# Shewhart Attribute and Variable Control Charts Using Modified Multiple Dependent State Sampling

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Attribute Chart Using Modified Multiple Dependent State (MDS) Sampling

**Step-1:**Count the number of defectives $D$ from $n$.

**Step-2:**If $LC{L}_{2}\le D\le UC{L}_{2}$, the process is in-control. If $D>UC{L}_{1}$ or $D<LC{L}_{1}$ out-of-control, otherwise, go to Step-3.

**Step-3:**Declare the process is in control if m proceeding subgroups are in-control except in one sample where $UC{L}_{2}\le D\le UC{L}_{1}$ or $LC{L}_{1}\le D\le LC{L}_{2}$.

#### 2.1. In-Control Process

_{0}) when $p={p}_{0}$ is finally obtained by

#### 2.2. Shifted (or Out-Of-Control) Process

_{1}) when $p={p}_{1}$ is finally obtained as

_{0}denoted by r

_{0}is obtained as the average number of samples when the false alarm indicates for the specified values of the MDS sample parameter i. Here, it is to be noted that the larger values of the ARL

_{0}are recommended as the process is in a situation of in-control, and no change occurs. For the specific values of ARL

_{0}= 200, 300, and 370, the values of $AR{L}_{1}$ have been computed by running the R-code program using the simulation approach. This type of simulation approach is employed when the mean and other measures of the proposed chart are unavailable. Several researchers of the statistical process control used the simulation approach for numerical calculations of the proposed methodology including, for example, [17].

## 3. ARLs of the Attribute Chart Using Modified MDS Sampling

_{1}is going to decrease. This means that the larger shifts are addressed quickly, for example a shift of size 1.01 is detected with 188.49 samples on the average for $i$ = 2, r

_{0}= 200 and ${p}_{0}=0.01$, while a shift of 1.80 is detected with 2.52 samples for the same process settings. The same pattern can be observed for ${p}_{0}$ = 0.05 and ${p}_{0}$ = 0.10.

## 4. Comparison of the Proposed Chart with Existing MDS chart

_{1}values from the proposed chart and [9], the results are reported in Table 4 for $i=3$ and $AR{L}_{0}$ = 300. A control chart having the smaller values of ARL

_{1}of the same parameters is considered as the more efficient chart. From Table 4, it can be noted that the proposed control chart has smaller ARL

_{1}as compared to the [9] chart. For instance, a shift of 1.20 is detected by an average of 50 samples, while the existing chart detects the same shift with an average of 43 samples. Figure 1 is presented for $i$ = 2, $AR{L}_{0}$ = 370 and ${p}_{0}=0.10$. From Figure 1, it is clear that the ARL curve for $n=100$ is higher than the ARL curve when $n=130$.

#### 4.1. Simulation Study

_{0}= 0.10. Then, the next 20 observations were generated from the shifted process with n = 205, p

_{1}= c*p

_{0}where c = 1.25 from Table 2, ARL

_{1}was 17.91, and the shift was detected on observation 17. The simulated data were 13, 25, 24, 21, 20, 19, 19, 22, 23, 22, 16, 26, 13, 19, 21, 24, 18, 18, 16, 18, 24, 32, 38, 18, 29, 20, 27, 22, 28, 28, 32, 25, 30, 25, 42, 24, 23, 24, 23, and 24. Figure 1 shows the simulated data of 40 observations, in which the process indicates an out-of-control situation after 20 + 15 = 35 samples. Thus, the proposed chart is effective in detecting a shift of 1.25 after 15 samples (a value of ARL

_{1}from Table 3 for r

_{0}= 370 and $i$ = 2). To compare the proposed control chart with the existing chart by [9], the values of statistic ${D}_{i}$ are also plotted on the control chart in Figure 2. By comparing Figure 2 with Figure 3, it can be noted that the existing control chart by [9] does not have the ability to detect the shift in the process. Figure 4 shows that the process is in-control. Similarly, the values of statistic ${D}_{i}$ are also plotted on the Shewhart control chart in Figure 4. From Figure 4, it can be observed that the Shewhart control chart does not detect the shift in the process. Therefore, the current chart is more efficient than two existing control charts in detecting quick shift in the manufacturing process.

#### 4.2. An Industrial Example

_{1}= 4.340957 and k

_{2}= 3.092937, and the control chart limits are calculated as LCL

_{1}= 0, LCL

_{2}= 4, UCL

_{2}= 27, and UCL

_{1}= 32. From Figure 5, it can be noted that although the process is in a state of control, the 3rd, 9th and 10th sample are near the control limits, which may cause the shift in the process.

## 5. Variable Chart Using Modified MDS Sampling

**Step-1:**Compute $\overline{X}$ from subgroup size $n$.

**Step-2:**If ${\mathit{LCL}}_{2}\le \overline{X}\le {\mathit{UCL}}_{2}$, the process is in-control. If $\overline{X}>UC{L}_{1}$ or $\overline{X}<LC{L}_{1}$ the process is out-of-control, otherwise, go to Step-3.

**Step-3:**Declare the process is in control if m proceeding subgroups declared the process as in-control except in one sample where $UC{L}_{2}<\overline{X}\le UC{L}_{1}$ or $LC{L}_{1}\le \overline{X}<LC{L}_{2}$.

#### 5.1. In-Control Process

#### 5.2. Shifted (or Out-Of-Control) Process

## 6. ARLs of Variable Chart Using Modified MDS Sampling

_{1}is going to decrease. This means that the larger shifts are addressed quickly, for example a shift of size 0.05 is detected with 188.89 samples on average for $i$ = 2, r

_{0}= 200, while a shift of 0.80 is detected with 4.25 samples for the same process settings. The same pattern can be observed for $n$ = 10.

## 7. Comparison of the Proposed Chart with Existing MDS Chart

_{1}values from the proposed chart and [8], the results are reported in Table 4 for $i=3$ and $AR{L}_{0}$ = 300. A control chart having the smaller values of ARL

_{1}of the same parameters is considered as the more efficient chart. From Table 4, it can be noted that the proposed control chart has a smaller ARL

_{1}as compared to the [8] chart. For instance, a shift of 1.20 is detected by an average of 109 samples, while the existing chart detects the same shift with an average of 89 samples. Figure 6 is presented for $i$ = 2 and $AR{L}_{0}$ = 370. From Figure 6, it is clear that the ARL curve for $n=5$ is higher than the ARL curve when $n=10$.

#### 7.1. Simulation Study

_{1}was 17.97, and the shift is detected on observation 17. The simulated data are not given here due to short space. Figure 7 shows the simulated data of 40 observations in which the process indicates an out-of-control situation after 20 + 17 = 37 samples. Thus, the proposed chart is effective in detecting a shift of 0.40 after 17 samples (a value of ARL

_{1}from Table 6 for r

_{0}= 200 and $i$ = 3).

#### 7.2. An industrial Example

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Control chart for the proposed chart using the simulated data for p

_{0}= 0.10 and r

_{0}= 370.

**Figure 3.**Control chart for the MDS sampling using the simulated data for p

_{0}= 0.10 and r

_{0}= 370.

**Figure 4.**Shewhart control chart for the single chart using the simulated data for p

_{0}= 0.10 and r

_{0}= 370.

**Figure 5.**Attribute control chart for the example [1].

**Figure 9.**Shewhart Control chart for the Single chart using the simulated data for n = 10 and r

_{0}= 370.

**Figure 10.**Variable control chart for the data given in [1].

${\mathit{r}}_{0}$ | $\mathit{i}$ = 2 | $\mathit{i}$ = 3 | ||||
---|---|---|---|---|---|---|

201.7800 | 301.5360 | 370.6202 | 200.9965 | 305.2319 | 373.4875 | |

${k}_{1}$ | 4.8498 | 3.4960 | 3.6292 | 5.1498 | 3.7202 | 3.9085 |

${k}_{2}$ | 2.9614 | 3.4723 | 3.5639 | 3.6066 | 3.4506 | 3.4507 |

n | 810 | 900 | 880 | 680 | 920 | 900 |

Shift | $AR{L}_{1}$ | |||||

1.00 | 201.78 | 301.54 | 370.62 | 201.00 | 305.23 | 373.49 |

1.01 | 188.49 | 273.78 | 339.37 | 209.70 | 276.88 | 342.11 |

1.03 | 163.20 | 224.71 | 281.99 | 225.50 | 226.49 | 283.89 |

1.05 | 140.34 | 183.98 | 232.62 | 237.95 | 184.58 | 233.43 |

1.07 | 120.23 | 150.71 | 191.33 | 245.73 | 150.38 | 191.13 |

1.10 | 95.16 | 112.31 | 142.83 | 246.62 | 111.07 | 141.53 |

1.13 | 75.49 | 84.47 | 107.28 | 234.44 | 82.76 | 105.36 |

1.15 | 64.88 | 70.29 | 89.09 | 220.41 | 68.43 | 86.97 |

1.17 | 55.92 | 58.79 | 74.32 | 203.29 | 56.88 | 72.11 |

1.20 | 45.01 | 45.42 | 57.16 | 174.96 | 43.54 | 54.96 |

1.25 | 31.90 | 30.32 | 37.84 | 129.61 | 28.64 | 35.85 |

1.30 | 23.11 | 20.89 | 25.83 | 93.29 | 19.46 | 24.14 |

1.40 | 12.90 | 10.83 | 13.14 | 48.19 | 9.84 | 11.97 |

1.50 | 7.76 | 6.25 | 7.43 | 26.04 | 5.57 | 6.63 |

1.70 | 3.45 | 2.74 | 3.12 | 9.04 | 2.40 | 2.73 |

1.80 | 2.52 | 2.05 | 2.28 | 5.81 | 1.80 | 2.00 |

2.00 | 1.60 | 1.38 | 1.48 | 2.85 | 1.25 | 1.33 |

${\mathit{r}}_{0}$ | $\mathit{i}$ = 2 | $\mathit{i}$ = 3 | ||||
---|---|---|---|---|---|---|

200.4457 | 301.5679 | 371.4451 | 200.1407 | 300.1609 | 370.1112 | |

${k}_{1}$ | 3.5377 | 4.5655 | 4.3772 | 6.1016 | 3.5814 | 5.6740 |

${k}_{2}$ | 2.9214 | 2.9908 | 3.1114 | 2.9065 | 3.0666 | 3.1167 |

n | 290 | 275 | 270 | 225 | 395 | 315 |

Shift | $AR{L}_{1}$ | |||||

1.00 | 200.45 | 301.57 | 371.45 | 200.14 | 300.16 | 370.11 |

1.01 | 177.09 | 266.99 | 328.98 | 180.06 | 259.69 | 324.69 |

1.03 | 138.23 | 209.93 | 258.33 | 146.05 | 193.00 | 250.51 |

1.05 | 108.23 | 165.99 | 203.62 | 118.95 | 143.30 | 194.32 |

1.07 | 85.20 | 132.11 | 161.37 | 97.33 | 106.92 | 151.76 |

1.10 | 60.28 | 95.04 | 115.23 | 72.75 | 69.98 | 106.19 |

1.13 | 43.37 | 69.47 | 83.54 | 55.03 | 46.82 | 75.56 |

1.15 | 35.17 | 56.86 | 67.99 | 45.98 | 36.27 | 60.76 |

1.17 | 28.73 | 46.85 | 55.71 | 38.61 | 28.39 | 49.20 |

1.20 | 21.52 | 35.47 | 41.82 | 29.98 | 20.04 | 36.29 |

1.25 | 13.79 | 22.99 | 26.73 | 20.12 | 11.78 | 22.51 |

1.30 | 9.23 | 15.43 | 17.71 | 13.87 | 7.35 | 14.47 |

1.40 | 4.68 | 7.65 | 8.57 | 7.12 | 3.40 | 6.60 |

1.50 | 2.76 | 4.27 | 4.69 | 4.05 | 1.96 | 3.46 |

1.70 | 1.43 | 1.87 | 1.98 | 1.81 | 1.13 | 1.50 |

1.80 | 1.20 | 1.44 | 1.50 | 1.42 | 1.04 | 1.21 |

2.00 | 1.04 | 1.10 | 1.12 | 1.09 | 1.00 | 1.03 |

${\mathit{r}}_{0}$ | $\mathit{i}$ = 2 | $\mathit{i}$ = 3 | ||||
---|---|---|---|---|---|---|

200.8375 | 301.9349 | 370.9957 | 200.0664 | 300.0649 | 374.3649 | |

${k}_{1}$ | 6.0760 | 3.7269 | 4.9422 | 3.9412 | 3.3411 | 3.6377 |

${k}_{2}$ | 2.7771 | 3.4279 | 2.9897 | 2.9031 | 3.3107 | 3.4403 |

n | 100 | 85 | 205 | 290 | 370 | 140 |

Shift | $AR{L}_{1}$ | |||||

1.00 | 200.84 | 301.93 | 371.00 | 200.07 | 300.06 | 374.36 |

1.01 | 182.99 | 275.08 | 327.06 | 177.38 | 249.31 | 322.69 |

1.03 | 151.61 | 226.98 | 250.60 | 134.45 | 167.37 | 241.28 |

1.05 | 125.60 | 186.56 | 190.56 | 99.38 | 111.46 | 182.06 |

1.07 | 104.27 | 153.23 | 145.03 | 72.94 | 74.92 | 138.68 |

1.10 | 79.38 | 114.47 | 97.35 | 46.14 | 42.58 | 93.83 |

1.13 | 61.04 | 86.18 | 66.57 | 29.74 | 25.26 | 64.82 |

1.15 | 51.54 | 71.72 | 52.25 | 22.48 | 18.27 | 51.22 |

1.17 | 43.72 | 59.97 | 41.39 | 17.17 | 13.47 | 40.83 |

1.20 | 34.48 | 46.28 | 29.66 | 11.71 | 8.83 | 29.52 |

1.25 | 23.76 | 30.80 | 17.74 | 6.56 | 4.81 | 17.91 |

1.30 | 16.83 | 21.13 | 11.14 | 3.97 | 2.93 | 11.40 |

1.40 | 9.11 | 10.83 | 5.02 | 1.88 | 1.51 | 5.30 |

1.50 | 5.39 | 6.17 | 2.70 | 1.24 | 1.11 | 2.93 |

1.70 | 2.42 | 2.65 | 1.30 | 1.01 | 1.00 | 1.41 |

1.80 | 1.82 | 1.96 | 1.11 | 1.00 | 1.00 | 1.17 |

2.00 | 1.27 | 1.32 | 1.01 | 1.00 | 1.00 | 1.02 |

$\mathit{n}=920$ | $\mathit{n}=88$ | |||
---|---|---|---|---|

Existing MDS Chart | Proposed Chart | Existing MDS Chart | Proposed Chart | |

Shift | $\mathit{A}\mathit{R}{\mathit{L}}_{1}$ | $\mathit{A}\mathit{R}{\mathit{L}}_{1}$ | $\mathit{A}\mathit{R}{\mathit{L}}_{1}$ | |

1.00 | 301.13 | 305.23 | 372.32 | 300.10 |

1.01 | 273.31 | 276.88 | 355.19 | 271.30 |

1.03 | 225.13 | 226.49 | 316.12 | 219.32 |

1.05 | 185.76 | 184.58 | 274.42 | 175.62 |

1.07 | 153.77 | 150.38 | 233.71 | 139.99 |

1.10 | 116.71 | 111.07 | 179.26 | 99.58 |

1.13 | 89.53 | 82.76 | 135.24 | 71.31 |

1.15 | 75.49 | 68.43 | 111.65 | 57.45 |

1.17 | 63.98 | 56.88 | 92.15 | 46.57 |

1.20 | 50.38 | 43.54 | 69.32 | 34.44 |

1.25 | 34.65 | 28.64 | 43.89 | 21.62 |

1.30 | 24.53 | 19.46 | 28.62 | 14.23 |

1.40 | 13.28 | 9.84 | 13.48 | 7.06 |

1.50 | 7.89 | 5.57 | 7.25 | 4.11 |

1.70 | 3.53 | 2.40 | 2.96 | 2.04 |

1.80 | 2.61 | 1.80 | 2.17 | 1.65 |

2.00 | 1.68 | 1.25 | 1.45 | 1.27 |

${\mathit{r}}_{0}$ | $\mathit{i}$ = 2 | $\mathit{i}$ = 3 | ||||
---|---|---|---|---|---|---|

201.3894 | 300.2447 | 370.9500 | 200.2556 | 301.5969 | 376.9614 | |

${k}_{1}$ | 3.2778 | 3.4290 | 3.3808 | 3.4319 | 3.3473 | 3.4785 |

${k}_{2}$ | 2.9806 | 3.0730 | 3.2267 | 2.9495 | 3.3291 | 3.2531 |

c | $AR{L}_{1}$ | |||||

0.00 | 201.39 | 300.24 | 370.95 | 200.26 | 301.60 | 376.96 |

0.01 | 200.86 | 299.41 | 369.86 | 199.72 | 300.69 | 375.80 |

0.03 | 196.72 | 292.88 | 361.31 | 195.54 | 293.63 | 366.77 |

0.05 | 188.89 | 280.56 | 345.24 | 187.64 | 280.34 | 349.81 |

0.07 | 178.14 | 263.71 | 323.38 | 176.79 | 262.31 | 326.82 |

0.10 | 158.56 | 233.27 | 284.21 | 157.08 | 230.07 | 285.86 |

0.13 | 137.46 | 200.79 | 242.87 | 135.88 | 196.18 | 242.97 |

0.15 | 123.66 | 179.74 | 216.35 | 122.04 | 174.49 | 215.64 |

0.17 | 110.59 | 159.94 | 191.57 | 108.95 | 154.28 | 190.23 |

0.20 | 92.83 | 133.26 | 158.48 | 91.20 | 127.36 | 156.51 |

0.25 | 68.68 | 97.39 | 114.54 | 67.12 | 91.76 | 112.12 |

0.30 | 50.73 | 71.09 | 82.78 | 49.30 | 66.13 | 80.34 |

0.40 | 28.23 | 38.64 | 44.19 | 27.07 | 35.15 | 42.20 |

0.50 | 16.36 | 21.87 | 24.61 | 15.45 | 19.52 | 23.13 |

0.70 | 6.33 | 8.05 | 8.82 | 5.79 | 7.01 | 8.06 |

0.80 | 4.25 | 5.25 | 5.69 | 3.83 | 4.55 | 5.14 |

0.90 | 3.01 | 3.61 | 3.87 | 2.69 | 3.13 | 3.47 |

${\mathit{r}}_{0}$ | $\mathit{i}=2$ | $\mathit{i}=3$ | ||||
---|---|---|---|---|---|---|

200.2658 | 300.8521 | 371.6013 | 200.6461 | 300.0891 | 370.2289 | |

${k}_{1}$ | 4.241692 | 3.865157 | 3.706407 | 4.389848 | 4.384344 | 3.524029 |

${k}_{2}$ | 2.81188 | 2.958272 | 3.082647 | 2.814882 | 2.941529 | 3.189481 |

c | $AR{L}_{1}$ | |||||

0 | 200.27 | 300.85 | 371.60 | 200.65 | 300.09 | 370.23 |

0.01 | 199.38 | 299.35 | 369.50 | 199.75 | 298.64 | 368.00 |

0.03 | 192.55 | 287.79 | 353.39 | 192.84 | 287.48 | 350.99 |

0.05 | 180.07 | 266.91 | 324.63 | 180.22 | 267.26 | 320.85 |

0.07 | 163.87 | 240.21 | 288.51 | 163.83 | 241.26 | 283.34 |

0.1 | 136.85 | 196.73 | 231.19 | 136.51 | 198.60 | 224.71 |

0.13 | 110.86 | 156.13 | 179.37 | 110.25 | 158.39 | 172.61 |

0.15 | 95.41 | 132.57 | 150.05 | 94.64 | 134.87 | 143.52 |

0.17 | 81.78 | 112.14 | 125.10 | 80.88 | 114.36 | 119.00 |

0.2 | 64.70 | 87.03 | 95.08 | 63.65 | 89.01 | 89.80 |

0.25 | 43.90 | 57.29 | 60.54 | 42.69 | 58.71 | 56.65 |

0.3 | 30.14 | 38.23 | 39.14 | 28.87 | 39.11 | 36.40 |

0.4 | 14.92 | 17.97 | 17.34 | 13.75 | 18.10 | 16.07 |

0.5 | 7.93 | 9.13 | 8.37 | 6.98 | 8.92 | 7.82 |

0.7 | 2.84 | 3.06 | 2.65 | 2.36 | 2.78 | 2.56 |

0.8 | 1.95 | 2.04 | 1.78 | 1.63 | 1.84 | 1.74 |

0.9 | 1.47 | 1.52 | 1.35 | 1.27 | 1.37 | 1.34 |

Existing MDS Chart | Proposed Chart | |
---|---|---|

c | $\mathit{A}\mathit{R}{\mathit{L}}_{1}$ | |

1.00 | 370.77 | 370.23 |

1.01 | 368.95 | 368.00 |

1.03 | 354.95 | 350.99 |

1.05 | 329.63 | 320.85 |

1.07 | 297.20 | 283.34 |

1.10 | 244.27 | 224.71 |

1.13 | 194.70 | 172.61 |

1.15 | 165.86 | 143.52 |

1.17 | 140.80 | 119.00 |

1.20 | 109.93 | 89.80 |

1.25 | 73.21 | 56.65 |

1.30 | 49.53 | 36.40 |

1.40 | 24.08 | 16.07 |

1.50 | 12.74 | 7.82 |

1.70 | 4.56 | 2.56 |

1.80 | 3.06 | 1.74 |

2.00 | 2.21 | 1.34 |

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**MDPI and ACS Style**

Aslam, M.; Khan, N.; Albassam, M.
Shewhart Attribute and Variable Control Charts Using Modified Multiple Dependent State Sampling. *Symmetry* **2019**, *11*, 53.
https://doi.org/10.3390/sym11010053

**AMA Style**

Aslam M, Khan N, Albassam M.
Shewhart Attribute and Variable Control Charts Using Modified Multiple Dependent State Sampling. *Symmetry*. 2019; 11(1):53.
https://doi.org/10.3390/sym11010053

**Chicago/Turabian Style**

Aslam, Muhammad, Nasrullah Khan, and Mohammed Albassam.
2019. "Shewhart Attribute and Variable Control Charts Using Modified Multiple Dependent State Sampling" *Symmetry* 11, no. 1: 53.
https://doi.org/10.3390/sym11010053