Design of the Exponential - Normal EWMA Convolution Chart

3.2 Design of the Exponential – Normal EWMA Convolution Chart

Monitoring the Ex-Gaussian random variable by applying the EWMA chart, the Exponential–Normal EWMA convolution chart, when the process is in-control,  1, the plotting statistic is

the control limits for the Exponential–Normal EWMA convolution chart is



Since Ex-Gaussian is not a symmetric distribution, the control limit parameter, L1 and L₂ needn‘t be set the same value.

3.3 Average Run Length of the Exponential – Normal EWMA Convolution Chart The Average Run Length (ARL) is used to measure the performance of a control chart. ARL is the average number of samples taken before finding an out-of-control signal. The Markov Chain approach was first introduced by Brooks and Evans (1972).

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The j^th subinterval‘s midpoint, m_j is

N The (N+1)^th state in which the point is not within the control limits, either greater than the upper limit or smaller than the lower limit, is designated as the absorbing state. The transition probability P_ij is the probability that given the state of statistic Z was i at time t - 1 but moved to the state j at time t, that is It could be derived that

)

ARL could be obtained by the following equation：

1 ) (  ^1

b I Q

ARL ^T (54)

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where Q stands for the transition probability matrix without the (N+1)^th column and row, I is the identity matrix, a 1xN matrix ‗one‘, and b is a vector with initial probability for each state, which could be obtained by satisfying the limitation of

b bQ  .

3.4 The ARL Calculation

(1). Determine (L₁,L₂) under Various ^ with ARL⁰ = 370

As mentioned earlier, we would like to investigate the impact of the measurement error on control charts. So we will calculate ARLs under different ratios of ²/². Note that ² is the variance of the measurement error and ² is the variance of the exponential random variable. First, we have to find the values of the parameters , L1 and L2 such that ARL0 = 370. In addition, set  100 and

2 1

 . The procedure to find the values of the parameters , L₁, and L₂ is：

Step 1：Set 0.05 and L₁ 0.5

Step 2：By eq. (51), set ARL₀ b^T(I Q)^11370, since L₁ 0.5 then we can get L2.

Step 3：Let L₁ L₁0.5 and L₁ 6, repeat Step 2.

Step 4：Let 0.1 and 0.9 repeat Step 1 - Step 3.

The ARLs under different values of , L1 and L2 are listed in Table 3.

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Table 4. The ARL under the combination of  ^with 0.05 and (L₁,L₂)(3,2.072) and ² 1

 ARL  ARL  ARL

0.1 1.009 1.1 227.663 2.1 6.192 0.2 1.191 1.2 101.974 2.2 5.519 0.3 1.680 1.3 53.145 2.3 4.989 0.4 2.702 1.4 32.120 2.4 4.562 0.5 4.924 1.5 21.641 2.5 4.211 0.6 10.215 1.6 15.778 2.6 3.920 0.7 24.095 1.7 12.196 2.7 3.674 0.8 63.981 1.8 9.852 2.8 3.464 0.9 184.525 1.9 8.233 2.9 3.283 1.0 370.535 2.0 7.065 3.0 3.126

Figure 4. The ARL under the combination of  ^with 0.05, (L₁,L₂)(3,2.702) and ² 1

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Table 5. The ARL under the combination of  ^with 0.1^, (L₁,L₂)(3.5,2.055) and ² 1

 ARL  ARL  ARL

0.1 1.028 1.1 258.059 2.1 8.124 0.2 1.358 1.2 132.038 2.2 7.150 0.3 2.185 1.3 72.800 2.3 6.386 0.4 3.984 1.4 44.938 2.4 5.774 0.5 7.996 1.5 30.400 2.5 5.275 0.6 17.260 1.6 22.052 2.6 4.862 0.7 39.422 1.7 16.875 2.7 4.516 0.8 94.544 1.8 13.462 2.8 4.223 0.9 226.106 1.9 11.098 2.9 3.971 1.0 370.101 2.0 9.394 3.0 3.753

Figure 5. The ARL under the combination of  with 0.1, (L₁,L₂)(3.5,2.055) and ²1

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Table 6. The ARL under the combination of  ^with 0.2^, (L₁,L₂)(4,1.925) and ² 1

 ARL  ARL  ARL

0.1 1.111 1.1 258.980 2.1 10.608 0.2 1.858 1.2 149.046 2.2 9.273 0.3 3.651 1.3 88.894 2.3 8.223 0.4 7.596 1.4 57.309 2.4 7.381 0.5 16.120 1.5 39.675 2.5 6.695 0.6 34.453 1.6 29.107 2.6 6.128 0.7 73.974 1.7 22.366 2.7 5.653 0.8 157.408 1.8 17.836 2.8 5.250 0.9 299.976 1.9 14.656 2.9 4.906 1.0 370.417 2.0 12.343 3.0 4.608

Figure 6. The ARL under the combination of  with 0.2, (L₁,L₂)(4,1.925) and ²1

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Table.7 The ARL under the combination of  with 0.3^, (L₁,L₂)(4.5,1.756) and ² 1

 ARL  ARL  ARL

0.1 1.248 1.1 283.636 2.1 13.350 0.2 2.501 1.2 178.080 2.2 11.592 0.3 5.390 1.3 111.276 2.3 10.212 0.4 11.497 1.4 73.299 2.4 9.107 0.5 23.932 1.5 51.169 2.5 8.209 0.6 48.680 1.6 37.591 2.6 7.468 0.7 96.864 1.7 28.813 2.7 6.849 0.8 185.454 1.8 22.872 2.8 6.325 0.9 312.233 1.9 18.686 2.9 5.879 1.0 370.402 2.0 15.637 3.0 5.494

Figure 7. The ARL under the combination of  with 0.3, (L₁,L₂)(4.5,1.7565) and ² 1

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Table 8. The ARL under the combination of  with 0.4^, (L₁,L₂)(5,1.6057) and ² 1

 ARL  ARL  ARL

0.1 1.444 1.1 314.295 2.1 16.717 0.2 3.289 1.2 214.983 2.2 14.414 0.3 7.348 1.3 140.366 2.3 12.611 0.4 15.495 1.4 94.188 2.4 11.172 0.5 31.062 1.5 66.129 2.5 10.007 0.6 59.731 1.6 48.544 2.6 9.048 0.7 110.640 1.7 37.057 2.7 8.250 0.8 195.033 1.8 29.245 2.8 7.578 0.9 306.713 1.9 23.734 2.9 7.007 1.0 370.418 2.0 19.721 3.0 6.516

Figure 8.The ARL under the combination of  with 0.4,(L₁,L₂)(5,1.6057) and ² 1

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Table 9. The ARL under the combination of  with 0.5^, (L₁,L₂)(5.5,1.4743) and ² 1

 ARL  ARL  ARL

0.1 1.719 1.1 344.994 2.1 20.960 0.2 4.282 1.2 257.320 2.2 17.946 0.3 9.648 1.3 176.364 2.3 15.594 0.4 19.853 1.4 120.896 2.4 13.724 0.5 38.153 1.5 85.478 2.5 12.214 0.6 69.507 1.6 62.736 2.6 10.978 0.7 120.966 1.7 47.708 2.7 9.952 0.8 199.878 1.8 37.435 2.8 9.091 0.9 300.394 1.9 30.181 2.9 8.361 1.0 370.874 2.0 24.903 3.0 7.737

Figure 9. The ARL under the combination of  ^with 0.5, (L₁,L₂)(5.5,1.4743), ² 1

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Table 10. The ARL under the combination of  ^with 0.05^, (L₁,L₂)(3,2.072), ² 1 1000000

 

 ARL  ARL  ARL

0.1 1.009 1.1 227.384 2.1 6.191 0.2 1.191 1.2 101.963 2.2 5.519 0.3 1.679 1.3 53.141 2.3 4.989 0.4 2.702 1.4 32.118 2.4 4.562 0.5 4.924 1.5 21.639 2.5 4.211 0.6 10.214 1.6 15.777 2.6 3.920 0.7 24.092 1.7 12.196 2.7 3.674 0.8 63.968 1.8 9.852 2.8 3.464 0.9 184.429 1.9 8.233 2.9 3.283 1.0 369.922 2.0 7.065 3.0 3.126

Figure 10. The ARL under the combination of  with 0.05, (L₁,L₂)(3,2.702),² 1and 1000000



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From Table 4 to Table 9, the ARL₁ increases as  increases to 1 from 0 but ARL1 decreases as  increases to 3 from 1; the ARL1 values are smaller than ARL0

values regardless of the size of the shift  . That is reasonable. The ARL values are very similar for small  (100) and large  (1000000).

Table 11. All reasonable combinations of , L₁ and L₂for the Exponential-Normal EWMA

convolution chart. (under ² 1^,  100⁾ 0001

. 0 / ²

2  



 L1 L2

0.05 3.0 2. 072

0.1 3.5 2. 055

0.2 4.0 1. 925

0.3 4.5 1. 7563

0.4 5.0 1. 6056

0.5 5.5 1. 4742

Table 11 showed an interesting phenomenon, as  gets larger, L₁ becomes larger and L2 becomes smaller.

(2). The Effect of the Value of ²/² on the Determination of L2

According to the discussion above, we found all reasonable combinations of )

,

(L₁ L₂ for l=0.05-0.5. To find out how the different ratio ²/² would affect

the choice of the combination of (L₁,L₂), we set l=0.05-0.5^、L1 = 3(0.5)5.5 with the ratio ²/² = 0.0001, 0.1, 0.25, 0.5 to find L2 that makes ARL0 = 370 under 100.

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 value gets larger, the amount of this increase is more obvious. For example, with

 small, say 0.05, L² increases from 2.072 to 2.173 as ²/² = 0.1 and 0.5 respectively, the difference between these two L2 values are slightly. But with larger value of , say 0.5, L₂ will be 1.4742 and 2.215 when ²/² changes from 0.1 to 0.5. The difference between these two L2 values is more obvious than the smaller  values. Hence, we should choose smaller  so as ²/² ratio varies, L₂ won‘t change too much and the performance of the control chart would be more stable.

(3). ARL0 under Different Values of ²/² and 

Next, we investigate the sensitivity of the Exponential–Normal EWMA convolution chart with six different combinations of , L1 and L2 from Table 12.

Since they‘re all reasonable choices, the only thing we need to focus on is the speed they reflect to the unusual causes, which means the level of ARL1 under the same amount of shift  . Smaller ARL₁ values mean that when unusual causes happened in

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the process, the control chart would catch it quickly, which in turn means better sensitivity. Table 13 to Table 16 list the ARLs under different size of the  in the process with different ²/² ratios. Note that in the case of 1, it indicates the ordinary convolution chart without EWMA approach. Besides, the column named

―ARL saved % ‖ is the ARL percentage saved between 0.05 and the 1 case.

05 . 0

05 . 0

% 1





 









ARL ARL saved ARL

ARL

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From Table 13 above, the Exponential–Normal EWMA convolution chart with 05

.

0

 has better performance in detecting the process mean shift on both sides compared to the ordinary convolution chart. In addition, with 1, the ordinary convolution chart without EWMA approach is the worst one. It is particularly obvious when the process deteriorates, say  0.9, ARL₁ = 361.869, almost twice as big as the case with 0.05. Besides, the ARL saved % by the EWMA approach is at least 35% and up to 99%. This shows the EWMA approach is extremely effective in monitoring the process.

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From Table 14 above, the Exponential–Normal EWMA convolution chart with 05

.

0

 has better performance in detecting the process mean shift on both sides compared to the ordinary convolution chart. And, the ordinary convolution chart without EWMA approach is the worst one. Again, compare to the convolution chart without EWMA approach, ARL % saved by EWMA approach is significant, at least 32% and could go up to 99%.

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From Table 15 above, the Exponential–Normal EWMA convolution chart with 05

.

0

 has better performance in detecting the process mean shift on both sides compared to the ordinary convolution chart. And, the ordinary convolution chart without EWMA approach is the worst one. Again, compare to the convolution chart without EWMA approach, ARL % saved by EWMA approach is significant, at least 29% and could go up to 99%.

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From Table 16 above, the Exponential–Normal EWMA convolution chart with 05

.

0

 has better performance in detecting the process mean shift on both sides compared to the ordinary convolution chart. And, the ordinary convolution chart without EWMA approach is the worst one. Comparing to the convolution chart without EWMA approach, the ARL % saved by EWMA approach is significant, at least 23% and could go up to 98%.

From Table 13 to Table 16, control charts with smaller  perform better than larger value of . Also, convolution charts with the EWMA approach perform better than the convolution chart without the EWMA approach. In general, the Exponential–Normal EWMA convolution chart with  0.05 performs the best among the seven charts.

(4). The Impact of ²/² on ARL0

Next, we study the impact of the measurement error by ARL. First assuming there‘s no measurement error, which means ² 0. Comparing its performance with other levels of the measurement error. Thus we have five charts with ²/²0 (no measurement error), along with other four levels：²/²0.0001, ²/² 0.1,

25 . 0 / ²

2  

 , ²/²0.5. Note that  100. We calculated the ARL saved % between the chart without the measurement error and the chart with the ratio 0.5 in the last column. The results are shown in Table 17.

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conclusion by the ARL % saved. Generally, if the chart is with the measurement error then its detection ability becomes poor. If error exists, we should keep ²/² 0.1 to keep the chart with the same detection ability; otherwise, the chart‘s detection ability would be poor than the charts with no measurement errors.

3.5 The Impact of the Mistaking of the Exponential - Normal EWMA Convolution Chart

We understand how the measurement error influences the detection ability of the Exponential-Normal EWMA convolution chart. Now, we wonder how seriously it would be when we misuse the Exponential-Normal EWMA convolution chart. We assume that the number of unit(s) X required to observe exactly 1 defect is an exponential random variable, and the measurement error is normally distributed. The observed value is Y X, where X and  are independent. In such situation, Y following the Ex-Gaussian distribution would be the best choice to construct the Exponential–Normal EWMA convolution chart. Somehow, the engineer misuses the Normal distribution. In spite of the distribution is wrong, the sample mean and

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The new random variable T  X₁ ₁ will have：

) ,

(

~N  ²²

T , when in-control

) ,

(

~N  ²²²

T , when out-of-control

Now we know the distribution is in the misused case, so the ARLs under different size of mean shifts could be calculated and are listed in Table 18. In Table 18, we set 0.05 since it performs better than the others. We would compare the ARLs between the Exponential–Normal EWMA convolution chart and the Normal-Normal EWMA chart with the case ²/² 0.1 and 0.5. Note that

100

 .

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convolution chart and misusing Normal-Normal EWMA chart

05 .

0



chart Exp–Nor EWMA Nor–Nor EWMA

2

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Table 18 shows that, if we misuse the Ex-Gaussian distribution to normal distribution, the ARL would increase significantly under the same ²/² ratio when  1 . For instance, when ² 0.9 and the ^2/² = 0.1, the Exponential-Normal EWMA chart‘s ARL1 is 192.956 but it increases to 318.862 when we misused the Normal-Normal EWMA chart. Similar result for other  values and ²/² ratios. On the other hand, When  1, if we misuse the Ex-Gaussian distribution to normal distribution, the ARL₁ would decrease, due to the skewness of the Ex-Gaussian distribution.

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Chapter 4. The Design of the EWMA Convolution Chart with Exponential Measurement Error

In this chapter, we assume the measurement error follows the exponential distribution, the same distribution with the X.

4.1 Measurement Error with Exponential Distribution

Now, assume the true value, X, of the product is exponentially distributed, and the observed value be U. The difference between these two could attribute to the measurement error, ₂. The relationship among these three is：

2



 X

U (55)

where X and ₂ are independent. We assume that the measurement error follows an exponential distribution with a mean  , that is：

) (

2~ 

 EXP (56) In this case, the measurement error is positive, so it only makes the observed value greater than the true value. Using the convolution of two exponential distributions, the p.d.f of U is：

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4.2 Design of the Exponential - Exponential EWMA Convolution Chart

Use U to construct an EWMA chart, the Exponential-Exponential EWMA convolution chart, the plotting statistic is：

) 1

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the control limits of the Exponential-Exponential EWMA convolution chart are：



Since the distribution of U is not symmetric, the control limit parameters shouldn‘t set to be equal here, we set L1 and L2 here.

4.3 Average Run Length of the Exponential-Exponential EWMA Convolution Chart

Similar to section 3.3 we use the same approach to evaluate the performance of the chart here, i.e. using Markov chain approximation to find the ARL.

(1). Determine (L₁,L₂) under various  _{with ARL}0 = 370

Again, we investigate the impact of the measurement error on control charts by calculating ARLs under different ratios of ²/². Note that ² and ² are the

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The combinations of L₁, L₂,  to make ARL₀ = 370 are shown in Table 19 ( 100).

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1.0 1971.000 166.469 -128938.000 370.739 1.5 1819.000 199.203 -118986.000 370.739 2.0 1667.000 231.937 -109035.000 370.739

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1.0 2684.000 174.384 -196861.000 370.243 1.5 2448.000 211.075 -179542.000 370.339 2.0 2212.000 247.767 -162224.000 370.435 2.5 1976.000 284.459 -144905.000 370.532 3.0 1740.000 321.151 -127586.000 370.628 3.5 1503.000 357.843 -110195.000 370.147 4.0 1266.000 394.534 -92802.600 369.667 4.5 1031.000 431.226 -75557.500 370.339

5.0 1.288 467.918 6.453 370.407

6.0 1.260 541.302 8.537 370.608

0.8 0.5 4458.000 141.827 -363911.000 370.120

1.0 4054.000 182.654 -330923.000 370.307 1.5 3649.000 223.481 -297853.000 370.120 2.0 3246.000 264.307 -264947.000 370.681 2.5 2840.000 305.134 -231796.000 370.120 3.0 2437.000 345.961 -198889.000 370.681 3.5 2031.000 386.788 -165738.000 370.120 4.0 1628.000 427.615 -132831.000 370.681 4.5 1223.000 468.442 -99761.500 370.494

5.0 1.190 509.269 3.808 370.433

6.0 1.167 590.922 5.725 370.099

0.9 0.5 8970.000 146.229 -811307.000 370.805

1.0 8060.000 191.458 -728990.000 370.713 1.5 7150.000 236.687 -646673.000 370.620 2.0 6240.000 281.916 -564356.000 370.528 2.5 5330.000 327.145 -482040.000 370.436 3.0 4420.000 372.374 -399723.000 370.344 3.5 3510.000 417.603 -317406.000 370.252 4.0 2600.000 462.832 -235090.000 370.160

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4.5 1690.000 508.061 -152773.000 370.068

5.0 1.095 553.290 1.921 370.084

6.0 1.079 643.748 3.396 370.465

From Table 19 we could pick any combinations of , L₁ and L₂ that makes ARL0 = 370, for example, set 0.05, L1 = 4, then L2 should be 1.901 to make ARL₀ = 370. Also, as the value of  gets bigger, the number of the combinations of

) ,

(L₁ L₂ is decreased. When 0.05, there‘re seven combinations of (L₁,L₂) which can make ARL0 = 370. But when  0.9, there are only two combinations.

Table 19 shows a similar result as Table 3.

Next, we have to check if these combinations are ―reasonable‖. Tables 20 to Table 25 and Figure 11 to Figure 16 show the reasonable combinations of , L₁ and L₂ under ARL₀ = 370 for l=0.05-0.5 _and  ^100. In addition, we add one more case by setting  1000000 with 0.05 and (L₁,L₂)(3,2.072) when ARL⁰ = 370. Table 26 and Figure 17 show the ARLs for various values of  ；Table 27 lists all reasonable combinations of , L1 and L2. Note that the ARL value under  1 is the ARL0, the rest of the ARLs are values of ARL1.

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Table 20. The ARLs under the combination of  ^with 0.05,(L₁,L₂)(3,2.072), _² _1

 ARL  ARL  ARL

0.1 1.009 1.1 227.662 2.1 6.192 0.2 1.191 1.2 101.974 2.2 5.519 0.3 1.680 1.3 53.145 2.3 4.989 0.4 2.702 1.4 32.120 2.4 4.562 0.5 4.924 1.5 21.641 2.5 4.211 0.6 10.215 1.6 15.779 2.6 3.920 0.7 24.094 1.7 12.196 2.7 3.674 0.8 63.979 1.8 9.852 2.8 3.464 0.9 184.519 1.9 8.233 2.9 3.283 1.0 370.526 2.0 7.065 3.0 3.126

Figure 11. The ARLs under the combination of  ^with 0.05, (L₁,L₂)(3,2.072), ² 1

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Table 21. The ARLs under the combination of  ^with 0.1, (L₁,L₂)(3.5,2.055), ²1

 ARL  ARL  ARL

0.1 1.028 1.1 258.040 2.1 8.124 0.2 1.358 1.2 132.035 2.2 7.150 0.3 2.185 1.3 72.800 2.3 6.386 0.4 3.984 1.4 44.938 2.4 5.774 0.5 7.994 1.5 30.399 2.5 5.275 0.6 17.257 1.6 22.052 2.6 4.862 0.7 39.413 1.7 16.875 2.7 4.516 0.8 94.520 1.8 13.462 2.8 4.223 0.9 226.041 1.9 11.098 2.9 3.971 1.0 370.021 2.0 9.394 3.0 3.753

Figure 12. The ARLs under the combination of 0.1, (L₁,L₂)(3.5,2.055), ² _1

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Table 22. The ARLs under the combination of  ^with 0.2, (L₁,L₂)(4,1.925), ²_1

 ARL  ARL  ARL

0.1 1.111 1.1 258.936 2.1 10.608 0.2 1.858 1.2 149.039 2.2 9.273 0.3 3.650 1.3 88.893 2.3 8.223 0.4 7.593 1.4 57.309 2.4 7.381 0.5 16.114 1.5 39.675 2.5 6.695 0.6 34.436 1.6 29.107 2.6 6.128 0.7 73.930 1.7 22.366 2.7 5.653 0.8 157.299 1.8 17.836 2.8 5.250 0.9 299.764 1.9 14.656 2.9 4.906 1.0 370.245 2.0 12.343 3.0 4.608

Figure 13. The ARLs under the combination of  ^with 0.2, (L₁,L₂)(4,1.925), _²_1

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Table 23. The ARLs under the combination of  ^with 0.3^, (L₁,L₂)(4.5,1.7561), ²_1

 ARL  ARL  ARL

0.1 1.247 1.1 283.516 2.1 13.350 0.2 2.500 1.2 178.053 2.2 11.592 0.3 5.385 1.3 111.270 2.3 10.212 0.4 11.483 1.4 73.297 2.4 9.107 0.5 23.898 1.5 51.169 2.5 8.209 0.6 48.605 1.6 37.590 2.6 7.468 0.7 96.706 1.7 28.813 2.7 6.849 0.8 185.142 1.8 22.871 2.8 6.325 0.9 311.755 1.9 18.686 2.9 5.879 1.0 370.033 2.0 15.637 3.0 5.494

Figure 14. The ARLs under the combination of 0.3, (L₁,L₂)(4.5,1.7561), ²1

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Table 24. The ARLs under the combination of 0.4, (L₁,L₂)(5,1.6057), ²1

 ARL  ARL  ARL

0.1 1.444 1.1 314.182 2.1 16.717 0.2 3.288 1.2 214.948 2.2 14.414 0.3 7.345 1.3 140.356 2.3 12.611 0.4 15.487 1.4 94.184 2.4 11.172 0.5 31.043 1.5 66.127 2.5 10.007 0.6 59.688 1.6 48.544 2.6 9.048 0.7 110.551 1.7 37.057 2.7 8.250 0.8 194.864 1.8 29.245 2.8 7.578 0.9 306.453 1.9 23.734 2.9 7.007 1.0 370.177 2.0 19.721 3.0 6.516

Figure 15. The ARLs under the combination of  ^with 0.4, (L₁,L₂)(5,1.6057), _² _1

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Table 25. The ARLs under the combination of 0.5, (L₁,L₂)(5.5,1.4744)^, ²1

 ARL  ARL  ARL

0.1 1.719 1.1 344.463 2.1 20.960 0.2 4.280 1.2 257.109 2.2 17.946 0.3 9.639 1.3 176.292 2.3 15.594 0.4 19.828 1.4 120.872 2.4 13.724 0.5 38.090 1.5 85.469 2.5 12.214 0.6 69.366 1.6 62.732 2.6 10.978 0.7 120.680 1.7 47.706 2.7 9.952 0.8 199.348 1.8 37.435 2.8 9.091 0.9 299.572 1.9 30.180 2.9 8.361 1.0 370.003 2.0 24.903 3.0 7.737

Figure 16. The ARLs under the combination of 0.5^, (L₁,L₂)(5.5,1.4744), ²1

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Table 26. The ARLs under the combination 0.05, (L₁,L₂)(3,2.072), _²_1, 1000000

 

 ARL  ARL  ARL

0.1 1.009 1.1 227.660 2.1 6.191 0.2 1.191 1.2 101.966 2.2 5.519 0.3 1.679 1.3 53.141 2.3 4.989 0.4 2.702 1.4 32.118 2.4 4.561 0.5 4.924 1.5 21.639 2.5 4.211 0.6 10.215 1.6 15.777 2.6 3.920 0.7 24.094 1.7 12.196 2.7 3.674 0.8 63.985 1.8 9.852 2.8 3.464 0.9 184.558 1.9 8.233 2.9 3.283 1 370.605 2 7.065 3 3.126

Figure 17. The ARLs under the combination 0.05^, (L₁,L₂)(3,2.072),

21

 ^, 1000000

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From Table 20 to Table 25, the ARL₁ values are always smaller than ARL₀ values regardless the size of the shift  . That is reasonable. Also the performance of the two cases with  100 and ^{1000000} (see Table 20 and Table 26) is almost the same.

Table 27. All reasonable combinations of 、L1、L2 for the Exponential-Exponential EWMA convolution chart (² 1,  100)

0001 . 0 / ²

2  



 L₁ L₂

0.05 3.0 2.072

0.1 3.5 2.055

0.2 4.0 1.925

0.3 4.5 1.7561

0.4 5.0 1.6057

0.5 5.5 1.4744

Table 27 shows a phenomenon that when  gets bigger, L¹ also becomes bigger but L₂ gets smaller.

(2). The Effect of the Values of ^2/2on the Determination of L2

From above discussion, we found all reasonable combinations of (L₁,L₂) for l=0.05-0.5. To find out how the ratio ²/² affects the choice of the

combination of (L₁,L₂), we set l=0.05-0.5^、L1 = 3(0,5)5.5 with the ratio ²/²

= 0.0001, 0.1, 0.25, 0.5 to find L₂ that makes ARL₀ = 370. Note that  100. The result is shown in Table 28.

‧

difference between these two L2 values is small. But with larger value of , say 0.5, L₂ are 1.474 and 1.744 when ²/² = 0.0001 and 0.5 respectively, the difference between these two L₂ values becomes bigger. In general, as  and L₁ are increase, L₂ decreases.

(3). ARL₀ under Different Values of ²/² and 

The six different combinations in Table 28 are all reasonable choices, the only thing we need to focus on is the speed they reflect to the unusual causes, which means the level of ARL₁ under the same amount of shift  . Table 29 to Table 32 list the

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From Table 29, the Exponential-Exponential EWMA convolution chart with 05

.

0

 has better performance in detecting the process mean shift on both sides compared to the ordinary convolution chart. In addition, with  1, the ordinary convolution chart without EWMA approach is the worst choice. It is trivial when process deteriorates, say  0.9, ARL₁ = 361.443, almost twice as big as the case with 0.05. Besides, the ARL % saved by the EWMA approach is at least 35%

and could go up to 99%. This shows the EWMA approach is extremely effective in monitoring the process.

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From Table 30, the Exponential-Exponential EWMA convolution chart with 05

.

0

 has better performance in detecting the process mean shift on both sides compared to the ordinary convolution chart. The ordinary convolution chart without EWMA approach is the worst one. The ARL saved % by the EWMA approach is significant, at least 33% and could go up to 99%.

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From Table 31, the Exponential-Exponential EWMA convolution chart with 05

.

0

 has better performance in detecting the process mean shift on both sides compared to the ordinary convolution chart. The ordinary convolution chart without EWMA approach is the worst one. The ARL % saved by the EWMA approach is significant, at least 30% and could go up to 98%.

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From the Table 32, same conclusion could be drawn just as previous tables.

From Table 29 to Table 32, control chart with smaller  performs better than larger value of . Also, convolution charts with the EWMA approach perform better than the convolution chart without the EWMA approach. In general, the Exponential-Exponential EWMA convolution chart with 0.05 has the best detection ability of the seven charts.

(4). The Impact of ²/² on ARL0 The results are shown in Table 33. Again, the ARL saved % is calculate as

0

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Table 33 shows the impact of the measurement error. When the measurement error exists, it affects the sensitivity of the control chart. As we could see, when

9 .

0

 and without the measurement error, the ARL¹ is 184.558, on the other side, with the same  value, but the ²/² ratio rises to 0.5, the ARL1 will increase to 208.347, obviously greater than the case without measurement error. We can get the same conclusion by the column of ARL saved %. Generally, the convolution chart with measurement error‘s detection ability becomes poorer. If error exists then we should control ²/² 0.1 to keep the convolution chart the same detection ability；otherwise, the convolution chart detection ability will be poorer than the charts with no measurement errors.

4.4 The Impact of the Mistaking of the Exponential-Exponential EWMA Convolution Chart

Now we understand how the measurement error influences the detection ability of the Exponential-Exponential EWMA convolution chart. Finally, we investigate how seriously it can be when we misuse the Exponential-Exponential EWMA convolution chart as a Normal-Normal EWMA chart. To find the result, again with the assumption that the number of unit(s) X required to observe exactly 1 defect and the measurement error are both exponential random variables. The observed value be Y will be U  X ₂, where X and ₂ are independent. In such situation, the Exponential-Exponential EWMA convolution chart will be the best choice. Somehow, the engineer mistook it as normal distribution. Despite the distribution is wrong, the sample mean and variance still remain the same. Hence, the distribution of X and ₂ are：

) , (

~ ²

2 N  

X , ₂ ~ N(,²), when in-control

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) , (

~ ^{2 2}

2 N  

X , ₂ ~ N(,²), when out-of-control and the new random variable, V  X₂ ₂：

) ,

(

~N   ²²

V , when in-control

) ,

(

~ N  ²²²

V , when out-of-control

so the ARLs under different size of mean shift could be calculated and listed in Table 34. In Table 34, we set 0.05 since it outperforms the others. We would compare the ARLs between Exponential-Exponential EWMA convolution chart and Normal-Normal EWMA chart with the case ²/² 0.1 and 0.5. Note that

100

 .

‧

EWMA convolution chart and the mistaking of Normal-Normal EWMA chart (0.05)

chart Exp - Exp EWMA Nor - Nor EWMA

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the Exponential-Exponential EWMA convolution chart is 191.143 but it increases to 318.862 when we mistook it as the Normal-Normal EWMA chart. when  1.1 and consider a special case by assuming the measurement error parameter also shifts when the process mean shifts. Which means the parameter  will change to  when the process is out-of-control. Assume the sample mean and variance remain the same. So the distributions of X2 and ₃ are：

The ARLs under different size of mean shifts were calculated and listed in Table 35 with ²/² 0.1 and 0.5. Note that  100.

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Table 35 shows that when we mistook the Exponential-Exponential EWMA convolution chart as a Normal-Normal EWMA chart, the ARL would increase significantly under the same ²/² ratio. For instance, when  0.9 and ²/²

= 0.1, the Exponential-Exponential EWMA chart‘s ARL is 191.143 but it increases to 318.862 when we mistook it as a Normal-Normal EWMA chart. Similar result for the other  values and ^2/2 ratios. When  also deviate to  , the ARL1 is smaller than the case which  won‘t deviate. For instance, if ^{ 0.9}, the ARL1 is 318.862 if  won‘t deviate but decrease to 231.392 when  deviate to  . Which

= 0.1, the Exponential-Exponential EWMA chart‘s ARL is 191.143 but it increases to 318.862 when we mistook it as a Normal-Normal EWMA chart. Similar result for the other  values and ^2/2 ratios. When  also deviate to  , the ARL1 is smaller than the case which  won‘t deviate. For instance, if ^{ 0.9}, the ARL1 is 318.862 if  won‘t deviate but decrease to 231.392 when  deviate to  . Which

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