2. The SDDS EWMA-AM Chart
2.3. Performance Comparison with Existing Control Charts
國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
14
2.3. Performance Comparison with Existing Control Charts
We compare the performance of the SDDS EWMA-AM chart with other existing mean control charts under different distributions of quality variables. The average time to signal (ATS) and the adjusted ATS (AATS) are considered to measure the performance of a control chart if the variable sampling intervals are adopted. For fixed sampling intervals chart, the ATS is equal to the ARL when the time between two successive samples equals one time unit and the relationship of AATS and ATS is (see Daudin (1992)). For performance comparison, or in-control AATS ( ) values of the competing charts are fixed at (or very close to) an acceptable value, such as 370 or 500, with a fixed sample size, , and then compare their out-of-control ARL or AATS.
If the in-control distribution of a quality variables is unknown, we consider different values of , which depends on the distribution, for comparison. Tables 8–10 show the detection performance of the SDDS EWMA-AM chart and the EWMA-AM chart in Yang and Arnold (2014), for 0.1–0.9, under , , and 0.1, 0.3, 0.5, respectively. It can be seen that the SDDS EWMA-AM chart has superior out-of-control detection performance than the EWMA-AM chart for small to large shifts in mean, 0.1–0.9, when 0.1, 0.3, 0.5.
‧
Table 8. Performance comparison of the SDDS EWMA-AM chart and the EWMA-AM chart with , and under
Table 9. Performance comparison of the SDDS EWMA-AM chart and the EWMA-AM chart with , and under
‧
Table 10. Performance comparison of SDDS EWMA-AM chart and EWMA-AM chart with , and under 0.5. the values of and corresponding under various in-control distributions.
For the IC symmetric normal distribution N(0, 1) of a quality variables with in-control mean 0 and variance 1, whose corresponding , Table 12 shows the detection performance of the SDDS EWMA-AM chart, the EWMA-AM chart, the NLE chart and the CEW chart (Zou and Tsung (2010)) with , and , Table 13 shows the detection performance of the SDDS EWMA-AM chart, the SL chart (Mukherjee and Chakraborti (2012)) and SC chart (Chowdhury et al. (2014)) with , and , Table 14 shows the detection performance of the SDDS EWMA-AM chart, Shewhart, DSVSI, VP, VSSI, VSS, VSI charts (Carot et al. (2002)) with , , and the shift scale . Table 15 shows the detection performance of the SDDS EWMA-AM
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
17
chart, Shewhart, DS, and VP charts (Torng and Lee (2009)) with , and .
It can be seen that under the standard normal distribution, (1) the SDDS EWMA-AM chart has superior out-of-control detection performance than the EWMA-AM, the NLE and the CEW charts with for (see Table 12); (2) the SDDS EWMA-AM chart has superior out-of-control detection performance than the SL and the SC charts with for (see Table 13); (3) the SDDS EWMA-AM chart has superior out-of-control detection performance than the Shewhart chart with for , superior than the DSVSI chart with , , , , , , , for , superior than the VP chart with , , , , , , , for , superior than the VSSI chart with , , , , , , , for , superior than the VSS chart with , , , , , , for , superior than the VSI chart with , , , , , , for (see Table 14); (4) the SDDS EWMA-AM chart has superior out-of-control detection performance than the Shewhart chart with for , superior than the DS chart with , , , , , for , superior than the DS chart with , , , , , for , superior than the VP chart with , , , , , , , for , superior than the VP chart with , , , , , ,
‧
‧
Table 12. Performance comparison of the SDDS EWMA-AM chart, the EWMA-AM chart, the NLE chart and the CEW chart with , and
Table 13. Performance comparison of the SDDS EWMA-AM chart, the SL charts and the SC charts with , and under N( , 1).
‧
Table 14. Performance comparison of the SDDS EWMA-AM chart and the Shewhart, DSVSI, VP, VSSI, VSS, and the VSI charts with ,
‧
Table 15. Performance comparison of the SDDS EWMA-AM chart and the Shewhart, DS, and the VP charts with , and
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
22
For the IC symmetric double exponential distribution DE(0, ) with mean 0 and variance 1 and IC symmetric DE(0, 1) with mean 0 and variance 2, and corresponding , Table 16 shows the detection performance of the SDDS EWMA-AM chart, the EWMA-AM chart and the SL charts (Mukherjee and Chakraborti (2012)) with , and under DE( , ), Table 17 shows the detection performance of the SDDS EWMA-AM chart, the SL chart and the SC chart with , and under DE( , 1).
It can be seen that under DE( , ), the SDDS EWMA-AM chart has superior out-of-control detection performance than the EWMA-AM and the SL charts for (see Table 16); under DE( , 1), the SDDS EWMA-AM chart has superior out-of-control detection performance than the SL and the SC charts for (see Table 17).
For the IC symmetric student's t distribution t(3), t(4), t(10), t(30) with degrees of freedom 3, 4, 10, 30, respectively, and corresponding but have heavier tails than the standard normal distribution. Table 18 shows the detection performance of the SDDS EWMA-AM chart, the NLE chart, the CEW chart and the EWM chart (Zou and Tsung (2010)) with , and under . Tables 19–21 show the detection performance of the SDDS EWMA-AM chart, the Shewhart, the DS, and the VP charts (Torng and Lee (2009)) with , and under t(4) , t(10) , t(30) , respectively.
It can be seen that under , the SDDS EWMA-AM chart has superior out-of-control detection performance than the NLE, the CEW and the EWM charts for (see Table 18); under t(4) , the SDDS EWMA-AM
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
23
chart has superior out-of-control detection performance than the Shewhart chart with for , superior than the DS chart with , , , , , for , superior than the DS chart with , , , , , for , superior than the VP chart with , , , , , , , for , superior than the VP chart with , , , , , , , for or (see Table 19);
under t(10) , the SDDS EWMA-AM chart has superior out-of-control detection performance than the Shewhart chart with for , superior than the DS chart with , , , , , for , superior than the DS chart with , , , , , for , superior than the VP chart with , , , , , , , for or , superior than the VP chart with , , , , , , , for or (see Table 20); under t(30) , the SDDS EWMA-AM chart has superior out-of-control detection performance than the Shewhart chart with for , superior than the DS chart with , , , , , for , superior than the DS chart with , , , ,
, for , superior than the VP chart with , , , , , , , for or , superior than the VP chart with , ,
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
24
, , , , , for or (see Table 21). The heavier tails for the t distribution, the superior out-of-control detection performance the SDDS EWMA-AM chart has.
Table 16. Performance comparison of the SDDS EWMA-AM chart, the EWMA-AM chart and the SL charts with , and under DE( , ).
SDDS EWMA-AM chart
EWMA-AM chart
SL chart m=30
SL chart m=50
E(n) ARL ARL ARL ARL
0.00 4.66 507.85 494.85 487.67 495.25
0.25 6.39 13.21 18.34 394.53 330.07
0.50 7.28 5.07 9.40 207.50 114.91
0.75 8.00 3.12 7.03 71.90 28.81
1.00 8.53 2.33 5.92 25.56 7.93
1.25 8.95 1.90 5.30 7.78 3.19
1.50 9.24 1.62 4.93 2.36 1.80
‧
Table 17. Performance comparison of the SDDS EWMA-AM chart, the SL chart and the SC chart with , and under DE( , 1).
Table 18. Performance comparison of the SDDS EWMA-AM chart, the NLE chart, the CEW chart and the EWM chart with , and
‧
Table 19. Performance comparison of the SDDS EWMA-AM chart, the Shewhart, the DS, and the VP charts with , and
‧
Table 20. Performance comparison of the SDDS EWMA-AM chart, the Shewhart, the DS, and the VP charts with , and
‧
Table 21. Performance comparison of the SDDS EWMA-AM chart, the Shewhart, the DS, and the VP charts with , and
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
29
For the IC symmetric uniform distribution Unif with mean 0, variance 1 and , and the IC exponential distribution Exp(1) with mean 1, variance 1 and , Table 22 shows the detection performance of the SDDS EWMA-AM chart and the EWMA-AM chart with , and under Unif and Exp(1) . It can be seen that under Unif and Exp(1) , the SDDS EWMA-AM chart has superior out-of-control detection performance than the EWMA-AM chart for (see Table 22).
For the chi-square distribution with degree of freedom 3, , and corresponding
, Table 23 shows the detection performance of the SDDS EWMA-AM chart, the NLE chart, the CEW chart and the EWM chart (Zou and Tsung (2010)) with , and under . It can be seen that under , the SDDS EWMA-AM chart has superior out-of-control detection performance than the NLE, the CEW and the EWM charts for (see Table 23).
‧
Table 22. Performance comparison of the SDDS EWMA-AM chart and the EWMA-AM chart with , and under
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
31
Table 23. Performance comparison of the SDDS EWMA-AM chart, the NLE chart, the CEW chart and the EWM chart with , and
under SDDS EWMA-AM
chart
NLE chart
CEW chart
EWM chart
E(n) ARL ARL ARL ARL
0.00 9.77 373.64 373.00 120.00 373.00
0.25 13.17 11.23 54.40 65.00 72.00
0.50 14.79 2.97 30.70 29.40 27.70
0.75 14.09 1.39 22.70 17.10 16.00
1.00 10.19 1.03 17.90 11.80 11.10
1.50 8.00 1.00 12.30 7.03 6.95
2.00 8.00 1.00 8.91 4.65 5.09
3.00 8.00 1.00 5.11 2.32 3.42
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
32