# Design of Sampling Plan Using Regression Estimator under Indeterminacy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design of the Proposed Plan

**Step #1.**Measure the bivariate quality characteristics $\left({y}_{N1},{x}_{N1}\right),\left({y}_{N2},{x}_{N2}\right),\dots ,\left({y}_{Nn},{x}_{Nn}\right)$ based on the sample size ${n}_{N}.$ Compute following the neutrosophic statistic ${M}_{Nr}$.

**Step #2**. Calculate ${V}_{Nr}=\frac{USL-\text{}{M}_{Nr}}{{\widehat{\sigma}}_{{M}_{Nr}}}$; ${\widehat{\sigma}}_{{M}_{Nr}}\text{}\mathsf{\u03f5}\text{}\left\{{\widehat{\sigma}}_{{M}_{y}},{\widehat{\sigma}}_{{M}_{x}}\right\}$

**Step #3.**Accept the lot if ${V}_{Nr}\ge {k}_{N}$, where ${k}_{N}\mathsf{\u03f5}\text{}\left\{{k}_{aL},{k}_{aU}\right\}$ is the neutrosophic acceptance number.

## 3. Comparative Study

## 4. Application of the Proposed Plan

_{N}) and the tensile strength (Y

_{N}). In the industry, the measurement of Brinell hardness is difficult and costly. The tensile strength is easy to measure and correlated with Brinell hardness. The observations of both variables will be obtained from the measurement process. According to [17] “observations include human judgments, and evaluations and decisions, a continuous random variable of a production process should include the variability caused by human subjectivity or measurement devices, or environmental conditions. These variability causes create vagueness in the measurement system”. Therefore, we expect that some observations of two variables are uncertain. Therefore, the existing sampling plan under the classical statistics cannot apply for the product inspection. As the tensile strength (Y

_{N}) is correlated with the main variable of study X

_{N}, therefore, the proposed neutrosophic regression model can be used for the inspection of the product. Similar data has been used by [27,28] using classical statistics. The data of the two variables with some uncertain observations are reported in the Table 5.

**Step 1.**Take a bivariate random sample of size 24 from the submitted lot and measure the quality characteristics {[143,143], [34.2,34.2]}, … {[194,198], [57.5,58.9]}

**Step 2.**Calculate ${V}_{Nr}\in \left\{7.33,49.50\right\}$

**Step 3.**Reject the lot ${V}_{Nr}\in \left\{7.33,49.50\right\}\le {k}_{N}\in \left\{17.4,20.1\right\}$. It is important to note that if the experimenter selects a sample of size 25, then a lot of the product will be accepted as $49.50>20.1$.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Authors | Year | Contributions |
---|---|---|

Aslam et al. [4] | 2017 | Introduced regression estimator in the sampling plan |

Smarandache [18] | 2010 | Introduced neutrosophic logic |

Smarandache [25] | 2014 | Introduced neutrosophic statistics |

Aslam [20] | 2018 | Introduced neutrosophic industrial statistics (NIS) |

**Table 2.**The plan parameters of the plan when $\alpha =0.05$, $\beta $ = 0.10 and ${r}_{Nxy}=\left\{0.7817,0.8319\right\}$.

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

0.001 | 0.004 | [321,323] | [93.9,105.2] | [0.9505,0.9512] | [0.0991,0.0999] |

0.006 | [184,186] | [69.7,77.9] | [0.9533,0.9539] | [0.0967,0.0968] | |

0.008 | [128,130] | [59.9,64.5] | [0.9503,0.9515] | [0.0929,0.0983] | |

0.010 | [102,106] | 50.1,57.5] | [0.9524,0.9544] | [0.0878,0.0973] | |

0.015 | [71,73] | [40.6,46.2] | [0.9596,0.9645] | [0.0968,0.0990] | |

0.020 | [55,77] | [35.1,40.1] | [0.9576,0.9633] | [0.0919,0.0954] | |

0.0025 | 0.030 | [62,65] | [34.9,40.0] | [0.9561,0.9655] | [0.0959,0.0966] |

0.050 | [38,40] | [26.2,30.3] | [0.9553,0.9587] | [0.0846,0.0955] | |

0.005 | 0.050 | [55,59] | [30.6,35.8] | [0.9515,0.9539] | [0.0788,0.0955] |

0.100 | [30,33] | [21.0,24.3] | [0.9639,0.9819] | [0.0900,0.0963] | |

0.140 | [23,25] | [17.4,20.1] | [0.9752,0.9865] | [0.0983, 0.0994] | |

0.01 | 0.030 | [233,235] | [63.5,71.7] | [0.9501,0.9518] | [0.0931,0.0978] |

0.03 | 0.090 | [134,136] | [40.6,45.9] | [0.9516,0.9576] | [0.0974,0.0986] |

0.05 | 0.100 | [285,290] | [56.3,63.9] | [0.9538,0.9552] | [0.0902,0.0998] |

0.150 | [115,118] | [33.7,38.3] | [0.9739,0.9788] | [0.0959,0.0989] |

**Table 3.**The plan parameters of the plan when $\alpha =0.05$, $\beta $ = 0.10 and ${r}_{Nxy}=\left\{0.88,0.90\right\}$.

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

0.001 | 0.004 | [264,299] | [104.6,121.0] | [0.9532,0.9682] | [0.0903,0.0846] |

0.006 | [161,172] | [79.3,89.6] | [0.9672,0.9690] | [0.0904,0.0769] | |

0.008 | [115,129] | [66.3,76.3] | [0.9568,0.9707] | [0.0705,0.0655] | |

0.01 | [99,103] | [60.2,66.3] | [0.9725,0.9831] | [0.0728,0.0858] | |

0.02 | [61,63] | [43.3,48.8] | [0.9947,0.9918] | [0.0977,0.0700] | |

0.0025 | 0.03 | [53,57] | [38.7,43.8] | [0.9685,0.9738] | [0.0746,0.0663] |

0.05 | [39,41] | [30.2,35.6] | [0.9945,0.9801] | [0.0995,0.0438] | |

0.005 | 0.05 | [47,58] | [33.7,40.9] | [0.9630,0.9766] | [0.0631,0.0425] |

0.1 | [24,33] | [21.6,27.3] | [0.9806,0.9955] | [0.0731,0.0514] | |

0.005 | 0.14 | [23,30] | [18.7,27.5] | [0.9989,0.9747] | [0.0908,0.0073] |

0.01 | 0.03 | [175,183] | [65.2,73.1] | [0.9590,0.9549] | [0.0873,0.0697] |

0.05 | 0.1 | [201,205] | [53.2,59.0] | [0.9654,0.9551] | [0.0690,0.0498] |

0.15 | [65,80] | [28.4,33.6] | [0.9564,0.9871] | [0.0631,0.0689] |

**Table 4.**Comparison of Proposed Plan with Aslam et al. [4] plan when $\alpha =0.05$, $\beta $ = 0.10 and ${r}_{Nxy}=\left\{0.7817,0.8319\right\}$.

${\mathit{p}}_{1}$ | ${\mathit{p}}_{2}$ | ${\mathit{n}}_{\mathit{N}}$ | $\mathit{n}$ |
---|---|---|---|

0.001 | 0.004 | [321,323] | 321 |

0.006 | [184,186] | 184 | |

0.008 | [128,130] | 128 | |

0.010 | [102,106] | 102 | |

0.015 | [71,73] | 71 | |

0.020 | [55,77] | 55 | |

0.0025 | 0.030 | [62,65] | 62 |

0.050 | [38,40] | 38 | |

0.005 | 0.050 | [55,59] | 55 |

0.100 | [30,33] | 30 | |

0.140 | [23,25] | 23 | |

0.01 | 0.030 | [233,235] | 233 |

0.03 | 0.090 | [134,136] | 134 |

0.05 | 0.100 | [285,290] | 285 |

0.150 | [115,118] | 115 |

Observations | ${\mathit{X}}_{\mathit{N}}$ | ${\mathit{Y}}_{\mathit{N}}$ | Observations | ${\mathit{X}}_{\mathit{N}}$ | ${\mathit{Y}}_{\mathit{N}}$ |
---|---|---|---|---|---|

1 | [143,143] | [34.2,34.2] | 13 | [187,191] | [58.2,64] |

2 | [200,200] | [57,57] | 14 | [186,186] | [57,57] |

3 | [168,175] | [47.5,50] | 15 | [172,172] | [49.4,49.4] |

4 | [181,181] | [53.4,53.4] | 16 | [182,182] | [57.2,57.2] |

5 | [148,148] | [47.8,47.8] | 17 | [177,180] | [50.6,45] |

6 | [178,178] | [51.5,51.5] | 18 | [204,204] | [55.1,55.1] |

7 | [162,168] | [45.9,50] | 19 | [178,178] | [50.9,50.9] |

8 | [215,215] | [59.1,59.1] | 20 | [198,200] | [57.9,60.9] |

9 | [161,161] | [48.4,48.4] | 21 | [160,160] | [45.5,45.5] |

10 | [141,141] | [47.3,47.3] | 22 | [183,187] | [53.9,55.8] |

11 | [175,177] | [57.3,59.6] | 23 | [179,179] | [51.2,51.2] |

12 | [187,187] | [58.5,58.5] | 24 | [194,198] | [57.5,58.9] |

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**MDPI and ACS Style**

Aslam, M.; AL-Marshadi, A.H.
Design of Sampling Plan Using Regression Estimator under Indeterminacy. *Symmetry* **2018**, *10*, 754.
https://doi.org/10.3390/sym10120754

**AMA Style**

Aslam M, AL-Marshadi AH.
Design of Sampling Plan Using Regression Estimator under Indeterminacy. *Symmetry*. 2018; 10(12):754.
https://doi.org/10.3390/sym10120754

**Chicago/Turabian Style**

Aslam, Muhammad, and Ali Hussein AL-Marshadi.
2018. "Design of Sampling Plan Using Regression Estimator under Indeterminacy" *Symmetry* 10, no. 12: 754.
https://doi.org/10.3390/sym10120754