# Design of Variable Sampling Plan for Pareto Distribution Using Neutrosophic Statistical Interval Method

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## Abstract

**:**

## 1. Introduction

## 2. Design of Variable Plan under Neutrosophic Statistics

**Step-1:**Select a neutrosophic random sample of size ${n}_{N}\in \left\{{n}_{L},{n}_{U}\right\}$ from the lot of the product and record the observations for ${X}_{Ni}\in \left\{{X}_{L},{X}_{U}\right\}=i=1,2,3,\dots ,n$.

**Step-2:**Accept a lot of the product if ${Y}_{N}={\overline{X}}_{N}+{k}_{N}{s}_{N}\le U$ or ${Y}_{N}={\overline{X}}_{N}+{k}_{N}{s}_{N}\ge L$, where ${k}_{N}\in \left\{{k}_{aL},{k}_{aU}\right\}$ is the neutrosophic acceptance number, where ${\overline{X}}_{L}={{\displaystyle \sum}}_{i=1}^{n}{x}_{i}^{L}/{n}_{L}$.

#### Neutrosophic Non-Linear Optimization

## 3. Comparative Study

## 4. Case Study

**Step-1:**Select a neutrosophic random sample of size ${n}_{N}\in \left\{38,89\right\}$ from the lot of the product and record the observations for ${X}_{Ni}\in \left\{{X}_{L},{X}_{U}\right\}=i=1,2,3,\dots ,55$.

**Step-2:**Accept the lot of the ball bearing product as ${Y}_{N}\in \left\{0.1269,0.1321\right\}\le U=14$.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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${\mathit{p}}_{1}$ | ${\mathit{p}}_{2}$ | ${\mathit{a}}_{\mathit{N}}$ | ${\mathit{\alpha}}_{\mathit{N}3}$ | ${\mathit{\alpha}}_{\mathit{N}4}$ | ${\mathit{n}}_{\mathit{N}}$ | ${\mathit{k}}_{\mathit{N}}$ | ${\mathit{L}}_{\mathit{N}}\left({\mathit{p}}_{1}\right)$ | ${\mathit{L}}_{\mathit{N}}\left({\mathit{p}}_{2}\right)$ |
---|---|---|---|---|---|---|---|---|

0.01 | 0.025 | [11,22] | [2.714,2.308] | [175.6,254.9] | [1706,1918] | [2.368,2.372] | [0.9845,0.09503] | [0.9793,0.094397] |

0.01 | 0.05 | [10,21] | [2.811,2.325] | [172.1,246.6] | [484,525] | [2.197,2.245] | [0.9854,0.08649] | [0.9641,0.060927] |

0.01 | 0.1 | [11,18] | [2.714,2.388] | [175.6,222.4] | [192,257] | [2.088,2.14] | [0.9644,0.05059] | [0.9591,0.023991] |

0.01 | 0.25 | [17,28] | [2.415,2.236] | [214.6,305.9] | [122,149] | [1.574,1.842] | [0.9994,0.08587] | [0.9869,0.01808] |

0.01 | 0.5 | [9,27] | [2.94,2.245] | [171,297.3] | [38,89] | [1.339,1.768] | [0.9945,0.08611] | [0.974,0.004256] |

0.02 | 0.05 | [6,14] | [3.81,2.525] | [218.7,192.8] | [1103,1288] | [2.114,2.148] | [0.9616,0.09953] | [0.9507,0.041457] |

0.02 | 0.1 | [21,23] | [2.325,2.293] | [246.6,263.3] | [368,419] | [1.939,1.951] | [0.971,0.07512] | [0.9727,0.058404] |

0.02 | 0.2 | [26,29] | [2.256,2.227] | [288.7,314.5] | [190,192] | [1.648,1.725] | [0.9936,0.08987] | [0.983,0.058857] |

0.02 | 0.25 | [10,11] | [2.811,2.714] | [172.1,175.6] | [154,160] | [1.65,1.817] | [0.9942,0.03101] | [0.9672,0.008] |

0.02 | 0.5 | [22,29] | [2.308,2.227] | [254.9,314.5] | [85,86] | [1.313,1.352] | [0.9979,0.03562] | [0.9952,0.035612] |

0.05 | 0.1 | [9,23] | [2.94,2.293] | [171,263.3] | [1898,2046] | [1.755,1.772] | [0.9955,0.08073] | [0.9871,0.065411] |

0.05 | 0.15 | [6,10] | [3.81,2.811] | [218.7,172.1] | [515,544] | [1.678,1.698] | [0.964,0.06447] | [0.9648,0.036618] |

0.05 | 0.2 | [11,24] | [2.714,2.28] | [175.6,271.7] | [326,521] | [1.619,1.686] | [0.9685,0.03263] | [0.9523,0.006865] |

0.05 | 0.25 | [13,19] | [2.576,2.365] | [186.3,230.4] | [159,234] | [1.497,1.502] | [0.9646,0.08775] | [0.9802,0.05685] |

0.05 | 0.5 | [11,12] | [2.714,2.637] | [175.6,180.5] | [100,152] | [1.352,1.376] | [0.9776,0.01274] | [0.9904,0.002472] |

${\mathit{p}}_{1}$ | ${\mathit{p}}_{2}$ | ${\mathit{a}}_{\mathit{N}}$ | ${\mathit{\alpha}}_{\mathit{N}3}$ | ${\mathit{\alpha}}_{\mathit{N}4}$ | ${\mathit{n}}_{\mathit{N}}$ | ${\mathit{k}}_{\mathit{N}}$ | ${\mathit{L}}_{\mathit{N}}\left({\mathit{p}}_{1}\right)$ | ${\mathit{L}}_{\mathit{N}}\left({\mathit{p}}_{2}\right)$ |
---|---|---|---|---|---|---|---|---|

0.01 | 0.025 | [17,27] | [2.415,2.245] | [214.6,297.3] | [2038,2297] | [2.404,2.411] | [0.9669,0.04046] | [0.9583,0.0379624] |

0.01 | 0.05 | [6,28] | [3.81,2.236] | [218.7,305.9] | [753,812] | [2.284,2.311] | [0.9736,0.01591] | [0.9526,0.0131754] |

0.01 | 0.1 | [12,13] | [2.637,2.576] | [180.5,186.3] | [270,364] | [2.153,2.187] | [0.9651,0.01454] | [0.9718,0.00389] |

0.01 | 0.25 | [14,15] | [2.525,2.483] | [192.8,199.8] | [90,136] | [1.779,1.812] | [0.9836,0.04614] | [0.9933,0.0161476] |

0.01 | 0.5 | [24,29] | [2.28,2.227] | [271.7,314.5] | [56,85] | [1.734,1.768] | [0.9512,0.01843] | [0.9694,0.0056095] |

0.02 | 0.05 | [9,11] | [2.94,2.714] | [171,175.6] | [1731,2067] | [2.108,2.161] | [0.9923,0.04951] | [0.9759,0.0082816] |

0.02 | 0.1 | [11,23] | [2.714,2.293] | [175.6,263.3] | [403,648] | [2.015,2.026] | [0.9554,0.02186] | [0.9698,0.0085848] |

0.02 | 0.2 | [6,11] | [3.81,2.714] | [218.7,175.6] | [153,288] | [1.776,1.972] | [0.9684,0.04723] | [0.9509,0.0006528] |

0.02 | 0.25 | [7,30] | [3.381,2.218] | [185.4,323.2] | [180,197] | [1.645,1.721] | [0.9965,0.02489] | [0.9843,0.0213151] |

0.02 | 0.5 | [15,16] | [2.483,2.446] | [199.8,207.1] | [81,103] | [1.368,1.44] | [0.997,0.02343] | [0.9972,0.0083661] |

0.05 | 0.1 | [13,27] | [2.576,2.245] | [186.3,297.3] | [1702,2210] | [1.818,1.818] | [0.9511,0.02239] | [0.9538,0.020496] |

0.05 | 0.15 | [6,17] | [3.81,2.415] | [218.7,214.6] | [731,781] | [1.668,1.696] | [0.9871,0.04121] | [0.9813,0.0218219] |

0.05 | 0.2 | [13,21] | [2.576,2.325] | [186.3,246.6] | [285,569] | [1.614,1.7] | [0.9594,0.04693] | [0.9541,0.0032972] |

0.05 | 0.25 | [10,25] | [2.811,2.267] | [172.1,280.2] | [329,339] | [1.468,1.551] | [0.9978,0.03336] | [0.9812,0.0209033] |

0.05 | 0.5 | [24,28] | [2.28,2.236] | [271.7,305.9] | [99,124] | [1.23,1.267] | [0.9882,0.0426] | [0.9894,0.0245211] |

${\mathit{p}}_{1}$ | ${\mathit{p}}_{2}$ | ${\mathit{a}}_{\mathit{N}}$ | ${\mathit{\alpha}}_{\mathit{N}3}$ | ${\mathit{\alpha}}_{\mathit{N}4}$ | ${\mathit{n}}_{\mathit{N}}$ | ${\mathit{k}}_{\mathit{N}}$ | ${\mathit{L}}_{\mathit{N}}\left({\mathit{p}}_{1}\right)$ | ${\mathit{L}}_{\mathit{N}}\left({\mathit{p}}_{2}\right)$ |
---|---|---|---|---|---|---|---|---|

0.01 | 0.025 | [20,30] | [2.344,2.218] | [238.5,323.2] | [1337,1825] | [2.399,2.429] | [0.9329,0.09126] | [0.9094,0.043424] |

0.01 | 0.05 | [25,27] | [2.267,2.245] | [280.2,297.3] | [447,462] | [2.25,2.306] | [0.9429,0.07925] | [0.9029,0.048256] |

0.01 | 0.1 | [10,23] | [2.811,2.293] | [172.1,263.3] | [143,223] | [2.052,2.062] | [0.955,0.09413] | [0.9691,0.064611] |

0.01 | 0.25 | [5,19] | [4.648,2.365] | [343.8,230.4] | [73,99] | [1.891,1.993] | [0.9172,0.06665] | [0.9308,0.015943] |

0.01 | 0.5 | [13,27] | [2.576,2.245] | [186.3,297.3] | [41,78] | [1.584,1.965] | [0.9721,0.03959] | [0.9043,0.002908] |

0.02 | 0.05 | [13,30] | [2.576,2.218] | [186.3,323.2] | [1120,1244] | [2.139,2.16] | [0.9484,0.0594] | [0.9078,0.055194] |

0.02 | 0.1 | [10,19] | [2.811,2.365] | [172.1,230.4] | [303,392] | [1.955,1.995] | [0.9635,0.06768] | [0.953,0.037956] |

0.02 | 0.2 | [8,24] | [3.118,2.28] | [173.9,271.7] | [159,191] | [1.756,1.939] | [0.9818,0.04094] | [0.9088,0.011826] |

0.02 | 0.25 | [11,25] | [2.714,2.267] | [175.6,280.2] | [101,126] | [1.652,1.793] | [0.9788,0.06578] | [0.938,0.031912] |

0.02 | 0.5 | [10,12] | [2.811,2.637] | [172.1,180.5] | [55,58] | [1.451,1.53] | [0.9812,0.03261] | [0.9694,0.022133] |

0.05 | 0.1 | [14,26] | [2.525,2.256] | [192.8,288.7] | [1206,1512] | [1.804,1.818] | [0.9354,0.06039] | [0.9184,0.04372] |

0.05 | 0.15 | [16,22] | [2.446,2.308] | [207.1,254.9] | [493,530] | [1.645,1.683] | [0.9781,0.094] | [0.9581,0.064101] |

0.05 | 0.2 | [13,18] | [2.576,2.388] | [186.3,222.4] | [221,308] | [1.634,1.637] | [0.9246,0.05986] | [0.9465,0.038161] |

0.05 | 0.25 | [22,23] | [2.308,2.293] | [254.9,263.3] | [276,280] | [1.458,1.504] | [0.9924,0.06826] | [0.9851,0.046336] |

0.05 | 0.5 | [10,19] | [2.811,2.365] | [172.1,230.4] | [62,100] | [1.35,1.41] | [0.9445,0.03897] | [0.9523,0.012875] |

p1 | p2 | Proposed Plan | Single Plan |
---|---|---|---|

${\mathit{n}}_{\mathit{N}}$ | n | ||

0.01 | 0.025 | [1337,1825] | [2733,2733] |

0.01 | 0.05 | [447,462] | [657,657] |

0.01 | 0.1 | [143,223] | [522,522] |

0.01 | 0.25 | [73,99] | [418,418] |

0.01 | 0.5 | [41,78] | [115,115] |

0.02 | 0.05 | [1120,1244] | [3621,3621] |

0.02 | 0.1 | [303,392] | [1260,1260] |

0.02 | 0.2 | [159,191] | [715,715] |

0.02 | 0.25 | [101,126] | [321,321] |

0.02 | 0.5 | [55,58] | [297,297] |

0.05 | 0.1 | [1206,1512] | [2078,2078] |

0.05 | 0.15 | [493,530] | [2499,2499] |

0.05 | 0.2 | [221,308] | [1518,1518] |

0.05 | 0.25 | [276,280] | [1085,1085] |

0.05 | 0.5 | [62,100] | [165,165] |

Observations | ||||
---|---|---|---|---|

[0.1154,0.1154] | [0.1171,0.1171] | [0.1055,0.1053] | [0.1157,0.1157] | [0.1044,0.10154] |

[0.1415,0.1088] | [0.1152,0.1234] | [0.11421,0.1142 | [0.1018,0.1095] | [0.1102,0.1010] |

[0.1032,0.1581] | [0.1065,0.1023] | [0.1072,0.1072] | [0.1043,0.1092] | [0.1018,0.11269] |

[0.1099,0.1152] | [0.1014,0.1016] | [0.1051,0.1087] | [0.1122,0.1043] | [0.1013,0.1013] |

[0.1031,0.1159] | [0.1070,0.1149] | [0.1017,0.1006] | [0.1308,0.11553] | [0.1019,0.1019] |

[0.1144,0.1144] | [0.1143,0.1022] | [0.1648,0.1197] | [0.1038,0.1428] | [0.1234,0.1234] |

[0.1095,0.1070] | [0.1184,0.1100] | [0.1231,0.1437] | [0.1166,0.1023] | [0.1015,0.1102] |

[0.1396,0.1071] | [0.1231,0.1143] | [0.1067,0.1067] | [0.1025,0.1208] | [0.1061,0.1112] |

[0.1089,0.1044] | [0.1015,0.1015] | [0.1012,0.1012] | [0.1007,0.1007] | [0.1040,0.1040] |

[0.1129,0.1012] | [0.1015,0.1012] | [0.1025,0.1025] | [0.1140,0.1126] | [0.1081,0.1081] |

[0.1251,0.15] | [0.1000,0.1223] | [0.1040,0.1040] | [0.10043,0.1004] | [0.1050,0.1050] |

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**MDPI and ACS Style**

Aslam, M.; Khan, N.; AL-Marshadi, A.H.
Design of Variable Sampling Plan for Pareto Distribution Using Neutrosophic Statistical Interval Method. *Symmetry* **2019**, *11*, 80.
https://doi.org/10.3390/sym11010080

**AMA Style**

Aslam M, Khan N, AL-Marshadi AH.
Design of Variable Sampling Plan for Pareto Distribution Using Neutrosophic Statistical Interval Method. *Symmetry*. 2019; 11(1):80.
https://doi.org/10.3390/sym11010080

**Chicago/Turabian Style**

Aslam, Muhammad, Nasrullah Khan, and Ali Hussein AL-Marshadi.
2019. "Design of Variable Sampling Plan for Pareto Distribution Using Neutrosophic Statistical Interval Method" *Symmetry* 11, no. 1: 80.
https://doi.org/10.3390/sym11010080