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The variable data is obtained from the measurement process which is not fully complete or clear in nature due to measurement error. The neutrosophic statistics which is the extension of classical statistics can be applied in the industry for the lot senescing when observations or parameters are uncertain or indeterminate or unclear. In this manuscript, a new sampling plan for the measurement error using the neutrosophic statistics is designed. The proposed sampling plan has two neutrosophic parameters, namely, sample size and acceptance number. The neutrosophic operating function is also given. The neutrosophic plan parameters will be determined through the neutrosophic optimization problem. Some tables are given for some specified parameters. From the comparison study, it is concluded that the proposed sampling plan is more flexible, adequate, and effective in the uncertainty environment as compared to the existing sampling plan under the classical statistics. A real example is given for the illustration purpose.

The achieving of the high quality of the product at a low cost is desired during the production process. The sampling plans have been widely used for inspecting the indeterminate or finished product. Every inspection system is based on a well-designed sampling plan and has been widely used in the industry for the lot sentencing. Using any inspection system, there is a chance for accepting a nonconforming unit or rejecting a confirming unit. Authors in [

In practice, usually, the experimenters are not certain about the proportion of defective product. In this case, for lot sentencing of the product, an approach called the fuzzy sampling plans can be applied for inspection of the product. The fuzzy sampling plans have been widely used in the industry for various situations. Several authors contributed in this area, including, for example, [

The existing sampling plans with measurement error are designed using the classical statistics. The classical statistics assumes precision or determinate and crisp observations in the measurement process. It was mentioned by [

By exploring the literature and to the best of our knowledge, there is no work on the designing of sampling plan with measurement error using the neutrosophic statistics. In this manuscript, a new sampling plan for the measurement error using the neutrosophic statistics is designed. The proposed sampling plan has two neutrosophic parameters, namely, sample size and acceptance number. The neutrosophic operating function is also given. The neutrosophic plan parameters will be determined through the neutrosophic optimization problem. Some tables are given for some specified parameters. A real example is given for the illustration purpose.

Suppose that a neutrosophic random variable

Note here that

The quality of interest beyond the upper specification limit (U) or the lower control limit (L) is called the nonconforming item. We suppose that the quality of interest has some unclear or indeterminate observations. Based on the above information, we propose the following sampling plan

Specify

Take a random sample

Accept the lot of the product of

The proposed neutrosophic plan with measurement error has two neutrosophic plan parameters, namely, sample size

The neutrosophic operating function (NOF) of the proposed neutrosophic plan is derived in the following steps.

According to the operational process of the plan, the lot of product will be accepted if

The plan parameters when

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0.001 | 0.002 | | | | |

0.003 | | | | | |

0.004 | | | | | |

0.008 | | | | | |

0.01 | | | | | |

0.015 | | | | | |

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0.005 | 0.05 | | | | |

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0.03 | 0.06 | | | | |

0.09 | | | | | |

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0.05 | 0.1 | | | | |

0.15 | | | | |

The plan parameters when

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0.001 | 0.002 | | | | |

0.003 | | | | | |

0.004 | | | | | |

0.008 | | | | | |

0.01 | | | | | |

0.015 | | | | | |

0.02 | | | | | |

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0.005 | 0.05 | | | | |

0.1 | | | | | |

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0.01 | 0.02 | | | | |

0.03 | | | | | |

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0.03 | 0.06 | | | | |

0.09 | | | | | |

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0.05 | 0.1 | | | | |

0.15 | | | | |

The plan parameters when

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0.001 | 0.002 | | | | |

0.003 | | | | | |

0.004 | | | | | |

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0.01 | | | | | |

0.015 | | | | | |

0.02 | | | | | |

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0.005 | 0.05 | | | | |

0.1 | | | | | |

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0.01 | 0.02 | | | | |

0.03 | | | | | |

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0.03 | 0.06 | | | | |

0.09 | | | | | |

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0.05 | 0.1 | | | | |

0.15 | | | | |

From Tables

The comparison of the proposed sampling plan under the neutrosophic statistics is compared with the sampling plan proposed by [

The comparison of plans when

| | Proposed Plan | Existing Plan |
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0.001 | 0.002 | | 359 |

0.003 | | 308 | |

0.004 | | 209 | |

0.008 | | 76 | |

0.01 | | 43 | |

0.015 | | 38 | |

0.02 | | 19 | |

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0.005 | 0.05 | | 31 |

0.1 | | 17 | |

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0.01 | 0.02 | | 266 |

0.03 | | 90 | |

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0.03 | 0.06 | | 148 |

0.09 | | 84 | |

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0.05 | 0.1 | | 142 |

0.15 | | 53 |

In this section, we will discuss the application of the proposed sampling plan in the steel industry. In this industry, the measurement of tensile strength (X) is very difficult and costly. It is noted that the hardness (Y) is correlated with tensile strength and easy to study with low cost. As the data is obtained from the measurement process some observations are not precise or unclear. Furthermore, for the testing purpose, the experimenters are not sure about the random sample selection from a lot of the product. Therefore, in the testing of steel product, there is indeterminacy in variable of interest and plan parameters. So, the proposed plan can be applied when there is indeterminacy either in plan parameters or in observations or in both. Suppose that, for this testing,

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The required statistics is given by

Specify

Take a random sample

Reject the product as

In this manuscript, a new sampling plan for the measurement error using the neutrosophic statistics is designed. The neutrosophic plan parameters are determined and reported for the industrial application. The application of the proposed plan is shown using the steel data. The proposed plan can be applied in the industry such as in the steel industry and building testing material where the measurement data is not precise. The existing sampling plan, in this case, cannot be applied for the testing of material when indeterminacy is in the observations or parameters or both. The proposed sampling plan using other sampling schemes can be extended for future research.

The data is given in the paper.

The author declares no conflict of interest regarding this paper.

The authors are deeply thankful to the editor and the reviewers for their valuable suggestions to improve the quality of this manuscript. This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, Muhammad Aslam, therefore, acknowledge with thanks DSR technical support.