# Testing of Grouped Product for the Weibull Distribution Using Neutrosophic Statistics

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design of Proposed Plan

**Step****1.**- Select a random sample ${n}_{N}\u03f5\left\{{n}_{L},{n}_{U}\right\}$ and distribute $r$ items into ${g}_{N}\u03f5\left\{{g}_{L},{g}_{U}\right\}$ groups.
**Step****2.**- Record first failure from ith group $\left(i=1,2,\dots {g}_{N}\right)$ and calculate neutrosophic fuzzy statistic ${v}_{N}={{\displaystyle \sum}}_{i=1}^{{g}_{N}}{Y}_{iN}^{m}$; ${g}_{N}\u03f5\left\{{g}_{L},{g}_{U}\right\}$, ${v}_{N}\u03f5\left\{{v}_{L},{v}_{U}\right\}$.
**Step****3.**- Accept lot of the product if ${v}_{N}\ge {k}_{N}{L}^{m}$, ${k}_{N}\u03f5\left\{{k}_{aL},{k}_{aU}\right\}$ is a neutrosophic fuzzy acceptance number.

#### Neutrosophic Fuzzy Non-Linear Optimization

**Step****1.**- Specify the value of r.
**Step****2.**- Determine the values of ${g}_{N}$ and ${k}_{N}$ using the search grid method through Equations (5)–(7).
**Step****3.**- Choose the parameters for the plan where indeterminacy interval in ${g}_{N}$ is minimum.

- For the fixed values of neutrosophic parameters, ${g}_{N}\u03f5\left\{{g}_{L},{g}_{U}\right\}$ and ${k}_{N}\u03f5\left\{{k}_{aL},{k}_{aU}\right\}$ decrease as $r$ increases from 5 to 10.
- For the fixed values of neutrosophic parameters, ${g}_{N}\u03f5\left\{{g}_{L},{g}_{U}\right\}$ and ${k}_{N}\u03f5\left\{{k}_{aL},{k}_{aU}\right\}$ decrease as LQL increases.

## 3. Application of Proposed Plan

**Step****1.**- Select a random sample 25 and distribute five items into five groups.
**Step****2.**- Perform sudden death testing and note down the first failure from each of the five groups $\left(i=1,2,\dots ,5\right)$. The number of first failures from the five groups are Y
_{1}= [220, 230], Y_{2}= [300, 320], Y_{3}= 285, Y_{4}= [155, 165] and Y_{5}= [365, 375]. The lifetime of ball bearing follows the neutrosophic Weibull distribution with parameter ${m}_{N}\u03f5\left\{2,2\right\}$ and lower specification limit $L$ = 200. The statistic ${v}_{N}$ is calculated as:$${v}_{N}={{\displaystyle \sum}}_{i=1}^{{g}_{N}}{Y}_{iN}^{m};\text{}{g}_{N}\u03f5\left\{3,5\right\},{v}_{N}\u03f5\left\{106.0,160.8\right\}.$$

^{2}, 230

^{2}] + [300

^{2}, 320

^{2}] + [285

^{2}, 285

^{2}] + [155

^{2}, 165

^{2}] + [365

^{2}, 375

^{2}] = [376875, 404375]. Now ${k}_{N}{L}^{{m}_{N}}\u03f5\left\{4240000,6432000\right\}$. As ${v}_{N}<{k}_{N}{L}^{{m}_{N}}$, so reject the lot of ball bearing product.

## 4. Comparison Study

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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p_{1} | p_{2} | $\text{}{\mathit{g}}_{\mathit{N}}\text{}$ | $\text{}{\mathit{k}}_{\mathit{N}}$ | $\mathit{L}\left({\mathit{C}}_{\mathit{A}\mathit{Q}\mathit{L}}\right)$ | $\mathit{L}\left({\mathit{C}}_{\mathit{L}\mathit{Q}\mathit{L}}\right)$ | |
---|---|---|---|---|---|---|

0.001 | 0.002 | Min | 19 | 2473.2 | 0.952163 | 0.099984 |

Max | 21 | 2717.5 | 0.962712 | 0.095018 | ||

0.003 | Min | 8 | 783.6 | 0.953490 | 0.099964 | |

Max | 10 | 953.3 | 0.975800 | 0.095052 | ||

0.004 | Min | 6 | 462.9 | 0.969175 | 0.099899 | |

Max | 8 | 592.6 | 0.988845 | 0.095073 | ||

0.006 | Min | 4 | 222.1 | 0.973434 | 0.099858 | |

Max | 6 | 311.3 | 0.994680 | 0.095147 | ||

0.008 | Min | 3 | 132.6 | 0.970167 | 0.099792 | |

Max | 5 | 201.2 | 0.996239 | 0.095123 | ||

0.010 | Min | 3 | 106.0 | 0.983213 | 0.099699 | |

Max | 5 | 160.8 | 0.998555 | 0.095117 | ||

0.015 | Min | 2 | 51.5 | 0.971999 | 0.099838 | |

Max | 4 | 89.4 | 0.998831 | 0.095418 | ||

0.020 | Min | 2 | 38.6 | 0.983592 | 0.099255 | |

Max | 4 | 66.9 | 0.999599 | 0.095298 | ||

0.0025 | 0.005 | Min | 19 | 987.8 | 0.952444 | 0.099978 |

Max | 21 | 1085.3 | 0.962975 | 0.095069 | ||

0.010 | Min | 6 | 184.6 | 0.969462 | 0.099904 | |

Max | 8 | 236.3 | 0.988989 | 0.095133 | ||

0.015 | Min | 4 | 88.5 | 0.973691 | 0.099564 | |

Max | 6 | 123.9 | 0.994785 | 0.095364 | ||

0.020 | Min | 3 | 52.7 | 0.983439 | 0.099924 | |

Max | 5 | 80.0 | 0.996321 | 0.095083 | ||

0.025 | Min | 3 | 42.1 | 0.983490 | 0.099512 | |

Max | 5 | 63.8 | 0.998600 | 0.095342 | ||

0.030 | Min | 2 | 25.6 | 0.958424 | 0.099282 | |

Max | 4 | 44.4 | 0.997442 | 0.095051 | ||

0.050 | Min | 2 | 15.2 | 0.984044 | 0.099320 | |

Max | 4 | 26.3 | 0.999624 | 0.096062 | ||

0.005 | 0.010 | Min | 19 | 492.7 | 0.952885 | 0.099908 |

Max | 21 | 541.3 | 0.963371 | 0.095048 | ||

0.015 | Min | 8 | 155.8 | 0.954335 | 0.099875 | |

Max | 10 | 189.5 | 0.976377 | 0.095086 | ||

0.020 | Min | 6 | 91.9 | 0.969849 | 0.099547 | |

Max | 8 | 117.5 | 0.989243 | 0.095381 | ||

0.030 | Min | 4 | 43.9 | 0.974240 | 0.099688 | |

Max | 6 | 61.5 | 0.994931 | 0.095194 | ||

0.040 | Min | 3 | 26.1 | 0.971200 | 0.099658 | |

Max | 5 | 39.5 | 0.996491 | 0.096118 | ||

0.050 | Min | 3 | 20.8 | 0.983945 | 0.099161 | |

Max | 5 | 31.5 | 0.998668 | 0.095215 | ||

0.100 | Min | 2 | 7.4 | 0.984787 | 0.099317 | |

Max | 4 | 12.8 | 0.999658 | 0.096182 | ||

0.01 | 0.020 | Min | 19 | 245.1 | 0.953835 | 0.099928 |

Max | 21 | 269.2 | 0.964273 | 0.095309 | ||

0.030 | Min | 8 | 77.3 | 0.955450 | 0.099925 | |

Max | 10 | 94.0 | 0.977123 | 0.095270 | ||

0.040 | Min | 5 | 39.2 | 0.950025 | 0.099569 | |

Max | 7 | 52.0 | 0.982409 | 0.095945 | ||

0.050 | Min | 4 | 26.1 | 0.955741 | 0.099193 | |

Max | 6 | 36.5 | 0.988705 | 0.095461 | ||

0.100 | Min | 3 | 10.2 | 0.984642 | 0.096525 | |

Max | 5 | 15.3 | 0.998813 | 0.096244 | ||

0.150 | Min | 2 | 4.8 | 0.975190 | 0.099150 | |

Max | 4 | 8.3 | 0.999095 | 0.096094 | ||

0.200 | Min | 2 | 3.5 | 0.986232 | 0.098790 | |

Max | 4 | 6.0 | 0.999729 | 0.099160 | ||

0.03 | 0.060 | Min | 19 | 80.1 | 0.957244 | 0.099165 |

Max | 21 | 87.9 | 0.967418 | 0.095266 | ||

0.090 | Min | 8 | 25.0 | 0.959516 | 0.099143 | |

Max | 10 | 30.3 | 0.980098 | 0.096447 | ||

0.120 | Min | 5 | 12.6 | 0.954365 | 0.096610 | |

Max | 7 | 16.6 | 0.985015 | 0.096118 | ||

0.150 | Min | 4 | 8.3 | 0.960407 | 0.096094 | |

Max | 6 | 11.5 | 0.990833 | 0.096298 | ||

0.300 | Min | 2 | 2.2 | 0.954964 | 0.097352 | |

Max | 5 | 4.5 | 0.999285 | 0.098199 | ||

0.05 | 0.100 | Min | 18 | 44.9 | 0.953778 | 0.098375 |

Max | 20 | 49.4 | 0.965643 | 0.096049 | ||

0.150 | Min | 8 | 14.5 | 0.963878 | 0.099439 | |

Max | 10 | 17.6 | 0.982585 | 0.095869 | ||

0.200 | Min | 5 | 7.2 | 0.960131 | 0.097749 | |

Max | 7 | 9.5 | 0.987505 | 0.096649 | ||

0.250 | Min | 4 | 4.7 | 0.965761 | 0.095135 | |

Max | 6 | 6.5 | 0.992691 | 0.096047 | ||

0.500 | Min | 2 | 1.2 | 0.961324 | 0.080608 | |

Max | 6 | 2.7 | 0.999915 | 0.095643 |

p_{1} | p_{2} | $\text{}{\mathit{g}}_{\mathit{N}}$ | ${\mathit{k}}_{\mathit{N}}$ | $\text{}\mathit{L}\left({\mathit{C}}_{\mathit{A}\mathit{Q}\mathit{L}}\right)\text{}$ | $\mathit{L}\left({\mathit{C}}_{\mathit{L}\mathit{Q}\mathit{L}}\right)$ | |
---|---|---|---|---|---|---|

0.001 | 0.002 | Min | 19 | 1236.6 | 0.952163 | 0.099984 |

Max | 21 | 1358.7 | 0.962724 | 0.095049 | ||

0.003 | Min | 8 | 391.8 | 0.953490 | 0.099964 | |

Max | 10 | 476.6 | 0.975815 | 0.095115 | ||

0.004 | Min | 6 | 231.5 | 0.969147 | 0.099791 | |

Max | 8 | 296.3 | 0.988845 | 0.095073 | ||

0.006 | Min | 4 | 111.1 | 0.973396 | 0.099670 | |

Max | 6 | 155.6 | 0.994688 | 0.095301 | ||

0.008 | Min | 3 | 66.3 | 0.970167 | 0.099792 | |

Max | 5 | 100.6 | 0.996239 | 0.095123 | ||

0.010 | Min | 3 | 53.0 | 0.983213 | 0.099699 | |

Max | 5 | 80.4 | 0.998555 | 0.095117 | ||

0.015 | Min | 2 | 25.8 | 0.971899 | 0.099239 | |

Max | 4 | 44.7 | 0.998831 | 0.095418 | ||

0.020 | Min | 2 | 19.3 | 0.983592 | 0.099255 | |

Max | 4 | 33.4 | 0.999602 | 0.095903 | ||

0.0025 | 0.005 | Min | 19 | 493.9 | 0.952444 | 0.099978 |

Max | 21 | 542.6 | 0.963004 | 0.095147 | ||

0.010 | Min | 6 | 92.3 | 0.969462 | 0.099904 | |

Max | 8 | 118.1 | 0.989014 | 0.095364 | ||

0.015 | Min | 4 | 44.3 | 0.973597 | 0.099096 | |

Max | 6 | 61.9 | 0.994805 | 0.095754 | ||

0.020 | Min | 3 | 26.4 | 0.970450 | 0.099229 | |

Max | 5 | 40.0 | 0.996321 | 0.095083 | ||

0.025 | Min | 3 | 21.1 | 0.983387 | 0.098644 | |

Max | 5 | 31.9 | 0.998600 | 0.095342 | ||

0.030 | Min | 2 | 12.8 | 0.958424 | 0.099282 | |

Max | 4 | 22.2 | 0.997442 | 0.095051 | ||

0.050 | Min | 2 | 7.6 | 0.984044 | 0.099320 | |

Max | 4 | 13.1 | 0.999629 | 0.097616 | ||

0.005 | 0.010 | Min | 19 | 246.4 | 0.952809 | 0.099739 |

Max | 21 | 270.6 | 0.963430 | 0.095205 | ||

0.015 | Min | 8 | 77.9 | 0.954335 | 0.099875 | |

Max | 10 | 94.7 | 0.976451 | 0.095405 | ||

0.020 | Min | 6 | 46.0 | 0.969713 | 0.099008 | |

Max | 8 | 58.7 | 0.989294 | 0.095848 | ||

0.030 | Min | 4 | 22.0 | 0.974054 | 0.098745 | |

Max | 6 | 30.7 | 0.994970 | 0.095980 | ||

0.040 | Min | 3 | 13.1 | 0.970920 | 0.098260 | |

Max | 5 | 19.7 | 0.996529 | 0.097257 | ||

0.050 | Min | 3 | 10.4 | 0.983945 | 0.099161 | |

Max | 5 | 15.7 | 0.998686 | 0.096635 | ||

0.100 | Min | 2 | 3.7 | 0.984787 | 0.099317 | |

Max | 4 | 6.4 | 0.999658 | 0.096182 | ||

0.01 | 0.020 | Min | 19 | 122.6 | 0.953686 | 0.099588 |

Max | 21 | 134.6 | 0.964273 | 0.095309 | ||

0.030 | Min | 8 | 38.7 | 0.955176 | 0.099196 | |

Max | 10 | 47.0 | 0.977123 | 0.095270 | ||

0.040 | Min | 5 | 19.6 | 0.950025 | 0.099569 | |

Max | 7 | 26.0 | 0.982409 | 0.095945 | ||

0.050 | Min | 4 | 13.1 | 0.955231 | 0.097616 | |

Max | 6 | 18.2 | 0.988843 | 0.096790 | ||

0.100 | Min | 3 | 5.1 | 0.984642 | 0.096525 | |

Max | 5 | 7.6 | 0.998847 | 0.099210 | ||

0.150 | Min | 2 | 2.4 | 0.975190 | 0.099150 | |

Max | 3 | 3.3 | 0.995249 | 0.097215 | ||

0.200 | Min | 2 | 1.8 | 0.985482 | 0.090371 | |

Max | 4 | 3.0 | 0.999729 | 0.099160 | ||

0.03 | 0.060 | Min | 19 | 40.1 | 0.956814 | 0.098132 |

Max | 21 | 43.9 | 0.967745 | 0.096233 | ||

0.090 | Min | 8 | 12.5 | 0.959516 | 0.099143 | |

Max | 10 | 15.1 | 0.980490 | 0.098475 | ||

0.120 | Min | 5 | 6.3 | 0.954365 | 0.096610 | |

Max | 7 | 8.3 | 0.985015 | 0.096118 | ||

0.150 | Min | 4 | 4.2 | 0.958944 | 0.091311 | |

Max | 7 | 6.5 | 0.995704 | 0.098411 | ||

0.300 | Min | 2 | 1.1 | 0.954964 | 0.097352 | |

Max | 3 | 1.5 | 0.988671 | 0.098094 | ||

0.05 | 0.100 | Min | 18 | 22.5 | 0.952981 | 0.096593 |

Max | 20 | 24.7 | 0.965643 | 0.096049 | ||

0.150 | Min | 8 | 7.3 | 0.962650 | 0.095621 | |

Max | 10 | 8.8 | 0.982585 | 0.095869 | ||

0.200 | Min | 5 | 3.6 | 0.960131 | 0.097749 | |

Max | 8 | 5.3 | 0.993116 | 0.097357 | ||

0.250 | Min | 4 | 2.4 | 0.963476 | 0.086889 | |

Max | 8 | 4.1 | 0.998501 | 0.098851 | ||

0.500 | Min | 2 | 0.6 | 0.961324 | 0.080608 | |

Max | 8 | 1.7 | 0.999996 | 0.099397 |

**Table 3.**The comparison of proposed plan with the plan in Reference [2].

p_{1} | p_{2} | Proposed Plan | Existing Plan |
---|---|---|---|

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

0.001 | 0.002 | [19, 21] | 231 |

0.003 | [8, 10] | 154 | |

0.004 | [6, 8] | 115 | |

0.006 | [4, 6] | 77 | |

0.008 | [3, 5] | 58 | |

0.05 | 0.100 | [18, 20] | 34 |

0.150 | [8, 10] | 16 | |

0.200 | [5, 7] | 8 |

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## Share and Cite

**MDPI and ACS Style**

Aslam, M.; Arif, O.H.
Testing of Grouped Product for the Weibull Distribution Using Neutrosophic Statistics. *Symmetry* **2018**, *10*, 403.
https://doi.org/10.3390/sym10090403

**AMA Style**

Aslam M, Arif OH.
Testing of Grouped Product for the Weibull Distribution Using Neutrosophic Statistics. *Symmetry*. 2018; 10(9):403.
https://doi.org/10.3390/sym10090403

**Chicago/Turabian Style**

Aslam, Muhammad, and Osama H. Arif.
2018. "Testing of Grouped Product for the Weibull Distribution Using Neutrosophic Statistics" *Symmetry* 10, no. 9: 403.
https://doi.org/10.3390/sym10090403