In this paper, a control chart scheme has been introduced for the mean monitoring using gamma distribution for belief statistics using multiple dependent (deferred) state sampling under the neutrosophic statistics. The coefficients of the control chart and the neutrosophic average run lengths have been estimated for specific false alarm probabilities under various process conditions. The offered chart has been compared with the existing classical chart through simulation and the real data. From the comparison, it is concluded that the performance of the proposed chart is better than that of the existing chart in terms of average run length under uncertain environment. The proposed chart has the ability to detect a shift quickly than the existing chart. It has been observed that the proposed chart is efficient in quick monitoring of the out-of-control process and a cherished addition in the toolkit of the quality control personnel.
The control chart is a key technique to statistical process control to ensure the quality of the production process. The technique of control chart requires the construction of a central line, and two control limits are known as the lower control limit (LCL) and upper control limit (UCL). The quality characteristic of interest is then plotted on this chart for the quick monitoring of an observation falling outside these two limits. The idea of control chart was floated by Shewhart A. Walter during the 1920s, and plenty of control chart techniques has been developed by researchers but remained unsuccessful to develop a robust control chart technique. Gamma distribution is the commonly employed probability distribution for estimating time between events in physical sciences. The gamma distribution is a well-fitted distribution to the failure between events and an excellent alternative to normal, log normal, and nonparametric approaches and routinely used in the control chart literature [
The technique of multiple dependent state (MDS) sampling was announced by Wortham and Baker [
There are many situations in the real world when the information may be determinate or indeterminate [
Aslam et al. [
Let
The random variable
Then, the approximately normal distribution of
Data are collected for belief statistics with the assumption of a single observation (
It is to be noted that the variable
Let the statistic proposed by Fallah Nezhad and Akhavan Niaki [
The above expression can be written as follows:
Let the specified starting value of
The technique of the planned chart is elaborated in the subsequent steps as follows: Step 1: choose an item randomly at the Step 2: if
The measures of the planned chart for the in-control and the out-of-control process are computed with the assumption that we change the scale parameter of underlying distribution while the shape parameter remains fixed during the entire process. Let
After some simplification, equation (
Similarly, the probability for in-decision can be written as
Here, it is worth noting that
After some simplification, equation (
The evaluation of the functioning of the developed control chart is judged by calculating the average run length (ARL) that is very commonly suggested by the quality control researchers [
It is very common that no process operates smoothly for a long time without any alteration in the process. So the evaluation of the changed process provides us the effectiveness of the proposed scheme in haste and prompt indication of the out-of-control process. Let the scale factor of the gamma distribution has been moved from
Thus, the mean and variance of
So, the probability of the out-of-control process for the shifted process at
The probability of in-decision for the shifted process is given by
The simplified form of equation (
The probability of in-control for the shifted process is given by
Similarly, the expression for the
Here,
So, the ARL of the shifted process is specified as follows:
Suppose Step 1: select a series of control chart coefficient Step 2: compute Step 3: for a constant value of Step 4: compute
The values of NARL for various parameters are presented in Tables
The values of NARL when
[3.2104, 3.2996] | [3.4042, 3.4512] | [3.1128, 3.2105] | |
[2.0148, 2.1321] | [2.0843, 2.2099] | [2.2992, 2.337] | |
Shift ( | ARL | ||
4.00 | [1.11, 1.02] | [1.12, 1.02] | [1.16, 1.03] |
3.00 | [1.34, 1.10] | [1.38, 1.11] | [1.47, 1.13] |
2.80 | [1.45, 1.14] | [1.50, 1.16] | [1.62, 1.19] |
2.50 | [1.71, 1.25] | [1.80, 1.28] | [2.00, 1.33] |
2.25 | [2.13, 1.42] | [2.29, 1.47] | [2.60, 1.55] |
2.00 | [2.99, 1.76] | [3.31, 1.85] | [3.85, 2.01] |
1.90 | [3.60, 2.01] | [4.05, 2.13] | [4.75, 2.34] |
1.80 | [4.49, 2.37] | [5.14, 2.55] | [6.10, 2.84] |
1.70 | [5.86, 2.94] | [6.86, 3.23] | [8.20, 3.66] |
1.60 | [8.09, 3.91] | [9.69, 4.41] | [11.68, 5.07] |
1.50 | [11.93, 5.7] | [14.71, 6.63] | [17.81, 7.78] |
1.40 | [19.03, 9.37] | [24.26, 11.34] | [29.38, 13.51] |
1.30 | [33.11, 17.9] | [43.88, 22.69] | [53.00, 27.34] |
1.20 | [62.51, 40.35] | [86.59, 54.10] | [104.3, 65.43] |
1.10 | [121.05, 100.93] | [175.52, 144.23] | [213, 175.21] |
1.00 | [200.56, 200.33] | [301.11, 301.00] | [371.94, 371.38] |
0.80 | [137.4, 72.63] | [201.93, 103.4] | [264.41, 134.97] |
0.75 | [96.92, 41.99] | [139.73, 58.2] | [192.24, 78.17] |
0.70 | [64.90, 23.81] | [91.62, 32.07] | [133.4, 43.87] |
0.60 | [26.44, 7.87] | [35.66, 9.91] | [57.36, 13.44] |
0.50 | [10.07, 3.05] | [12.88, 3.53] | [21.57, 4.47] |
0.40 | [3.86, 1.57] | [4.62, 1.69] | [7.36, 1.92] |
0.30 | [1.71, 1.10] | [1.88, 1.13] | [2.57, 1.18] |
0.25 | [1.29, 1.02] | [1.36, 1.03] | [1.66, 1.05] |
0.15 | [1.01, 1.00] | [1.02, 1.00] | [1.04, 1.00] |
0.10 | [1, 1] | [1, 1] | [1, 1] |
0.05 | [1, 1] | [1, 1] | [1, 1] |
The values of NARL when
[3.022, 3.7678] | [3.6019, 4.2229] | [3.3592, 3.6508] | |
[2.0893, 2.1015] | [2.0616, 2.1804] | [2.1514, 2.2392] | |
Shift ( | ARL | ||
4.00 | [1, 1] | [1, 1] | [1, 1] |
3.00 | [1.02, 1.01] | [1.02, 1.01] | [1.03, 1.01] |
2.80 | [1.04, 1.01] | [1.04, 1.01] | [1.04, 1.02] |
2.50 | [1.08, 1.03] | [1.08, 1.04] | [1.09, 1.04] |
2.25 | [1.17, 1.08] | [1.17, 1.09] | [1.19, 1.1] |
2.00 | [1.37, 1.19] | [1.38, 1.21] | [1.42, 1.23] |
1.90 | [1.52, 1.27] | [1.54, 1.31] | [1.6, 1.33] |
1.80 | [1.76, 1.40] | [1.8, 1.45] | [1.88, 1.49] |
1.70 | [2.14, 1.60] | [2.22, 1.68] | [2.35, 1.74] |
1.60 | [2.82, 1.96] | [2.98, 2.1] | [3.19, 2.18] |
1.50 | [4.08, 2.64] | [4.47, 2.91] | [4.84, 3.06] |
1.40 | [6.77, 4.14] | [7.74, 4.76] | [8.5, 5.09] |
1.30 | [13.29, 8.13] | [16.17, 9.89] | [18.04, 10.8] |
1.20 | [31.95, 21.57] | [42.16, 28.11] | [47.94, 31.51] |
1.10 | [90.67, 76.19] | [130.59, 107.63] | [153.49, 125.65] |
1.00 | [201.15, 202.12] | [301.9, 303.82] | [373.1, 374.48] |
0.80 | [47.94, 26.73] | [61.2, 35.94] | [75.35, 42.64] |
0.75 | [25.69, 12.79] | [30.64, 16.47] | [37.74, 19.25] |
0.70 | [13.77, 6.55] | [15.52, 8.06] | [18.97, 9.22] |
0.60 | [4.34, 2.34] | [4.53, 2.62] | [5.31, 2.84] |
0.50 | [1.81, 1.31] | [1.82, 1.37] | [1.99, 1.43] |
0.40 | [1.14, 1.04] | [1.13, 1.05] | [1.17, 1.06] |
0.30 | [1.01, 1] | [1.01, 1] | [1.01, 1] |
0.25 | [1, 1] | [1, 1] | [1, 1] |
0.15 | [1, 1] | [1, 1] | [1, 1] |
0.10 | [1, 1] | [1, 1] | [1, 1] |
0.05 | [1, 1] | [1, 1] | [1, 1] |
The values of NARL when
[3.4983, 3.6534] | [3.449, 3.8141] | [3.6313, 4.0783] | |
[1.9746, 2.1033] | [2.0774, 2.185] | [2.1067, 2.2228] | |
Shift ( | ARL | ||
4.00 | [1.55, 1.18] | [1.62, 1.2] | [1.65, 1.21] |
3.00 | [2.27, 1.47] | [2.44, 1.52] | [2.54, 1.54] |
2.80 | [2.59, 1.59] | [2.81, 1.66] | [2.95, 1.69] |
2.50 | [3.36, 1.89] | [3.70, 1.99] | [3.94, 2.05] |
2.25 | [4.50, 2.34] | [5.06, 2.52] | [5.46, 2.63] |
2.00 | [6.69, 3.25] | [7.73, 3.6] | [8.49, 3.82] |
1.90 | [8.16, 3.88] | [9.54, 4.37] | [10.57, 4.68] |
1.80 | [10.22, 4.82] | [12.13, 5.53] | [13.56, 5.98] |
1.70 | [13.21, 6.26] | [15.95, 7.33] | [18.02, 8.02] |
1.60 | [17.71, 8.59] | [21.81, 10.32] | [24.93, 11.45] |
1.50 | [24.75, 12.62] | [31.16, 15.58] | [36.08, 17.56] |
1.40 | [36.14, 20.02] | [46.75, 25.56] | [54.85, 29.32] |
1.30 | [55.12, 34.6] | [73.6, 45.9] | [87.6, 53.65] |
1.20 | [86.83, 64.46] | [120.38, 89.28] | [145.3, 106.53] |
1.10 | [136.74, 122.13] | [197.6, 177.02] | [241.52, 215.44] |
1.00 | [200.78, 200.53] | [301.32, 301.36] | [371.79, 372.22] |
0.80 | [235.2, 154.47] | [362.15, 229.88] | [447.56, 281.83] |
0.75 | [206.85, 111.91] | [318.34, 164.28] | [391.37, 199.76] |
0.70 | [171.25, 76.55] | [263.51, 110.72] | [321.6, 133.47] |
0.60 | [101.35, 32.33] | [156.15, 45.21] | [187.25, 53.49] |
0.50 | [51.53, 12.72] | [79.21, 17] | [93.29, 19.66] |
0.40 | [22.91, 4.98] | [34.67, 6.23] | [40.1, 6.98] |
0.30 | [8.84, 2.17] | [12.8, 2.49] | [14.5, 2.67] |
0.25 | [5.21, 1.57] | [7.26, 1.72] | [8.1, 1.8] |
0.15 | [1.78, 1.06] | [2.15, 1.08] | [2.29, 1.09] |
0.10 | [1.18, 1] | [1.28, 1] | [1.32, 1.01] |
0.05 | [1.01, 1] | [1.01, 1] | [1.01, 1] |
The values of NARL when
[3.1112, 3.779] | [3.7071, 4.0566] | [3.522, 3.7427] | |
[2.0442, 2.1009] | [2.0576, 2.1815] | [2.1181, 2.2313] | |
Shift ( | ARL | ||
4.00 | [1.07, 1.02] | [1.07, 1.03] | [1.08, 1.03] |
3.00 | [1.24, 1.11] | [1.25, 1.12] | [1.27, 1.13] |
2.80 | [1.32, 1.15] | [1.34, 1.17] | [1.36, 1.18] |
2.50 | [1.52, 1.27] | [1.57, 1.3] | [1.61, 1.32] |
2.25 | [1.85, 1.44] | [1.95, 1.49] | [2.01, 1.53] |
2.00 | [2.53, 1.8] | [2.75, 1.9] | [2.86, 1.96] |
1.90 | [3.02, 2.05] | [3.34, 2.2] | [3.49, 2.28] |
1.80 | [3.73, 2.44] | [4.23, 2.65] | [4.45, 2.76] |
1.70 | [4.85, 3.04] | [5.64, 3.38] | [5.97, 3.55] |
1.60 | [6.68, 4.07] | [8.02, 4.65] | [8.56, 4.93] |
1.50 | [9.9, 5.99] | [12.32, 7.05] | [13.29, 7.57] |
1.40 | [16, 9.91] | [20.8, 12.14] | [22.71, 13.23] |
1.30 | [28.58, 19.01] | [39.04, 24.38] | [43.33, 27.09] |
1.20 | [56.37, 42.68] | [81.05, 57.77] | [92.08, 65.84] |
1.10 | [115.95, 104.55] | [173.78, 150.2] | [204.15, 177.31] |
1.00 | [200.13, 201.81] | [305.58, 303.31] | [371.16, 370.54] |
0.80 | [117.09, 75.62] | [168.59, 107.95] | [204.29, 129.77] |
0.75 | [78.76, 43.72] | [108.53, 60.82] | [131.47, 72.62] |
0.70 | [50.62, 24.77] | [66.61, 33.53] | [80.64, 39.68] |
0.60 | [19.34, 8.19] | [23.42, 10.38] | [28.09, 11.95] |
0.50 | [7.13, 3.15] | [8.07, 3.69] | [9.44, 4.08] |
0.40 | [2.80, 1.60] | [3.00, 1.74] | [3.36, 1.83] |
0.30 | [1.40, 1.11] | [1.43, 1.14] | [1.51, 1.16] |
0.25 | [1.14, 1.03] | [1.15, 1.04] | [1.18, 1.05] |
0.15 | [1, 1] | [1, 1] | [1, 1] |
0.10 | [1, 1] | [1, 1] | [1, 1] |
0.05 | [1, 1] | [1, 1] | [1, 1] |
In this section, the benefits of the neutrosophic control chart under MDS sampling for the belief statistics will be discussed. Aslam et al. [
We will compare the planned neutrosophic chart using MDS sampling for the belief statistics with the existing neutrosophic chart for belief statistics provided by Aslam et al. [
The NARL values of the proposed and the existing charts when
Existing | Proposed | Existing | Proposed | Existing | Proposed | |
[2.8071, 2.8141] | [2.9354, 2.9416] | [3.0003, 3.0012] | ||||
[3.2104, 3.2996] | [3.4042, 3.4512] | [3.1128, 3.2105] | ||||
[2.0148, 2.1321] | [2.0843, 2.2099] | [2.2992, 2.337] | ||||
Shift ( | ARL | |||||
4.00 | [1.28, 1.06] | [1.11, 1.02] | [1.32, 1.07] | [1.12, 1.02] | [1.34, 1.07] | [1.16, 1.03] |
3.00 | [1.76, 1.24] | [1.34, 1.10] | [1.88, 1.28] | [1.38, 1.11] | [1.95, 1.31] | [1.47, 1.13] |
2.80 | [1.98, 1.34] | [1.45, 1.14] | [2.13, 1.4] | [1.50, 1.16] | [2.22, 1.43] | [1.62, 1.19] |
2.50 | [2.5, 1.58] | [1.71, 1.25] | [2.75, 1.68] | [1.80, 1.28] | [2.89, 1.73] | [2.00, 1.33] |
2.25 | [3.27, 1.97] | [2.13, 1.42] | [3.67, 2.13] | [2.29, 1.47] | [3.91, 2.22] | [2.60, 1.55] |
2.00 | [4.75, 2.74] | [2.99, 1.76] | [5.48, 3.05] | [3.31, 1.85] | [5.92, 3.22] | [3.85, 2.01] |
1.90 | [5.74, 3.28] | [3.60, 2.01] | [6.71, 3.7] | [4.05, 2.13] | [7.29, 3.93] | [4.75, 2.34] |
1.80 | [7.13, 4.05] | [4.49, 2.37] | [8.47, 4.66] | [5.14, 2.55] | [9.28, 4.98] | [6.10, 2.84] |
1.70 | [9.18, 5.24] | [5.86, 2.94] | [11.1, 6.12] | [6.86, 3.23] | [12.26, 6.61] | [8.20, 3.66] |
1.60 | [12.32, 7.12] | [8.09, 3.91] | [15.19, 8.51] | [9.69, 4.41] | [16.96, 9.28] | [11.68, 5.07] |
1.50 | [17.4, 10.34] | [11.93, 5.7] | [21.93, 12.67] | [14.71, 6.63] | [24.76, 13.98] | [17.81, 7.78] |
1.40 | [26.08, 16.27] | [19.03, 9.37] | [33.74, 20.51] | [24.26, 11.34] | [38.61, 22.94] | [29.38, 13.51] |
1.30 | [41.89, 28.25] | [33.11, 17.9] | [55.90, 36.86] | [43.88, 22.69] | [64.99, 41.90] | [53.00, 27.34] |
1.20 | [72.12, 54.92] | [62.51, 40.35] | [99.88, 74.68] | [86.59, 54.10] | [118.35, 86.58] | [104.3, 65.43] |
1.10 | [127.96, 115.71] | [121.05, 100.93] | [185.00, 165.6] | [175.52, 144.23] | [224.15, 196.69] | [213, 175.21] |
1.00 | [200.02, 204.41] | [200.56, 200.33] | [300.17, 306.26] | [301.11, 301.00] | [370.82, 371.83] | [371.94, 371.38] |
0.80 | [152.58, 98.67] | [137.4, 72.63] | [227.95, 143.72] | [201.93, 103.4] | [281.15, 172.31] | [264.41, 134.97] |
0.75 | [119.34, 67.59] | [96.92, 41.99] | [177.29, 97.32] | [139.73, 58.2] | [218.11, 116.09] | [192.24, 78.17] |
0.70 | [90.73, 45.44] | [64.90, 23.81] | [134.07, 64.61] | [91.62, 32.07] | [164.56, 76.64] | [133.4, 43.87] |
0.60 | [49.26, 19.71] | [26.44, 7.87] | [71.88, 27.17] | [35.66, 9.91] | [87.73, 31.78] | [57.36, 13.44] |
0.50 | [24.63, 8.22] | [10.07, 3.05] | [35.26, 10.86] | [12.88, 3.53] | [42.65, 12.45] | [21.57, 4.47] |
0.40 | [11.21, 3.44] | [3.86, 1.57] | [15.56, 4.26] | [4.62, 1.69] | [18.55, 4.75] | [7.36, 1.92] |
0.30 | [4.64, 1.6] | [1.71, 1.10] | [6.1, 1.82] | [1.88, 1.13] | [7.09, 1.95] | [2.57, 1.18] |
0.25 | [2.91, 1.22] | [1.29, 1.02] | [3.68, 1.32] | [1.36, 1.03] | [4.19, 1.37] | [1.66, 1.05] |
0.15 | [1.27, 1] | [1.01, 1.00] | [1.4, 1.01] | [1.02, 1.00] | [1.49, 1.01] | [1.04, 1.00] |
0.10 | [1.03, 1] | [1, 1] | [1.05, 1] | [1, 1] | [1.07, 1] | [1, 1] |
0.05 | [1, 1] | [1, 1] | [1, 1] | [1, 1] | [1, 1] | [1, 1] |
The comparison of the planned chart with the existing chart has also been presented by simulation data given in Tables
The simulated data for the proposed control chart.
Sr. no. | ln ( | ||
---|---|---|---|
1 | [0.007, 0.569] | [0.007, 1.323] | [−5.016, 0.28] |
2 | [0.374, 0.246] | [0.598, 0.326] | [−0.515, −1.122] |
3 | [0.308, 0.888] | [0.445, 7.959] | [−0.811, 2.074] |
4 | [0.702, 0.991] | [2.359, 105.995] | [0.858, 4.663] |
5 | [0.454, 0.006] | [0.832, 0.006] | [−0.184, −5.084] |
6 | [0.744, 0.447] | [2.906, 0.808] | [1.067, −0.214] |
7 | [0.64, 0.269] | [1.781, 0.368] | [0.577, −1.001] |
8 | [0.035, 0.972] | [0.036, 34.967] | [−3.327, 3.554] |
9 | [0.896, 0.565] | [8.659, 1.3] | [2.159, 0.263] |
10 | [0.155, 0.493] | [0.183, 0.973] | [−1.698, −0.027] |
11 | [0.273, 0.882] | [0.375, 7.501] | [−0.981, 2.015] |
12 | [0.917, 0.727] | [11.026, 2.656] | [2.4, 0.977] |
13 | [0.499, 0.964] | [0.997, 26.938] | [−0.003, 3.294] |
14 | [0.732, 0.818] | [2.735, 4.492] | [1.006, 1.502] |
15 | [0.023, 0.261] | [0.023, 0.353] | [−3.754, −1.041] |
16 | [0.465, 0.988] | [0.868, 79.71] | [−0.141, 4.378] |
17 | [0.146, 0.841] | [0.171, 5.303] | [−1.768, 1.668] |
18 | [0.873, 0.972] | [6.844, 34.4] | [1.923, 3.538] |
19 | [0.95, 0.021] | [19.136, 0.021] | [2.952, −3.85] |
20 | [0.605, 0.723] | [1.531, 2.606] | [0.426, 0.958] |
21 | [0.277, 0.02] | [0.383, 0.02] | [−0.96, −3.89] |
22 | [0.954, 0.853] | [20.717, 5.789] | [3.031, 1.756] |
23 | [0.875, 0.97] | [7.015, 32.67] | [1.948, 3.486] |
24 | [0.577, 0.876] | [1.364, 7.06] | [0.311, 1.954] |
25 | [0.808, 0.952] | [4.216, 19.804] | [1.439, 2.986] |
26 | [0.902, 0.995] | [9.252, 206.617] | [2.225, 5.331] |
27 | [0.535, 0.645] | [1.15, 1.816] | [0.14, 0.597] |
28 | [0.44, 0.221] | [0.786, 0.283] | [−0.24, −1.262] |
29 | [0.965, 0.999] | [27.261, 933.686] | [3.305, 6.839] |
30 | [0.532, 0.992] | [1.135, 131.102] | [0.126, 4.876] |
31 | [0.425, 0.999] | [0.74, 1913.741] | [−0.301, 7.557] |
32 | [0.592, 0.819] | [1.450, 4.517] | [0.371, 1.508] |
33 | [0.995, 0.977] | [193.48, 43.177] | [5.265, 3.765] |
34 | [0.456, 0.479] | [0.838, 0.919] | [−0.177, −0.085] |
35 | [0.672, 1.000] | [2.053, 2577.057] | [0.719, 7.854] |
36 | [0.683, 0.968] | [2.159, 30.231] | [0.770, 3.409] |
37 | [0.635, 0.288] | [1.738, 0.404] | [0.553, −0.907] |
38 | [0.358, 0.991] | [0.559, 111.696] | [−0.582, 4.716] |
39 | [0.927, 0.930] | [12.662, 13.193] | [2.539, 2.580] |
40 | [0.634, 0.847] | [1.733, 5.526] | [0.550, 1.710] |
The simulated data for the existing chart.
Sr. no. | ln ( | ||
---|---|---|---|
1 | [0.352, 0.515] | [0.543, 1.063] | [−0.61, 0.061] |
2 | [0.237, 0.825] | [0.31, 4.718] | [−1.171, 1.551] |
3 | [0.25, 0.424] | [0.334, 0.735] | [−1.096, −0.308] |
4 | [0.767, 0.87] | [3.291, 6.721] | [1.191, 1.905] |
5 | [0.569, 0.042] | [1.322, 0.044] | [0.279, −3.125] |
6 | [0.877, 0.019] | [7.107, 0.019] | [1.961, −3.939] |
7 | [0.359, 0.834] | [0.56, 5.039] | [−0.58, 1.617] |
8 | [0.806, 0.968] | [4.153, 29.856] | [1.424, 3.396] |
9 | [0.02, 0.881] | [0.021, 7.412] | [−3.88, 2.003] |
10 | [0.662, 0.071] | [1.958, 0.077] | [0.672, −2.57] |
11 | [0.909, 0.963] | [9.933, 26.211] | [2.296, 3.266] |
12 | [0.959, 0.142] | [23.368, 0.166] | [3.151, −1.797] |
13 | [0.232, 0.99] | [0.302, 95.866] | [−1.199, 4.563] |
14 | [0.063, 0.965] | [0.067, 27.253] | [−2.701, 3.305] |
15 | [0.918, 0.146] | [11.154, 0.172] | [2.412, −1.763] |
16 | [0.086, 0.109] | [0.094, 0.122] | [−2.364, −2.105] |
17 | [0.936, 0.944] | [14.55, 16.968] | [2.678, 2.831] |
18 | [0.198, 0.265] | [0.248, 0.36] | [−1.396, −1.02] |
19 | [0.214, 0.605] | [0.271, 1.533] | [−1.304, 0.427] |
20 | [0.507, 0.411] | [1.028, 0.699] | [0.027, −0.359] |
21 | [0.98, 0.998] | [49.554, 414.687] | [3.903, 6.028] |
22 | [0.984, 0.278] | [60.027, 0.385] | [4.095, −0.954] |
23 | [0.752, 0.979] | [3.034, 46.245] | [1.11, 3.834] |
24 | [0.857, 0.713] | [5.976, 2.482] | [1.788, 0.909] |
25 | [0.267, 0.946] | [0.364, 17.361] | [−1.01, 2.854] |
26 | [0.744, 0.166] | [2.904, 0.199] | [1.066, −1.614] |
27 | [0.577, 0.974] | [1.363, 36.773] | [0.31, 3.605] |
28 | [0.79, 0.589] | [3.761, 1.432] | [1.325, 0.359] |
29 | [0.841, 0.657] | [5.275, 1.914] | [1.663, 0.649] |
30 | [0.969, 0.808] | [31.497, 4.206] | [3.45, 1.436] |
31 | [0.557, 0.959] | [1.257, 23.411] | [0.229, 3.153] |
32 | [0.763, 0.998] | [3.228, 572.344] | [1.172, 6.35] |
33 | [0.94, 0.998] | [15.578, 631.372] | [2.746, 6.448] |
34 | [0.165, 0.848] | [0.197, 5.575] | [−1.625, 1.718] |
35 | [0.873, 0.451] | [6.898, 0.822] | [1.931, −0.196] |
36 | [0.836, 0.688] | [5.091, 2.207] | [1.628, 0.792] |
37 | [0.983, 0.928] | [56.62, 12.949] | [4.036, 2.561] |
38 | [0.783, 0.956] | [3.61, 21.757] | [1.284, 3.08] |
39 | [0.631, 0.883] | [1.712, 7.528] | [0.538, 2.019] |
40 | [0.361, 0.998] | [0.565, 488.171] | [−0.571, 6.191] |
The proposed control chart using simulated data.
The existing chart proposed by Aslam et al. [
The existing Shewhart chart under classical statistics.
In this section, we will discuss the application of the proposed chart using the Urinary Tract Infection (UTI) data from a big hospital. According to Santiago and Smith [
The UTI data.
Sr. no. | ln ( | ||
---|---|---|---|
1 | [0.712, 0.448] | [2.471, 0.813] | [0.905, −0.207] |
2 | [0.201, 0.89] | [0.251, 8.101] | [−1.382, 2.092] |
3 | [0.099, 0.897] | [0.11, 8.668] | [−2.211, 2.16] |
4 | [0.245, 0.087] | [0.325, 0.095] | [−1.123, −2.354] |
5 | [0.417, 0.655] | [0.715, 1.896] | [−0.335, 0.64] |
6 | [0.083, 0.033] | [0.09, 0.034] | [−2.405, −3.373] |
7 | [0.555, 0.397] | [1.245, 0.658] | [0.219, −0.419] |
8 | [0.719, 0.114] | [2.555, 0.129] | [0.938, −2.051] |
9 | [0.893, 0.915] | [8.361, 10.723] | [2.124, 2.372] |
10 | [0.697, 0.097] | [2.296, 0.108] | [0.831, −2.23] |
11 | [0.573, 0.193] | [1.341, 0.239] | [0.294, −1.432] |
12 | [0.234, 0.701] | [0.305, 2.346] | [−1.187, 0.853] |
13 | [0.932, 0.072] | [13.773, 0.078] | [2.623, −2.55] |
14 | [0.533, 0.165] | [1.143, 0.197] | [0.133, −1.622] |
15 | [0.134, 0.015] | [0.155, 0.016] | [−1.867, −4.157] |
16 | [0.674, 0.955] | [2.071, 21.464] | [0.728, 3.066] |
17 | [0.008, 0.563] | [0.008, 1.286] | [−4.799, 0.251] |
18 | [0.241, 0.939] | [0.317, 15.27] | [−1.149, 2.726] |
19 | [0.481, 0.123] | [0.928, 0.14] | [−0.074, −1.967] |
20 | [0.044, 0.433] | [0.046, 0.764] | [−3.071, −0.269] |
21 | [0.109, 0.928] | [0.123, 12.969] | [−2.098, 2.563] |
22 | [0.56, 0.236] | [1.275, 0.308] | [0.243, −1.176] |
23 | [0.025, 0.844] | [0.026, 5.411] | [−3.666, 1.688] |
24 | [0.027, 0.153] | [0.028, 0.181] | [−3.574, −1.71] |
25 | [0.826, 0.59] | [4.739, 1.438] | [1.556, 0.363] |
26 | [0.125, 0.872] | [0.143, 6.795] | [−1.942, 1.916] |
27 | [0.011, 0.361] | [0.011, 0.565] | [−4.488, −0.571] |
28 | [0.025, 0.198] | [0.026, 0.247] | [−3.667, −1.399] |
29 | [0.567, 0.69] | [1.311, 2.23] | [0.271, 0.802] |
30 | [0.909, 0.175] | [9.999, 0.211] | [2.303, −1.554] |
31 | [0.325, 0.8] | [0.482, 4.006] | [−0.731, 1.388] |
32 | [0.126, 0.998] | [0.144, 519.999] | [−1.941, 6.254] |
33 | [0.474, 0.728] | [0.901, 2.671] | [−0.104, 0.982] |
34 | [0.35, 0.022] | [0.539, 0.022] | [−0.619, −3.809] |
35 | [0.425, 0.049] | [0.74, 0.051] | [−0.301, −2.971] |
36 | [0.876, 0.723] | [7.063, 2.614] | [1.955, 0.961] |
37 | [0.177, 0.068] | [0.215, 0.072] | [−1.539, −2.625] |
38 | [0.371, 0.99] | [0.591, 100.78] | [−0.526, 4.613] |
39 | [0.043, 0.173] | [0.045, 0.209] | [−3.108, −1.565] |
40 | [0.117, 0.191] | [0.132, 0.235] | [−2.022, −1.446] |
Using the given information, the value of
The neutrosophic statistic ln (
The proposed chart for UTI data.
Aslam et al. [
Shewhart chart for UTI data.
In this article, the planning of a control chart for gamma-distributed belief statistic using MDS sampling under the neutrosophic statistic has been offered. The parameters of the planned chart have been estimated for vague data using code programming for R language. Neutrosophic average run lengths for indeterminacy intervals under different process settings for various shift levels have been calculated. The comparison of the planned scheme with the existing chart has been made which shows the better identifying skill of the out-of-control process. It has been perceived that the proposed scheme is a valuable accumulation in the toolkit of the quality control professionals for the monitoring of neutrosophic data. A real-world example has been added for the practical application of the planned scheme by the quality control workers. The proposed chart ensures the producer/customer that the product manufactured using the proposed control chart will be according to the given specification limits and good quality product as mentioned in ISO 9001: 2015 Quality Management Systems (
The data used to support the findings of this study are included within the article.
The authors declare that they have no conflicts of interest.
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. (G-1407-130-1440). The authors, therefore, gratefully acknowledge the DSR technical and financial support.