 1.14] |
3 | [2.82, 1.00] | [3.30, 1.00] | [4.23, 1.00] |
4 | [1.40, 1.00] | [1.54, 1.00] | [1.82, 1.00] |
5 | [1.08, 1.00] | [1.12, 1.00] | [1.21, 1.00] |
6 | [1.01, 1.00] | [1.02, 1.00] | [1.04, 1.00] |
Neutrosophic Control Limits | |||
---|---|---|---|
[0.121, 1.36] | [0.109, 0.815] | [0.0908, 1.21] | |
[12.3, 19.9] | [15.135, 18.198] | [13.5992, 21.39] | |
0.1 | [1.00, 1.00] | [1.00, 1.00] | [1.00, 1.00] |
0.2 | [1.01, 1.00] | [1.02, 1.00] | [1.02, 1.00] |
0.3 | [1.10, 1.00] | [1.15, 1.00] | [1.12, 1.00] |
0.4 | [1.37, 1.01] | [1.55, 1.00] | [1.45, 1.02] |
0.5 | [2.08, 1.11] | [2.72, 1.07] | [2.35, 1.14] |
0.6 | [4.02, 1.53] | [6.42, 1.39] | [4.97, 1.68] |
0.7 | [10.11, 3.18] | [20.95, 2.55] | [14.12, 3.89] |
0.75 | [17.71, 5.54] | [42.43, 4.10] | [26.61, 7.31] |
0.8 | [32.83, 11.15] | [89.79, 7.50] | [53.45, 16.03] |
0.85 | [62.65, 25.8] | [181.82, 15.73] | [110.80, 40.93] |
0.9 | [114.54, 66.39] | [294.62, 37.93] | [216.44, 117.07] |
0.92 | [140.01, 96.61] | [323.83, 56.07] | [268.18, 176.84] |
0.95 | [175.93, 156.06] | [337.46, 104.54] | [337.22, 294.34] |
0.98 | [199.53, 203.24] | [322.35, 200.44] | [375.37, 375.25] |
0.99 | [203.48, 208.65] | [313.57, 248.88] | [379.59, 379.24] |
1 | [205.45, 208.09] | [303.79, 307.56] | [379.96, 371.89] |
1.1 | [163.83, 98.63] | [203.67, 744.86] | [286.61,161.48] |
1.2 | [111.83, 41.49] | [136.53, 321.09] | [193.27, 65.86] |
1.3 | [77.73, 19.68] | [94.63, 136.05] | [133.63, 30.25] |
1.4 | [55.66, 10.43] | [67.61, 63.56] | [95.17, 15.48] |
1.5 | [40.93, 6.11] | [49.61, 32.46] | [69.59, 8.73] |
1.6 | [30.82, 3.92] | [37.26, 17.98] | [52.07, 5.38] |
1.7 | [23.70, 2.73] | [28.58, 10.71] | [39.77, 3.59] |
1.8 | [18.58, 2.04] | [22.33, 6.83] | [30.94, 2.58] |
1.9 | [14.81, 1.63] | [17.75, 4.64] | [24.47, 1.98] |
2 | [12.00, 1.38] | [14.33, 3.34] | [19.65, 1.61] |
2.5 | [5.08, 1.02] | [5.95, 1.29] | [7.92, 1.04] |
3 | [2.77, 1.00] | [3.17, 1.02] | [4.06, 1.00] |
4 | [1.39, 1.00] | [1.51, 1.00] | [1.77, 1.00] |
5 | [1.07, 1.00] | [1.11, 1.00] | [1.19, 1.00] |
6 | [1.01, 1.00] | [1.01, 1.00] | [1.04, 1.00] |
Neutrosophic Control Limits | Proposed Control Chart | Existing Control Chart | ||||
---|---|---|---|---|---|---|
ARL = 200 | ARL = 300 | ARL = 370 | ||||
NARL1 | ARL1 | |||||
0.1 | [10.28, 1.45] | [13.43, 1.49] | [9.81, 1.43] | 1.45 | 1.49 | 1.43 |
0.2 | [28.02, 2.91] | [40.55, 3.11] | [26.63, 2.81] | 2.91 | 3.11 | 2.81 |
0.3 | [57.00, 6.20] | [88.45, 6.86] | [55.42, 5.87] | 6.20 | 6.86 | 5.87 |
0.4 | [95.77, 13.22] | [155.28, 15.12] | [98.58, 12.32] | 13.22 | 15.1 | 12.32 |
0.5 | [137.44, 27.61] | [226.82, 32.64] | [155.05, 25.52] | 27.61 | 32.6 | 25.52 |
0.6 | [172.70, 55.32] | [283.97, 67.72] | [218.65, 51.63] | 55.32 | 67.72 | 51.63 |
0.7 | [195.74, 102.02] | [316.96, 130.22] | [279.38, 100.34] | 102.02 | 130.22 | 100.34 |
0.75 | [202.48, 131.52] | [324.90, 172.01] | [305.64, 136.51] | 131.52 | 172.01 | 136.51 |
0.8 | [206.48, 161.66] | [328.31, 216.90] | [328.06, 181.27] | 161.66 | 216.90 | 181.27 |
0.85 | [208.20, 188.06] | [328.22, 258.76] | [346.29, 233.11] | 188.06 | 258.76 | 233.11 |
0.9 | [208.11, 206.64] | [325.53, 290.79] | [360.32, 287.91] | 206.64 | 290.79 | 287.91 |
0.92 | [207.68, 211.35] | [323.91, 299.73] | [364.80, 309.24] | 211.35 | 299.73 | 309.24 |
0.95 | [206.66, 215.43] | [321.01, 308.62] | [370.39, 339.15] | 215.43 | 308.62 | 339.15 |
0.98 | [205.29, 216.16] | [317.68, 312.24] | [374.70, 365.18] | 216.16 | 312.24 | 365.18 |
0.99 | [204.76, 215.74] | [316.49, 312.36] | [375.88, 372.78] | 215.74 | 312.36 | 372.78 |
1 | [204.20, 215.02] | [315.26, 311.99] | [376.92, 379.77] | 215.02 | 311.99 | 376.92 |
1.1 | [197.40, 196.27] | [301.78, 288.42] | [381.40, 413.47] | 196.27 | 288.42 | 381.40 |
1.2 | [189.38, 168.95] | [287.42, 248.89] | [377.72, 392.50] | 168.95 | 248.89 | 377.7 |
1.3 | [181.06, 142.89] | [273.34, 210.15] | [369.13, 347.01] | 142.89 | 210.15 | 347.01 |
1.4 | [172.94, 120.90] | [260.06, 177.26] | [357.92, 298.46] | 120.90 | 177.26 | 298.46 |
1.5 | [165.25, 103.03] | [247.76, 150.53] | [345.55, 255.18] | 103.03 | 150.53 | 255.18 |
1.6 | [158.07, 88.61] | [236.47, 129.01] | [332.90, 218.90] | 88.61 | 129.01 | 218.90 |
1.7 | [151.42, 76.92] | [226.14, 111.61] | [320.47, 189.04] | 76.92 | 111.61 | 189.04 |
1.8 | [145.30, 67.36] | [216.70, 97.42] | [308.54, 164.53] | 67.36 | 97.42 | 164.53 |
1.9 | [139.65, 59.47] | [208.06, 85.73] | [297.23, 144.29] | 59.47 | 85.73 | 144.29 |
2 | [134.44, 52.89] | [200.13, 76.01] | [286.60, 127.46] | 52.89 | 76.01 | 127.46 |
2.5 | [113.69, 32.17] | [168.76, 45.55] | [242.88, 75.01] | 32.17 | 45.55 | 75.01 |
3 | [99.01, 21.79] | [146.74, 30.45] | [211.32, 49.30] | 21.79 | 30.45 | 49.30 |
4 | [79.63, 12.21] | [117.73, 16.65] | [169.38, 26.17] | 12.21 | 16.65 | 26.17 |
5 | [67.30, 8.05] | [99.31, 10.76] | [142.69, 16.46] | 8.05 | 10.76 | 16.46 |
6 | [58.70, 5.87] | [86.47, 7.69] | [124.09, 11.50] | 5.87 | 7.69 | 11.50 |
Sample No. | Sample No. | ||
---|---|---|---|
1 | [3.01, 2.76] | 21 | [0.36, 16.32] |
2 | [2.61, 7.57] | 22 | [0.39, 4.44] |
3 | [7.74, 4.57] | 23 | [0.98, 10.49] |
4 | [0.84, 4.25] | 24 | [6.06, 6.49] |
5 | [1.19, 10.66] | 25 | [7.31, 2.59] |
6 | [2.55, 10.25] | 26 | [2.54, 9.34] |
7 | [3.73, 7.66] | 27 | [6.18, 11.58] |
8 | [3.37, 6.82] | 28 | [1.78, 6.91] |
9 | [6.19, 5.43] | 29 | [2.61, 7.28] |
10 | [6.12, 4.31] | 30 | [5.05, 4.73] |
11 | [3.03, 9.71] | 31 | [3.67, 8.89] |
12 | [0.33, 2.64] | 32 | [4.74, 6.45] |
13 | [3.46, 5.18] | 33 | [0.36, 13.32] |
14 | [3.11, 11.91] | 34 | [3.00, 9.29] |
15 | [1.67, 5.78] | 35 | [0.67, 11.71] |
16 | [1.09, 4.59] | 36 | [0.61, 3.30] |
17 | [2.29, 8.98] | 37 | [2.01, 1.69] |
18 | [2.47, 4.16] | 38 | [0.29, 15.05] |
19 | [1.22, 11.02] | 39 | [1.00, 12.85] |
20 | [3.13, 7.45] | 40 | [2.55, 9.40] |
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Aslam, M.; Khan, N.; Albassam, M. Control Chart for Failure-Censored Reliability Tests under Uncertainty Environment. Symmetry 2018, 10, 690. https://doi.org/10.3390/sym10120690
Aslam M, Khan N, Albassam M. Control Chart for Failure-Censored Reliability Tests under Uncertainty Environment. Symmetry. 2018; 10(12):690. https://doi.org/10.3390/sym10120690
Chicago/Turabian StyleAslam, Muhammad, Nasrullah Khan, and Mohammed Albassam. 2018. "Control Chart for Failure-Censored Reliability Tests under Uncertainty Environment" Symmetry 10, no. 12: 690. https://doi.org/10.3390/sym10120690
APA StyleAslam, M., Khan, N., & Albassam, M. (2018). Control Chart for Failure-Censored Reliability Tests under Uncertainty Environment. Symmetry, 10(12), 690. https://doi.org/10.3390/sym10120690