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3.4 Discussion and Comparison of proposed chart with EWMA median –
57
scheme. This can be demonstrated from a case in Table 3.4 and Table 3.6. While the n=5, δ=0.75 and λ=0.25, the MRL value of RSS based EWMA median chart is 20 which is higher than the MRL values of the RSS based proposed chart for all the incidents of L excluding when L=1. Besides that, in section 3.3.1, we established that large value of L could be selected for detecting small shifts in a process whereas small value of L could be chosen for detecting large shifts in a process when λ is fixed. This is consistent to the observation here as the sensitivity of Synthetic EWMA median chart is higher than the EWMA median chart in detecting the small shift when L is large and in detecting the large shift when L is small.
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Table 3.6: Dictionary of the parameter K based on the desired in-control ARL (ARL0) and the run-length profiles for the EWMA median chart under RSS when n=5,
0.05, 0.25, 0.50, 0.75
and
0.00, 0.25, 0.50, 0.75,1.00,1.50, 2.00
.Percentiles of Run Length Distribution
λ, K δ ARL 5th 10th 20th 30th 40th 50th 60th 70th 80th 90th 95th
(0.05, 0.7601) 0.00 370.73 33 52 94 141 198 260 340 442 582 828.5 1098.5
0.25 94.54 17.5 24 34 45 57 71 89 111 143 198 250
0.50 34.35 11 13 17 21 25 29 33 40 48 63 77
0.75 19.55 8 9 12 13 15 17 20 23 26 33 39
1.00 13.30 6 7 9 10 11 12 14 15 17 21 24
1.50 8.29 5 5 6 7 7 8 9 9 10 12 13
2.00 6.02 4 4 5 5 5 6 6 7 7 8 9
(0.25, 2.0909) 0.00 370.19 22 40 84 133 191 258 338 441.5 590 857 1107.5
0.25 175.96 13 22 43 65 92 123 161 213 282 396 513
0.50 59.09 7 10 17 24 32 42 55 70 91 130 170
0.75 26.39 5 6 9 12 16 20 25 31 39 55 71
1.00 14.69 4 5 6 8 10 11 14 17 22 29 36
1.50 6.97 3 3 4 5 5 6 7 8 9 12 15
2.00 4.45 2 2 3 3 4 4 5 5 6 7 8
(0.50, 3.2837) 0.00 370.73 20 40 84 133 187.5 255 337 442 586 854 1112
0.25 230.70 14 26 53 84 121 162 211 275 371 527 679
0.50 102.55 7 12 23 37 53 72 95 124 163 232 307
0.75 46.40 5 7 12 18 24 32 42 54 74 104 135
1.00 24.10 3 4 7 10 13 17 22 28 37 52 68
1.50 9.10 2 3 4 5 6 7 9 11 14 18 23
2.00 4.81 2 2 2 3 3 4 5 6 7 9 11
(0.75, 4.4472) 0.00 370.81 20 41 85 136 189 256 336 444 592 859 1121
0.25 276.58 15 30 62 100 142 191 254 336 450 633 817
0.50 144.98 8 16 33 52 75 101 134 175 233 334 421
0.75 76.51 5 9 17 28 39 53 70 93 124 175 228
1.00 41.58 3 5 10 15 22 29 38 50 65 95 123
1.50 14.56 2 3 4 6 8 10 13 17 23 32 41
2.00 6.81 1 2 2 3 4 5 6 8 10 14 18
59 3.5 An illustrative example of application
Under this section, an instance in which the complete data set in a hard- bake process has been reported in Montgomery (2007) is employed to clarify the implementation and construction of the proposed Synthetic EWMA median chart based on RSS scheme. The dataset acquired from Montgomery (2007) is listed in Appendix H and it involves 45 subgroups, each of size, n = 5 in the semiconductor production. From the dataset, our main concern is to create statistical control of the flow width of the resist in the process (measured in microns) using proposed Synthetic EWMA median chart under RSS. Note that the competing RSS based EWMA median chart is also created for the sake of comparison. The complete data set is pooled to attain the total of 225 measurements. From these pooled measurements, ranked set sampling scheme is applied to bring in 30 subgroups, each of size n = 5 with replacement selection.
For both charts, assuming the in-control ARL, ARL0 is fixed at a preferred value, which is 370 in our study. The (n, λ, L, K) values for the proposed Synthetic EWMA median chart is obtained from Table 3.1 whereas the (λ, K) values for the RSS based EWMA median chart is obtained from Table 3.6. The control limits and test statistics of both control charting structures are then determined using the 30 subgroups drawn and the parameters. Next, we suppose that an undefined shift in the process is present after a certain period. To assess the shift, an additional of 20 subgroups is drawn from the data set, each of size n = 5, under the RSS schemes and all sample values are added with 0.05. The subsequent test statistics for both charts have been computed from these added subgroups.
60
In our study, only two selected situations are considered to assess the sensitivity of the RSS based proposed Synthetic EWMA median chart and EWMA median chart. In the first case, we set λ=0.05, L=10 and K=0.4466 for the RSS based proposed chart whereas we opt for λ=0.05 and K=0.7601 for EWMA median chart under RSS. In the second case, we choose λ=0.25, L=5 and K=1.4161 for the RSS based proposed chart whereas we select λ=0.25 and K=2.0909 for EWMA median chart under RSS. The control charts of both cases are displayed in Figure 3.1 to Figure 3.4.
In the first case, from the Figure 3.1, the 42nd sample exceeds the control limits of the Synthetic EWMA median chart under RSS. The 42nd sample will be treated as a non-conforming sample and this gives a CRL1 = 42 – 0 = 42. This CRL1 value is compared with the lower limit of conforming run length chart where L=10. When the CRL1 value is bigger than L, the process is resumed by resetting the test statistic to the initial value and the subsequent statistics is plotted until the next nonconforming sample is detected. The 45th sample fell beyond the control limit of the proposed chart. The 45th sample is treated as a nonconforming sample again and this gives a CRL2 = 45 – 42 = 3. At this time, we can observe that the CRL2 value is less than the L, the first out-of-control signals is triggered. On the other hand, from the Figure 3.2, it is observed that all samples stayed between the control limits of the EWMA median chart under RSS. In other words, no out-of-control event is discovered. Moreover, in the second case, we can monitor that the 35th sample exceed the control limits of the proposed chart from the illustration in Figure 3.3. The 35th sample can be regarded as a non-conforming sample and this gives a CRL1 = 35 – 0 = 35. The
61
CRL1 value obtained is compared with the lower limit of conforming run length chart where L=5. Since L is smaller than the CRL1 value, the process is resumed by setting the test statistic to the initial mean and the plotting of subsequent statistics is recommenced until the next nonconforming sample is discovered.
Next, it is observed that the 46th sample lied beyond the control limit of the proposed chart. The 46th sample is treated as a nonconforming sample again and this gives a CRL2 = 46 – 35 = 11. Similarly, the process is resumed as the CRL value is greater than L. The 50th sample is considered as a non-conforming sample because it exceeds the control limits of the chart. While the CRL3 = 50 – 46 = 4, which is less than the L, the first out-of-control signals is triggered at the 50th sample. From Figure 3.4, it is clearly shown that all the test statistics lied between the control limits of the EWMA median chart under RSS, which means that uncontrollable event cannot be discovered when the process is unstable.
With regards to the out-of-control detection efficacy, the findings indicate that the proposed chart is more efficient in identifying a process shift than its counterpart EWMA median chart under RSS.
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Figure 3.1: Synthetic EWMA median chart under RSS in the first case when n=5, λ=0.05, K=0.4466 and L=10.
Figure 3.2: EWMA median chart under RSS in the first case when n=5, λ=0.05 and K=0.7601.
CRL₁=42 CRL₂=3 CRL₃=1 CRL₄=1 CRL₅=1 CRL₆=1
1.475 1.485 1.495 1.505 1.515 1.525 1.535 1.545
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
Synthetic EWMA Median chart-RSS
1.48 1.49 1.5 1.51 1.52 1.53 1.54 1.55 1.56
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
EWMA median chart -RSS
63
Figure 3.3: Synthetic EWMA median chart under RSS in the second case when n=5, λ=0.25, K=1.4161 and L=5.
Figure 3.4: EWMA median chart under RSS in the second case when n=5, λ=0.25 and K=2.0909
CRL₁=35 CRL₂=11 CRL₃=4
1.43 1.45 1.47 1.49 1.51 1.53 1.55 1.57 1.59 1.61
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
Synthetic EWMA Median chart-RSS
1.43 1.48 1.53 1.58 1.63
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
EWMA median chart -RSS
64 CHAPTER 4
CONCLUSION