# EWMA Control Chart Using Repetitive Sampling for Monitoring Blood Glucose Levels in Type-II Diabetes Patients

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

_{t}; t = 1, 2, …, which follows the normal distribution with mean μ and variance σ

^{2}. Based on this assumption, we propose the following steps in the repetitive sampling EWMA chart [25]:

_{t}≥ UCL

_{1}or EWMA

_{t}≤ LCL

_{1}. Declare the process as in-control if LCL

_{2}≤ EWMA

_{t}≤ UCL

_{2}. Otherwise, go to Step 1 and repeat the process.

_{1}, UCL

_{2}and LCL

_{1}, LCL

_{2}are given as:

_{1}and k

_{2}are control chart coefficients, $\overline{X}$ is the sample mean, and s is the sample standard deviation.

_{0}) is given by Equation (9), which is the expected number of subgroups to be examined until the process is declared to be out of control when the process is truly in control:

_{1}) is obtained as follows:

_{0}be the assumed in-control ARL. We estimated the control constants k

_{1}and k

_{2}using Monte Carlo simulation such that $AR{L}_{0}\ge {r}_{0}.$ We noted first out-of-control (run length) and repeated the process 10,000 times. We developed programming to obtain the estimates of control constants. The program is available with authors upon request. Then using Equation (13), we obtain ARL

_{1}based on the determined values of k

_{1}and k

_{2}for various shift values of ${\mu}_{1}=\mu +c\sigma $. From Table 1, Table 2 and Table 3, wherein the ARL for r

_{0}= 370, c = 0 to 1.0, and $\lambda $ = 0.10, 0.20, and 0.30, respectively, we observe the following behavior of ARL

_{1}:

- The case of $\mu =0,\sigma =1$, that is when the process is in-control, ARL value obtained is very close to the target r
_{0}values. - As the shift $c$ increases (i.e., the process mean increases), the out-of-control ARLs decrease rapidly. A similar trend can be observed from Table 2 and Table 3 whereas decreasing speed seems to get faster after c = 0.1. When sample size increases, the values of ARL
_{1}decrease. It means that at the large sample size, we have a quick indication about the shift in the sugar level (see Figure 1).

## 3. Case Study Results

## 4. Conclusions and Recommendations

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The proposed chart ARL1 performance for different values of n at $\lambda $ = 0.10 and ${r}_{0}=370$.

c | k_{1} = 3.0066; k_{2} = 2.2356 | |||||
---|---|---|---|---|---|---|

n | ||||||

5 | 10 | 20 | 30 | 50 | 100 | |

ARL_{1} | ||||||

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

0.01 | 353.12 | 337.55 | 309.80 | 285.82 | 246.48 | 180.10 |

0.02 | 309.80 | 264.89 | 202.54 | 161.53 | 111.35 | 56.53 |

0.03 | 255.40 | 190.77 | 121.26 | 85.18 | 49.32 | 19.43 |

0.05 | 157.38 | 91.57 | 43.43 | 25.36 | 11.59 | 3.56 |

0.08 | 71.94 | 31.58 | 11.15 | 5.60 | 2.39 | 1.15 |

0.1 | 43.43 | 16.54 | 5.22 | 2.64 | 1.37 | 1.02 |

0.15 | 13.75 | 4.26 | 1.53 | 1.12 | 1.01 | 1.00 |

0.2 | 5.22 | 1.75 | 1.06 | 1.01 | 1.00 | 1.00 |

0.25 | 2.48 | 1.17 | 1.00 | 1.00 | 1.00 | 1.00 |

0.3 | 1.53 | 1.03 | 1.00 | 1.00 | 1.00 | 1.00 |

0.4 | 1.06 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

0.5 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

c | k_{1} = 3.0134; k_{2} = 1.9885 | |||||
---|---|---|---|---|---|---|

n | ||||||

5 | 10 | 20 | 30 | 50 | 100 | |

ARL_{1} | ||||||

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

0.01 | 369.06 | 368.13 | 366.28 | 364.45 | 360.83 | 352.05 |

0.02 | 366.28 | 362.63 | 355.52 | 348.65 | 335.58 | 306.40 |

0.03 | 361.73 | 353.78 | 338.77 | 324.84 | 299.79 | 249.85 |

0.05 | 347.81 | 327.85 | 293.42 | 264.81 | 220.03 | 150.24 |

0.08 | 317.53 | 276.78 | 217.76 | 177.24 | 125.62 | 66.17 |

0.1 | 293.42 | 240.66 | 173.06 | 131.95 | 85.20 | 38.94 |

0.15 | 229.96 | 160.99 | 94.11 | 62.43 | 33.31 | 11.56 |

0.2 | 173.06 | 104.58 | 51.38 | 30.48 | 14.06 | 4.19 |

0.25 | 127.94 | 67.88 | 28.79 | 15.61 | 6.52 | 2.00 |

0.3 | 94.11 | 44.52 | 16.66 | 8.44 | 3.40 | 1.32 |

0.4 | 51.38 | 20.02 | 6.25 | 3.04 | 1.47 | 1.03 |

0.5 | 28.79 | 9.65 | 2.84 | 1.58 | 1.09 | 1.00 |

0.6 | 16.66 | 5.06 | 1.66 | 1.16 | 1.01 | 1.00 |

0.7 | 10.00 | 2.94 | 1.23 | 1.04 | 1.00 | 1.00 |

0.8 | 6.25 | 1.94 | 1.08 | 1.01 | 1.00 | 1.00 |

0.9 | 4.10 | 1.45 | 1.02 | 1.00 | 1.00 | 1.00 |

1 | 2.84 | 1.22 | 1.01 | 1.00 | 1.00 | 1.00 |

c | k_{1} = 3.0105; k_{2} = 2.0796 | |||||
---|---|---|---|---|---|---|

n | ||||||

5 | 10 | 20 | 30 | 50 | 100 | |

ARL_{1} | ||||||

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

0.01 | 361.77 | 353.86 | 338.92 | 325.05 | 300.11 | 250.36 |

0.02 | 338.92 | 312.15 | 268.43 | 234.28 | 184.55 | 115.17 |

0.03 | 306.03 | 259.12 | 195.20 | 153.91 | 104.29 | 51.39 |

0.05 | 230.54 | 161.76 | 94.95 | 63.23 | 33.98 | 11.98 |

0.08 | 136.81 | 74.79 | 32.92 | 18.30 | 7.87 | 2.38 |

0.1 | 94.95 | 45.26 | 17.17 | 8.80 | 3.59 | 1.36 |

0.15 | 39.02 | 14.24 | 4.31 | 2.20 | 1.24 | 1.01 |

0.2 | 17.17 | 5.31 | 1.73 | 1.19 | 1.02 | 1.00 |

0.25 | 8.20 | 2.47 | 1.16 | 1.02 | 1.00 | 1.00 |

0.3 | 4.31 | 1.51 | 1.03 | 1.00 | 1.00 | 1.00 |

0.4 | 1.73 | 1.06 | 1.00 | 1.00 | 1.00 | 1.00 |

0.5 | 1.16 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

0.6 | 1.03 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

Week | Blood Sugar Level (mg/dL) | $\overline{\mathit{X}}$ | $\mathit{E}\mathit{W}\mathit{M}{\mathit{A}}_{\mathit{t}}$ | Week | Blood Sugar Level (mg/dL) | $\overline{\mathit{X}}$ | $\mathit{E}\mathit{W}\mathit{M}{\mathit{A}}_{\mathit{t}}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 370 | 175 | 193 | 192 | 197 | 225.4 | 197.6 | 21 | 149 | 157 | 126 | 160 | 137 | 145.8 | 174.0 |

2 | 313 | 255 | 170 | 294 | 203 | 247.0 | 207.5 | 22 | 132 | 203 | 229 | 184 | 123 | 174.2 | 174.0 |

3 | 270 | 205 | 190 | 203 | 194 | 212.4 | 208.5 | 23 | 126 | 190 | 237 | 187 | 139 | 175.8 | 174.4 |

4 | 190 | 221 | 177 | 173 | 171 | 186.4 | 204.1 | 24 | 143 | 204 | 200 | 245 | 187 | 195.8 | 178.6 |

5 | 185 | 242 | 278 | 202 | 189 | 219.2 | 207.1 | 25 | 117 | 219 | 170 | 197 | 158 | 172.2 | 177.4 |

6 | 190 | 228 | 184 | 165 | 268 | 207.0 | 207.1 | 26 | 114 | 201 | 264 | 169 | 178 | 185.2 | 178.9 |

7 | 177 | 166 | 173 | 224 | 234 | 194.8 | 204.6 | 27 | 122 | 179 | 235 | 167 | 226 | 185.8 | 180.3 |

8 | 175 | 239 | 268 | 198 | 176 | 211.2 | 205.9 | 28 | 134 | 213 | 182 | 137 | 269 | 187.0 | 181.6 |

9 | 165 | 176 | 196 | 201 | 246 | 196.8 | 204.1 | 29 | 132 | 284 | 180 | 207 | 235 | 207.6 | 186.8 |

10 | 183 | 150 | 243 | 188 | 172 | 187.2 | 200.7 | 30 | 110 | 246 | 110 | 272 | 117 | 171.0 | 183.7 |

11 | 185 | 165 | 164 | 188 | 231 | 186.6 | 197.9 | 31 | 107 | 234 | 212 | 201 | 141 | 179.0 | 182.7 |

12 | 177 | 189 | 178 | 186 | 186 | 183.2 | 195.0 | 32 | 125 | 220 | 225 | 113 | 214 | 179.4 | 182.1 |

13 | 165 | 274 | 248 | 183 | 179 | 209.8 | 197.9 | 33 | 105 | 190 | 196 | 187 | 252 | 186.0 | 182.9 |

14 | 169 | 177 | 159 | 269 | 207 | 196.2 | 197.6 | 34 | 107 | 232 | 209 | 257 | 225 | 206.0 | 187.5 |

15 | 170 | 218 | 197 | 140 | 186 | 182.2 | 194.5 | 35 | 116 | 234 | 241 | 214 | 182 | 197.4 | 189.5 |

16 | 155 | 170 | 206 | 155 | 176 | 172.4 | 190.1 | 36 | 118 | 189 | 194 | 183 | 164 | 169.6 | 185.5 |

17 | 160 | 231 | 228 | 220 | 241 | 216.0 | 195.3 | 37 | 116 | 207 | 271 | 213 | 219 | 205.2 | 189.4 |

18 | 152 | 161 | 179 | 162 | 168 | 164.4 | 189.1 | 38 | 105 | 173 | 179 | 226 | 165 | 169.6 | 185.5 |

19 | 162 | 173 | 111 | 153 | 200 | 159.8 | 183.2 | 39 | 108 | 215 | 246 | 259 | 236 | 212.8 | 190.9 |

20 | 165 | 196 | 173 | 168 | 158 | 172.0 | 181.0 | 40 | 109 | 281 | 134 | 200 | 232 | 191.2 | 191.0 |

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## Share and Cite

**MDPI and ACS Style**

Aslam, M.; Rao, G.S.; Khan, N.; Al-Abbasi, F.A.
EWMA Control Chart Using Repetitive Sampling for Monitoring Blood Glucose Levels in Type-II Diabetes Patients. *Symmetry* **2019**, *11*, 57.
https://doi.org/10.3390/sym11010057

**AMA Style**

Aslam M, Rao GS, Khan N, Al-Abbasi FA.
EWMA Control Chart Using Repetitive Sampling for Monitoring Blood Glucose Levels in Type-II Diabetes Patients. *Symmetry*. 2019; 11(1):57.
https://doi.org/10.3390/sym11010057

**Chicago/Turabian Style**

Aslam, Muhammad, Gadde Srinivasa Rao, Nasrullah Khan, and Fahad A. Al-Abbasi.
2019. "EWMA Control Chart Using Repetitive Sampling for Monitoring Blood Glucose Levels in Type-II Diabetes Patients" *Symmetry* 11, no. 1: 57.
https://doi.org/10.3390/sym11010057