# Control Charts for Monitoring Process Capability Index Using Median Absolute Deviation for Some Popular Distributions

^{1}

^{2}

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^{*}

## Abstract

**:**

## 1. Introduction

_{0}[21]. It is considered that the value of ARL

_{0}should be larger, as it indicates the process is in the form of in control. The expected number of samples from the out-of-control process should be smaller, as it indicates the out-of-control process as quickly as possible, which is denoted by ARL

_{1}. References [21,22] used a martingale approach for the derivation of the ARL formula for the normal as well as for the non-normal distributions. A numerical integration approach was applied by [23] to evaluate the performance of the proposed control chart. Several basic and commonly used methods for the calculation of the ARL for the exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts were reviewed by [24]. Many researchers of quality control charts have used the ARL for the performance evaluation of the proposed charts, including [22,25,26,27,28,29,30,31,32,33,34].

## 2. Process Capability Indices (PCIs) Using Median Absolute Deviation

## 3. Design of MAD Control Charts for Non-Normal Distributions

_{0}) and out-of-control ARL (ARL

_{1}). ARL

_{0}is the average number of samples that triggers a control chart signals, when the process is in control. ARL

_{0}should be as large as possible, since the process is in control. ARL

_{1}is the average number of samples until a control chart signals when the process is out of control. The performance of the control chart is assessed by ARL

_{1}, with smaller values of ARL

_{1}indicating the supremacy of the chart.

_{0}) is obtained by:

_{1}and given by:

_{0}, ARL

_{1}, and control chart coefficient k. Suppose that ${r}_{0}$ represents the specified in-control ARL

_{0}. The summary of an algorithm is as follows:

## 4. Example

**Simulation Results for Weibull Distribution**

_{0}at 370 and the specific in-control shape = 1.8 and scale = 2.0, we found the control chart lower limit LCL = 0.69859 for the proposed chart at $n=25$. The lists of the ${\widehat{C}}_{pk}$ values for these 60 simulations are given below. The graphical display of the proposed control chart is presented in Figure 1.

**Simulation Results for Gamma Distribution**

_{0}at 370 and the specific in-control shape = 3 and scale = 0.75, we found LCL = 0.86988 for the proposed chart at $n=25$. The lists of the ${\widehat{C}}_{pk}$ values for these 62 simulations are given below. The graphical display of the proposed control chart is presented in Figure 2.

**Simulation Results for Log-Normal Distribution**

_{0}at 370 and the specific in-control mean = 0.5 and standard deviation = 1.00, we found LCL = 0.782699 for the proposed chart at $n=25$. The lists of the ${\widehat{C}}_{pk}$ values for these 58 simulations are given below. The graphical display of the proposed control chart is presented in Figure 3.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**The average run lengths (ARLs) for the proposed chart when n = 25, WD(2.8,3.5), upper specification limit (USL) = 6.3488, and lower specification limit (LSL) = 0.5280.

LCL | 1.4553 | 1.4755 | 1.4947 | 1.5207 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.0 | 370.78 | 300.28 | 250.18 | 200.14 |

0.1 | 223.91 | 186.82 | 156.87 | 125.04 |

0.2 | 136.13 | 114.78 | 97.39 | 78.75 |

0.3 | 83.33 | 71.67 | 61.72 | 51.22 |

0.4 | 53.12 | 45.53 | 39.49 | 33.28 |

0.5 | 33.89 | 29.66 | 26.00 | 22.11 |

0.6 | 22.12 | 19.67 | 17.48 | 15.11 |

0.7 | 14.93 | 13.40 | 12.07 | 10.40 |

0.8 | 10.17 | 9.15 | 8.36 | 7.47 |

0.9 | 7.26 | 6.65 | 6.12 | 5.54 |

1.0 | 5.29 | 4.87 | 4.55 | 4.17 |

LCL | 1.7175 | 1.7364 | 1.7526 | 1.7729 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.0 | 370.60 | 300.72 | 250.33 | 200.09 |

0.1 | 187.45 | 154.39 | 129.15 | 106.36 |

0.2 | 97.57 | 80.81 | 68.88 | 58.18 |

0.3 | 51.82 | 44.05 | 38.34 | 32.26 |

0.4 | 28.88 | 25.04 | 22.25 | 19.33 |

0.5 | 17.06 | 15.03 | 13.45 | 11.81 |

0.6 | 10.35 | 9.22 | 8.39 | 7.49 |

0.7 | 6.63 | 5.97 | 5.52 | 5.01 |

0.8 | 4.44 | 4.08 | 3.81 | 3.53 |

0.9 | 3.18 | 3.00 | 2.84 | 2.66 |

1.0 | 2.38 | 2.26 | 2.16 | 2.04 |

LCL | 0.69859 | 0.70748 | 0.71529 | 0.72491 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.0 | 370.36 | 300.49 | 250.21 | 200.39 |

0.1 | 149.43 | 122.68 | 105.38 | 87.26 |

0.2 | 65.43 | 55.67 | 47.95 | 40.50 |

0.3 | 29.45 | 25.56 | 22.68 | 19.97 |

0.4 | 15.01 | 13.33 | 12.07 | 10.66 |

0.5 | 8.14 | 7.44 | 6.86 | 6.24 |

0.6 | 4.83 | 4.53 | 4.26 | 3.93 |

0.7 | 3.16 | 2.99 | 2.84 | 2.69 |

0.8 | 2.20 | 2.12 | 2.04 | 1.95 |

0.9 | 1.64 | 1.59 | 1.55 | 1.51 |

1.0 | 1.33 | 1.31 | 1.29 | 1.27 |

LCL | 0.80405 | 0.81194 | 0.81829 | 0.82695 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.0 | 370.35 | 300.15 | 250.19 | 200.23 |

0.1 | 113.80 | 93.09 | 80.28 | 66.39 |

0.2 | 40.16 | 34.11 | 29.98 | 25.61 |

0.3 | 16.07 | 14.18 | 12.84 | 11.26 |

0.4 | 7.33 | 6.64 | 6.17 | 5.51 |

0.5 | 3.81 | 3.55 | 3.37 | 3.19 |

0.6 | 2.38 | 2.25 | 2.15 | 2.04 |

0.7 | 1.62 | 1.57 | 1.53 | 1.48 |

0.8 | 1.27 | 1.24 | 1.22 | 1.20 |

0.9 | 1.11 | 1.10 | 1.10 | 1.09 |

1.0 | 1.04 | 1.03 | 1.03 | 1.02 |

LCL | 0.41605 | 0.41785 | 0.41785 | 0.4216 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.0 | 370.19 | 300.56 | 251.14 | 200.65 |

0.1 | 30.52 | 27.65 | 25.69 | 23.34 |

0.2 | 7.99 | 7.61 | 7.29 | 6.94 |

0.3 | 3.47 | 3.37 | 3.28 | 3.18 |

0.4 | 2.00 | 1.96 | 1.93 | 1.89 |

0.5 | 1.40 | 1.38 | 1.37 | 1.36 |

0.6 | 1.15 | 1.15 | 1.15 | 1.14 |

0.7 | 1.05 | 1.05 | 1.04 | 1.04 |

0.8 | 1.01 | 1.01 | 1.01 | 1.01 |

0.9 | 1.00 | 1.00 | 1.00 | 1.00 |

1.0 | 1.00 | 1.00 | 1.00 | 1.00 |

LCL | 0.44240 | 0.44479 | 0.44679 | 0.44938 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.0 | 370.35 | 300.49 | 300.49 | 250.17 |

0.1 | 28.01 | 24.83 | 24.83 | 22.58 |

0.2 | 5.77 | 5.38 | 5.38 | 5.07 |

0.3 | 2.30 | 2.22 | 2.22 | 2.14 |

0.4 | 1.38 | 1.36 | 1.36 | 1.34 |

0.5 | 1.10 | 1.10 | 1.10 | 1.09 |

0.6 | 1.02 | 1.02 | 1.02 | 1.02 |

0.7 | 1.00 | 1.00 | 1.00 | 1.00 |

0.8 | 1.00 | 1.00 | 1.00 | 1.00 |

0.9 | 1.00 | 1.00 | 1.00 | 1.00 |

1.0 | 1.00 | 1.00 | 1.00 | 1.00 |

LCL | 1.75078 | 1.77519 | 1.79798 | 1.82517 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.00 | 370.93 | 300.11 | 250.02 | 200.23 |

0.05 | 28.23 | 24.42 | 21.70 | 18.68 |

0.10 | 4.17 | 3.87 | 3.63 | 3.38 |

0.15 | 1.46 | 1.42 | 1.39 | 1.36 |

0.20 | 1.03 | 1.03 | 1.03 | 1.02 |

0.25 | 1.00 | 1.00 | 1.00 | 1.00 |

0.30 | 1.00 | 1.00 | 1.00 | 1.00 |

0.35 | 1.00 | 1.00 | 1.00 | 1.00 |

0.40 | 1.00 | 1.00 | 1.00 | 1.00 |

LCL | 2.04145 | 2.06135 | 2.08025 | 2.10258 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.00 | 370.93 | 300.32 | 250.19 | 200.47 |

0.05 | 28.23 | 12.49 | 11.20 | 9.86 |

0.10 | 4.17 | 1.93 | 1.86 | 1.77 |

0.15 | 1.46 | 1.06 | 1.05 | 1.05 |

0.20 | 1.03 | 1.00 | 1.00 | 1.00 |

0.25 | 1.00 | 1.00 | 1.00 | 1.00 |

0.30 | 1.00 | 1.00 | 1.00 | 1.00 |

0.35 | 1.00 | 1.00 | 1.00 | 1.00 |

0.40 | 1.00 | 1.00 | 1.00 | 1.00 |

LCL | 0.86988 | 0.88014 | 0.88914 | 0.90178 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.00 | 370.76 | 300.08 | 250.40 | 200.10 |

0.05 | 100.36 | 83.44 | 72.36 | 59.28 |

0.10 | 30.83 | 26.95 | 24.03 | 20.64 |

0.15 | 11.09 | 10.04 | 9.27 | 8.24 |

0.20 | 4.89 | 4.52 | 4.27 | 3.89 |

0.25 | 2.60 | 2.47 | 2.36 | 2.24 |

0.30 | 1.68 | 1.63 | 1.60 | 1.55 |

0.35 | 1.26 | 1.24 | 1.23 | 1.21 |

0.40 | 1.08 | 1.08 | 1.07 | 1.06 |

0.45 | 1.02 | 1.02 | 1.01 | 1.01 |

0.50 | 1.00 | 1.00 | 1.00 | 1.00 |

**Table 10.**The ARLs for the proposed chart when n = 50, GD(3.0,0.75), USL = 6.9553, and LSL = 0.2534.

LCL | 0.99658 | 1.00516 | 1.01279 | 1.02297 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.00 | 370.29 | 300.27 | 250.02 | 200.65 |

0.05 | 65.99 | 55.71 | 48.96 | 40.57 |

0.10 | 15.76 | 13.92 | 12.69 | 11.32 |

0.15 | 5.25 | 4.90 | 4.57 | 4.16 |

0.20 | 2.36 | 2.25 | 2.16 | 2.05 |

0.25 | 1.47 | 1.43 | 1.40 | 1.36 |

0.30 | 1.13 | 1.12 | 1.11 | 1.10 |

0.35 | 1.03 | 1.02 | 1.02 | 1.02 |

0.40 | 1.00 | 1.00 | 1.00 | 1.00 |

0.45 | 1.00 | 1.00 | 1.00 | 1.00 |

0.50 | 1.00 | 1.00 | 1.00 | 1.00 |

**Table 11.**The ARLs for the proposed chart when n = 25, GD(0.5,1.00), USL = 3.9397, and LSL = 0.0100.

LCL | 0.12151 | 0.12293 | 0.124032 | 0.125462 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.00 | 370.28 | 300.29 | 250.09 | 200.07 |

0.05 | 109.93 | 86.44 | 71.08 | 54.67 |

0.10 | 26.35 | 20.01 | 15.66 | 11.47 |

0.15 | 5.69 | 4.53 | 3.86 | 3.29 |

0.20 | 2.36 | 2.16 | 2.02 | 1.88 |

0.25 | 1.59 | 1.52 | 1.47 | 1.42 |

0.30 | 1.28 | 1.25 | 1.23 | 1.21 |

0.35 | 1.14 | 1.13 | 1.12 | 1.10 |

0.40 | 1.06 | 1.06 | 1.06 | 1.05 |

0.45 | 1.03 | 1.03 | 1.03 | 1.02 |

0.50 | 1.01 | 1.01 | 1.01 | 1.01 |

**Table 12.**The ARLs for the proposed chart when n = 50, GD(0.5,1.00), USL = 3.9397, and LSL = 0.0100.

LCL | 0.133825 | 0.134377 | 0.134881 | 0.135491 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.00 | 370.02 | 300.43 | 250.56 | 200.24 |

0.05 | 22.95 | 19.07 | 16.14 | 13.25 |

0.10 | 3.53 | 3.27 | 3.07 | 2.84 |

0.15 | 1.71 | 1.66 | 1.62 | 1.58 |

0.20 | 1.25 | 1.23 | 1.21 | 1.20 |

0.25 | 1.09 | 1.08 | 1.08 | 1.08 |

0.30 | 1.03 | 1.03 | 1.03 | 1.02 |

0.35 | 1.01 | 1.01 | 1.01 | 1.01 |

0.40 | 1.00 | 1.00 | 1.00 | 1.00 |

0.45 | 1.00 | 1.00 | 1.00 | 1.00 |

0.50 | 1.00 | 1.00 | 1.00 | 1.00 |

**Table 13.**The ARLs for the proposed chart when n = 25, LND(0.45,1.50), USL = 74.7197, and LSL = 0.0329.

LCL | 0.97622 | 0.97914 | 0.98198 | 0.985647 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.00 | 370.32 | 300.13 | 250.23 | 200.42 |

0.05 | 30.25 | 27.39 | 25.08 | 22.50 |

0.10 | 8.34 | 7.90 | 7.45 | 6.97 |

0.15 | 3.95 | 3.82 | 3.71 | 3.57 |

0.20 | 2.46 | 2.41 | 2.36 | 2.31 |

0.25 | 1.81 | 1.79 | 1.77 | 1.75 |

0.30 | 1.50 | 1.48 | 1.48 | 1.46 |

0.35 | 1.31 | 1.31 | 1.30 | 1.29 |

0.40 | 1.20 | 1.20 | 1.19 | 1.19 |

**Table 14.**The ARLs for the proposed chart when n = 50, LND(0.45,1.50), USL = 74.7197, and LSL = 0.0329.

LCL | 1.01494 | 1.01788 | 1.020871 | 1.02412 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.00 | 370.37 | 300.28 | 250.39 | 200.74 |

0.05 | 25.34 | 22.81 | 20.54 | 18.31 |

0.10 | 5.99 | 5.67 | 5.35 | 5.00 |

0.15 | 2.67 | 2.58 | 2.50 | 2.42 |

0.20 | 1.71 | 1.68 | 1.65 | 1.61 |

0.25 | 1.33 | 1.31 | 1.30 | 1.29 |

0.30 | 1.16 | 1.15 | 1.15 | 1.14 |

0.35 | 1.08 | 1.07 | 1.07 | 1.07 |

0.40 | 1.04 | 1.03 | 1.03 | 1.03 |

**Table 15.**The ARLs for the proposed chart when n = 25, LND(0.50,1.00), USL = 21.6678, and LSL = 0.1255.

LCL | 0.782699 | 0.79418 | 0.79418 | 0.80091 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.00 | 370.09 | 300.25 | 250.16 | 200.07 |

0.05 | 87.76 | 75.80 | 63.66 | 52.91 |

0.10 | 28.65 | 25.39 | 22.72 | 19.74 |

0.15 | 12.44 | 11.41 | 10.33 | 9.32 |

0.20 | 6.64 | 6.22 | 5.74 | 5.29 |

0.25 | 4.02 | 3.79 | 3.59 | 3.38 |

0.30 | 2.75 | 2.62 | 2.50 | 2.40 |

0.35 | 2.07 | 2.02 | 1.95 | 1.89 |

0.40 | 1.70 | 1.67 | 1.63 | 1.59 |

0.45 | 1.45 | 1.43 | 1.40 | 1.38 |

**Table 16.**The ARLs for the proposed chart when n = 50, LND(0.50,1.00), USL = 21.6678, and LSL = 0.1255.

LCL | 0.85217 | 0.85708 | 0.861658 | 0.866851 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.00 | 370.42 | 300.10 | 250.15 | 200.19 |

0.05 | 64.64 | 55.22 | 48.30 | 41.41 |

0.10 | 17.78 | 15.83 | 14.33 | 12.80 |

0.15 | 7.03 | 6.45 | 6.01 | 5.56 |

0.20 | 3.56 | 3.38 | 3.20 | 3.03 |

0.25 | 2.19 | 2.11 | 2.04 | 1.97 |

0.30 | 1.62 | 1.58 | 1.55 | 1.51 |

0.35 | 1.33 | 1.31 | 1.29 | 1.27 |

0.40 | 1.17 | 1.16 | 1.15 | 1.14 |

0.45 | 1.09 | 1.08 | 1.08 | 1.07 |

**Table 17.**The ARLs for the proposed chart when n = 25, LND(0.95,0.40), USL = 7.2452, and LSL = 0.9228.

LCL | 1.08958 | 1.10396 | 1.11754 | 1.13415 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.00 | 370.45 | 300.50 | 250.14 | 200.29 |

0.05 | 102.92 | 86.53 | 71.75 | 59.16 |

0.10 | 34.52 | 29.03 | 25.59 | 21.93 |

0.15 | 13.71 | 12.10 | 10.91 | 9.62 |

0.20 | 6.53 | 5.98 | 5.50 | 4.93 |

0.25 | 3.66 | 3.42 | 3.24 | 3.03 |

0.30 | 2.39 | 2.28 | 2.19 | 2.10 |

0.35 | 1.78 | 1.72 | 1.67 | 1.62 |

0.40 | 1.42 | 1.39 | 1.37 | 1.33 |

0.45 | 1.22 | 1.20 | 1.19 | 1.17 |

0.50 | 1.11 | 1.10 | 1.09 | 1.08 |

**Table 18.**The ARLs for the proposed chart when n = 50, LND(0.95,0.40), USL = 7.2452, and LSL = 0.9228.

LCL | 1.25597 | 1.26777 | 1.27836 | 1.29156 |
---|---|---|---|---|

c | ARL_{0} = 370 | ARL_{0} = 300 | ARL_{0} = 250 | ARL_{0} = 200 |

0.00 | 370.24 | 300.17 | 250.52 | 200.20 |

0.05 | 67.86 | 57.26 | 49.22 | 41.82 |

0.10 | 18.08 | 15.95 | 14.14 | 12.61 |

0.15 | 6.47 | 5.91 | 5.45 | 4.98 |

0.20 | 3.09 | 2.89 | 2.73 | 2.55 |

0.25 | 1.87 | 1.81 | 1.75 | 1.68 |

0.30 | 1.36 | 1.32 | 1.30 | 1.27 |

0.35 | 1.15 | 1.14 | 1.13 | 1.12 |

0.40 | 1.05 | 1.05 | 1.04 | 1.04 |

0.45 | 1.01 | 1.01 | 1.01 | 1.01 |

0.50 | 1.00 | 1.00 | 1.00 | 1.00 |

© 2019 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**

Aslam, M.; Rao, G.S.; AL-Marshadi, A.H.; Ahmad, L.; Jun, C.-H.
Control Charts for Monitoring Process Capability Index Using Median Absolute Deviation for Some Popular Distributions. *Processes* **2019**, *7*, 287.
https://doi.org/10.3390/pr7050287

**AMA Style**

Aslam M, Rao GS, AL-Marshadi AH, Ahmad L, Jun C-H.
Control Charts for Monitoring Process Capability Index Using Median Absolute Deviation for Some Popular Distributions. *Processes*. 2019; 7(5):287.
https://doi.org/10.3390/pr7050287

**Chicago/Turabian Style**

Aslam, Muhammad, G. Srinivasa Rao, Ali Hussein AL-Marshadi, Liaquat Ahmad, and Chi-Hyuck Jun.
2019. "Control Charts for Monitoring Process Capability Index Using Median Absolute Deviation for Some Popular Distributions" *Processes* 7, no. 5: 287.
https://doi.org/10.3390/pr7050287