Chapter 3. Constructions of the Median Loss, EWMA and Optimal VSI Median Loss
3.1 Construction of the Median Loss Control Chart
3.1.2 Performance Measurement of the Median Loss chart
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3.1.2 Performance Measurement of the Median Loss chart
3.1.2.1 Biased ARL
In this section, we use Average Run Length (ARL) to measure the performance of ML chart. ARL is the average number of samples before the control chart produces a signal, which is the most popular performance measure for a control chart.
The in-control process ARL (ARL0) is always fixed in a request level, for example 370.4, while the out-of-control process ARL (ARL1) is as small as better.
The ARL0 for the ML chart is
1
0
ARL
, (3-2)where
P ( LCL ML UCL | in control ML )
is the false alarm rate.We are going to derive the out-of-control distribution of the sample median loss to calculate ARL1 values of the ML chart. Suppose that
X is the quality
* characteristic when process going out-of-control, andX
* ~SN
(
*,a
*,b
) with expectation
1
0
1
0,
1
0, and standard deviation
1
2
0,
2
1, then we have:2 2
0 2 0
1 0
*
2 ) 1 (
2
b b
b
,) 1 (
1 2 2
2 0
* 2
b b a
.‧
Following the same proving procedure in section 3.1 and 3.2, we can obtain the pdf and cdf of
ML as
*average number of samples before the control chart produces a signal when process is out-of-control, that is
‧
that ML chart performs better when the distribution of X is left-skewed. In most of the cases the ARL1 of the ML chart increases when
1 and/or
2 increase. When large shift of
1 (
1 2
) and moderate to large shift of
2 (
2 1 . 5
), the ARL1 values have no much difference no matter the values ofb
.Table 3-2.1 ARL
1 of ML Chart‧
Table 3-2.1 (Continued)
delta3
‧
Table 3-2.1 (Continued)
delta3
‧
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Fig. 3-1 to Fig. 3-3 show the control limits of the ML chart and various pdfs of
ML for
*
1 1 , 2 , 3
and
2 1
under 0 . 0027
, n=5,
0
1,
3 =1 and500 , 0 ,
500
b
respectively. The black curve is the in-control pdf of ML. Fig. 3-1 to Fig. 3-3 show that the mode off
ML*( )
shifts to right and downward when
1 goes from 1 to 3 under
2 1
, no matter what values ofb
. It indicates that the probability ofML falls outside the control limits increases or the detection ability of
* the chart increases.Fig. 3-1 b 500
,
2=1Fig. 3-2 b 0
,
2=1‧ 國
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Fig. 3-3 b 500
,
2=1Fig. 3-4 to Fig. 3-6 show that the mode of
f
ML*( )
shifts to right and downwardwhen
2 shift from 1.5 to 3 for
1 0
, 0 . 0027
, n=5,
0
1,
3=1 and500
, 0 ,
500
b
. It indicates that the probability ofML falls outside the control
* limits increases or the detection ability of the chart increases. Note that whenb 500
the shape off
ML*( )
becomes less smooth thanb 500
andb 0
.Fig. 3-4 b 500
,
1=0‧ 國
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Fig. 3-5 b 0
,
1=0Fig. 3-6 b 500
,
1=0Fig. 3-7 to Fig. 3-9 show that the mode of
f
ML*( )
shifts rightward and downward when
1 shifts from 1 to 3 for
2 3
, 0 . 0027
, n=5,
0
1,
3=1 andb 500 , 0 , 500
. It indicates that the probability ofML fall outside the control
* limits increases, hence increases the detection ability.‧ 國
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Fig. 3-7 b 500
,
2 3
Fig. 3-8 b 0
,
2=3Fig. 3-9 b 500
,
2 3
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Fig. 3-10 to Fig. 3-12 show that the mode of
f
ML*( )
shifts leftward and downward when
2 shifts from 1.5 to 3 for
1 3
, 0 . 0027
,n 5
,
0
1,3
1
andb 500 , 0 , 500
. However, since the detection ability of the ML chart increases when the mode off
ML*( )
goes rightward rather than leftward. It causes some ARL1 values not strictly decreasing while
2 increases. When either
1 or
2 is large, the ARL1 values are all close to 1 and have no much difference.In summary for Fig 3-1 to Fig 3-12, we conclude that:
(1) For
1 1 , 2 , 3
,
2 1
under
3
1 andb 500 , 0 , 500
the ARL1 is strictly decreasing when
1 increases.(2) For
1 0
,
2 1 . 5 , 2 , 2 . 5 , 3
under
3
1 andb 500 , 0
the ARL1 is strictly decreasing when
2 increases.(3) For
1 1 , 2 , 3
,
2 3
under
3
1 andb 500 , 0
the ARL1 is strictly decreasing when
1 increases.(4) In other situations the ARL1 of the ML chart may not be strictly decreasing when
1 and/or
2 increases under
3
1 andb 500 , 0 , 500
. When either
1 or
2 is large, the ARL1 values are all very close to 1.‧ 國
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Fig. 3-10 b 500
,
1=3Fig. 3-11 b 0
,
1=3Fig. 3-12 b 500
,
1=3‧ 國
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To find how the ARL1 is influenced by
1,
2,
3 orb
, the response tableand diagram of average values of ARL1 (
ARL
1) verses
1,
2,
3 orb
are obtained in Table 3-3 and in Figure 3-13 based on the ARL1s for various combinations of
1,
2,
3 orb
in Table 3-2. We found that, the ARL1 decreases when
1 or
2 increases, and increases whenb
increases. The ARL1 is sensitive to the change of
1 and
2, but is not sensitive to the change of
3.Table 3-3 Response table for ARL
1 of ML chartlevel delta1 delta2 delta3 b 1 6.484 4.278 2.767 1.460 2 3.747 2.774 2.803 2.555 3 2.429 2.282 2.950 4.493 4 1.759 2.009 2.913 -
5 1.397 - 2.746 -
6 1.198 - - -
Difference 5.286 2.270 0.204 3.034
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3.1.2.2 Unbiased ARL Calculation for the Median Loss Chart
Pignatiello, et al. (1995) defined the concept of an ARL-unbiased control chart, as they state “the control chart is said to be ARL-unbiased if its ARL curve achieves its maximum when the process parameter is equal to its in-control value”. That is, the unbiased-ARL1 is always smaller than ARL0 even for asymmetric distributed monitoring statistics. However, the biased ARL1 do not have the reasonable property.
Knoth and Morais (2013) proposed an ARL-unbiased
S
2 chart and a EWMA-S
2 chart to monitor process dispersion, based on the concept of Pignatiello, et al. (1995).Pascual (2013) proposes combined individual and moving range charts based on ARL-unbiasedness for monitoring changes in either the Weibull scale or shape parameter. So far using a single chart to joint monitor process mean and variance based on unbiased ARL has yet not been studied.
Fig 3.14 shows the ARLs of the ML chart in (3-1). We can see that the ARL1s of the ML chart are larger than ARL0 =370.4 for
2
1 2
and
2 1
. Hence the ARL1s are biased.‧ 國
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Fig. 3-14 Biased ARL of the ML chart
( 0 . 0027
,
0
0,
0
1,
3
0 andb 500
)To let the ML chart have reasonable ARL1 when
1 0
and/or
2 1
, we need to adjust its control limits using the ARL-unbiased approach proposed by Sven and Manuel (2013). Let
ML and
ML be the in-control mean and standard deviation of ML, we redefine the control limits of the ML chart asML ML
ML ML
L LCL
L UCL
1 2
. (3-5)The
ML and
ML in (3-5) can be calculated by letting
0t fML(t| 0,a0,b,n, 3)dt
ML
and
0 0 0 32
2 (t ML) fML(t| ,a ,b,n, )dt
ML
.Consider the out-of-control process with
1
0
1
0,
1 0 and
1
2
0,2 1
. Therefore the ARL-unbiased control limits coefficientsL
1 andL
2 in (3-5)‧
The Second Partial Derivative Test, for example see J. Stewart (2005), is applied to check whether the (L1,L2) values let ARL0 be a local maximum. Let
To check that the ARL0 is a local maximum, the Second Partial Derivative Test asserts
If
( 0 , 1 ) 0
‧
approach of these constraint is to satisfy
3
‧
values which can satisfy the following condition0
(the Second Partial Derivative Test represents that the ARL0 is a local maximum when
1 within
1 and
2 within1
2,
1,
2 0
). 500 respectively. The results are shown in Table 3-14. We found that the solutions of (L1,L2) can be obtained only whenb 0
and
3
0. However, in the cases with eitherb 500 , 500
or
3
1there is no solution.‧ 國
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Table 3-14 Control Limits of the Unbiased-ARL
1 ML chart underb 0
and
3
0n ML ML L1 L2 LCL UCL ARL0
3 0.705 0.766 0.918 7.659 0.001 6.571 370.370 5 0.616 0.561 1.087 6.767 0.006 4.411 370.370 7 0.574 0.458 1.222 6.218 0.014 3.420 370.370 11 0.533 0.351 1.424 5.568 0.033 2.488 370.370 15 0.513 0.294 1.569 5.186 0.051 2.039 370.370 31 0.483 0.198 1.898 4.488 0.108 1.371 370.370 51 0.472 0.152 2.099 4.141 0.153 1.102 370.370
Step 6. Find the interval of
1 and
2 such that ARL1 < ARL0, which is given in Table 3-15.The unbiased ARL1 property is checked by graphical approach, as in Figure 3-15.
We can see that the adjusted control limits (L1,L2) ensure that the ARL1 is unbiased for either mean (
1 0
) or variance (
2 1
) change, shown as the red line and the yellow line in the figure. However, the ARL0 may not be the global maximum, although ithas ensured to be a local maximum by the Second Partial Derivative Test.For example, if
2 1
and
1 lies within 2
the ARL1 may exceed ARL0.‧ 國
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(a)
n 5
(b)n 51
Fig. 3-15 The ARL of ARL
1-unbiased ML chart ( 0 . 0027
,
0
0,
0
1,
3
0 andb 0
)Table 3-15 lists the (L1,L2) obtained in Table 3-14 and the corresponding interval of
1 such that ARL1 be unbiased under
0
0,
0
1,
3
0,b 0
and2 , 5 . 1 , 0 . 1 , 8 . 0 , 5 .
2
0
respectively. We found the intervals of
1 in Table 3-15 becomes narrower when sample size increases.‧ 國
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Table 3-15 Values of
(L1,L2) and the combination of
1 and
2 such that ARL1 < ARL0n
3 b L1 L2
20.5 0.8 1.0 1.5 2
Intervals of
1 such that ARL1 < ARL03 0 0 0.918 7.659
(–∞, –1.626), (–0.561, 0.561),
(1.626, ∞)
(–∞, –1.003), (–0.467, 0.467),
(1.003, ∞)
(–∞, ∞) (–∞, ∞) (–∞, ∞)
5 0 0 1.087 6.767
(–∞, –1.351), (–0.563, 0.563),
(1.351, ∞),
(–∞, –0.847) (–0.474, 0.474)
(0.847, ∞)
(–∞, ∞) (–∞, ∞) (–∞, ∞)
7 0 0 1.222 6.218
(–∞, –1.208) (–0.566, 0.566)
(1.208, ∞)
(–∞, –0.774) (–0.48, 0.48) (0.774, ∞)
(–∞, ∞) (–∞, ∞) (–∞, ∞)
11 0 0 1.424 5.568
(–∞, –1.06) (–0.572, 0.572)
(1.06, ∞)
(–∞, –0.705) (–0.488, 0.488)
(0.705, ∞)
(–∞, ∞) (–∞, ∞) (–∞, ∞)
15 0 0 1.569 5.186
(–∞, -0.984) (–0.577, 0.577)
(0.984, ∞)
(–∞, –0.672) (–0.493, 0.493)
(0.672, ∞)
(–∞, ∞) (–∞, ∞) (–∞, ∞)
31 0 0 1.898 4.488
(–∞, –0.86) (–0.591, 0.591)
(0.86, ∞)
(–∞, –0.622) (–0.505, 0.505)
(0.622, ∞)
(–∞, ∞) (–∞, ∞) (–∞, ∞)
51 0 0 2.099 4.141
(–∞, –0.807) (–0.602, 0.602)
(0.807, ∞)
(–∞, –0.601) (–0.512, 0.512)
(0.601, ∞)
(–∞, ∞) (–∞, ∞) (–∞, ∞)
Recall that when
b 0
or
3
0 there is no solution of (L1,L2) satisfies (3-6), (3-7) and (3-8).We may consider only the process mean change to preserve the unbiasedness of ARL1 of the ML chart, only (3-6) and (3-8) are applied. The new solutions of coefficients of control limits are called (L1,L2). The Second Derivative Test is applied to check ARL0 is a local maximum if only process mean change is considered.
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The Second Derivative Test asserts that if 0
2 1, 2 0
1
2
L L
ARL , then ARL0 is a local maximum when only mean change.
Table 3-16 lists the feasible values of (L1,L2) by solving (3-6) and (3-8) and the corresponding intervals of
1 such that the ARL1 be unbiased for specified
2 under the combinations ofn 3 , 5 , 7 , 11 , 15 , 31 , 51
,
0
0,
0
1, 0 . 0027
,1 , 5 . 0 ,
3
0
andb 500
, 0, 500 respectively.We found that the solutions of (L1,L2) cannot be obtained when
b 500
and 51
n
, or when
3
0 andb 500
.‧
‧
Table 3-16 (Continued)
n
3 b L1’ L2’‧ 國
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Fig. 3-16 (a) shows the ARL of the biased-ARL1 ML chart and the unbiased-ARL1 ML chart with adjusted (L1,L2) for
2
1
2,
2 1
,n 5
,0
0
,
0
1, 0 . 0027
,
3
0.5 andb 500
. The solid red line represents the unbiased ARL1, and the dark-blue dashed line represents the biased ARL. We found that the unbiased ARL1 is smaller than ARL0=370.4 but biased ARL1 does not when only process mean change.Figure 3-16 (b) shows the ARL1s of the ML chart with coefficients of the control limits (L1,L2) under
3
1 3
and0 . 1
2 3
,n 5
,
0
0 ,
0
1,0027 .
0
,
3
0.5 andb 500
. We can see that when
1 0
and
2 0 . 5
the ARL1 may be biased. Since the coefficients (L1,L2) of the control limits do not satisfy (3-7).
(a) ARL1 when
2
1
2,
2 1
(b) ARL1 when 3
1 3
,0 . 1
2 3
Fig. 3-16 The ARL of the adjusted ML chart parameters
(L1,L2) under ,n 5
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3.1.2.3 ARL
1Comparison between the Biased ARL and Unbiased ARL of the Median Loss Charts
The ARL values comparing among these three methods are given in Table 3-17, the three methods are (1) The ARL1-unbiased method with partial derivative of
1 and
2 (2) The ARL1-unbiased method with only partial derivative of
1 and fixed2
1
(3) The ARL1-biased ML chart.From Table 3-17 we can see that, “method 1” ensures that ARL1 smaller than ARL0 for either mean or variance change; “method 2” ensures that unbiased-ARL1 smaller than ARL0 if we consider only process mean change; “method 3” leads the biased-ARL1 to be larger than ARL0 if either mean and/or variance changes. On the other hand, if
1 1
and
2 1 . 5
, the detection ability or ARL1 of these three methods have no much difference. A more detailed discussion of comparing these three methods is listed in Table 3-18.‧ 國
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Table 3-17 The ARL
1 comparison between ARL1-biased and unbiased ML charts (n 5
,
0
0,
0
1, 0 . 0027
)1
2 b0,30 b0,3 1 b500,30.5 b500,3 1Method 1 Method 2 Method 3 Method 2 Method 3 Method 2 Method 3 Method 2 Method 3 0 1 370.37 370.37 370.37 370.37 370.37 370.37 370.37 370.37 370.37 0.2 1 367.930 295.189 332.322 265.741 204.742 67.558 43.311 68.922 39.014 1 1 47.457 16.196 24.152 10.279 8.146 2.365 2.182 2.276 2.059
0 0.2 5.787 - 8.836 - 1.50e+15 - 2252.547 - 2.63e+08
0 0.5 61.494 - 101.295 - 7435.591 - 293.793 - 1884.877
0 0.8 231.401 - 382.283 - 806.521 - 491.949 - 735.808
0 1.5 30.070 - 17.962 - 37.716 - 5.600 - 7.494
1 1.5 9.529 5.290 6.591 5.137 4.528 1.805 1.727 1.885 1.781 2 1.5 2.193 1.687 1.854 1.540 1.465 1.149 1.135 1.152 1.135 3 1.5 1.150 1.078 1.101 1.055 1.045 1.022 1.019 1.021 1.018 1 2 4.202 2.880 3.317 3.656 3.355 1.510 1.467 1.639 1.576 2 2 1.998 1.630 1.757 1.621 1.554 1.161 1.148 1.185 1.168 3 2 1.238 1.147 1.178 1.124 1.109 1.043 1.039 1.047 1.042 1 3 1.944 1.620 1.734 2.277 2.163 1.237 1.220 1.342 1.313 2 3 1.577 1.386 1.454 1.590 1.541 1.122 1.113 1.166 1.153 3 3 1.276 1.186 1.218 1.233 1.213 1.057 1.053 1.075 1.069 2 0.5 2.829 1.361 1.687 1.144 1.085 1.01 1.007 1.008 1.005 2 1 2.35 1.604 1.829 1.395 1.313 1.088 1.077 1.081 1.067 3 0.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3 1 1.049 1.015 1.024 1.008 1.005 1.003 1.003 1.003 1.002 -3 0.5 1.000 1.000 1.000 1.008 95.804 1.000 1.000 9.539 7.008 -3 1 1.049 1.015 1.024 1.306 8.146 1.202 1.164 4.853 4.251 -2 1 2.350 1.604 1.829 5.454 370.370 5.219 4.716 19.420 27.231 -1 1 47.457 16.196 24.152 130.092 175.217 47.213 53.277 57.786 106.498 -0.2 1 367.930 295.189 332.322 300.810 374.662 246.226 390.035 241.236 461.946
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Table 3-18 The Comparison Among the Three ARL-unbiased and -unbiased Methods.
(Method 1) The ARL1 un-biased method with partial derivative of
1 and
2 used.Advantage
The ARL1 are all smaller than ARL0 with all the combina-tions of (1,2) under
b 0
and 3 0 considered in Table 3-17. It is the most appropriate among these three methods, which can ensure the unbiasedness of ARL for either mean and/or variance change.
Disadvantage
Can be applied only when
b 0
and 30. The out-of-control detection performance is not as good as the other two methods in many of the out-of-control situa-tion.
Appropriate situ-ation for use
When
b 0
and 30 for changes in mean and/or var-iance.(Method 2) The ARL1 un-biased method with partial derivative of
1 used.Advantage
An appropriate method to monitor only process mean shifts.
Can be applied when
b 0
or 500
and
0 T
, which Method1 cannot.Disadvantage
Cannot be applied when
b 500
andn 51
, and when 500
b
and
0 T
. Cannot ensure the unbiasedness when process variance change. The performance may be poor when the process variance becomes smaller than its in-control value.
Appropriate situ-ation for use
When
b 0
or 500
, with
X T . Or if we know the out-of-control process only occurs mean change.
(Method 3) The ARL1 bi-ased ML chart.
Advantage
Can be applied with all the real values of
b
and 3. The simplest method. The out-of-control detection perfor-mance among these three methods are likely to be the same when 12 and21.5.
Disadvantage
The ARL1 may larger than ARL0 in those combinations of
) ,
(1 2 considered in Table 3-17.
In most of the cases when 10, its performance is poorer than method 2.
Appropriate situ-ation for use
Moderate to large shifts in mean and/or variance.
When both method 1 and method 2 cannot be applied, like for