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 [
An EWMA chart for monitoring the sample mean
In the EEr scheme, EWMA mean chart and the EWMA variance chart are combined by plotting the EWMA of
and for a point in which et is less than
Simulated data set were used for comparing the performance of the two combined schemes. For this simulation, the in-control mean (µ0) 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 |
---|---|---|---|
EEr | Rectangular | lE = 0.134, E0 =0 HE = 0.345, hE = −0.345 le = 0.106, e0 = −0.270 He = 0.215, he = −0.867 | lE = 0.120, E0 = 0 HE = 0.3385, hE = −0.3385 le = 0.100, e0 = −0.270 He = 0.2205, he = −0.8772 |
EEe | Elliptical | lE = 0.134, E0 = 0 HE = 0.372, hE = −0.372 le = 0.106, e0 = −0.270 He = 0.250, he = −0.92 | lE = 0.120, E0 = 0 HE = 0.3722, hE = −0.3722 le = 0.100, e0 = −0.270 He = 0.2540, he = −0.8994 |
where
where,
EEr | EEe | EEr | EEe | |||||
---|---|---|---|---|---|---|---|---|
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 |
EEr | EEe | EEr | EEe | |||||
---|---|---|---|---|---|---|---|---|
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 EEe scheme is faster than the EEr scheme in detecting the shifts in process mean and variance but the design procedure of EEe scheme is little complex due to its complex equations. Further less variance is signaled slowly in the EEe schemes compare to EEr scheme which is a preferred characteristic. Therefore EEe scheme outperformed the EEr 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].