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This paper will introduce the neutrosophic COM-Poisson (NCOM-Poisson) distribution. Then, the design of the attribute control chart using the NCOM-Poisson distribution is given. The structure of the control chart under the neutrosophic statistical interval method will be given. The algorithm to determine the average run length under neutrosophic statistical interval system will be given. The performance of the proposed control chart is compared with the chart based on classical statistics in terms of neutrosophic average run length (NARL). A simulation study and a real example are also added. From the comparison of the proposed control chart with the existing chart, it is concluded that the proposed control chart is more efficient in detecting a shift in the process. Therefore, the proposed control chart will be helpful in minimizing the defective product. In addition, the proposed control chart is more adequate and effective to apply in uncertainty environment.

Control chart is an important tool of the statistical process control (SPC) that has been widely used in the industry and service company for the monitoring of the manufacturing process. In the industry, specifications are set to manufacture the product. The manufacturing process away from the target causes the production of the nonconforming items. Therefore, an increase in the nonconforming items causes the minimizing of the profit of the company. The control chats provide the signal when the process shifted from the target parameters. A timely signal about the shift in the process helps the industrial engineers to sort out the problem and bring back the process to the in-control state. The operational procedure of the control chart is decided based on data obtained from the production process. The production data is either discrete data or continuous data. The discrete data is obtained from the counting process while the continuous data is obtained from the measurement process. The control charts using both data have been widely used in the industry for the monitoring of the process. Although the control charts based on the variable data are more informative than the control charts based on the attribute data, the variable control chart cannot be applied when the purpose is to monitor the number of nonconforming items. According to [

Usually, the attribute control charts are designed when the proportion defective parameter is determined or crisp value. In practice, it is not always possible that the industrial engineers know about the proportion defective parameter. In this situation, the attribute control charts based on fuzzy approach are applied for the monitoring of nonconformities. Reference [

The existing control chart based on the COM-Poisson distribution is designed under the classical statistics. The classical statistics assumed no indeterminacy in the proportion defective parameters or the observations. Fuzzy logic is based on the degree of truth/false, rather than “false or true.” Reference [

The existing control charts to monitor the nonconforming items can be applied only when all observations in the data are precise, exact, and determined. Therefore, the existing control charts using COM-Poisson distribution under classical statistics cannot be applied for the monitoring of the process when uncertain observations are in the data. By exploring the literature, and according to the best of our knowledge, there is no work on the design of attribute control charts based on COM-Poisson distribution under the neutrosophic statistical interval method. In this paper, we will first introduce the neutrosophic COM-Poisson (NCOM-Poisson) distribution. Then, the design of the attribute control chart using the NCOM-Poisson distribution will be given. We will present the structure of the control chart under the neutrosophic statistical interval method. We expect that the proposed control chart will be more effective, informative, flexible, and adequate in uncertainty environment. The algorithm to determine the average run length under neutrosophic statistical interval system will be given. The performance of the proposed control chart is compared with the chart based on classical statistics in terms of neutrosophic average run length (NARL). A simulation study and a real example are also added. The findings of this current study will redound to the benefit of industry where the statistical quality control plays an important role. Thus, the industries that apply the proposed control chart will be able to produce a high-quality product. For the researcher, the current study will help them uncover areas in the neutrosophic statistics that many researchers are not able to explore. Thus, a new methodology and its application on control chart using neutrosophic COM-Poisson in uncertainty may be arrived at. It is hoped that the proposed chart using neutrosophic COM-Poisson will be more efficient in detecting a shift in the process. It is expected that the proposed chart will be more flexible and informative under uncertainty than the existing competitor’s chart. The rest of the paper is set as follows: a brief introduction about NCOM-Poisson distribution is given in Section

In this section, the introduction of the NCOM-Poisson Distribution is given, which is the generalization of the classical COM-Poisson distribution proposed by [

This section presents the design of the proposed control chart for the NCOM-Poisson distribution. The proposed control chart to monitor the number of nonconformities under the neutrosophic statistics is stated as follows.

The proposed control chart for the NCOM-Poisson distribution under the neutrosophic statistics is the extension of the control chart for the COM-Poisson distribution under the classical statistics proposed by [

The performance of any control chart is measured with the average run length (ARL) which is the indication when on the average the process will be out of control. The smaller the values of ARL, the better the performance of the control chart. The ARL under the neutrosophic statistics is termed as neutrosophic average run length (NARL) which is defined by

The NARLs of proposed chart when

| [2.8071,2.8079] | [2.9355,2.9373] | [2.9997,3.0019] |

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| NARL | ||

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1 | [200.02,200.55] | [300.24,302.03] | [370.03,372.73] |

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1.0125 | [172.25,181.78] | [255.28,271.45] | [312.54,333.53] |

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1.025 | [144.95,162.56] | [211.69,240.42] | [257.22,293.95] |

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1.0375 | [119.88,143.74] | [172.33,210.39] | [207.70,255.84] |

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1.05 | [98.03,125.99] | [138.64,182.38] | [165.69,220.54] |

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1.0625 | [79.67,109.73] | [110.85,157.05] | [131.38,188.80] |

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1.075 | [64.63,95.16] | [88.50,134.65] | [104.03,160.92] |

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1.125 | [28.95,53.30] | [37.41,72.18] | [42.70,84.35] |

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1.15 | [20.12,40.26] | [25.36,53.43] | [28.58,61.81] |

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1.2 | [10.62,23.88] | [12.83,30.57] | [14.14,34.72] |

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1.25 | [6.28,15.03] | [7.31,18.65] | [7.92,20.84] |

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1.3 | [4.09,10.02] | [4.63,12.10] | [4.93,13.34] |

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1.35 | [2.90,7.05] | [3.20,8.31] | [3.37,9.05] |

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1.4 | [2.21,5.20] | [2.38,6.00] | [2.48,6.46] |

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1.45 | [1.78,4.00] | [1.89,4.53] | [1.95,4.83] |

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1.5 | [1.51,3.19] | [1.58,3.55] | [1.62,3.76] |

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1.6 | [1.22,2.23] | [1.25,2.41] | [1.26,2.51] |

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1.8 | [1.03,1.43] | [1.03,1.49] | [1.04,1.52] |

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1.9 | [1.01,1.25] | [1.01,1.29] | [1.01,1.31] |

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2 | [1.00,1.15] | [1.00,1.17] | [1.00,1.18] |

The NARLs of proposed chart when

| [2.807,2.814] | [2.935,2.938] | [3,3.004] |

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| NARL | ||

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0 | [200.14,204.36] | [300.03,303.05] | [370.59,374.97] |

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1.0125 | [187.97,194.09] | [280.28,286.60] | [345.22,353.8] |

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1.025 | [175.81,183.70] | [260.66,270.04] | [320.10,332.53] |

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1.0375 | [163.84,173.34] | [241.46,253.58] | [295.59,311.46] |

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1.05 | [152.20,163.11] | [222.91,237.43] | [271.99,290.83] |

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1.0625 | [141.01,153.12] | [205.19,221.74] | [249.53,270.85] |

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1.075 | [130.35,143.45] | [188.44,206.64] | [228.38,251.68] |

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1.125 | [93.92,108.98] | [132.20,153.59] | [158.01,184.87] |

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1.15 | [79.49,94.57] | [110.43,131.83] | [131.08,157.75] |

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1.2 | [57.2,71.22] | [77.50,97.20] | [90.81,115.00] |

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1.25 | [41.75,54.04] | [55.29,72.31] | [64.02,84.64] |

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1.3 | [31.07,41.53] | [40.3,54.56] | [46.16,63.24] |

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1.35 | [23.62,32.40] | [30.05,41.86] | [34.09,48.09] |

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1.4 | [18.34,25.68] | [22.93,32.67] | [25.78,37.23] |

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1.45 | [14.52,20.68] | [17.88,25.94] | [19.94,29.33] |

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1.5 | [11.72,16.91] | [14.23,20.93] | [15.74,23.50] |

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1.6 | [8.04,11.78] | [9.50,14.25] | [10.38,15.80] |

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1.8 | [4.46,6.58] | [5.06,7.66] | [5.42,8.32] |

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1.9 | [3.55,5.20] | [3.96,5.96] | [4.20,6.42] |

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2 | [2.92,4.24] | [3.22,4.80] | [3.39,5.13] |

The NARLs of proposed chart when

| [2.807,2.811] | [2.935,2.936] | [3,3.003] |

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| NARL | ||

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0 | [200.10,202.48] | [300.21,301.13] | [370.11,374.02] |

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1.0125 | [167.01,169.90] | [246.73,249.01] | [301.76,306.82] |

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1.025 | [132.70,136.05] | [192.32,195.81] | [232.89,238.88] |

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1.0375 | [102.10,105.59] | [144.89,148.98] | [173.56,179.79] |

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1.05 | [77.27,80.61] | [107.35,111.45] | [127.19,133.03] |

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1.0625 | [58.21,61.21] | [79.22,82.99] | [92.89,98.01] |

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1.075 | [44.01,46.61] | [58.74,62.01] | [68.19,72.50] |

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1.125 | [15.99,17.25] | [19.95,21.51] | [22.38,24.28] |

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1.15 | [10.42,11.29] | [12.65,13.71] | [13.98,15.25] |

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1.2 | [5.15,5.59] | [5.95,6.47] | [6.42,7.02] |

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1.25 | [3.03,3.274] | [3.37,3.65] | [3.56,3.88] |

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1.3 | [2.05,2.20] | [2.22,2.38] | [2.31,2.49] |

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1.35 | [1.56,1.65] | [1.64,1.75] | [1.69,1.80] |

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1.4 | [1.30,1.36] | [1.34,1.41] | [1.37,1.44] |

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1.45 | [1.15,1.19] | [1.18,1.22] | [1.19,1.23] |

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1.5 | [1.07,1.10] | [1.09,1.11] | [1.09,1.12] |

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1.6 | [1.01,1.02] | [1.02,1.02] | [1.02,1.03] |

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1.8 | [1.00,1.00] | [1.00,1.00] | [1.00,1.00] |

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1.9 | [1.00,1.00] | [1.00,1.00] | [1.00,1.00] |

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2 | [1.00,1.00] | [1.00,1.00] | [1.00,1.00] |

To determine the neutrosophic control chart coefficients

Predefine the values of

Determine the suitable indeterminacy interval of

Repeat the process 10,000 times and select values of

Find indeterminacy interval of

A control chart having the smaller values of NARL is said to be the more efficient control chart. This section presents the comparison of the proposed control chart with the control chart under the classical statistics in terms of NARL. The values of NARL for the proposed control chart under the neutrosophic statistics and the existing control chart proposed by [

Comparison of NARLs when

Proposed Chart | Existing Chart | |||||
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| NARL | ARL | ||||

1 | [200.0,200.5] | [300.2,302.0] | [370.0,372.7 | 200.0 | 300.0 | 370.0 |

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1.0125 | [172.2,181.7] | [255.2,271.4] | [312.5,333.5] | 177.5 | 263.7 | 323.5 |

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1.025 | [144.9,162.5] | [211.6,240.4] | [257.2,293.9] | 154.9 | 227.4 | 277.3 |

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1.0375 | [119.8,143.7] | [172.3,210.3] | [207.7,255.8] | 133.2 | 193.1 | 234.0 |

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1.05 | [98.0,125.9] | [138.6,182.3] | [165.6,220.5] | 113.4 | 162.3 | 195.3 |

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1.0625 | [79.6,109.7] | [110.8,157.0] | [131.3,188.8] | 95.9 | 135.4 | 161.8 |

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1.075 | [64.6,95.1] | [88.5,134.6] | [104.0,160.9] | 80.8 | 112.5 | 133.6 |

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1.125 | [28.9,53.3] | [37.4,72.1] | [42.7,84.3] | 40.9 | 54.2 | 62.7 |

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1.15 | [20.1,40.2] | [25.3,53.4] | [28.5,61.8] | 29.7 | 38.5 | 44.0 |

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1.2 | [10.6,23.8] | [12.8,30.5] | [14.1,34.7] | 16.6 | 20.7 | 23.2 |

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1.25 | [6.2,15.0] | [7.3,18.6] | [7.9,20.8] | 10.0 | 12.1 | 13.4 |

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1.3 | [4.0,10.0] | [4.6,12.1] | [4.9,13.3] | 6.6 | 7.7 | 8.4 |

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1.35 | [2.9,7.0] | [3.2,8.3] | [3.3,9.0] | 4.6 | 5.2 | 5.6 |

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1.4 | [2.2,5.2] | [2.3,6.0] | [2.4,6.4] | 3.4 | 3.8 | 4.0 |

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1.45 | [1.7,4.0] | [1.8,4.5] | [1.9,4.8] | 2.6 | 2.9 | 3.0 |

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1.5 | [1.5,3.1] | [1.5,3.5] | [1.6,3.7] | 2.1 | 2.3 | 2.4 |

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1.6 | [1.2,2.2] | [1.2,2.4] | [1.2,2.5] | 1.6 | 1.6 | 1.7 |

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1.8 | [1.0,1.4] | [1.0,1.4] | [1.0,1.5] | 1.1 | 1.1 | 1.1 |

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1.9 | [1.0,1.2] | [1.0,1.2] | [1.0,1.3] | 1.0 | 1.0 | 1.0 |

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2 | [1.0,1.1] | [1.0,1.1] | [1.0,1.18] | 1.0 | 1.0 | 1.0 |

The efficiency of the proposed control chart over the chart proposed by [

The proposed control chart for simulated data.

The exiting control chart for simulated data.

This section presents the application of the proposed control chart in a well-known electrical company in Saudi Arabia. This company manufactured the printed circuits boards (PCB) which have been used in several electronic goods, electrical items, and computers. The main function of PCB is to connect the features with each other and to provide the mechanical support to the electrical product. The company is not sure about the proportion defective parameter for the monitoring of PCB product. In addition, due to the complex system of PCB, there is uncertainty in a number of nonconformities in a sample. Due to uncertainty in proportion defective parameter, it is not possible to apply the control chart designed under the classical statistics. The company is interested to apply the proposed control chart for the monitoring of a number of nonconformities. Let the company decide

Number of nonconformities in 100 samples.

Sample# | nonconformities | Sample# | nonconformities |
---|---|---|---|

1 | | 16 | |

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2 | | 17 | |

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3 | | 18 | |

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4 | | 19 | |

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5 | | 20 | |

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6 | | 21 | |

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7 | | 22 | |

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8 | | 23 | |

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9 | | 24 | |

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10 | | 25 | |

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11 | | 26 | |

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12 | | 27 | |

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13 | | 28 | |

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14 | | 29 | |

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15 | | 30 | |

The data presented in Table

The proposed control chart for the real data.

The existing control chart for the real data.

This paper introduced the NCOM-Poisson distribution first. Then, we proposed the control chart using this distribution under the neutrosophic statistics. The proposed chart is the extension of the control chart using the COM-Poisson under the classical statistics. The NARLs are derived under the neutrosophic statistical method. From the comparison, it is concluded that the proposed control chart performs better than the existing control chart in detecting the shift in the process. The proposed control chart can be applied when observations are unclear, fuzzy, and imprecise. The proposed control chart is more adequate and is an effective method under the uncertainty environment. The proposed control chart can be only applied when the data follows the NCOM-Poisson distribution. The proposed control chart can be applied in the electronics industry, the food industry, and automobile industry. The results of the proposed control chart can be improved using the repetitive sampling and the multiple dependent state sampling as future research.

Statistical process control

Conway and Maxwell distribution

Neutrosophic Conway and Maxwell distribution

Neutrosophic statistics

Neutrosophic average run length

Neutrosophic random variable

Determinate part

Indeterminate part

Lower value

Upper value

Neutrosophic scale parameter

Neutrosophic dispersion parameter

Neutrosophic probability mass function

Neutrosophic normalizing constant

Neutrosophic mean

Neutrosophic variance

Neutrosophic lower control limit

Neutrosophic upper control limit

Neutrosophic control chart coefficient

The probability of in-control process

The probability of out-of-control process.

The data is given in the paper.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, Muhammad Aslam, therefore, thanks DSR technical support. The authors are deeply thankful to the editor and reviewers for their valuable suggestions to improve the presentation and quality of the paper.