# Monitoring the Variability in the Process Using Neutrosophic Statistical Interval Method

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## Abstract

**:**

## 1. Introduction

## 2. Design of Proposed Control Chart

**Step-1:**Select a random sample of size ${n}_{N}$ from the production process and compute ${S}_{N}^{2}$.

**Step**

**-2:**Declare the process is in-control state if $LC{L}_{N}\le {S}_{N}^{2}\le UC{L}_{N}$; where $LC{L}_{N}\text{}\u03f5\left\{LC{L}_{L},LC{L}_{U}\right\}$ and $UC{L}_{N}\text{}\u03f5\left\{UC{L}_{L},UC{L}_{U}\right\}$ are neutrosophic interval control limits.

- For the fixed values of ${n}_{N}\text{}\u03f5\left\{{n}_{L},{n}_{U}\right\}$ and $c$, the range in indeterminacy interval of NARL increases as $NAR{L}_{0N}$ decreases from 300 to 370.
- For the fixed values of $NAR{L}_{0N}$ and $c$, the range in indeterminacy interval of NARL decreases as ${n}_{N}\text{}\u03f5\left\{{n}_{L},{n}_{U}\right\}$ increases.

**Step-1:**- Specify the indeterminacy interval of ${n}_{N}\u03f5\left\{{n}_{L},{n}_{U}\right\}$ and $c$.
**Step-2:**- Determine the indeterminacy interval of ${k}_{N}\u03f5\left\{{k}_{L},{k}_{U}\right\}$ such that $NAR{L}_{0N}\ge {r}_{0N}$.
**Step-3:**- Find indeterminacy interval of $NAR{L}_{1N}$ using ${k}_{N}\text{}\u03f5\left\{{k}_{L},{k}_{U}\right\}$ selected in Step-2.

## 3. Comparison Studies

#### 3.1. Comparison by NARL

#### 3.2. Comparison by Simulation

^{th}sample. The same values statistic ${S}^{2}$ under the classical statistic is plotted in Figure 2. Figure 2 indicates that the process is an in-control state. By comparing Figure 1 with Figure 2, it is concluded that the proposed control chart has the ability to detect a shift in the process. Also, the proposed control chart is more effective in the uncertainty environment.

## 4. Case Study

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**The neutrosophic plan parameters when ${n}_{N}\text{}\u03f5\text{}\left\{3,4\right\}$ and ${r}_{0N}$ = 300,370.

Parameters | ${\mathit{r}}_{0\mathit{N}}\text{}=\text{}300$ | ${\mathit{r}}_{0\mathit{N}}\text{}=\text{}370$ |
---|---|---|

${n}_{N}$ | [3,4] | [3,4] |

${k}_{N}$ | [4.716,4.784] | [4.921,4.925] |

$c$ | $NAR{L}_{1N}$ | |

1 | [303.571,482.268] | [372.916,567.398] |

1.1 | [180.551,257.614] | [217.685,298.395] |

1.2 | [117.098,153.218] | [138.998,175.183] |

1.3 | [81.177,98.952] | [95.096,111.902] |

1.4 | [59.298,68.163] | [68.685,76.363] |

1.5 | [45.169,49.433] | [51.809,54.931] |

1.6 | [35.597,37.374] | [40.482,41.237] |

1.7 | [28.85,29.239] | [32.562,32.06] |

1.8 | [23.935,23.534] | [26.833,25.662] |

1.9 | [20.251,19.399] | [22.567,21.048] |

2 | [17.423,16.317] | [19.311,17.625] |

3 | [6.721,5.593] | [7.198,5.874] |

4 | [4.174,3.353] | [4.394,3.474] |

**Table 2.**The neutrosophic plan parameters when ${n}_{N}\text{}\u03f5\text{}\left\{4,6\right\}$ and ${r}_{0N}$ = 300,370.

Parameters | ${\mathit{r}}_{0\mathit{N}}\text{}=\text{}300$ | ${\mathit{r}}_{0\mathit{N}}\text{}=\text{}370$ |
---|---|---|

${n}_{N}$ | [4,6] | [4,6] |

${k}_{N}$ | [4.37095,4.38408] | [4.56277,4.60236] |

$c$ | $NAR{L}_{1N}$ | |

1 | [300.03049,490.72292] | [373.89462,521.39944] |

1.1 | [167.75455,236.62553] | [204.67051,261.86897] |

1.2 | [103.63777,130.01298] | [124.23386,148.39858] |

1.3 | [69.11578,78.91434] | [81.62574,92.2282] |

1.4 | [48.93744,51.76218] | [57.06262,61.60346] |

1.5 | [36.34426,36.10675] | [41.91335,43.57537] |

1.6 | [28.05528,26.46593] | [32.04372,32.28357] |

1.7 | [22.35463,20.20032] | [25.31664,24.84157] |

1.8 | [18.28768,15.94213] | [20.55525,19.72459] |

1.9 | [15.29499,12.93761] | [17.07608,16.07855] |

2 | [13.03379,10.74914] | [14.46377,13.40067] |

3 | [4.84897,3.5816] | [5.18032,4.44486] |

4 | [3.02663,2.21472] | [3.17374,2.68331] |

**Table 3.**The neutrosophic plan parameters when ${n}_{N}\text{}\u03f5\text{}\left\{9,10\right\}$ and ${r}_{0N}$ = 300,370.

Parameters | ${\mathit{r}}_{0\mathit{N}}\text{}=\text{}300$ | ${\mathit{r}}_{0\mathit{N}}\text{}=\text{}370$ |
---|---|---|

${n}_{N}$ | [9,10] | [9,10] |

${k}_{N}$ | [3.77774,3.87857] | [3.90143,3.92448] |

$c$ | $NAR{L}_{1N}$ | |

1 | [310.11015,398.93766] | [374.85685,429.14801] |

1.1 | [140.42988,169.87263] | [166.07453,181.17044] |

1.2 | [73.89048,85.19126] | [85.81325,90.2116] |

1.3 | [43.56058,48.35266] | [49.82352,50.89613] |

1.4 | [28.04099,30.19755] | [31.65868,31.6243] |

1.5 | [19.345,20.33186] | [21.59801,21.19929] |

1.6 | [14.10561,14.53629] | [15.59629,15.09889] |

1.7 | [10.75721,10.90994] | [11.79341,11.29453] |

1.8 | [8.51099,8.52027] | [9.26122,8.79481] |

1.9 | [6.94206,6.87633] | [7.504,7.0795] |

2 | [5.80835,5.70385] | [6.24147,5.85883] |

3 | [2.16011,2.05723] | [2.24008,2.0842] |

4 | [1.48785,1.42243] | [1.51913,1.43261] |

$\mathit{c}$ | Proposed Chart | Existing Chart |
---|---|---|

$\mathit{N}\mathit{A}\mathit{R}{\mathit{L}}_{1\mathit{N}}$ | ||

1 | [300.03049,490.72292] | 300.0044 |

1.1 | [167.75455,236.62553] | 167.7414 |

1.2 | [103.63777,130.01298] | 103.6304 |

1.3 | [69.11578,78.91434] | 69.11124 |

1.4 | [48.93744,51.76218] | 48.93448 |

1.5 | [36.34426,36.10675] | 36.34222 |

1.6 | [28.05528,26.46593] | 28.05381 |

1.7 | [22.35463,20.20032] | 22.35353 |

1.8 | [18.28768,15.94213] | 18.28684 |

1.9 | [15.29499,12.93761] | 15.29432 |

2 | [13.03379,10.74914] | 13.03325 |

3 | [4.84897,3.5816] | 4.84884 |

4 | [3.02663,2.21472] | 3.02657 |

Sample No. | Sample Observation | ${\mathit{S}}_{\mathit{N}}^{2}$ | ||||
---|---|---|---|---|---|---|

1 | [74.03,74.03] | [74.002,73.991] | [74.019,74.019] | [73.992,73.992] | [74.008,74.001] | [0.014772,0.017242] |

2 | [73.995,73.995] | [73.992,74.003] | [74.001,74.001] | [74.011,74.011] | [74.004,74.004] | [0.007503,0.005762] |

3 | [73.988,74.017] | [74.024,74.024] | [74.021,74.021] | [74.005,74.005] | [74.002,73.995] | [0.014748,0.012116] |

4 | [74.002,74.002] | [73.996,73.996] | [73.993,73.993] | [74.015,74.015] | [74.009,74.009] | [0.009083,0.009083] |

5 | [73.992,73.992] | [74.007,74.007] | [74.015,74.015] | [73.989,73.989] | [74.014,73.998] | [0.012219,0.010756] |

6 | [74.009,74.009] | [73.994,74.001] | [73.997,73.997] | [73.985,73.985] | [73.993,73.993] | [0.008706,0.008944] |

7 | [73.995,73.998] | [74.006,74.006] | [73.994,73.994] | [74,74] | [74.005,74.005] | [0.005523,0.00498] |

8 | [73.985,73.985] | [74.003,74.01] | [73.993,73.993] | [74.015,74.015] | [73.988,73.988] | [0.012256,0.01348] |

9 | [74.008,74.005] | [73.995,73.995] | [74.009,74.009] | [74.005,74.005] | [74.004,74.004] | [0.005541,0.005177] |

10 | [73.998,73.998] | [74,74] | [73.99,73.99] | [74.007,74.007] | [73.995,73.995] | [0.006285,0.006285] |

11 | [73.994,73.998] | [73.998,73.998] | [73.994,73.994] | [73.995,73.995] | [73.99,74.001] | [0.002864,0.002775] |

12 | [74.004,74.004] | [74,74.002] | [74.007,74.005] | [74,74.001] | [73.996,73.996] | [0.004219,0.003507] |

13 | [73.983,73.993] | [74.002,74.002] | [73.998,73.998] | [73.997,73.997] | [74.012,74.005] | [0.010455,0.004637] |

14 | [74.006,74.006] | [73.967,73.985] | [73.994,73.994] | [74,74] | [73.984,73.996] | [0.015304,0.007759] |

15 | [74.012,74.012] | [74.014,74.012] | [73.998,73.998] | [73.999,73.999] | [74.007,74.007] | [0.007314,0.006804] |

16 | [74,74] | [73.984,73.984] | [74.005,74.005] | [73.998,73.998] | [73.996,73.996] | [0.007797,0.007797] |

17 | [73.994,73.994] | [74.012,74.012] | [73.986,73.986] | [74.005,74.005] | [74.007,74.007] | [0.010569,0.010569] |

18 | [74.006,74.006] | [74.01,74.011] | [74.018,74.018] | [74.003,74.003] | [74,74.001] | [0.006986,0.006834] |

19 | [73.984,73.984] | [74.002,74.002] | [74.003,74.003] | [74.005,74.005] | [73.997,73.997] | [0.008468,0.008468] |

20 | [74,74] | [74.01,74.01] | [74.013,74.009] | [74.02,74.015] | [74.003,74.003] | [0.007981,0.005941] |

21 | [73.982,73.982] | [74.001,74.001] | [74.015,74.015] | [74.005,74.005] | [73.996,73.996] | [0.012153,0.012153] |

22 | [74.004,74.004] | [73.999,73.999] | [73.99,73.99] | [74.006,74.006] | [74.009,74.002] | [0.007436,0.006261] |

23 | [74.01,74.01] | [73.989,73.989] | [73.99,73.99] | [74.009,74.005] | [74.014,74.011] | [0.011929,0.010747] |

24 | [74.015,74.011] | [74.008,74.008] | [73.993,73.993] | [74,74] | [74.01,74.011] | [0.008701,0.007893] |

25 | [73.982,73.982] | [73.984,73.989] | [73.995,73.995] | [74.017,74.012] | [74.013,74.01] | [0.016177,0.013088] |

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**MDPI and ACS Style**

Aslam, M.; Khan, N.; Khan, M.Z.
Monitoring the Variability in the Process Using Neutrosophic Statistical Interval Method. *Symmetry* **2018**, *10*, 562.
https://doi.org/10.3390/sym10110562

**AMA Style**

Aslam M, Khan N, Khan MZ.
Monitoring the Variability in the Process Using Neutrosophic Statistical Interval Method. *Symmetry*. 2018; 10(11):562.
https://doi.org/10.3390/sym10110562

**Chicago/Turabian Style**

Aslam, Muhammad, Nasrullah Khan, and Muhammad Zahir Khan.
2018. "Monitoring the Variability in the Process Using Neutrosophic Statistical Interval Method" *Symmetry* 10, no. 11: 562.
https://doi.org/10.3390/sym10110562