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The exponential weighted moving average technique used in process mean and variance monitoring charts was combined by Gan in 1997 and proposed two combined joint monitoring schemes one with rectangular control region and the other with elliptical control region. Performance of these two schemes may very depend on the shifts in mean or variance to be detected quickly. In this paper, performances of these two schemes are evaluated with respect to the average run length properties. The results reveal that elliptical scheme is little faster in detecting the shifts in process mean and increase in variance within a limit.

Exponential weighted moving average (EWMA) chart for monitoring a shift in process mean was introduced by Roberts in 1959 and the design procedure of EWMA mean charts was reported by Crowder in 1989 [_{r}) and an elliptical control region (EE_{e}) were proposed by Gan [_{r} and EE_{e} schemes is evaluated different scenarios based on average run length (ARL) properties. ARL is defined as the average number of samples taken until an out-of-control signal is issued in quality control schemes.

An EWMA chart for monitoring the sample mean _{0} is usually set at target mean µ_{0}, l_{E} is a constant such that _{E} and h_{E} are the upper control limit and lower control limit respectively. The EWMA chart for monitoring the sample variance _{0} is usually set at_{e} is a positive constant such that _{t} is greater than the H_{e} or e_{t} is less than the h_{e}. H_{e} and h_{e} are the upper and lower control limits respectively for the EWMA variance chart.

In the EE_{r} scheme, EWMA mean chart and the EWMA variance chart are combined by plotting the EWMA of _{e} scheme, the distance from the point^{2} for each sample. The quantity _{t} is greater than

and for a point in which e_{t} is less than

_{r} scheme, but it is an out-of control point with respect to the EE_{e} scheme. Similarly, point A is an out-of-control point with respect to the EE_{r} scheme, but it is an in-control point with respect to the EE_{e} scheme. In process monitoring, both the magnitude of the shift and the direction are important and therefore it advisable to plot the individual samples.

Simulated data set were used for comparing the performance of the two combined schemes. For this simulation, the in-control mean (µ_{0}) and variance

The performances of the schemes are compared based on out of control ARLs when there is a shift in mean or variance or in both. Schemes detect various magnitudes of shifts in mean and variance based on their sensitivities. The scheme which gives smaller ARL when there is a shift in mean or variance is the better scheme. The shifts in mean and variance investigated are given by

and

Scheme | Control Region | Control Chart Parameters ARL = 250 | Control Chart Parameters ARL = 370 |
---|---|---|---|

EE_{r} | Rectangular | l_{E} = 0.134, E_{0} =0 H_{E} = 0.345, h_{E} = −0.345 l_{e} = 0.106, e_{0} = −0.270 H_{e} = 0.215, h_{e} = −0.867 | l_{E} = 0.120, E_{0} = 0 H_{E} = 0.3385, h_{E} = −0.3385 l_{e} = 0.100, e_{0} = −0.270 H_{e} = 0.2205, h_{e} = −0.8772 |

EE_{e} | Elliptical | l_{E} = 0.134, E_{0} = 0 H_{E} = 0.372, h_{E} = −0.372 l_{e} = 0.106, e_{0} = −0.270 H_{e} = 0.250, h_{e} = −0.92 | l_{E} = 0.120, E_{0} = 0 H_{E} = 0.3722, h_{E} = −0.3722 l_{e} = 0.100, e_{0} = −0.270 H_{e} = 0.2540, h_{e} = −0.8994 |

where

where,

_{r} and EE_{e} schemed for a ∆ shift in mean and a δ shift in variance for particular in-control ARL.

_{r} scheme performs very similar with EE_{e} scheme. _{e} scheme detect the shifts quicker than the EE_{r} scheme. However this property is lost when ∆ > 1.5 but it does not harm because after ∆ > 1.5 both schemes detect the shifts very quickly. When δ < 1, EE_{e} scheme gives larger out-of-control ARLs compare to the EE_{r} scheme and therefore _{e} scheme is smaller compare to EE_{r} scheme. This is a good property because decrease in variance is always favored. The design procedure of these schemes can be introduced through software for industrial use so that it can be a readymade scheme.

EE_{r} | EE_{e} | EE_{r} | EE_{e} | |||||
---|---|---|---|---|---|---|---|---|

0 | 0.5 | 5.80 | 6.33 | 1 | 0.5 | 5.70 | 4.95 | |

0 | 0.75 | 21.90 | 24.57 | 1 | 0.75 | 9.66 | 8.58 | |

0 | 0.95 | 235.44 | 252.32 | 1 | 0.95 | 10.19 | 10.64 | |

0 | 1 | 250.00 | 250.00 | 1 | 1 | 10.14 | 10.55 | |

0 | 1.05 | 136.53 | 129.33 | 1 | 1.05 | 10.04 | 10.25 | |

0 | 1.1 | 67.70 | 63.44 | 1 | 1.1 | 9.86 | 9.78 | |

0 | 1.25 | 18.89 | 18.13 | 1 | 1.25 | 8.74 | 8.04 | |

0 | 1.5 | 8.17 | 7.9 | 1 | 1.5 | 6.40 | 5.83 | |

0 | 3 | 2.57 | 2.47 | 1 | 3 | 2.52 | 2.41 | |

0.2 | 0.5 | 5.79 | 6.27 | 1.5 | 0.5 | 4.89 | 4.05 | |

0.2 | 0.75 | 21.75 | 22.6 | 1.5 | 0.75 | 5.66 | 5.49 | |

0.2 | 0.95 | 134.57 | 135.1 | 1.5 | 0.95 | 5.74 | 6.03 | |

0.2 | 1 | 128.94 | 128.88 | 1.5 | 1 | 5.75 | 6.03 | |

0.2 | 1.05 | 88.34 | 83.03 | 1.5 | 1.05 | 5.76 | 5.97 | |

0.2 | 1.1 | 53.72 | 49.33 | 1.5 | 1.1 | 5.76 | 5.88 | |

0.2 | 1.25 | 18.02 | 16.96 | 1.5 | 1.25 | 5.66 | 5.46 | |

0.2 | 1.5 | 8.08 | 7.75 | 1.5 | 1.5 | 5.05 | 4.6 | |

0.2 | 3 | 2.57 | 2.46 | 1.5 | 3 | 2.46 | 2.34 | |

0.4 | 0.5 | 5.79 | 6.05 | 3 | 0.5 | 2.59 | 2.48 | |

0.4 | 0.75 | 20.46 | 18.43 | 3 | 0.75 | 2.60 | 2.71 | |

0.4 | 0.95 | 51.56 | 52.17 | 3 | 0.95 | 2.63 | 2.77 | |

0.4 | 1 | 48.54 | 49.24 | 3 | 1 | 2.64 | 2.77 | |

0.4 | 1.05 | 41.48 | 39.89 | 3 | 1.05 | 2.64 | 2.77 | |

0.4 | 1.1 | 32.82 | 30.27 | 3 | 1.1 | 2.65 | 2.77 | |

0.4 | 1.25 | 15.83 | 14.56 | 3 | 1.25 | 2.68 | 2.75 | |

0.4 | 1.5 | 7.83 | 7.42 | 3 | 1.5 | 2.71 | 2.68 | |

0.4 | 3 | 2.56 | 2.46 | 3 | 3 | 2.16 | 2.03 | |

0.6 | 0.5 | 5.79 | 5.74 | |||||

0.6 | 0.75 | 16.90 | 14.07 | |||||

0.6 | 0.95 | 24.43 | 25.13 | |||||

0.6 | 1 | 23.69 | 24.41 | |||||

0.6 | 1.05 | 22.20 | 21.92 | |||||

0.6 | 1.1 | 20.08 | 18.95 | |||||

0.6 | 1.25 | 13.17 | 11.98 | |||||

0.6 | 1.5 | 7.44 | 6.95 | |||||

0.6 | 3 | 2.55 | 2.44 |

EE_{r} | EE_{e} | EE_{r} | EE_{e} | |||||
---|---|---|---|---|---|---|---|---|

0 | 0.5 | 6.23 | 6.45 | 1 | 0.5 | 6.16 | 5.18 | |

0 | 0.75 | 24.37 | 24.63 | 1 | 0.75 | 10.54 | 9.31 | |

0 | 0.95 | 331.51 | 310.88 | 1 | 0.95 | 11.09 | 11.81 | |

0 | 1 | 370.07 | 370.0 | 1 | 1 | 10.98 | 11.84 | |

0 | 1.05 | 186.01 | 185.27 | 1 | 1.05 | 10.88 | 11.53 | |

0 | 1.1 | 84.16 | 81.51 | 1 | 1.1 | 10.75 | 10.99 | |

0 | 1.25 | 21.22 | 20.56 | 1 | 1.25 | 9.62 | 8.94 | |

0 | 1.5 | 8.89 | 8.67 | 1 | 1.5 | 6.98 | 6.39 | |

0 | 3 | 2.76 | 2.67 | 1 | 3 | 2.72 | 2.6 | |

0.2 | 0.5 | 6.24 | 6.38 | 1.5 | 0.5 | 5.3 | 4.31 | |

0.2 | 0.75 | 24.35 | 22.98 | 1.5 | 0.75 | 6.12 | 6.01 | |

0.2 | 0.95 | 176.66 | 169.82 | 1.5 | 0.95 | 6.21 | 6.68 | |

0.2 | 1 | 166.54 | 175.29 | 1.5 | 1 | 6.23 | 6.7 | |

0.2 | 1.05 | 112.41 | 110.38 | 1.5 | 1.05 | 6.23 | 6.64 | |

0.2 | 1.1 | 65.66 | 61.8 | 1.5 | 1.1 | 6.23 | 6.56 | |

0.2 | 1.25 | 20.14 | 19.15 | 1.5 | 1.25 | 6.15 | 6.04 | |

0.2 | 1.5 | 8.83 | 8.53 | 1.5 | 1.5 | 5.5 | 5.04 | |

0.2 | 3 | 2.77 | 2.68 | 1.5 | 3 | 2.64 | 2.52 | |

0.4 | 0.5 | 6.27 | 6.2 | 3 | 0.5 | 2.85 | 2.75 | |

0.4 | 0.75 | 22.89 | 19.06 | 3 | 0.75 | 2.83 | 2.99 | |

0.4 | 0.95 | 59.98 | 62.22 | 3 | 0.95 | 2.84 | 3.05 | |

0.4 | 1 | 56.58 | 61.13 | 3 | 1 | 2.84 | 3.06 | |

0.4 | 1.05 | 47.87 | 48.81 | 3 | 1.05 | 2.85 | 3.05 | |

0.4 | 1.1 | 38.21 | 35.85 | 3 | 1.1 | 2.86 | 3.05 | |

0.4 | 1.25 | 17.7 | 16.33 | 3 | 1.25 | 2.88 | 3.01 | |

0.4 | 1.5 | 8.52 | 8.13 | 3 | 1.5 | 2.93 | 2.91 | |

0.4 | 3 | 2.77 | 2.66 | 3 | 3 | 2.34 | 2.2 | |

0.6 | 0.5 | 6.23 | 5.9 | |||||

0.6 | 0.75 | 18.84 | 14.94 | |||||

0.6 | 0.95 | 27.13 | 29.11 | |||||

0.6 | 1 | 26.29 | 28.42 | |||||

0.6 | 1.05 | 24.81 | 25.55 | |||||

0.6 | 1.1 | 22.6 | 21.85 | |||||

0.6 | 1.25 | 14.63 | 13.36 | |||||

0.6 | 1.5 | 8.12 | 7.57 | |||||

0.6 | 3 | 2.75 | 2.64 |

In overall the EE_{e} scheme is faster than the EE_{r} scheme in detecting the shifts in process mean and variance but the design procedure of EE_{e} scheme is little complex due to its complex equations. Further less variance is signaled slowly in the EE_{e} schemes compare to EE_{r} scheme which is a preferred characteristic. Therefore EE_{e} scheme outperformed the EE_{r} scheme.

This work was supported by the South Eastern University of Sri Lanka and the Higher Education for the Twenty First Century Project, Sri Lanka [SEUSL/O-AS/N1].