# Inspection Plan Based on the Process Capability Index Using the Neutrosophic Statistical Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design of a Neutrosophic Plan Based on PCI

**Step****1:**- Select a random sample of size ${n}_{N}\u03f5\left\{{n}_{L},{n}_{U}\right\}$ from the product lot. Compute the statistic ${\widehat{C}}_{{N}_{pk}}\u03f5Min\left\{\frac{USL-{\overline{X}}_{N}}{3{s}_{N}},\frac{{\overline{X}}_{N}-LSL}{3{s}_{N}}\right\}$; ${\overline{X}}_{N}\u03f5\left\{{\overline{X}}_{L},{\overline{X}}_{U}\right\}$,${s}_{N}\u03f5\left\{{s}_{L},{s}_{U}\right\}$.
**Step****2:**- Accept a product lot of ${\widehat{C}}_{{N}_{pk}}\ge {k}_{N}$; ${k}_{N}\u03f5\left\{{k}_{aL},{k}_{aU}\right\}$, otherwise reject a product lot, where ${k}_{N}\u03f5\left\{{k}_{aL},{k}_{aU}\right\}$ is the neutrosophic acceptance number. An acceptance number is also called the action number/boundary number. A product lot is rejected if the statistic ${\widehat{C}}_{{N}_{pk}}$ is smaller than ${k}_{N}$, otherwise, the product lot is accepted.

#### Research Methodology

- Specify the values of AQL, LQL, $\alpha $ and $\beta $.
- Specify the suitable ranges for ${n}_{N}\u03f5\left\{{n}_{L},{n}_{U}\right\}$ such that ${n}_{L}<{n}_{U}$ and ${k}_{N}\u03f5\left\{{k}_{aL},{k}_{aU}\right\}$ such that ${k}_{aL}<{k}_{aU}$.
- Perform the simulation by the grid search method and select those values of the neutrosophic plan parameters where ${n}_{N}\u03f5\left\{{n}_{L},{n}_{U}\right\}$ and satisfy the conditions given in Equations (7)–(9).

## 3. Comparison Study

## 4. Application of the Proposed Plan

**Step****1:**- Select a random sample of size ${n}_{N}$ = $\text{}\left\{128,133\right\}$ from a product lot. Compute the statistic ${\widehat{C}}_{{N}_{pk}}\u03f5\left[1.7377,2.0639\right]$.
**Step****2:**- Accept a product lot as $\left[1.7377,2.0639\right]\ge \left\{1.022,1.024\right\}$.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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p_{1} | p_{2} | ${\mathit{n}}_{\mathit{N}}$ | ${\mathit{k}}_{\mathit{N}}$ | ${\mathit{L}}_{\mathit{N}}\left({\mathit{p}}_{1}\right)$ | ${\mathit{L}}_{\mathit{N}}\left({\mathit{p}}_{2}\right)$ |
---|---|---|---|---|---|

0.001 | 0.002 | [602, 643] | [1.093, 1.095] | [0.9500, 0.95033] | [0.0441, 0.0891] |

0.003 | [218, 228] | [1.052, 1.054] | [0.9500, 0.9505] | [0.06223, 0.0898] | |

0.004 | [128, 133] | [1.022, 1.024] | [0.9506, 0.9513] | [0.0700, 0.0914] | |

0.006 | [69, 71] | [0.978, 0.980] | [0.9513, 0.9517] | [0.0807, 0.0969] | |

0.008 | [47, 49] | [0.946, 0.948] | [0.9506, 0.9528] | [0.0848, 0.0977] | |

0.010 | [36, 38] | [0.921, 0.923] | [0.9502, 0.9504] | [0.0849, 0.0958] | |

0.015 | [24, 28] | [0.874, 0.876] | [0.9541, 0.9675] | [0.0914, 0.0959] | |

0.020 | [18, 20] | [0.842, 0.844] | [0.9521, 0.9614] | [0.0761, 0.0823] | |

0.0025 | 0.030 | [21, 23] | [0.793, 0.795] | [0.9529, 0.9606] | [0.0923, 0.0995] |

0.050 | [13, 15] | [0.731, 0.735] | [0.9567, 0.9674] | [0.0607, 0.0754] | |

0.005 | 0.050 | [19, 21] | [0.730, 0.732] | [0.9512, 0.9599] | [0.0897, 0.0967] |

0.100 | [9, 11] | [0.631, 0.633] | [0.9575, 0.9740] | [0.0957, 0.0961] | |

0.01 | 0.020 | [274, 290] | [0.854, 0.856] | [0.9500, 0.9504] | [0.0513, 0.0881] |

0.030 | [95, 99] | [0.803, 0.805] | [0.9504, 0.9512] | [0.0696, 0.0918] | |

0.03 | 0.060 | [165, 174] | [0.718, 0.720] | [0.9503, 0.9509] | [0.0581, 0.0903] |

0.090 | [55, 57] | [0.659, 0.661] | [0.9505, 0.9511] | [0.0756, 0.0950] | |

0.05 | 0.100 | [123, 129] | [0.647, 0.649] | [0.9502, 0.9505] | [0.0690, 0.0986] |

0.150 | [41, 43] | 0.584, 0.586 | 0.9509, 0.9530 | 0.0736, 0.0911 |

p_{1} | p_{2} | Proposed Plan | Plan Based on Classical Statistics |
---|---|---|---|

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

0.001 | 0.002 | [602, 643] (R = 41) | 1134 (R = 1134) |

0.003 | [218, 228] (R = 10) | 351 (R = 351) | |

0.004 | [128, 133] (R = 5) | 161 (R = 161) | |

0.006 | [69, 71] (R = 2) | 74 (R = 74) | |

0.008 | [47, 49] (R = 2) | 47 (R = 47) | |

0.01 | 0.020 | [274, 290] (R = 16) | 449 (R = 449) |

0.030 | [95, 99] (R = 4) | 132 (R = 132) | |

0.03 | 0.060 | [165, 174] (R = 9) | 240 (R = 240) |

0.090 | [55, 57] (R = 2) | 68 (R = 68) | |

0.05 | 0.100 | [123, 129] (R = 6) | 167 (R = 167) |

0.150 | [41, 43] (R = 2) | 46 (R = 46) |

**Table 3.**Indeterminate data of Amplified Sensors from [40].

[1.9422,1.9422] | [1.9651, 1.9651] | [2.0230, 2.0435] | [1.9712,1.9712] | [1.9975,1.9975] | [2.0164,2.0164] | [1.9927,1.9927] | [1.9566,1.9566] |

[1.9738, 1.9738] | [1.9541, 1.9541] | [1.9800, 1.9800] | [1.9596, 1.9596] | [1.9811, 1.9811] | [2.0088, 2.0088] | [1.9858, 1.9858] | [1.9677, 1.9677] |

[2.0001, 2.0001] | [1.9659, 1.9659] | [1.9955, 1.9955] | [1.9842, 1.9842] | [1.9909,2.0512] | [1.9829, 1.9829] | [1.9684, 1.9684] | [1.9942, 1.9942] |

[1.9897, 1.9897] | [1.9836, 1.9836] | [1.9891, 1.9891] | [1.9608, 1.9608] | [2.0109, 2.0109] | [1.9912, 1.9912] | [2.0077, 2.0077] | [1.9803, 1.9803] |

[2.0106, 2.0106] | [1.9885, 1.9885] | [1.9704, 1.9704] | [1.9882, 1.9882] | [1.9689, 1.9689] | [1.9553, 1.9553] | [1.9741, 1.9741] | [1.9825, 1.9825] |

[1.9640, 1.9640] | [2.0187, 2.0187] | [1.9616, 1.9616] | [1.9865, 1.9865] | [1.9556, 1.9556] | [1.9817, 1.9817] | [1.9774, 1.9774] | [1.9316, 1.9316] |

[1.9841, 1.9841] | [1.9919, 1.9919] | [1.9737, 1.9737] | [1.9958, 1.9958] | [2.0121, 2.0121] | [2.0021, 2.0521] | [1.9665, 1.9665] | [1.9773, 1.9773] |

[1.9841, 1.9841] | [1.9570, 1.9875] | [1.9610, 1.9610] | [2.0015, 2.0015] | [1.9750, 1.9750] | [1.9825, 1.9825] | [1.9758, 1.9758] | [1.9682, 1.9682] |

[1.9668, 1.9668] | [1.9696, 1.9696] | [2.0334, 2.0334] | [1.9656, 1.9656] | [1.9819, 1.9819] | [2.0116, 2.0116] | [1.9754, 1.9754] | [1.9986, 1.9986] |

[2.0114, 2.0114] | [1.9861, 1.9861] | [1.9743, 1.9743] | [1.9594, 1.9594] | [1.9712,1.9914] | [1.9849, 1.9849] | [1.9711, 1.9711] | [1.9486, 1.9486] |

[1.9837, 1.9837] | [1.9424, 1.9424] | [1.9744, 1.9744] | [1.9605, 1.9605] | [1.9719, 1.9719] | [1.9656, 1.9656] | [1.9549, 1.9549] | [2.0174, 2.0174] |

[1.9779, 1.9779] | [2.0072, 2.0072] | [1.9875, 1.9875] | [1.9781, 1.9781] | [1.9834, 1.9834] | [1.9893, 1.9893] | [1.9276, 1.9276] | [1.9513, 1.9513] |

[1.9971, 1.9971] | [1.9963, 1.9963] | [1.9375, 1.9375] | [1.9941, 1.9941] | [1.9763, 1.9763] | [2.0108, 2.0108] | [1.9687, 1.9687] | [1.9559, 1.9559] |

[1.9611, 1.9611] | [1.9729, 1.9729] | [1.9992, 1.9992] | [1.9925, 1.9925] | [2.0073, 2.0073] | [1.9742, 1.9742] | [1.9557, 1.9557] | [1.9726, 1.9726] |

[1.9964, 1.9964] | [1.9614, 1.9614] | [1.9768, 1.9768] | [1.9991, 1.9991] | [1.9832, 1.9832] | [1.9847, 1.9847] | [1.9849, 1.9849] | [1.9918, 1.9918] |

[1.9748, 1.9748] | [1.9664, 1.9664] | [2.0035, 2.0245] | [1.9822, 1.9822] | [1.9882,1.9999] | [1.9809, 1.9809] | [1.9920, 1.9920] | [1.9994,2.0512] |

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**MDPI and ACS Style**

Aslam, M.; Albassam, M.
Inspection Plan Based on the Process Capability Index Using the Neutrosophic Statistical Method. *Mathematics* **2019**, *7*, 631.
https://doi.org/10.3390/math7070631

**AMA Style**

Aslam M, Albassam M.
Inspection Plan Based on the Process Capability Index Using the Neutrosophic Statistical Method. *Mathematics*. 2019; 7(7):631.
https://doi.org/10.3390/math7070631

**Chicago/Turabian Style**

Aslam, Muhammad, and Mohammed Albassam.
2019. "Inspection Plan Based on the Process Capability Index Using the Neutrosophic Statistical Method" *Mathematics* 7, no. 7: 631.
https://doi.org/10.3390/math7070631