# Design of a New Variable Shewhart Control Chart Using Multiple Dependent State Repetitive Sampling

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## Abstract

**:**

## 1. Introduction

## 2. Design of Proposed Chart

#### 2.1. Measures for in-Control Process

#### 2.2. Measures for Out-of-Control Process

- For fixed values of all other parameters, ${\mathrm{ARL}}_{1}$ decreases as $\mathrm{i}$ increases from 2 to 3.
- For fixed values of all other parameters, ${\mathrm{ARL}}_{1}$ decreases as ${r}_{0}$ increases from 300 to 370.

## 3. Advantages of Proposed Chart

## 4. Industrial Application

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The [18] chart for simulated data.

$\mathit{A}\mathit{R}{\mathit{L}}_{\mathbf{0}}\mathbf{=}\mathbf{300}$ | ||||||

k_{1} | 2.9352 | 2.9352 | 2.9352 | 2.9352 | 2.9352 | 2.9352 |

k_{2} | 2.7865 | 2.7833 | 2.7919 | 2.7812 | 2.5991 | 2.6161 |

n | 5 | 10 | 20 | 30 | 40 | 50 |

c | ARL_{1} | |||||

0 | 300.00 | 300.00 | 300.01 | 300.00 | 300.00 | 300.00 |

0.01 | 288.30 | 282.96 | 277.10 | 269.77 | 225.11 | 222.28 |

0.02 | 276.24 | 265.44 | 253.13 | 239.19 | 175.50 | 170.96 |

0.05 | 239.51 | 213.87 | 184.82 | 159.02 | 94.43 | 88.13 |

0.1 | 182.86 | 143.06 | 103.74 | 78.02 | 41.06 | 35.77 |

0.15 | 137.06 | 94.69 | 58.83 | 39.95 | 20.27 | 16.70 |

0.2 | 102.38 | 63.35 | 34.61 | 21.78 | 10.98 | 8.72 |

0.25 | 76.79 | 43.17 | 21.24 | 12.66 | 6.48 | 5.05 |

0.3 | 58.06 | 30.04 | 13.61 | 7.84 | 4.14 | 3.22 |

0.4 | 34.18 | 15.51 | 6.33 | 3.60 | 2.12 | 1.71 |

0.5 | 21.00 | 8.72 | 3.46 | 2.07 | 1.40 | 1.22 |

0.6 | 13.47 | 5.32 | 2.19 | 1.44 | 1.13 | 1.05 |

0.7 | 9.01 | 3.52 | 1.58 | 1.17 | 1.03 | 1.01 |

0.8 | 6.28 | 2.50 | 1.28 | 1.06 | 1.01 | 1.00 |

0.9 | 4.56 | 1.90 | 1.12 | 1.02 | 1.00 | 1.00 |

1 | 3.44 | 1.54 | 1.05 | 1.00 | 1.00 | 1.00 |

$\mathit{A}\mathit{R}{\mathit{L}}_{\mathbf{0}}\mathbf{=}\mathbf{300}$ | ||||||

k_{1} | 2.9352 | 2.9352 | 2.9352 | 2.9352 | 2.9352 | 2.9352 |

k_{2} | 2.7467 | 2.7797 | 2.7912 | 2.7145 | 2.5794 | 2.6090 |

n | 5 | 10 | 20 | 30 | 40 | 50 |

c | ARL_{1} | |||||

0 | 300.00 | 300.00 | 300.00 | 300.00 | 300.00 | 300.00 |

0.01 | 284.79 | 282.54 | 276.98 | 257.00 | 220.62 | 220.58 |

0.02 | 269.85 | 264.70 | 252.95 | 219.69 | 170.06 | 168.94 |

0.05 | 227.73 | 212.66 | 184.57 | 137.75 | 90.35 | 86.70 |

0.1 | 169.13 | 141.91 | 103.57 | 66.37 | 39.26 | 35.20 |

0.15 | 125.11 | 93.86 | 58.73 | 34.30 | 19.47 | 16.47 |

0.2 | 92.98 | 62.80 | 34.56 | 18.98 | 10.60 | 8.62 |

0.25 | 69.70 | 42.81 | 21.21 | 11.20 | 6.29 | 5.00 |

0.3 | 52.78 | 29.80 | 13.59 | 7.04 | 4.04 | 3.20 |

0.4 | 31.27 | 15.41 | 6.33 | 3.33 | 2.08 | 1.70 |

0.5 | 19.35 | 8.67 | 3.46 | 1.96 | 1.39 | 1.22 |

0.6 | 12.50 | 5.30 | 2.19 | 1.40 | 1.13 | 1.05 |

0.7 | 8.42 | 3.50 | 1.58 | 1.15 | 1.03 | 1.01 |

0.8 | 5.92 | 2.49 | 1.28 | 1.05 | 1.01 | 1.00 |

0.9 | 4.32 | 1.90 | 1.12 | 1.01 | 1.00 | 1.00 |

1 | 3.28 | 1.54 | 1.05 | 1.00 | 1.00 | 1.00 |

$\mathit{A}\mathit{R}{\mathit{L}}_{\mathbf{0}}\mathbf{=}\mathbf{370}$ | ||||||

k_{1} | 2.9996 | 2.9996 | 2.9996 | 2.9996 | 2.9996 | 2.9997 |

k_{2} | 2.7784 | 2.7951 | 2.7591 | 2.7491 | 2.7632 | 2.6391 |

n | 5 | 10 | 20 | 30 | 40 | 50 |

c | ARL_{1} | |||||

0 | 370.00 | 370.00 | 370.00 | 370.00 | 370.00 | 370.00 |

0.01 | 346.38 | 339.78 | 319.86 | 307.08 | 302.49 | 257.71 |

0.02 | 324.14 | 311.40 | 277.17 | 256.48 | 248.23 | 191.57 |

0.05 | 265.41 | 238.15 | 183.39 | 154.64 | 141.26 | 94.73 |

0.1 | 190.88 | 152.18 | 97.22 | 72.72 | 60.63 | 37.77 |

0.15 | 138.65 | 98.79 | 54.57 | 37.16 | 28.76 | 17.49 |

0.2 | 101.93 | 65.54 | 32.17 | 20.38 | 14.92 | 9.07 |

0.25 | 75.87 | 44.47 | 19.85 | 11.93 | 8.43 | 5.21 |

0.3 | 57.16 | 30.87 | 12.79 | 7.44 | 5.17 | 3.31 |

0.4 | 33.60 | 15.88 | 6.03 | 3.47 | 2.45 | 1.74 |

0.5 | 20.66 | 8.90 | 3.33 | 2.02 | 1.53 | 1.23 |

0.6 | 13.26 | 5.42 | 2.13 | 1.42 | 1.18 | 1.06 |

0.7 | 8.89 | 3.57 | 1.55 | 1.16 | 1.05 | 1.01 |

0.8 | 6.20 | 2.53 | 1.26 | 1.05 | 1.01 | 1.00 |

0.9 | 4.51 | 1.92 | 1.11 | 1.01 | 1.00 | 1.00 |

1 | 3.40 | 1.55 | 1.05 | 1.00 | 1.00 | 1.00 |

$\mathit{A}\mathit{R}{\mathit{L}}_{\mathbf{0}}\mathbf{=}\mathbf{370}$ | ||||||

k_{1} | 2.9996 | 2.9996 | 2.9996 | 2.9997 | 2.9997 | 2.9997 |

k_{2} | 2.7569 | 2.7578 | 2.7089 | 2.6637 | 2.5650 | 2.5805 |

n | 5 | 10 | 20 | 30 | 40 | 50 |

c | ARL_{1} | |||||

0 | 370.00 | 370.00 | 370.00 | 370.00 | 370.00 | 370.00 |

0.01 | 343.64 | 333.49 | 308.33 | 284.36 | 245.12 | 239.94 |

0.02 | 319.36 | 300.96 | 260.24 | 226.10 | 178.94 | 172.41 |

0.05 | 257.47 | 223.12 | 165.07 | 127.63 | 89.32 | 82.64 |

0.1 | 182.55 | 139.49 | 86.02 | 59.17 | 38.12 | 33.10 |

0.15 | 131.80 | 90.09 | 48.43 | 30.66 | 18.92 | 15.58 |

0.2 | 96.70 | 59.84 | 28.78 | 17.14 | 10.34 | 8.22 |

0.25 | 71.98 | 40.76 | 17.92 | 10.23 | 6.15 | 4.80 |

0.3 | 54.28 | 28.42 | 11.66 | 6.51 | 3.97 | 3.10 |

0.4 | 32.02 | 14.77 | 5.59 | 3.14 | 2.06 | 1.67 |

0.5 | 19.76 | 8.36 | 3.14 | 1.89 | 1.38 | 1.20 |

0.6 | 12.74 | 5.13 | 2.04 | 1.36 | 1.12 | 1.05 |

0.7 | 8.57 | 3.41 | 1.51 | 1.14 | 1.03 | 1.01 |

0.8 | 6.01 | 2.44 | 1.24 | 1.04 | 1.01 | 1.00 |

0.9 | 4.38 | 1.87 | 1.10 | 1.01 | 1.00 | 1.00 |

1 | 3.32 | 1.52 | 1.04 | 1.00 | 1.00 | 1.00 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Aldosari, M.S.; Aslam, M.; Khan, N.; Jun, C.-H.
Design of a New Variable Shewhart Control Chart Using Multiple Dependent State Repetitive Sampling. *Symmetry* **2018**, *10*, 641.
https://doi.org/10.3390/sym10110641

**AMA Style**

Aldosari MS, Aslam M, Khan N, Jun C-H.
Design of a New Variable Shewhart Control Chart Using Multiple Dependent State Repetitive Sampling. *Symmetry*. 2018; 10(11):641.
https://doi.org/10.3390/sym10110641

**Chicago/Turabian Style**

Aldosari, Mansour Sattam, Muhammad Aslam, Nasrullah Khan, and Chi-Hyuck Jun.
2018. "Design of a New Variable Shewhart Control Chart Using Multiple Dependent State Repetitive Sampling" *Symmetry* 10, no. 11: 641.
https://doi.org/10.3390/sym10110641