# The Design of GLR Control Chart for Monitoring the Geometric Observations Using Sequential Sampling Scheme

^{*}

## Abstract

**:**

## 1. Introduction

_{0}), including susceptibility for capturing the parameter changes in an out-of-control situation.

## 2. The SS GLR Chart for Geometric Observations

- Calculate ${R}_{m,k,l}$.
- If ${R}_{m,k,l}$ ≤ g, then at sampling point k, we will stop sampling and await till sampling point $k+1$ to be sampled accordingly. So, here the sample range at k sampling point is ${n}_{k}=l$ = l.
- In case $\mathrm{g}<{R}_{m,k,l}<h$ then we have two options which need to be followed.
- (i)
- Draw further analysis at point $k$.
- (ii)
- Move to stage (1) as addressed above, with the existing set of data points at the k sampling point specified as $l=l+1$.

- When ${R}_{m,k,l}>h$, subsequently indicates that there has been a shift in the parameter $\theta $.

## 3. Performance Measures

#### 3.1. In-Control Performance Measurements

#### 3.2. Out-of-Control Performance Measurements

#### 3.3. The Extra Quadratic Loss

## 4. Choosing the Parameters of the SS GLR Chart

#### 4.1. The Window Size of the SS GLR Chart

#### 4.2. The Control Limits of SS GLR Chart

_{10}ANOS:

## 5. Performance Comparisons with Other Charts

#### 5.1. Comparison with the Geometric GLR Chart

#### 5.2. Comparison with the CUSUM Geometric Chart

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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SS GLR Charts, ASN = 1.5, d = 1.5 | |||||||
---|---|---|---|---|---|---|---|

m= | 1 | 4 | 10 | 15 | 50 | 100 | 250 |

δ | |||||||

1 | 1401.10 | 1401.10 | 1400.36 | 1400.22 | 1400.22 | 1400.05 | 1400.06 |

1.1 | 684.35 | 649.06 | 621.78 | 612.98 | 580.32 | 563.19 | 551.37 |

1.2 | 235.67 | 212.02 | 189.44 | 180.50 | 162.09 | 157.70 | 156.29 |

1.3 | 104.81 | 89.79 | 78.79 | 75.18 | 68.79 | 68.35 | 69.05 |

1.8 | 14.21 | 12.29 | 11.62 | 11.62 | 11.98 | 12.19 | 12.40 |

2 | 9.75 | 8.19 | 7.91 | 7.98 | 8.32 | 8.61 | 8.83 |

3 | 3.36 | 3.00 | 3.11 | 3.23 | 3.50 | 3.59 | 3.53 |

4 | 2.06 | 1.92 | 2.03 | 2.14 | 2.32 | 2.33 | 2.41 |

6 | 1.30 | 1.28 | 1.35 | 1.39 | 1.49 | 1.53 | 1.55 |

10 | 1.03 | 1.00 | 1.03 | 1.05 | 1.09 | 1.11 | 1.11 |

15 | 0.97 | 0.94 | 0.95 | 0.95 | 0.97 | 0.98 | 0.98 |

20 | 0.95 | 0.90 | 0.92 | 0.91 | 0.95 | 0.95 | 0.98 |

30 | 0.93 | 0.87 | 0.90 | 0.90 | 0.91 | 0.94 | 0.94 |

EQL= | 106.48 | 99.45 | 97.90 | 97.06 | 95.99 | 96.07 | 96.08 |

h= | 6.8373 | 6.8645 | 6.8773 | 6.8848 | 6.8853 | 6.8833 | 6.8763 |

g= | 0.9290 | 1.1990 | 1.3633 | 1.4313 | 1.5945 | 1.6668 | 1.7428 |

SS GLR Charts, ASN = 5, d = 5 | |||||||
---|---|---|---|---|---|---|---|

m= | 1 | 5 | 10 | 20 | 50 | 100 | 150 |

δ | |||||||

1 | 1401.33 | 1401.28 | 1401.23 | 1401.28 | 1401.10 | 1401.28 | 1401.55 |

1.1 | 361.71 | 351.00 | 345.32 | 343.02 | 343.42 | 343.10 | 343.92 |

1.2 | 128.78 | 108.02 | 104.86 | 103.55 | 105.14 | 105.21 | 103.47 |

1.3 | 74.96 | 51.56 | 51.73 | 54.12 | 56.24 | 54.43 | 53.77 |

1.8 | 13.91 | 13.83 | 13.38 | 15.47 | 15.21 | 14.70 | 14.57 |

2 | 15.31 | 9.53 | 10.25 | 12.01 | 12.63 | 12.82 | 12.55 |

3 | 6.85 | 4.64 | 4.42 | 4.24 | 4.17 | 4.29 | 4.16 |

4 | 3.33 | 3.27 | 3.13 | 3.41 | 3.41 | 3.49 | 3.31 |

6 | 2.72 | 2.28 | 2.23 | 2.25 | 2.38 | 2.29 | 2.38 |

10 | 2.59 | 2.06 | 1.98 | 2.03 | 2.11 | 2.04 | 2.11 |

15 | 2.49 | 1.98 | 1.93 | 1.95 | 2.03 | 2.03 | 2.03 |

20 | 2.45 | 1.97 | 1.88 | 1.91 | 2.00 | 2.00 | 2.00 |

30 | 2.40 | 1.86 | 1.81 | 1.89 | 1.92 | 1.99 | 1.99 |

EQL= | 173.83 | 139.48 | 135.76 | 139.33 | 142.96 | 144.48 | 144.54 |

h= | 5.6545 | 5.6016 | 5.5997 | 5.5937 | 5.5902 | 5.5897 | 5.5890 |

g= | 0.3373 | 0.4309 | 0.4545 | 0.4804 | 0.5069 | 0.5151 | 0.5238 |

SS Geometric GLR Chart | Geometric GLR Chart | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

ASN = n = 5 | ||||||||||

m= | 1 | 5 | 10 | 50 | 100 | 100 | 160 | 180 | 200 | 300 |

δ | ||||||||||

1 | 1401.33 | 1401.28 | 1401.23 | 1401.1 | 1401.28 | 1400.6 | 1400.45 | 1400.05 | 1400.9 | 1400.9 |

1.1 | 361.71 | 351 | 345.32 | 343.42 | 343.1 | 421.35 | 353.6 | 356.3 | 355.7 | 354.05 |

1.2 | 128.78 | 108.02 | 104.86 | 105.14 | 105.21 | 174.95 | 175.35 | 175.45 | 175.6 | 175.75 |

1.3 | 74.96 | 51.56 | 51.73 | 56.24 | 54.43 | 129.1 | 127.3 | 128.5 | 128.9 | 129 |

1.8 | 13.91 | 13.83 | 13.38 | 15.21 | 14.7 | 30.95 | 31.05 | 31.55 | 31.7 | 31.8 |

2 | 15.31 | 9.53 | 10.25 | 12.63 | 12.82 | 24.25 | 24.35 | 24.55 | 24.55 | 24.65 |

3 | 6.85 | 4.64 | 4.42 | 4.17 | 4.29 | 10.85 | 10.9 | 10.8 | 10.85 | 10.9 |

4 | 3.33 | 3.27 | 3.13 | 3.41 | 3.49 | 7.95 | 7.95 | 7.95 | 8 | 8.05 |

6 | 2.72 | 2.28 | 2.23 | 2.38 | 2.29 | 5.55 | 5.65 | 5.65 | 5.65 | 5.7 |

10 | 2.59 | 2.06 | 1.98 | 2.11 | 2.04 | 3.55 | 3.55 | 3.55 | 3.6 | 3.6 |

15 | 2.49 | 1.98 | 1.93 | 2.03 | 2.03 | 2.55 | 2.55 | 2.55 | 2.55 | 2.55 |

20 | 2.45 | 1.97 | 1.88 | 2 | 2 | 2.55 | 2.55 | 2.55 | 2.55 | 2.55 |

30 | 2.4 | 1.86 | 1.81 | 1.92 | 1.99 | 2.5 | 2.5 | 2.5 | 2.5 | 2.5 |

EQL= | 173.83 | 139.48 | 135.76 | 142.96 | 144.48 | 248.31 | 245.56 | 245.79 | 246.05 | 246.13 |

h= | 5.6545 | 5.6016 | 5.5997 | 5.5902 | 5.5897 | 5.576 | 5.592 | 5.608 | 5.624 | 5.64 |

g= | 0.3373 | 0.4309 | 0.4545 | 0.5069 | 0.5151 | - | - | - | - | - |

SS Geometric GLR Chart | CUSUM Geometric Chart | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

ASN = n = 5 | ||||||||||

m= | 1 | 5 | 10 | 50 | 100 | δ_{1} = 1.1 | 1.5 | 1.7 | 1.8 | 2 |

δ | ||||||||||

1 | 1401.33 | 1401.28 | 1401.23 | 1401.1 | 1401.28 | 1400.4 | 1400.45 | 1400.1 | 1400.7 | 1400.1 |

1.2 | 128.78 | 108.02 | 104.86 | 105.14 | 105.21 | 148.6 | 135.15 | 167 | 198.65 | 227.25 |

1.3 | 74.96 | 51.56 | 51.73 | 56.24 | 54.43 | 89.2 | 92 | 113.6 | 121.4 | 146.95 |

1.8 | 13.91 | 13.83 | 13.38 | 15.21 | 14.7 | 37.2 | 27.2 | 25.6 | 25.85 | 27.3 |

2 | 15.31 | 9.53 | 10.25 | 12.63 | 12.82 | 31.35 | 22.05 | 20.3 | 19.7 | 20.4 |

3 | 6.85 | 4.64 | 4.42 | 4.17 | 4.29 | 22.3 | 13.55 | 12.1 | 11.35 | 10.5 |

4 | 3.33 | 3.27 | 3.13 | 3.41 | 3.49 | 18.95 | 11.7 | 10.15 | 9.55 | 8.8 |

6 | 2.72 | 2.28 | 2.23 | 2.38 | 2.29 | 17.6 | 10.05 | 7.8 | 7.2 | 6.8 |

10 | 2.59 | 2.06 | 1.98 | 2.11 | 2.04 | 15.4 | 9.85 | 6.9 | 6.9 | 6.75 |

15 | 2.49 | 1.98 | 1.93 | 2.03 | 2.03 | 16.45 | 9.8 | 7.1 | 7.05 | 7.05 |

20 | 2.45 | 1.97 | 1.88 | 2 | 2 | 15.2 | 8.95 | 7 | 7.05 | 6.85 |

30 | 2.4 | 1.86 | 1.81 | 1.92 | 1.99 | 14.65 | 7.2 | 6.75 | 6.8 | 6.7 |

h= | 5.6545 | 5.6016 | 5.5997 | 5.5902 | 5.5897 | 2.112 | 3.864 | 4 | 4.096 | 4.264 |

g= | 0.3373 | 0.4309 | 0.4545 | 0.5069 | 0.5151 | - | - | - | - | - |

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**MDPI and ACS Style**

Shahzad, F.; Huang, Z.; Shafqat, A.
The Design of GLR Control Chart for Monitoring the Geometric Observations Using Sequential Sampling Scheme. *Symmetry* **2020**, *12*, 1964.
https://doi.org/10.3390/sym12121964

**AMA Style**

Shahzad F, Huang Z, Shafqat A.
The Design of GLR Control Chart for Monitoring the Geometric Observations Using Sequential Sampling Scheme. *Symmetry*. 2020; 12(12):1964.
https://doi.org/10.3390/sym12121964

**Chicago/Turabian Style**

Shahzad, Faisal, Zhensheng Huang, and Ambreen Shafqat.
2020. "The Design of GLR Control Chart for Monitoring the Geometric Observations Using Sequential Sampling Scheme" *Symmetry* 12, no. 12: 1964.
https://doi.org/10.3390/sym12121964