Chapter 4. Process Variance Change Investigation for Weibull Process
4.2. Monte-Carlo Simulation for Determining UCL and LCL
The main purpose of individuals control chart is assisting on identifying shifts or drifts in processes and it’s easily to be implemented. In this paper we study the effects on the capability estimates when the process output obeys gamma distribution with process variance change is remained unknown, so the
S control chart is a convention tool to monitor process variability and can help 2
us quickly determine whether the process is stable or not. But, when we adopt the control chart, some assumptions should be satisfied, such as the process characteristics must follow normal distribution. However, since our study is based on the gamma processes, violating the assumption, we will need to replace the traditional upper and lower control limits,
(
S n2 −1) χα22, 1n− and
(
S n2 −1) χ12−(α2 , 1)n− , as quantiles of the cumulative distribution function from
different parameters of Weibull(α β
, ), where S is an unbiased estimator of 2 σ
2.
In order to calculate the probability of misjudgment, one will first need to know the upper and lower control limits (UCL and LCL, respectively) of the process run. It is extremely difficult, if not impossible, to obtain the explicit formula of the distribution of sample variance when data follow gamma distribution. In this paper, Monte-Carlo simulation method was performed to investigate the behavior of sampling distribution of variance for gamma data and determine the estimated upper and lower control limits. So, in our study, the UCL and LCL are estimated through Monte-Carlo simulation method. The steps of Monte-Carlo algorithm to determine the control limits of S control 2 chart are summarized as follows:
Step1: We generate N preliminary samples from Weibull(α ,
β
), each of size k, and let Si be the variance of the ith sample.Step2: To sort S , we obtain i S <S <...<S , let (1) (2) (N)
ˆ t
p be the percentile for S . i For example N=10 , then ˆ6 t =S0.1 (10 )6 , soˆ t =S
p (N*p).Step3: The upper and lower control limits for Weibull(α ,
β
) can be estimated by ˆt0.99865and ˆt0.00135.4.3. Detection Power of
S for Weibull Data 2Utilizing the UCL and LCL obtained by Monte-Carlo approach, we derived the power of S for Weibull process data. Since the Type II error 2
β
is( )
β
=P LCL S≤ 2 ≤UCL|σ
=kσ
empirical cumulative distribution function of sample variance from Weibull distribution with that variance has changed and
σ
1 is the standard deviation after process change (σ
0 is the standard deviation of the original process). The control limits LCL and UCL are calculated as F0.00135 and F0.99865 respectively.We develop a Matlab program to compute the probability of process variance out of control limits. When process variance changes from
σ
2to k2×σ
2, and mean is fixed, the parameters α andβ
will change to new parameters α and '' β
, then we can obtain the detection power under the situation that the process variance changes. The parameters α and '' β
can be found by untilizing the following steps: adjustments. From Table 5, we observe the detection power gets larger as sample size (n) increasing.Table 5. Detection power for various weibull distributions.
Weibull(1,3)
Subgroup Size n Magnitude of
change in σ 9 10 11 12 13
1 0.00272 0.00286 0.00268 0.00266 0.00276 1.5 0.23732 0.26349 0.28855 0.31660 0.34155 2 0.58041 0.62368 0.66314 0.70087 0.73517 2.5 0.72655 0.76466 0.80016 0.83068 0.85645
3 0.78041 0.81628 0.84627 0.87217 0.89399 3.5 0.80037 0.83348 0.86098 0.88494 0.90451
Weibull(1,4)
Subgroup Size n Magnitude of
change in σ 9 10 11 12 13
1 0.00264 0.00277 0.00259 0.00273 0.00275 1.5 0.23987 0.27072 0.29748 0.32572 0.35763 2 0.64447 0.69275 0.73354 0.77036 0.80625 2.5 0.81486 0.85297 0.88250 0.90716 0.92722 3 0.87588 0.90519 0.92758 0.94490 0.95854 3.5 0.89843 0.92341 0.94229 0.95637 0.96773
Weibull(1,5)
Subgroup Size n Magnitude of
change in σ 9 10 11 12 13
1 0.00262 0.00261 0.00267 0.00271 0.00275 1.5 0.21636 0.24175 0.27207 0.29781 0.32945 2 0.65080 0.69954 0.74586 0.78218 0.81768 2.5 0.84854 0.88345 0.91227 0.93255 0.95023 3 0.91675 0.93968 0.95792 0.97026 0.97916 3.5 0.94173 0.95917 0.97233 0.98074 0.98706
4.4. Modified Standard Deviation Adjustment for Weibull Process
We set a given sample size (n) α
= 1
and givenβ
, then sampling large data (107) which are from Weibull distribution to estimate the control limits and compute the detection power of S for Weibull data with the given change 2 magnitude and n.From the mentioned above, we fix power =P LCL S
(
≤ 2 ≤UCL|σ
1 =kσ
0)
=0.5 to find k. We develop a Matlab program to compute the standard deviation change adjustment AS . The standard deviation adjustment means that
of =
β
1(1)11 and n=8(1)35. For example, ifβ
is 3 with n=10, the standard deviation change adjustment AS50 is 1.785. Whenβ
=1, AS50 are all greater than 5. From Figure 4, we find the shape is extraordinarily unlike the normal distribution whenβ
=1. When shape parameter is smaller than 1.5 (see Figure 5), we note that the distribution is a long-tail right skewed distribution.Why not discuss the relationship between AS50 and scale parameter α ? By Lu (2003), we can compute the probability of Xn when X1,...,Xn is a random of that, we may infer the standard deviation adjustments AS50 would not be affected by the scale parameter α . But, we can not prove this result theoretically.
Figure 6 depicts the AS curves of the Weibull process with scale parameter 50 α =1 and α =3 for subgroup sizes n=10, 15, and 20. It can be seen that the magnitude of standard deviation change would not change for α values
Table 6. AS50 values for several subgroup size n and various
β
values.AS50 Weibull distribution(1,
β
)n (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8) (1,9) (1,10) (1,11) 8 11.865 2.672 1.934 1.854 1.861 1.906 1.943 1.992 2.031 2.072 2.109 9 11.797 2.383 1.848 1.787 1.797 1.836 1.873 1.914 1.957 2.000 2.031 10 11.563 2.195 1.785 1.729 1.748 1.779 1.824 1.861 1.896 1.934 1.967 11 11.250 2.078 1.727 1.688 1.699 1.740 1.771 1.814 1.847 1.884 1.904 12 11.094 1.984 1.689 1.646 1.670 1.695 1.736 1.770 1.799 1.834 1.861 13 10.660 1.914 1.656 1.618 1.635 1.665 1.699 1.731 1.764 1.795 1.824 14 10.195 1.857 1.617 1.592 1.605 1.637 1.670 1.705 1.734 1.759 1.785 15 9.758 1.809 1.592 1.566 1.584 1.613 1.643 1.677 1.706 1.732 1.756 16 9.854 1.768 1.568 1.543 1.559 1.590 1.620 1.649 1.679 1.705 1.729 17 9.703 1.734 1.549 1.527 1.543 1.570 1.600 1.627 1.654 1.682 1.705 18 9.047 1.703 1.523 1.508 1.525 1.550 1.580 1.604 1.633 1.657 1.680 19 8.500 1.670 1.508 1.494 1.509 1.536 1.563 1.589 1.611 1.638 1.659 20 8.227 1.648 1.492 1.478 1.497 1.518 1.544 1.574 1.598 1.618 1.643 21 8.063 1.627 1.479 1.465 1.484 1.503 1.531 1.557 1.580 1.601 1.620 22 7.297 1.607 1.467 1.456 1.467 1.495 1.518 1.543 1.563 1.586 1.606 23 6.422 1.586 1.453 1.442 1.457 1.480 1.506 1.528 1.552 1.574 1.590 24 6.094 1.575 1.446 1.432 1.449 1.471 1.491 1.515 1.541 1.558 1.573 25 5.656 1.552 1.434 1.426 1.438 1.459 1.482 1.503 1.527 1.548 1.563 26 5.109 1.540 1.422 1.413 1.428 1.449 1.473 1.494 1.514 1.535 1.550 27 4.445 1.529 1.412 1.405 1.420 1.438 1.463 1.483 1.504 1.522 1.540 28 3.953 1.516 1.406 1.397 1.411 1.432 1.453 1.476 1.496 1.515 1.532 29 3.748 1.504 1.398 1.391 1.402 1.425 1.444 1.465 1.484 1.503 1.521 30 3.516 1.494 1.392 1.382 1.397 1.414 1.437 1.457 1.476 1.493 1.508 31 3.270 1.480 1.384 1.377 1.389 1.408 1.427 1.446 1.47 1.484 1.500 32 3.254 1.479 1.380 1.368 1.377 1.401 1.428 1.447 1.469 1.467 1.490 33 3.133 1.464 1.375 1.365 1.381 1.395 1.412 1.438 1.454 1.473 1.484 34 2.907 1.455 1.366 1.362 1.371 1.395 1.410 1.429 1.442 1.462 1.479 35 2.859 1.451 1.360 1.347 1.366 1.387 1.406 1.411 1.432 1.453 1.465
Figure 7 presents the power curves of estimated S2control chart for various sample size. The mean of power curve is detection power with various vary magnitude units for standard deviation. For small change in S all curves are 2 close to zero. As the magnitude of change creasing, so does the power of chart to detect it. The horizontal line drawn on this graph shows that is a 50% chance of missing a 1.777 times the size of standard deviation when n is 11, where as σ must change 1.873 times to have this same probability when n is 9.
Figure 6(a). The AS curves of 50 the Weibull process with α
= 1
for different n values on the horizontal.Figure 6(b). The AS curves of 50 the Weibull process with α
= 3
for different n values on the horizontal.Figure 6. AS values with different α values. 50
Figure 7. Power curves of estimated S2control chart for subgroup size 9, 10, and 11 when (
α β
, ) = (1,7).4.5. Capability Adjustment for Weibull Process
The index Cpk has been referred to as a yield-based index since it provides bound on the process on the process yield for a normality distribution process with a fixed value of Cpk.This index Cpk
is defined in chapter 1. The proper use
of process capability, is based on several assumptions. One of the most important assumption is that the process monitored is supposed to be stable and the output is approximately normal distribution.When the distribution of a process characteristic is non-normal, PCIs calculated using existing method often lead to erroneous and misleading interpretation of the process capability. Several approaches to the problems of PCIs for the non-normal populations have been suggested. Chen and Pearn (1997) consider come generalizations of these basic capability indices to cover non-normal distribution. In the non-normal case, if we are able to find a better
0.5
Acknowledging that a process will experience change in X0.99865−X0.5 or
0.5 0.00135
X −X of various magnitudes and knowing that not all of these will be discovered, some allowance for them must be made when estimating outgoing quality so customers are not disappointed. Because change ranging in times from 0 up to AS are the likely to remain undetected, a conservative approach it to 50 assume that every missed change it as large as AS50.
When utilizing dynamic Cpk to estimate process capability, we replace
0.99865 0.5 and 0.5 0.00135
X −X X −X withAS50(X0.99865−X0.5) and AS X( 0.5−X0.00135)in the Cpk formula just mentioned above, respective. Both of these adjustments are incorporated into the Cpk formula, now called the “dynamic” Cpk index, by making the following modifications:
Dynamic 0.5 0.5
50 0.5 0.00135 50 0.99865 0.5
LSL USL
50 0.5 50 0.00135 50 0.99865 50 0.5
LSL USL
min{ − , − }
= ⋅ − ⋅ ⋅ − ⋅
X X
AS X AS X AS X AS X
By including the adjustment in this assessment for undetected variance change, the estimate of capability decreases and the number nonconforming parts measured (calculated) will increase.
Chapter 5. Application
Surface-mount technology (SMT) is a method for constructing electronic circuits in which the components (SMC, or Surface Mounted Components) are mounted directly onto the surface of printed circuit boards (PCBs). Electronic devices so made are called surface-mount devices or SMDs. In the industry it has largely replaced the through-hole technology construction method of fitting components with wire leads into holes in the circuit board.
An SMT component is usually smaller than its through-hole counterpart because it has either smaller leads or no leads at all. It may have short pins or leads of various styles, flat contacts, a matrix of solder balls (BGAs), or terminations on the body of the component.
The SMD resistors come into several possible case sizes. Each size is described as a 4 digits number. The first 2 digits indicate the length; the last 2 indicate the width (in 0.01", or 10 mils units).Figure 9 display a view on common SMDs.For example, the three most popular sizes are “0603”, “0805”, and “1206”.
That mean 1.6×0.3mm, 1.8×0.5mm, and 11.2×0.6mm.
Figure 8. A view on common SMDs.
At SMT process, one of the most important factors is the size of the SMD.
The SMD resistor “0603” as shown in figure 9, we let the LSL and USL of the length for line segment “H” are 0.1mm and 0.5mm. This company utilize
S2control chart to monitor the process. Generally, S2charts are preferable to their more familiar counterparts, x
−
R charts, when either1. The sample size n is moderately large-say, n
> 10
or12.2. The sample size n is variable.
Figure 9. Dimension of SMD.
This company use n=10 to monitor the process. The collected sample data (a part of historical data) are shown in Table 7. From Figure 10, we use Minitab program to conclude the data collected from the factory are not normal distributed. The data analysis results justify that the process is significantly away from the normal distribution. By the goodness-of-fit tests as shown in figure 11, the historical data indicates that the process pretty approximates to be distributed as Weibull distribution. The parameters α and
β
of this Weibull process could be estimated from the historical data, giving αˆ 0.304 =
andβ
ˆ 6.299 by MLE =Using the maximum likelihood Equations (2) and (3), we can estimate
β
and α parameter when data are from Weibull distribution with the samples.Then we use the bootstrap to compute confidence intervals for shape parameter.
data
Figure 10. Histogram plot of the historical data.
C1
Figure 11. Weibull probability plot of the historical data.
Table 7. The 100 observations are collected from the historical data.
0.1795 0.2641 0.2689 0.3114 0.3333 0.2827 0.3735 0.2584 0.3206 0.2433 0.2018 0.3194 0.259 0.3329 0.2876 0.2795 0.2866 0.1837 0.3523 0.3727 0.3154 0.2916 0.3195 0.2989 0.2545 0.3281 0.2697 0.2405 0.3196 0.3498 0.3191 0.2816 0.2758 0.2636 0.3037 0.2802 0.3008 0.3152 0.2396 0.2844 0.3018 0.2514 0.3949 0.2572 0.3235 0.3631 0.3398 0.2659 0.2357 0.2052 0.3122 0.3035 0.2447 0.3932 0.3259 0.31 0.3268 0.2792 0.3152 0.2646
We utilize this control chart to monitor the process, and collect another new dara are shown in Table 8. By the goodness-of-fit tests as shown in figure 12, the new data indicates that the process pretty approximates to be distributed as Weibull distribution. The parameters α and
β
of this Weibull process could be estimated from the historical data, giving αˆ 0.3041 =
andβ
ˆ 0.5677 by MLE. =Figure 12. Weibull probability plot of the new data.
Table 8. The 100 observations of the new data.
0.3114 0.2577 0.3851 0.2976 0.2509 0.2785 0.2032 0.2085 0.3123 0.2167 0.2172 0.2125 0.3085 0.2436 0.3209 0.2847 0.297 0.159 0.3274 0.2532 0.2908 0.2597 0.2209 0.2545 0.3907 0.272 0.2348 0.3345 0.2425 0.2379 0.2448 0.3157 0.3358 0.1581 0.3013 0.2311 0.2884 0.3055 0.1951 0.3037 0.1431 0.3473 0.2516 0.3034 0.2848 0.2636 0.3981 0.307 0.4135 0.2855 0.3433 0.3175 0.2944 0.3351 0.1861 0.3503 0.2198 0.2506 0.3348 0.2551 0.2448 0.3551 0.308 0.2301 0.1826 0.3463 0.2598 0.3072 0.3279 0.2644 0.1497 0.2452 0.383 0.2449 0.3383 0.3208 0.3235 0.2054 0.3257 0.2866 0.3644 0.3269 0.286 0.2341 0.2872 0.2883 0.2513 0.3035 0.347 0.3135 0.2634 0.2871 0.33 0.3247 0.2325 0.3333 0.3359 0.1721 0.3007 0.2539
Therefore, it is appropriate to use this approach and we can obtain more accurate measures of three quantile: X0.00135 =0.102105, X0.5 =0.282017, and
0.99865 =0.411047
X under consideration. Then “dynamic” Cpk index can be calculate as follows:
0.5 0.5
50 0.99865 0.5 50 0.5 0.00135
USL LSL
dynamic min ,
( ) ( )
0.5 281962 0.281962 0.1
min ,
1.779 (0.411047 0.281962) 1.779 (0.281962 0.102105)
probability to be detected. For example, if n=15, the AS would be 1.613 for 50 Weibull distribution (α= 0.3
andβ
= 6 ) then0.5 0.5
50 0.99865 0.5 50 0.5 0.00135
USL LSL
dynamic min ,
( ) ( )
0.5 0.281962 0.281962 0.1
min ,
1.613 (0.411047 0.281962) 1.613 (0.281962 0.102105)
Chapter 6. Conclusion
This paper has considered the problem for adjusting estimates of process capability by including a variance change when data is from non-normal distribution. In the Bothe’ studies, statistically derived adjustments are proposed under the data assumed to be approximately normally distributed. But the case of non-normal processes occurs frequently in practice. We also provide tables for the engineers to use for their in-plant applications. However, this “Dynamic” C pk index assume
μ
remain stable when σ change. Ifμ
and σ subjected to undetected increases and decreases? Further studies are need to determine how those change would affect estimates of outgoing quality.References
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Appendix A. AS
50values for several subgroup sizes and various shape parameter.
Table 9. AS50 values for n=8(1)35 and
β
=12(1)21 values.AS50 Weibull distribution(1,
β
)n (1,12) (1,13) (1,14) (1,15) (1,16) (1,17) (1,18) (1,19) (1,20) (1,21) N(0,1) 8 2.154 2.167 2.198 2.231 2.255 2.267 2.285 2.309 2.319 2.334 1.928 9 2.062 2.093 2.110 2.140 2.156 2.176 2.206 2.219 2.229 2.243 1.858 10 1.998 2.021 2.051 2.063 2.082 2.104 2.122 2.137 2.152 2.171 1.802 11 1.941 1.957 1.988 2.008 2.029 2.045 2.057 2.075 2.090 2.096 1.755 12 1.887 1.914 1.936 1.951 1.973 1.983 2.004 2.017 2.032 2.052 1.716 13 1.848 1.869 1.887 1.906 1.926 1.941 1.954 1.970 1.983 1.995 1.682 14 1.807 1.826 1.852 1.873 1.885 1.899 1.916 1.933 1.937 1.959 1.652 15 1.777 1.801 1.820 1.834 1.852 1.863 1.882 1.893 1.904 1.910 1.626 16 1.752 1.771 1.791 1.805 1.822 1.832 1.842 1.855 1.867 1.880 1.602 17 1.728 1.741 1.762 1.777 1.791 1.801 1.812 1.826 1.839 1.849 1.581 18 1.699 1.720 1.736 1.754 1.770 1.773 1.789 1.803 1.810 1.816 1.562 19 1.677 1.699 1.711 1.726 1.741 1.754 1.765 1.769 1.784 1.790 1.545 20 1.656 1.678 1.688 1.705 1.719 1.729 1.744 1.756 1.763 1.769 1.529 21 1.641 1.656 1.671 1.688 1.695 1.715 1.722 1.738 1.744 1.749 1.514 22 1.625 1.639 1.652 1.670 1.682 1.693 1.698 1.715 1.721 1.728 1.501 23 1.610 1.622 1.639 1.648 1.659 1.678 1.681 1.692 1.705 1.712 1.488 24 1.592 1.610 1.623 1.637 1.644 1.661 1.669 1.680 1.689 1.693 1.477 25 1.578 1.595 1.608 1.622 1.631 1.644 1.648 1.660 1.672 1.680 1.466 26 1.564 1.582 1.592 1.605 1.619 1.625 1.634 1.643 1.656 1.663 1.456 27 1.557 1.567 1.580 1.595 1.607 1.617 1.626 1.631 1.638 1.650 1.446 28 1.547 1.555 1.572 1.580 1.594 1.604 1.611 1.619 1.625 1.633 1.438 29 1.535 1.549 1.558 1.570 1.582 1.592 1.598 1.607 1.612 1.623 1.429 30 1.521 1.534 1.551 1.561 1.571 1.578 1.585 1.594 1.601 1.608 1.421 31 1.521 1.526 1.545 1.544 1.556 1.571 1.573 1.583 1.590 1.600 1.414 32 1.510 1.520 1.528 1.542 1.545 1.560 1.568 1.566 1.571 1.599 1.406 33 1.496 1.507 1.521 1.524 1.539 1.543 1.546 1.564 1.575 1.566 1.400 34 1.488 1.504 1.506 1.522 1.532 1.540 1.548 1.551 1.557 1.572 1.393 35 1.481 1.492 1.507 1.513 1.537 1.528 1.523 1.537 1.552 1.558 1.387
Table 10. AS50 values for n=8(1)35 and
β
=22(1)31 values.AS50 Weibull distribution(1,
β
)n (1,22) (1,23) (1,24) (1,25) (1,26) (1,27) (1,28) (1,29) (1,30) (1,31) N(0,1) 8 2.346 2.373 2.375 2.390 2.388 2.413 2.423 2.436 2.435 2.438 1.928 9 2.253 2.277 2.277 2.284 2.298 2.313 2.313 2.331 2.343 2.345 1.858 10 1.998 2.021 2.051 2.063 2.082 2.104 2.122 2.137 2.152 2.171 1.802 11 1.941 1.957 1.988 2.008 2.029 2.045 2.057 2.075 2.090 2.096 1.755 12 1.887 1.914 1.936 1.951 1.973 1.983 2.004 2.017 2.032 2.052 1.716 13 1.848 1.869 1.887 1.906 1.926 1.941 1.954 1.970 1.983 1.995 1.682 14 1.807 1.826 1.852 1.873 1.885 1.899 1.916 1.933 1.937 1.959 1.652 15 1.777 1.801 1.820 1.834 1.852 1.863 1.882 1.893 1.904 1.910 1.626 16 1.752 1.771 1.791 1.805 1.822 1.832 1.842 1.855 1.867 1.880 1.602 17 1.728 1.741 1.762 1.777 1.791 1.801 1.812 1.826 1.839 1.849 1.581 18 1.699 1.720 1.736 1.754 1.770 1.773 1.789 1.803 1.810 1.816 1.562 19 1.677 1.699 1.711 1.726 1.741 1.754 1.765 1.769 1.784 1.790 1.545 20 1.656 1.678 1.688 1.705 1.719 1.729 1.744 1.756 1.763 1.769 1.529 21 1.641 1.656 1.671 1.688 1.695 1.715 1.722 1.738 1.744 1.749 1.514 22 1.625 1.639 1.652 1.670 1.682 1.693 1.698 1.715 1.721 1.728 1.501 23 1.610 1.622 1.639 1.648 1.659 1.678 1.681 1.692 1.705 1.712 1.488 24 1.592 1.610 1.623 1.637 1.644 1.661 1.669 1.680 1.689 1.693 1.477 25 1.578 1.595 1.608 1.622 1.631 1.644 1.648 1.660 1.672 1.680 1.466 26 1.564 1.582 1.592 1.605 1.619 1.625 1.634 1.643 1.656 1.663 1.456 27 1.557 1.567 1.580 1.595 1.607 1.617 1.626 1.631 1.638 1.650 1.446 28 1.547 1.555 1.572 1.580 1.594 1.604 1.611 1.619 1.625 1.633 1.438 29 1.535 1.549 1.558 1.570 1.582 1.592 1.598 1.607 1.612 1.623 1.429 30 1.521 1.534 1.551 1.561 1.571 1.578 1.585 1.594 1.601 1.608 1.421 31 1.615 1.605 1.618 1.612 1.619 1.641 1.625 1.612 1.661 1.651 1.414 32 1.583 1.593 1.602 1.611 1.617 1.624 1.620 1.611 1.638 1.624 1.406 33 1.571 1.595 1.602 1.592 1.594 1.604 1.613 1.592 1.620 1.612 1.400 34 1.569 1.571 1.590 1.584 1.589 1.595 1.611 1.584 1.610 1.611 1.393 35 1.561 1.559 1.579 1.575 1.582 1.583 1.598 1.575 1.601 1.610 1.387
Appendix B. The Average Run Length of Weibull Distributions.
Table 11. Average run length of Weibull with 1.5 times standard deviation change.
ARL Weibull(1,
β
) Normaln 1 2 3 3.6 4 5 6 7 8 9 10 N(0,1) 7 20.964 7.457 5.467 5.147 5.671 6.186 7.251 8.622 9.547 11.531 12.376 7.222 8 19.033 6.660 4.664 4.616 4.757 5.211 6.068 7.179 8.497 9.633 10.723 6.175 9 17.762 6.044 4.158 4.033 4.200 4.612 5.450 6.115 7.350 7.700 8.971 5.402 10 15.941 5.191 3.905 3.582 3.694 4.084 4.446 5.132 5.959 6.854 7.792 4.768 11 15.340 4.683 3.457 3.240 3.359 3.707 4.209 5.050 5.393 6.096 6.671 4.272 12 14.51 4.475 3.179 3.002 3.06 3.392 3.775 4.381 4.592 5.414 5.849 3.840 13 14.055 4.166 2.912 2.826 2.827 3.025 3.436 3.937 4.285 4.78 5.552 3.542 14 13.300 3.826 2.774 2.627 2.648 2.789 3.212 3.674 3.866 4.557 4.854 3.222 5 12.781 3.618 2.527 2.482 2.448 2.720 2.847 3.287 3.712 4.136 4.732 2.992 16 12.021 3.421 2.421 2.296 2.288 2.514 2.801 2.914 3.450 3.835 4.113 2.781 17 11.639 3.283 2.278 2.113 2.134 2.193 2.524 2.908 3.062 3.273 3.835 2.594 18 11.312 3.033 2.154 1.988 2.070 2.178 2.388 2.695 2.935 3.270 3.540 2.450 19 11.018 2.897 2.053 1.983 1.927 2.020 2.234 2.442 2.882 3.003 3.497 2.328 20 10.416 2.749 1.983 1.851 1.848 1.986 2.135 2.313 2.576 2.808 3.173 2.203 21 9.966 2.651 1.875 1.782 1.776 1.895 1.981 2.284 2.365 2.813 2.964 2.094 22 9.838 2.545 1.784 1.731 1.714 1.815 1.929 2.127 2.314 2.618 2.802 2.000 23 9.532 2.340 1.738 1.69 1.658 1.746 1.859 2.028 2.299 2.456 2.705 1.924 24 9.231 2.346 1.679 1.611 1.612 1.647 1.772 1.986 2.209 2.194 2.500 1.842 25 8.797 2.348 1.645 1.564 1.568 1.606 1.658 1.876 2.066 2.156 2.380 1.777 26 8.365 2.184 1.57 1.498 1.528 1.538 1.646 1.860 1.962 2.064 2.255 1.711 27 8.103 2.152 1.528 1.478 1.480 1.496 1.624 1.719 1.923 1.998 2.174 1.668 28 7.983 2.089 1.514 1.448 1.446 1.445 1.575 1.664 1.759 1.942 2.039 1.619 29 7.508 2.038 1.480 1.411 1.387 1.430 1.504 1.632 1.746 1.827 1.980 1.560 30 7.432 2.007 1.449 1.359 1.399 1.379 1.464 1.599 1.686 1.812 1.908 1.526 31 7.268 1.898 1.406 1.349 1.350 1.368 1.418 1.524 1.651 1.782 1.907 1.494 1.710 1.846 1.451
Table 12. Average run length of Weibull with 2 times standard deviation change.
ARL Weibull(1,
β
) Normaln 1 2 3 3.6 4 5 6 7 8 9 10 N(0,1) 7 8.388 3.027 2.101 1.952 1.891 1.883 1.948 2.075 2.246 2.36 2.564 2.046 8 7.722 2.65 1.884 1.766 1.711 1.688 1.739 1.868 1.949 2.114 2.184 1.814 9 7.179 2.466 1.736 1.607 1.545 1.523 1.578 1.652 1.704 1.814 1.947 1.635 10 6.722 2.299 1.614 1.497 1.433 1.434 1.481 1.543 1.589 1.705 1.800 1.510 11 6.629 2.103 1.512 1.394 1.380 1.361 1.375 1.427 1.473 1.567 1.638 1.416 12 6.360 1.993 1.422 1.323 1.297 1.277 1.297 1.341 1.380 1.455 1.514 1.335 13 5.543 1.889 1.377 1.282 1.246 1.216 1.244 1.262 1.339 1.398 1.445 1.276 14 5.423 1.780 1.304 1.230 1.205 1.188 1.195 1.219 1.264 1.321 1.349 1.142 5 5.333 1.685 1.265 1.184 1.164 1.146 1.161 1.196 1.223 1.256 1.286 1.187 16 5.128 1.643 1.221 1.155 1.132 1.119 1.136 1.157 1.181 1.213 1.245 1.155 17 4.853 1.569 1.199 1.139 1.110 1.098 1.107 1.121 1.181 1.165 1.208 1.126 18 4.638 1.542 1.170 1.113 1.090 1.078 1.088 1.106 1.126 1.139 1.177 1.105 19 4.543 1.449 1.146 1.093 1.079 1.067 1.073 1.085 1.094 1.12 1.140 1.087 20 4.480 1.417 1.125 1.074 1.070 1.053 1.059 1.069 1.086 1.099 1.118 1.072 21 4.236 1.397 1.112 1.069 1.053 1.044 1.049 1.056 1.066 1.083 1.096 1.059 22 4.024 1.360 1.096 1.058 1.044 1.037 1.037 1.048 1.055 1.066 1.083 1.049 23 3.919 1.319 1.078 1.047 1.039 1.032 1.031 1.041 1.045 1.060 1.070 1.041 24 3.865 1.300 1.072 1.041 1.031 1.023 1.027 1.030 1.037 1.046 1.06 1.034 25 3.576 1.274 1.061 1.033 1.027 1.02 1.021 1.024 1.03 1.037 1.048 1.028 26 3.703 1.255 1.056 1.029 1.021 1.017 1.017 1.021 1.026 1.032 1.042 1.022 27 3.528 1.232 1.045 1.024 1.019 1.013 1.015 1.016 1.021 1.028 1.036 1.018 28 3.403 1.208 1.041 1.021 1.014 1.011 1.012 1.014 1.018 1.024 1.031 1.015 29 3.277 1.199 1.034 1.017 1.012 1.008 1.008 1.011 1.014 1.019 1.025 1.013 30 3.111 1.181 1.028 1.015 1.010 1.006 1.006 1.009 1.012 1.015 1.021 1.011 31 3.097 1.171 1.027 1.012 1.009 1.005 1.005 1.007 1.010 1.014 1.016 1.008 32 3.050 1.156 1.022 1.010 1.007 1.004 1.004 1.006 1.008 1.011 1.013 1.007 33 2.963 1.145 1.021 1.009 1.005 1.003 1.004 1.005 1.006 1.008 1.011 1.006 34 2.854 1.132 1.017 1.007 1.004 1.003 1.003 1.003 1.005 1.006 1.009 1.004 35 2.781 1.129 1.015 1.006 1.004 1.002 1.002 1.003 1.004 1.006 1.007 1.004
Table 13. Average run length of Weibull with 2.5 times standard deviation change.
ARL Weibull(1,
β
) Normaln 1 2 3 3.6 4 5 6 7 8 9 10 N(0,1) 7 6.064 2.286 1.597 1.458 1.403 1.339 1.327 1.34 1.384 1.44 1.485 1.338 8 5.551 2.085 1.466 1.35 1.313 1.247 1.231 1.253 1.264 1.299 1.333 1.239 9 5.067 1.926 1.384 1.273 1.224 1.184 1.162 1.174 1.194 1.228 1.24 1.173 10 4.786 1.798 1.307 1.211 1.172 1.133 1.125 1.126 1.145 1.157 1.175 1.126 11 4.56 1.674 1.259 1.169 1.127 1.094 1.088 1.098 1.105 1.115 1.134 1.092 12 4.343 1.618 1.201 1.129 1.101 1.072 1.062 1.069 1.073 1.081 1.103 1.066 13 4.171 1.525 1.173 1.103 1.08 1.051 1.047 1.048 1.054 1.064 1.077 1.047 14 3.896 1.467 1.137 1.083 1.065 1.039 1.033 1.035 1.038 1.047 1.052 1.035 5 3.895 1.413 1.115 1.069 1.049 1.03 1.025 1.027 1.027 1.034 1.039 1.026 16 3.616 1.373 1.093 1.053 1.036 1.022 1.017 1.019 1.02 1.025 1.03 1.019 17 3.545 1.318 1.08 1.042 1.03 1.018 1.012 1.012 1.015 1.016 1.019 1.013 18 3.434 1.302 1.068 1.033 1.022 1.012 1.01 1.01 1.01 1.013 1.015 1.01 19 3.319 1.267 1.057 1.025 1.017 1.008 1.007 1.008 1.007 1.008 1.012 1.007 20 3.175 1.24 1.045 1.02 1.013 1.007 1.004 1.005 1.006 1.007 1.008 1.005 21 3.044 1.215 1.038 1.016 1.01 1.004 1.003 1.003 1.004 1.005 1.006 1.003 22 3.041 1.196 1.033 1.013 1.007 1.003 1.003 1.002 1.002 1.003 1.004 1.003 23 2.921 1.174 1.026 1.01 1.006 1.002 1.002 1.002 1.002 1.002 1.003 1.002 24 2.765 1.16 1.021 1.009 1.005 1.002 1.001 1.001 1.001 1.002 1.002 1.001 25 2.72 1.146 1.019 1.006 1.003 1.002 1.001 1.001 1.001 1.001 1.001 1.001 26 2.662 1.135 1.016 1.005 1.003 1.001 1.001 1.001 1.001 1.001 1.001 1.001 27 2.6 1.119 1.012 1.004 1.002 1.001 1 1 1 1.001 1.001 1 28 2.521 1.108 1.011 1.003 1.001 1 1 1 1 1 1 1 29 2.446 1.103 1.008 1.002 1.001 1 1 1 1 1 1 1 30 2.4 1.087 1.008 1.002 1.001 1 1 1 1 1 1 1 31 2.393 1.08 1.006 1.001 1.001 1 1 1 1 1 1 1 32 2.289 1.072 1.005 1.001 1 1 1 1 1 1 1 1 33 2.279 1.067 1.004 1.001 1 1 1 1 1 1 1 1
Table 14. Average run length of Weibull with 3 times standard deviation change.
ARL Weibull(1,
β
) Normaln 1 2 3 3.6 4 5 6 7 8 9 10 N(0,1) 7 4.773 2.081 1.449 1.315 1.267 1.192 1.155 1.157 1.161 1.168 1.178 1.139 8 4.494 1.896 1.352 1.24 1.193 1.131 1.109 1.105 1.1 1.115 1.122 1.09 9 4.303 1.747 1.278 1.18 1.143 1.091 1.072 1.066 1.07 1.073 1.079 1.059 10 3.993 1.642 1.224 1.137 1.107 1.062 1.047 1.042 1.044 1.045 1.051 1.037 11 3.825 1.563 1.18 1.105 1.078 1.041 1.032 1.029 1.029 1.03 1.033 1.024 12 3.739 1.495 1.145 1.081 1.058 1.031 1.022 1.019 1.019 1.02 1.021 1.016 13 3.515 1.422 1.121 1.064 1.043 1.022 1.014 1.013 1.012 1.013 1.014 1.009 14 3.411 1.377 1.097 1.049 1.034 1.014 1.01 1.007 1.008 1.008 1.009 1.007 5 3.334 1.334 1.084 1.038 1.025 1.012 1.006 1.005 1.005 1.005 1.006 1.004 16 3.125 1.301 1.065 1.029 1.018 1.008 1.004 1.003 1.003 1.003 1.004 1.002 17 2.996 1.265 1.054 1.023 1.014 1.005 1.003 1.002 1.002 1.002 1.002 1.002 18 2.933 1.239 1.046 1.017 1.01 1.003 1.002 1.001 1.001 1.001 1.002 1.001 19 2.836 1.216 1.035 1.014 1.007 1.002 1.001 1.001 1.001 1.001 1.001 1.001 20 2.735 1.192 1.03 1.011 1.006 1.002 1.001 1.001 1.001 1 1.001 1 21 2.661 1.173 1.025 1.008 1.004 1.001 1 1 1 1 1 1 22 2.578 1.153 1.02 1.006 1.003 1.001 1 1 1 1 1 1 23 2.507 1.138 1.016 1.005 1.002 1 1 1 1 1 1 1 24 2.479 1.124 1.014 1.004 1.002 1 1 1 1 1 1 1 25 2.419 1.115 1.011 1.003 1.001 1 1 1 1 1 1 1 26 2.307 1.105 1.009 1.002 1.001 1 1 1 1 1 1 1 27 2.344 1.09 1.007 1.002 1.001 1 1 1 1 1 1 1 28 2.231 1.08 1.006 1.001 1.001 1 1 1 1 1 1 1 29 2.181 1.073 1.005 1.001 1.001 1 1 1 1 1 1 1
30 2.164 1.07 1.004 1.001 1 1 1 1 1 1 1 1
31 2.104 1.061 1.003 1.001 1 1 1 1 1 1 1 1
32 2.078 1.054 1.003 1 1 1 1 1 1 1 1 1
33 2.025 1.05 1.002 1 1 1 1 1 1 1 1 1
34 2.016 1.046 1.002 1 1 1 1 1 1 1 1 1
35 1.971 1.04 1.001 1 1 1 1 1 1 1 1 1
Table 15. Average run length of Weibull with 3.5 times standard deviation change.
ARL Weibull(1,
β
) Normaln 1 2 3 3.6 4 5 6 7 8 9 10 N(0,1) 7 4.198 1.98 1.406 1.274 1.216 1.141 1.109 1.091 1.085 1.082 1.083 1.064 8 3.941 1.813 1.318 1.199 1.153 1.094 1.067 1.053 1.05 1.053 1.051 1.037 9 3.722 1.679 1.255 1.15 1.113 1.061 1.041 1.034 1.029 1.029 1.031 1.022 10 3.605 1.589 1.202 1.116 1.082 1.042 1.026 1.019 1.018 1.017 1.017 1.013 11 3.442 1.52 1.159 1.087 1.064 1.028 1.017 1.013 1.011 1.01 1.011 1.007 12 3.263 1.452 1.131 1.067 1.044 1.02 1.01 1.007 1.006 1.006 1.006 1.004 13 3.153 1.398 1.102 1.053 1.035 1.013 1.007 1.004 1.004 1.003 1.003 1.002 14 3.040 1.349 1.088 1.039 1.026 1.009 1.004 1.003 1.002 1.002 1.002 1.001 5 2.962 1.313 1.068 1.031 1.019 1.006 1.003 1.002 1.001 1.001 1.001 1.001 16 2.855 1.275 1.056 1.024 1.014 1.004 1.002 1.001 1.001 1.001 1.001 1.001 17 2.791 1.248 1.048 1.019 1.01 1.003 1.001 1 1 1 1 1 18 2.674 1.214 1.04 1.015 1.008 1.002 1.001 1 1 1 1 1 19 2.578 1.194 1.032 1.011 1.005 1.001 1 1 1 1 1 1 20 2.508 1.176 1.025 1.009 1.004 1.001 1 1 1 1 1 1 21 2.459 1.156 1.021 1.006 1.003 1.001 1 1 1 1 1 1 22 2.39 1.147 1.019 1.005 1.002 1 1 1 1 1 1 1 23 2.37 1.127 1.015 1.004 1.002 1 1 1 1 1 1 1 24 2.247 1.114 1.012 1.003 1.001 1 1 1 1 1 1 1 25 2.249 1.105 1.01 1.003 1.001 1 1 1 1 1 1 1 26 2.175 1.092 1.008 1.002 1.001 1 1 1 1 1 1 1 27 2.129 1.086 1.007 1.001 1 1 1 1 1 1 1 1 28 2.089 1.075 1.006 1.001 1 1 1 1 1 1 1 1
29 2.035 1.07 1.005 1.001 1 1 1 1 1 1 1 1
30 2.012 1.062 1.003 1.001 1 1 1 1 1 1 1 1 31 1.994 1.058 1.003 1.001 1 1 1 1 1 1 1 1
32 1.938 1.052 1.003 1 1 1 1 1 1 1 1 1
33 1.911 1.045 1.002 1 1 1 1 1 1 1 1 1