Construction of the Average Loss Chart

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chart by assuming the distribution of the quality characteristic is skew normal, and discuss the performance of the AL chart either when the distribution is left-skewed, symmetric or right-skewed respectively.

4.1 Construction of the Average Loss Chart

4.1.1 Approximate Distribution of Average Loss by Using Edgeworth Expansion Method

The first step to construct the AL chart is to find the distribution of AL when X follows skew-normal distribution. Edgeworth expansion, for example see Peter Hall (1992), is used in this study to approximate the distribution of AL. Edgeworth (1905) derived Edgeworth expansion that relates the cdf of a random variable having expectation zero and variance 1 to the cdf of the standard normal distribution, using the Chebyshev-Hermite polynomials.

Since the in-control X follows

SN

(



₀,

a

₀,

b

). We can obtain the r^th moments of

)

2

( X T L

_i



_i



by





    



0 0

0 0

2 0 0

) (

) ( ) 2 (

a dx b x a

T x a x

M

_r ^r

  

, r = 1,2,… , (4-1)

where



() and

 ( )

denote the pdf and cdf of the standard normal distribution. Then the expectation and the standard deviation of

L

can be obtained by



_LM₁ and



_L

 M

₂

 M

₁² . Define

Z

_n

 n

(

AL  

_L)/



_L , then we can approximate the cdf of

Z

_n by Edgeworth expansion, which is expressed in (4-2)



 



   





 



 





 ( )

72 ) 1 24 (

1 ) 1

6 ( 1 ) 1

( )

( ₃ ⁽³⁾ ₄ ⁽⁴⁾ z ²₃ ⁽⁶⁾ z

z n z n

z

FZn

  

(4-2)

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where ^(r)() ^r_r() dz

d and



_rbe the r^th cumulant of

L

L

L







.

Note that



^(r)

( z )     1

^r1

He

_r1

( z )  ( z )

, where He_r1(z) is the Chebyshev-Hermite polynomial. And one can obtain He_r1() by equation

 

 



 

 



_

 ( ) ( )

) ( ) 1

( ₂

1

He z z

dz d z z

He

_{r r}





. (4-3)

The Hermite polynomials obtained from (4-3) are shown in Table 4-1.

Table 4-1 The Hermite polynomial

Order Hermite polynomial

)

0(

z He

1

)

1

( z

He z

)

2

( z

He _z

²

_

₁ )

3(

z

He z

(

z

²



3)

)

4

( z

He

z⁴6z²3 )

5(

z

He z 

(

z

⁴



10

z

²



15)

To obtain



_r , one can use the relation



_r

 

_r

/ 

_L^r, where



_r is the r^th cumulant of

L

. From Peter Hall (1992), the cumulants of

L

can be obtained from the moments of

L

, as shown in Table 4-2.

Table 4-2 The cumulants of L 

(

X  T

)²

Order Cumulants of

L

 M

‧

The approximate pdf of

Z

_n can be obtained by differentiating (4-2) as

 

The accuracy of this approximation is examined by Pearson



² goodness-of-fit test, for example see Gerald Keller (2004). Consider

n  5 , 11

,



₃



1,



₀



0,



₀



1 approximated cdf is not different from the distribution of the samples. Moreover, the

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Table 4-3 P-value of the Pearson 

² goodness-of-fit test for Edgeworth expansion

b n Number of samples

50000 200 100

-500 5 0.000 0.0712 0.2493

-500 11 0.000 0.6475 0.3345

0 5 0.000 0.1816 0.3838

0 11 0.2737 0.1866 0.0519

500 5 0.000 0.0554 0.2622

500 11 0.000 0.7499 0.9357

A graphical approach is given in Fig. 4-1. Set



₃



1;



₀



0 ;



₀



1,

 500



b

and

n  5 , 11

respectively. The black curve is the cumulated proportions by 50000 random samples of AL from

X ~ SN ( 

₀

, a

₀

, b )

with sample size

n , which

is used to simulate the exact cdf of AL. The blue curve is the approximated cdf by Edgeworth expansion. We can see that when sample size goes larger (

n  11

) the blue curve attends to be more close to the black curve, representing more accuracy of the cdf approximation of the AL statistic.

(a)

n  5

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(b)

n  11

Fig. 4-1 The approximated cdf of AL by Edgeworth expansion

(

m  50000

for the simulated cdf of AL)

4.1.2 Control Limits of the Average Loss Chart

An AL chart with false alarm rate



is in the form of

) 2 / (

) 2 / 1 (

1 1















AL AL

F LCL

F

UCL

(4-7)

Table 4-4.1 lists the UCLs and LCLs of AL chart under various combinations of

51

..., , 5 ,

 3

n

,



₃



0,0.5,...,2 and

b   500 , 0 , 500

under

  0 . 0027

,



₀



0 and



₀



1.

Table 4-4.2 lists the UCLs and LCLs of AL chart under various combinations of

51

..., , 5 ,

 3

n

,



₃



0,0.5,...,2 and

b   2 , 0 , 2

under

  0 . 0027

,



₀



0 and

0



1



.

‧

From both Table 4-4.1 and Table 4-4.2 we can see that the UCLs decrease and LCLs increase when n goes larger. On the other hand, when



₃



0 the control limits are wider when

b  500 (  2 )

than

b   500 ( 2 )

. The control limits of AL chart have no obvious trend when



₃ change.

Table 4-4.1 Control limits of the AL Chart

delta3 (0.000, 10.456)

(0.000, 7.449) (0.000, 11.000) (0.000, 13.817)

(0.009, 10.347) (0.404, 14.570) (0.483, 17.700)

5 (0.260, 11.139)

(0.950, 9.275) (1.062, 12.117) (1.112, 14.469)

7 (1.475, 12.796)

9 (1.739, 10.072)

(1.725, 11.74) (1.913, 11.000)

13 (2.063, 10.446) -500 (0.343, 2.860) (0.616, 2.688) (0.785, 3.463) (1.479, 5.166) (2.531, 7.542)

‧

Table 4-4.1 (Continued)

delta3

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Table 4-4.2 Control limits of AL Chart

delta3

n b 0 1 2

(LCL, UCL) (LCL, UCL) (LCL, UCL)

5 -2

0 2

(0.083, 3.984) (0.024, 3.665) (0.083, 3.984)

(0.167, 5.647) (0.114, 6.431) (0.040, 7.377)

(0.991, 11.009) (1.062, 12.117) (1.190, 13.315)

11 -2

0 2

(0.262, 2.902) (0.224, 2.692) (0.262, 2.902)

(0.568, 4.312) (0.509, 4.806) (0.443, 5.413)

(2.023, 8.869) (1.958, 9.523) (1.984, 10.265)

21 -2

0 2

(0.383, 2.309) (0.359, 2.163) (0.383, 2.309)

(0.86, 3.600) (0.783, 3.932) (0.713, 4.347)

(2.736, 7.721) (2.621, 8.143) (2.562, 8.630)

4.1.3 Out-of-Control Detection Performance Measurement of the AL Chart

Table 4-5.1 gives the ARL1s of the AL chart for all the combinations of



₁=0.5, 1.0, 1.5, 2.0, 3.0,



₂=1.5, 2.0, 2.5, 3.0,

b   500 , 0 , 500

and



₃=0.0, 0.5, 1.0, 1.5, 2.0 under

  0 . 0027

,

n  5

,



₀



0 and



₀



1.

Table 4-5.2 gives the ARL₁s of the AL chart for all the combinations of



₁=0.5, 1.0, 1.5, 2.0, 3.0,



₂=1.5, 2.0, 2.5, 3.0,

b  

2,0,2 and



₃=0.0, 0.5, 1.0, 1.5, 2.0 under

  0 . 0027

,

n  5

,



₀



0 and



₀



1.

From both Table 4-5.1 and Table 4-5.2 we can see that the ARL1s are smaller

‧

‧

Table 4-5.1 (Continued)



1



₂ ^b

^

³

‧

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The response table and diagram for ARL1 are shown in Table 4-6 and in Fig. 4-2.

We can see that, in average, the ARL1 of the AL chart decreases when



₁ or



₂ increases, and increases when

b

increases, but almost no change when



₃ increases.

Table 4-6 The response table for ARL

₁ of the AL chart

delta1 delta2 delta3 b level 1 3.102 2.457 1.873 1.421 level 2 1.916 1.532 1.482 1.510 level 3 1.415 1.310 1.521 1.950 level 4 1.198 1.209 1.594 - level 5 1.092 - 1.665 -

level 6 1.040 - - -

Difference 2.062 1.249 0.391 0.528

Fig. 4-2 Response diagram for ARL

₁ of the AL chart

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4.2. An EWMA Average Loss Chart

In this section our objective is to construct an EWMA average loss (EWMA-AL) chart to compare with the EWMA-ML chart.

4.2.1 Construction of the EWMA-AL Chart

Define

EWMA

_AL,t as the plotting statistic of the EWMA-AL chart at time t, which is in the form of

1 ,

,_t

 

_t

 ( 1  )

_ALt

AL

AL EWMA

EWMA  

(4-8)

where



(0,1) is the smoothing parameter. Note that when

  1

the EWMA-AL chart will reduce to the AL chart proposed in section 4.1.1. The control limits when





t

is expressed by

2 , 2 ,

1 2

AL AL

AL AL

k UCL

k UCL

 

 

 

 

 



 



(4-9)

where

k

₁ and

k

₂ are the control chart coefficients which can be determined by Markov Chain approach such that

ARL

₀



370.4, the Markov chain procedure steps are the same as described in section 3.2.1.

Table 4-7 lists the control parameters

( k

₁

, k

₂

)

and the corresponding control limits of EWMA-AL chart when

t  

by setting

n  5

and

ARL

₀



370.4. We found that the control limits be more symmetric when



is small, for example

05 .

 0



.

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Table 4-7 The control limits of the EWMA-AL chart ( n  5

,

ARL

₀



370.4)

b  k1 k2 UCL LCL

-500

1 2.678 3.577 4.718 0.000 0.4 2.741 3.335 3.267 0.958 0.2 2.701 3.086 2.782 1.316 0.05 2.420 2.566 2.312 1.706

-2

1 2.799 3.849 5.647 -0.653 0.4 2.250 3.548 3.681 0.934 0.2 2.448 3.239 3.023 1.227 0.05 2.322 2.663 2.404 1.648

0

1 3.776 4.045 6.431 0.000 0.4 2.512 3.736 4.046 0.624 0.2 2.293 3.373 3.232 1.163 0.05 2.328 2.638 2.463 1.592

2

1 4.087 4.219 7.377 -3.208 0.4 3.550 3.940 4.511 -0.262 0.2 2.068 3.535 3.502 1.122 0.05 2.245 2.721 2.555 1.542

500

1 4.189 4.293 8.324 0.000 0.4 3.653 4.038 4.974 0.000 0.2 1.942 3.613 3.774 1.046 0.05 2.234 2.715 2.640 1.473

4.2.2 Out-of-Control Detection Performance Measurement of the EWMA-AL Chart

We use the ARL₁ to measure the out-of-control detection performance of the EWMA-AL chart. The ARL1s were calculated by Markov chain approach, which is same as we described in section 3.2.2.

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4.2.3 Out-of-Control Detection Performance Comparison between the AL and the EWMA-AL Charts

Table 4-8 gives the ARL₁s of the EWMA-AL chart for various of combinations of



₁=0.5, 1.0, 1.5, 2.0, 3.0 and



₂=1.5, 2.0, 2.5, 3.0 under

ARL

₀



370.4,

n  5

,



3=1,



₀



1 and

  0 . 05 , 0 . 2

respectively. The ARL1s are compared with those of the AL chart proposed in section 4.1.1.

From Table 4-8 we can see that, under

b   500

, 0 or 500, the ARL₁ of the EWMA-AL chart increases when



₁ and/or



₂ increases. On the other hand, the EWMA-AL chart performs better when

b   500

rather than

b  0

or

500

.

Comparing the ARL₁s of the EWMA-AL chart and the AL chart, we found that the EWMA-AL chart with

  0 . 2

performs better when process has small change in mean and/or variance, like



₁

 2

and



₂

 1 . 5

. In most of the cases, the AL chart performs better than the EWMA-AL chart. We conclude that there is no need to use EWMA technique on the AL chart when the distribution of the quality characteristic X follows skew-normal distribution.

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Table 4-8.1 The ARL

1s comparison of the EWMA-AL and the AL charts



1



₂ ^{EWMA-AL (}

 ^{ 0 . 05}

) EWMA-AL (

  0 . 2

) AL chart

b=-500 b=0 b=500 b=-500 b=0 b=500 b=-500 b=0 b=500 0.5 1 6.382 9.715 14.249 4.953 9.540 18.381 10.687 31.243 40.169 1 1 2.737 3.830 5.182 1.916 2.865 4.347 1.754 4.718 11.795 1.5 1 1.731 2.316 2.990 1.222 1.644 2.209 1.103 1.641 3.321

2 1 1.213 1.660 2.083 1.034 1.173 1.481 1.012 1.095 1.504 3 1 1.003 1.032 1.224 1.000 1.001 1.001 1.000 1.000 1.000 0 1.5 6.653 9.931 14.149 5.294 8.826 15.441 9.393 14.463 9.606 0 2 2.960 4.125 5.584 2.075 3.215 5.769 2.194 3.719 4.398 0 2.5 1.827 2.569 3.507 1.428 2.057 3.537 1.335 2.036 2.721 0 3 1.405 1.928 2.763 1.243 1.598 2.580 1.117 1.499 2.033 0.5 1.5 3.269 4.76 6.597 2.361 3.707 5.828 2.355 5.692 7.961 1 1.5 2.08 2.895 3.885 1.499 2.149 3.043 1.333 2.549 4.056 1.5 1.5 1.486 2.022 2.625 1.166 1.503 1.978 1.09 1.517 2.273 2 1.5 1.179 1.536 1.947 1.047 1.192 1.462 1.023 1.155 1.487 3 1.5 1.013 1.074 1.265 1.002 1.010 1.047 1.001 1.006 1.027 0.5 2 2.145 3.015 4.054 1.542 2.315 3.676 1.376 2.607 3.333 1 2 1.624 2.257 2.971 1.251 1.739 2.520 1.15 1.789 2.496 1.5 2 1.305 1.762 2.268 1.109 1.388 1.861 1.062 1.363 1.873 2 2 1.134 1.436 1.808 1.044 1.185 1.469 1.024 1.151 1.442 3 2 1.019 1.096 1.277 1.005 1.025 1.098 1.003 1.016 1.075 0.5 2.5 1.556 2.195 3.02 1.196 1.758 2.762 1.117 1.754 2.264 1 2.5 1.326 1.84 2.503 1.109 1.497 2.171 1.058 1.457 1.89 1.5 2.5 1.179 1.558 2.067 1.064 1.301 1.756 1.037 1.255 1.604

2 2.5 1.093 1.348 1.73 1.034 1.169 1.471 1.02 1.131 1.39 3 2.5 1.021 1.103 1.294 1.008 1.039 1.152 1.004 1.025 1.113 0.5 3 1.239 1.75 2.571 1.070 1.471 2.204 1.023 1.406 1.793 1 3 1.144 1.569 2.261 1.032 1.343 1.880 1.013 1.28 1.594 1.5 3 1.093 1.406 1.952 1.030 1.233 1.626 1.016 1.179 1.439 2 3 1.06 1.273 1.69 1.023 1.149 1.434 1.014 1.106 1.315 3 3 1.02 1.099 1.322 1.008 1.050 1.187 1.005 1.029 1.137

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Table 4-8.2 The ARL

1s comparison of the EWMA-AL and the AL charts



1



₂ ^{EWMA-AL (}

 ^{ 0 . 05}

) EWMA-AL (

  0 . 2

) AL chart b=-2 b=0 b=2 b=-2 b=0 b=2 b=-2 b=0 b=2 1 1 3.333 3.830 4.458 2.403 2.865 3.522 3.002 4.718 7.874 2 1 1.460 1.660 1.868 1.089 1.173 1.307 1.039 1.095 1.234 3 1 1.012 1.032 1.090 1.000 1.001 1.000 1.000 1.000 1.000 0 1.5 8.370 9.931 12.015 6.932 8.826 11.688 11.226 14.463 15.003 0 2 3.579 4.125 4.839 2.684 3.215 4.202 2.910 3.719 4.716 0 3 1.720 1.928 2.239 1.424 1.598 1.987 1.321 1.499 1.827 1 1.5 2.538 2.906 3.370 1.842 2.150 2.556 1.815 2.368 3.201 2 1.5 1.372 1.539 1.736 1.110 1.192 1.309 1.065 1.136 1.272 3 1.5 1.036 1.075 1.149 1.005 1.010 1.019 1.002 1.004 1.008 1 2 1.976 2.251 2.589 1.504 1.739 2.079 1.416 1.712 2.155 2 2 1.292 1.436 1.611 1.105 1.185 1.303 1.068 1.139 1.263 3 2 1.051 1.098 1.171 1.012 1.025 1.048 1.007 1.015 1.032 1 3 1.400 1.584 1.865 1.202 1.343 1.590 1.144 1.266 1.454 2 3 1.169 1.285 1.457 1.076 1.149 1.273 1.050 1.107 1.207 3 3 1.055 1.108 1.192 1.022 1.050 1.101 1.013 1.031 1.068

4.3. An Optimal Variable Sampling Interval Average Loss Chart

4.3.1 Construction of the Optimal VSI Average Loss Chart

The VSI-AL chart is composed of UCL, warning control limits (WL) and LCL, which is the form of

, 0

, ,











LCL

w WL

k UCL

AL AL

AL AL









(4-10)

where w denote the warning factor of WL with

0  w  k

, and the plotting statistic is

ML .

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AL AL

k UCL    

AL

AL

w

WL    



0

LCL

The region between LCL and WL is called the “central region” (CR), the region between WL and LCL is called the “warning region”(WR), and the region above UCL is called the “action region”(AR), as in Figure 4-17. Two sampling intervals

t

₁ and

t

2 are adopted. The long sampling interval (

t

₁) is adopted when AL fell into CR; the short sampling interval (

t

₂) is adopted then the statistic AL fell into WR, where









_{2 1}

0 t t

.

Fig. 4-3 The visualization of the VSI-AL chart

When

t

₁

 t

₂

 t

₀ and

w  0

, the VSI-AL chart is reduced to one-sided AL chart with fixed sampling interval (FSI)

t .

₀

(1) To construct the specified VSI-AL chart

To determine

k

and w of the VSI-AL chart with specified

t

₁ and

t

₂, the procedure steps are

Step 1. Specify the values of

n

,



,



₀,



₀,



₃,

t

₀,

t

₁ and

t

₂.

Step 2. Determine k by solving the equation

P  AL    k    1  

, then

CR (t

1

)

WR (t

2

)

AR

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Step 3. Determine w by solving equation

P

_C

t

₁

 P

_W

t

₂

 P

_A



0

 t

₀(1

 

)



0

 

, where

P

_C

 P

(

AL  WL

),

P

_W

 P

(

LCL  AL  UCL

)and

P

_A

 

. Then

AL

AL

AL t t

t F t

w





_





















2 1

2

1 (0 )(1 )

(4-12)

(2) To construct the optimal VSI-AL chart

The optimal VSI-AL chart is to find the optimal coefficient of the warning limits (

w

^*) and the optimal VSIs (

t

₁^*,

t

₂^*) that may minimize the ATS1.The procedure steps of finding the optimal parameters

w

^* and (

t

₁^*,

t

₂^*) are the same as described in section 3.2.1.

4.3.2 Out-of-control Detection Performance Measurement of the Optimal VSI Average Loss Chart

We use the ATS₁ to measure the out-of-control performance of the VSI-AL chart.

The ATS1 of the chart can be calculated by Markov chain approach, which has the same procedure as described in section 3.2.2.

4.3.3 ATS

1

s Comparison among the AL Chart, specified VSI-AL Chart and the Optimal VSI-AL Chart

To construct the optimal VSI-AL chart, we considered all the combinations of



1=0.5, 1.0, 2.0, 3.0,



₂=1.5, 2.0, 3.0 and

b   500

, 0, 500 under

ATS

₀



370.4,

 5

n

,



₃ =1,



₀



0,



₀



1,

t =1. The optimal values of

₀ (

t

₁^*,

t

₂^*) with the corresponding w^*, WL^* and the minimal ATS1s of the optimal VSI-AL chart are shown in Table 4-10.

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From Table 4-10, we found that the ATS1s of the optimal VSI-AL chart are smaller than those of the specified VSI-AL and FSI-AL charts. Most ARL1s of the FSI-AL chart are smaller than those of the specified VSI-AL chart unless



₁ and



₂ are small. The reason is the inappropriate choice of the specified VSIs.

The calculation of the saved out-of-control detection time (%) in the last three columns of Table 4-10 are the same as described in Table 3-20.

Using the optimal VSI-AL chart to replace of the FSI-AL chart may save the out-of-control detection time from 0 to 78.55%.

Using the optimal VSI-AL chart to replace of the specified VSI-AL chart may save the out-of-control detection time from 7.41% to 50.275%.

Using the specified VSI-AL chart to replace of the FSI-AL chart may save the out-of-control detection time from 0% to 71.742%

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Table 4-10 ATS

1s of optimal VSI-AL chart, specified VSI-AL chart and the AL chart

b



₁



₂

Optimal VSI-AL chart VSI-AL FSI-AL Save time 1

(%)

Save time 2

(%)

Save time 3 t1* t2* w^* WL^* ATS1 (t₁,t₂)(2,0.1) t0=1 (%)

-500 0.5 1 1.682 0.000 0.169 2.128 3.558 4.221 10.687 66.707 60.503 15.707 -500 1 1 1.067 0.000 1.574 3.196 1.251 2.105 1.754 28.677 -20.011 40.570 -500 2 1 1.002 0.016 3.101 4.357 1.009 2.001 1.012 0.296 -97.727 49.575 -500 3 1 1.000 0.072 3.297 4.506 1.000 2.000 1.000 0.000 -100.000 50.000 -500 0 1.5 1.682 0.000 0.169 2.128 3.927 4.489 9.347 57.987 51.974 12.519 -500 0 2 1.197 0.000 0.935 2.711 1.433 2.108 2.194 34.686 3.920 32.021 -500 0 3 1.034 0.000 1.957 3.488 1.040 1.881 1.117 6.893 -68.397 44.710 -500 0.5 1.5 1.197 0.000 0.935 2.711 1.533 2.309 2.355 34.904 1.953 33.608 -500 1 1.5 1.043 0.000 1.829 3.390 1.161 2.067 1.333 12.903 -55.064 43.832 -500 2 1.5 1.002 0.006 3.103 4.358 1.018 2.004 1.023 0.489 -95.894 49.202 -500 3 1.5 1.000 0.072 3.297 4.506 1.001 2.000 1.001 0.000 -99.800 49.950 -500 0.5 2 1.034 0.000 1.957 3.488 1.175 2.064 1.376 14.608 -50.000 43.072 -500 1 2 1.021 0.000 2.211 3.680 1.092 2.039 1.15 5.043 -77.304 46.444 -500 2 2 1.002 0.008 3.103 4.358 1.020 2.006 1.024 0.391 -95.898 49.153 -500 3 2 1.000 0.084 3.297 4.506 1.002 2.000 1.003 0.100 -99.402 49.900 -500 0.5 3 1.012 0.001 2.467 3.875 1.000 1.929 1.023 2.248 -88.563 48.160 -500 1 3 1.002 0.005 3.103 4.358 1.010 1.993 1.013 0.296 -96.742 49.323 -500 2 3 1.000 0.010 3.238 4.461 1.014 2.008 1.014 0.000 -98.028 49.502 -500 3 3 1.000 0.084 3.297 4.506 1.005 2.002 1.005 0.000 -99.204 49.800

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Table 4-10 (Continued)

b



₁



₂

Optimal VSI-AL chart VSI-AL chart FSI-AL Save time 1

(%)

Save time 2

(%)

Save time 3 t1* t2* w* WL* ATS1 (%)

) 1 . 0 , 2 ( ) , (t₁ t₂ 

t0=1

0 0.5 1 1.912 0.000 -0.109 1.881 9.568 10.867 31.241 69.374 65.216 11.954 0 1 1 1.250 0.000 0.755 2.828 1.787 2.439 4.718 62.124 48.304 26.732 0 2 1 1.015 0.000 2.773 5.038 1.037 2.007 1.095 5.297 -83.288 48.331 0 3 1 1.000 0.102 3.780 6.141 1.000 2.000 1 0.000 -100.000 50.000 0 0 1.5 1.912 0.000 -0.109 1.881 8.095 8.742 14.463 44.030 39.556 7.401 0 0 2 1.217 0.000 0.852 2.933 2.366 3.040 3.719 36.381 18.258 22.171 0 0 3 1.069 0.000 1.620 3.774 1.290 2.164 1.499 13.943 -44.363 40.388 0 0.5 1.5 1.380 0.000 0.467 2.512 2.704 3.281 5.107 47.053 35.755 17.586 0 1 1.5 1.164 0.000 1.043 3.143 1.520 2.266 2.368 35.811 4.307 32.921 0 2 1.5 1.020 0.000 2.579 4.825 1.068 2.009 1.136 5.986 -76.849 46.839 0 3 1.5 1.000 0.046 3.735 6.091 1.003 2.000 1.004 0.100 -99.203 49.850 0 0.5 2 1.164 0.000 1.043 3.143 1.703 2.458 2.448 30.433 -0.408 30.716 0 1 2 1.069 0.000 1.620 3.774 1.347 2.179 1.712 21.320 -27.278 38.183 0 2 2 1.023 0.001 2.483 4.720 1.084 2.015 1.139 4.829 -76.910 46.203 0 3 2 1.001 0.014 3.634 5.981 1.011 1.997 1.015 0.394 -96.749 49.374 0 0.5 3 1.047 0.000 1.908 4.091 1.229 2.120 1.379 10.877 -53.735 42.028 0 1 3 1.032 0.000 2.195 4.405 1.170 2.079 1.266 7.583 -64.218 43.723 0 2 3 1.015 0.001 2.773 5.037 1.079 2.021 1.107 2.529 -82.565 46.611 0 3 3 1.002 0.007 3.539 5.876 1.025 1.997 1.031 0.582 -93.695 48.673

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Table 4-10 (Continued)

b



₁



₂

Optimal VSI-AL chart VSI-AL chart FSI-AL Save time 1

(%)

Save time 2

(%)

Save time 3 t1^* t2^* w^* WL ATS1 (t₁,t₂)(2,0.1) t₀=1 (%)

500 0.5 1 1.901 0.000 -0.175 1.743 15.485 17.224 40.17 61.451 57.122 10.096 500 1 1 1.790 0.000 -0.097 1.857 2.530 3.333 11.795 78.550 71.742 24.092 500 2 1 1.057 0.000 1.765 4.600 1.069 2.034 1.503 28.876 -35.329 47.443 500 3 1 1.000 0.351 4.054 7.971 0.995 2.001 0.996 0.100 -100.904 50.275 500 0 1.5 1.696 0.000 -0.019 1.971 11.73 14.874 11.73 0.000 -26.803 7.913 500 0 2 1.313 0.000 0.524 2.771 3.948 4.774 4.679 15.623 -2.030 17.302 500 0 3 1.100 0.000 1.300 3.915 1.717 2.588 2.064 16.812 -25.388 33.655 500 0.5 1.5 1.799 0.000 -0.104 1.847 4.813 5.419 7.961 39.543 31.931 11.183 500 1 1.5 1.322 0.000 0.503 2.741 2.196 2.816 4.056 45.858 30.572 22.017 500 2 1.5 1.063 0.000 1.657 4.441 1.148 2.017 1.487 22.798 -35.642 43.084 500 3 1.5 1.007 0.000 3.538 7.212 1.013 1.995 1.027 1.363 -94.255 49.223 500 0.5 2 1.179 0.000 0.912 3.343 2.579 3.373 3.333 22.622 -1.200 23.540 500 1 2 1.159 0.000 0.989 3.457 1.819 2.609 2.496 27.123 -4.527 30.280 500 2 2 1.073 0.000 1.532 4.257 1.202 2.061 1.442 16.644 -42.926 41.679 500 3 2 1.018 0.001 3.085 6.544 1.047 1.978 1.075 2.605 -84.000 47.068 500 0.5 3 1.073 0.000 1.532 4.257 1.616 2.544 1.793 9.872 -41.885 36.478 500 1 3 1.073 0.000 1.532 4.257 1.485 2.432 1.594 6.838 -52.572 38.939 500 2 3 1.061 0.000 1.687 4.485 1.256 2.193 1.315 4.487 -66.768 42.727 500 3 3 1.045 0.001 2.074 5.055 1.117 2.047 1.137 1.759 -80.035 45.432

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Chapter 5. ATS

1

Comparison among all Proposed Loss Con-trol Charts and Other Existed ConCon-trol Charts

5.1 Introduction of Some Existed Control Charts

(1) Weighted Variance

X

and R Control Chart

The weighted variance (WV) method, like the Shewhart control charts, uses the standard deviation to set the limits of the control chart. However, its standard deviation is multiplied by two different weights. One weight is used in the UCL, while the other is used in the LCL. Let P_X P(X 



_X), where



_X is the expected value of X. Then the UCL weight is 2P_X , and the LCL weight is 2(1P_X). There is no assumption on the underlying population for the WV method. The control limits of the WV

X

chart are expressed as

, ) 1 ( 2 3

, 2 3

X X

X X

X X X X

n P LCL

n P UCL











 

 

where



_X is the standard deviation of X. Similarly, the control limits of the WV R chart are

 ^{ 3 3 2 2 ( 1  , )} 

^

^,







X R

R R

X R R R

P LCL

P UCL









where [a denotes ]^ max(0,a) and



_R is the standard deviation of R.

If the underlying population is symmetric, then

P  0 . 5

. That is, the WV

X

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population is skewed to the right, then

P

_X is greater than 0.5. Similarly, if the underlying population is skewed to the left, then

P

_X is less than 0.5.

Table 5-1 lists the parameters of the WV

X

and R chart when n=5 and the underlying distribution of X is skew-normal with



_X =0 and



_X =1 and

2 , 500 





b

, 0, 2, 500 respectively.

Table 5-1 Chart parameters for WV Method

b Px



_R



_R

-500 0.425 2.246 0.920 -5 0.434 2.274 0.914 0 0.500 2.326 0.864 5 0.566 2.274 0.914 500 0.575 2.246 0.920

(2) Skewness Correction

X

Control Chart

The Skewness Correction (SC) method proposed by Chan and Cui (2007) is based in Cornish-Fisher expansion. It provides asymmetric control limits using

 3

standard deviations plus the same skewness correction.

The control limits of the SC

X

chart are

, ) 3 (

, ) 3 (

* 4

* 4

n c LCL

c n UCL

X X X

X X X

 

 















‧

Similarly, the control limits of the SC R chart are

 ^{( ( 3 3 )}

4^*

^{) ,}  ^,

When X follows is skew-normal distribution, the skewness of X is given by



²



^3/2

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Table 5-2 Chart parameters for SC Method

b



_R



_R k3⁽X⁾ k₃(R) _{c4* d4*} -500 2.246 0.920 -0.445 0.628 -0.571 0.776

-5 2.274 0.914 -0.381 0.616 -0.493 0.764 -2 2.311 0.892 -0.203 0.555 -0.268 0.697 0 2.326 0.864 0.000 0.466 0.000 0.595 2 2.311 0.892 0.203 0.555 0.268 0.697 5 2.274 0.914 0.381 0.616 0.493 0.764 500 2.246 0.920 0.445 0.628 0.571 0.776

5.2 ATS

1

Comparison among all Proposed Loss Control Charts and Other Existed Control Charts

The ATS1s of the FSI-ML, EWMA-ML, optimal VSI-ML, FSI-AL, EWMA-AL, optimal VSI-AL and the WV and SC

X

and R charts are given in Table 5-3 to 5-8 for

ATS

₀



185,

n  5

,



₀



0,



₀



1,

t =1,

₀



₃



0,1 and

b   500

, 0, 500 respectively.

Table 5-3 gives the ATS1s among all the methods when

b   500

and



₃



0. The response diagram is shown in Fig. 5-1. We can see that the ATS₀ of the WV

X

and R charts is much higher than 185, which implies a much higher false alarm rate.

However, the ATS0 of the SC

X

and R charts is lower than 185.

From Table 5-3 or Fig. 5-1, we found that the optimal VSI charts perform better than other FSI charts for either small or large shift in process mean and variance. The optimal VSI-ML chart performs better than the optimal VSI-AL chart.

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For FSI charts, from Table 5-3 or Fig. 5-3 we can see that the performance of EWMA-ML, WV and SC

X

and R charts are almost the same, and all are better than other FSI charts when small shift of



₁ and



₂. For the rest of the FSI charts, the FSI-ML chart performs better than EWMA-AL chart, and the EWMA-AL chart performs better than the FSI-AL chart. However, when



₁ and



₂ are moderate to large, the performance of FSI-ML, EWMA-ML, FSI-AL, EWMA-AL, WV and SC

X

and R charts have no much difference.

Table 5-3 The ATS

_1s comparison among the 8 methods when

b   500

and



₃



0

b



₁



₂ WV SC FSI-AL EWMA-AL

(

  0 . 2

)

Optimal

VSI-AL FSI-ML EWMA-ML (



0.2)

Optimal VSI-ML -500 0 1 80.577 318.818 185.195 185.185 173.282 185.185 185.185 182.281 -500 0.5 1.5 3.510 3.886 21.300 5.570 1.529 4.729 3.003 0.866 -500 1.0 1.5 1.877 1.865 6.347 3.034 1.431 2.469 1.994 0.308 -500 2.0 1.5 1.079 1.076 1.237 1.380 1.067 1.281 1.236 0.044 -500 3.0 1.5 1.004 1.004 1.016 1.036 1.014 1.047 1.042 0.005 -500 0.5 2.0 1.665 1.972 2.914 2.179 1.306 2.192 1.860 0.268 -500 1.0 2.0 1.317 1.442 1.821 1.779 1.064 1.710 1.554 0.150 -500 2.0 2.0 1.056 1.073 1.110 1.198 1.057 1.239 1.206 0.043 -500 3.0 2.0 1.006 1.009 1.018 1.035 1.016 1.070 1.063 0.011 -500 0.5 3.0 1.095 1.164 1.237 1.386 1.031 1.353 1.299 0.071 -500 1.0 3.0 1.060 1.104 1.075 1.136 0.978 1.267 1.231 0.053 -500 2.0 3.0 1.019 1.033 1.007 1.020 1.006 1.144 1.128 0.027 -500 3.0 3.0 1.007 1.010 1.011 1.020 1.012 1.071 1.064 0.012

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Fig. 5-1 The response diagram of the ATS

₁s for the eight charts when

b   500

and



₃



0

Table 5-4 gives the ATS1s among all the charts when

b  0

and



₃



0. The response diagram is shown in Fig. 5-2. We can see that the ATS0 of the WV

X

and R charts is higher than 185, and the ATS₀ of the SC

X

and R charts is lower than 185.

2 4 6 8

0123456

0 1 2 3 4 5 6

ATS₁

delta1 delta2

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

(delta3=0, b=-500) WV SC FSI-AL EWMA-AL opt.VSI-AL FSI-ML EWMA-ML opt.VSI-ML

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Table 5-4 The ATS

1s comparison among the 8 charts when

b  0

and



₃



0

b



₁



₂ WV SC FSI-AL EWMA-AL Optimal

VSI-AL FSI-ML EWMA-ML Optimal VSI-ML 0 0 1 136.771 246.538 185.137 185.185 174.755 185.185 185.185 181.813 0 0.5 1.5 4.329 5.693 4.575 3.411 2.608 9.876 5.821 3.063 0 1.0 1.5 2.478 2.771 2.663 2.324 1.743 5.278 3.584 1.203 0 2.0 1.5 1.167 1.181 1.230 1.254 1.125 1.687 1.535 0.086 0 3.0 1.5 1.006 1.006 1.013 1.018 1.010 1.078 1.070 0.003 0 0.5 2.0 1.913 2.273 1.881 1.851 1.444 3.642 2.776 0.688 0 1.0 2.0 1.623 1.813 1.592 1.618 1.315 2.878 2.336 0.445 0 2.0 2.0 1.163 1.194 1.166 1.200 1.114 1.630 1.515 0.104 0 3.0 2.0 1.019 1.022 1.025 1.036 1.021 1.147 1.132 0.015 0 0.5 3.0 1.177 1.240 1.176 1.226 1.131 1.702 1.576 0.127 0 1.0 3.0 1.141 1.196 1.151 1.197 1.117 1.620 1.515 0.113 0 2.0 3.0 1.075 1.102 1.085 1.114 1.071 1.386 1.335 0.063 0 3.0 3.0 1.024 1.032 1.032 1.048 1.026 1.185 1.168 0.025

Fig. 5-2 The response diagram of the ATS

₁s for the eight charts when

b  0

and



₃



0

2 4 6 8

0123456

0 1 2 3 4 5 6

ATS₁

delta1 delta2

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

(delta3= 0, b= 0) WV SC FSI-AL EWMA-AL opt.VSI-AL FSI-ML EWMA-ML opt.VSI-ML

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and R charts is higher than 185, and the ATS0 of the SC

X

and R charts is lower than 185.

Table 5-5 The ATS

1s among the 8 charts when

b  500

and



₃



0

b



₁



₂ WV SC FSI-AL EWMA-AL

(

  0 . 2

)

Optimal

VSI-AL FSI-ML EWMA-ML (

  0 . 2

)

Optimal VSI-ML 500 0 1 147.616 322.125 185.195 185.185 173.282 185.185 185.185 182.281 500 0.5 1.5 6.277 9.503 4.652 5.361 4.102 17.299 9.737 7.349 500 1.0 1.5 3.579 4.861 3.029 3.656 2.454 7.331 6.358 3.995 500 2.0 1.5 1.263 1.404 1.581 1.686 1.227 2.045 1.800 0.267 500 3.0 1.5 1.001 1.005 1.056 1.103 1.035 1.083 1.065 0.000 500 0.5 2.0 2.650 3.320 2.508 3.078 1.953 4.312 2.889 0.816 500 1.0 2.0 2.305 2.848 1.967 2.648 1.681 5.613 3.598 1.323 500 2.0 2.0 1.374 1.518 1.396 1.626 1.235 2.191 2.026 0.662 500 3.0 2.0 1.024 1.045 1.103 1.170 1.062 1.233 1.195 0.015 500 0.5 3.0 1.277 1.362 1.428 1.799 1.220 1.575 1.465 0.125 500 1.0 3.0 1.328 1.435 1.345 1.701 1.256 1.710 1.559 0.158 500 2.0 3.0 1.269 1.363 1.207 1.410 1.218 2.002 1.750 0.233 500 3.0 3.0 1.106 1.145 1.111 1.207 1.129 1.506 1.461 0.224

Fig. 5-3 The response diagram of the ATS

₁s for the eight charts when

b  500

and



₃



0

2 4 6 8

02468

0 2 4 6 8

ATS₁

delta1 delta2

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

(delta3= 0, b= 500) WV SC FSI-AL EWMA-AL opt.VSI-AL FSI-ML EWMA-ML opt.VSI-ML

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Tables 5-6 gives the ATS1s among all the methods when

b   500

and



₃



0. The response diagram is shown in Fig. 5-4.

Table 5-6 The ATS

1s comparison among the 8 charts when

b   500

and



₃



1

b



₁



₂ WV SC FSI-AL EWMA-AL

(



0.2)

Optimal

VSI-AL FSI-ML EWMA-ML (0.2)

Optimal VSI-ML -500 0 1 80.577 318.818 185.185 185.185 182.828 185.185 185.185 182.828 -500 0.5 1.5 3.510 3.886 2.105 2.170 1.039 2.865 3.444 1.039 -500 1.0 1.5 1.877 1.865 1.280 1.416 0.328 1.737 2.044 0.328 -500 2.0 1.5 1.079 1.076 1.019 1.037 0.035 1.127 1.218 0.035 -500 3.0 1.5 1.004 1.004 1.001 1.002 0.003 1.017 1.035 0.003 -500 0.5 2.0 1.665 1.972 1.314 1.451 0.563 2.005 2.366 0.563 -500 1.0 2.0 1.317 1.442 1.128 1.208 1.088 1.549 1.775 0.275 -500 2.0 2.0 1.056 1.073 1.021 1.036 1.017 1.160 1.248 0.064 -500 3.0 2.0 1.006 1.009 1.002 1.004 1.002 1.040 1.069 0.013 -500 0.5 3.0 1.095 1.164 1.013 1.054 0.998 1.420 1.586 0.222 -500 1.0 3.0 1.060 1.104 1.010 1.023 1.007 1.301 1.427 0.152 -500 2.0 3.0 1.019 1.033 1.012 1.020 1.012 1.147 1.215 0.068 -500 3.0 3.0 1.007 1.010 1.005 1.007 1.004 1.066 1.100 0.027

2 4 6 8

0.00.51.01.52.02.53.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

ATS₁

delta1 delta2

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

(delta3=1, b=-500) WV SC FSI-AL EWMA-AL opt.VSI-AL FSI-ML EWMA-ML opt.VSI-ML

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

Table 5-7 gives the ATS1s among all the eight charts when

b  0

and



₃



1. The response diagram is shown in Fig. 5-5.

Table 5-7 The ATS

1s comparison among the 8 methods when

b  0

and



₃



1

b



₁



₂ WV SC FSI-AL EWMA-AL

(0.2)

Optimal

VSI-AL FSI-ML EWMA-ML (

  0 . 2

)

Optimal VSI-ML 0 0 1 136.771 246.538 185.256 185.185 178.881 185.185 185.185 182.204 0 0.5 1.5 4.329 5.693 4.467 3.351 2.513 9.022 5.704 2.954 0 1.0 1.5 2.478 2.771 2.177 1.997 1.481 3.730 2.783 0.579 0 2.0 1.5 1.167 1.181 1.115 1.153 1.063 1.363 1.327 0.019 0 3.0 1.5 1.006 1.006 1.003 1.006 1.002 1.032 1.035 0.000 0 0.5 2.0 1.913 2.273 2.280 2.173 1.653 4.827 3.721 1.404 0 1.0 2.0 1.623 1.813 1.631 1.651 1.329 2.933 2.484 0.559 0 2.0 2.0 1.163 1.194 1.122 1.155 1.079 1.460 1.417 0.065 0 3.0 2.0 1.019 1.022 1.012 1.019 1.009 1.087 1.090 0.005 0 0.5 3.0 1.177 1.240 1.344 1.421 1.220 2.345 2.127 0.413 0 1.0 3.0 1.141 1.196 1.242 1.305 1.164 1.996 1.864 0.281 0 2.0 3.0 1.075 1.102 1.097 1.131 1.075 1.467 1.436 0.104 0 3.0 3.0 1.024 1.032 1.028 1.043 1.023 1.183 1.181 0.030

Fig. 5-5 The response diagram of the ATS

₁ for the eight methods when

b  0

and



₃



1

2 4 6 8

0123456

0 1 2 3 4 5 6

ATS₁

delta1 delta2

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

(delta3=1, b=0) WV SC FSI-AL EWMA-AL opt.VSI-AL FSI-ML EWMA-ML opt.VSI-ML

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

Table 5-8 gives the ATS1 comparison among all the methods when

b  500

and

3



1



. The response diagram of ATS₁ is shown in Fig. 5-6.

Table 5-8 The ATS

1s among the 8 charts when

b  500

and



₃



1

b



₁



₂ WV SC FSI-AL EWMA-AL

(

  0 . 2

)

Optimal

VSI-AL FSI-ML EWMA-ML (

  0 . 2

)

Optimal VSI-ML 500 0 1 147.616 322.125 185.185 185.185 179.960 185.185 185.185 182.463 500 0.5 1.5 6.277 9.503 6.993 5.213 4.294 17.426 10.395 7.619 500 1.0 1.5 3.579 4.861 3.690 2.799 2.106 7.418 4.091 1.101 500 2.0 1.5 1.263 1.404 1.423 1.388 1.141 2.045 1.573 0.000 500 3.0 1.5 1.001 1.005 1.020 1.030 1.011 1.083 1.049 0.000 500 0.5 2.0 2.650 3.320 3.127 3.421 2.468 10.220 7.525 3.882 500 1.0 2.0 2.305 2.848 2.362 2.376 1.772 5.622 4.070 1.953 500 2.0 2.0 1.374 1.518 1.399 1.414 1.194 2.195 1.772 0.039 500 3.0 2.0 1.024 1.045 1.065 1.080 1.044 1.233 1.161 0.000 500 0.5 3.0 1.277 1.362 1.737 2.074 1.594 6.266 3.700 0.852 500 1.0 3.0 1.328 1.435 1.552 1.785 1.469 4.357 3.432 0.671 500 2.0 3.0 1.269 1.363 1.293 1.389 1.249 2.363 2.083 0.436 500 3.0 3.0 1.106 1.145 1.127 1.167 1.114 1.506 1.397 0.042

2 4 6 8

02468

0 2 4 6 8

ATS₁

delta1 delta2

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

0.5 1 2 3 1.5 2 3

Response diagram

(delta3=1, b=500) WV SC FSI-AL EWMA-AL opt.VSI-AL FSI-ML EWMA-ML opt.VSI-ML

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

From the response diagrams of Fig. 5-1 to 5-6, Table 5-9 gives ARL1s comparison of these eight control charts under b500,0,500 and



₃



0,1 respectively. We can see that the optimal VSI charts are better than FSI charts at all combinations of (

b

,



₃).

In most of the cases the optimal VSI-ML chart performs better than the optimal VSI-AL chart. However, when

b  500

and



₁ and



₂ are small the optimal VSI-AL chart performs better than the optimal VSI-ML chart.

The FSI-ML chart seems to have poorer performance than other charts. However, the FSI-ML chart performs better than the FSI-AL chart and the EWMA-AL chart when

b   500

and



₃



0. When

b   500

and large



₁ and



₂, the FSI-ML chart has performance at least as good as the WV and SC

X

and R charts.

The EWMA-ML chart performs better than the FSI-ML chart in most of the combinations of (

b

,



₃). However, when

b   500

and



₃



1 the EWMA-ML chart has no advantage to the FSI-ML chart no matter



₁ and



₂ are small or large.

The EWMA-ML chart seems to have poorer performance than WV

X

and R charts, SC

X

and R charts, FSI-AL chart and EWMA-AL chart. However, when

b   500

and



₃



0, the EWMA-ML chart is comparable to the WV

X

and R charts, SC

X

and R charts, FSI-AL chart and EWMA-AL chart. The EWMA-ML chart performs better than the FSI-AL and EWMA-AL charts in small



₁ and



₂ when

 500



b

and



₃



0.

The FSI-AL chart seems to have better detection ability than the EWMA-AL

 

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

than the WV

X

and R charts, SC

X

and R charts, ML chart and EWMA-ML chart in most of the cases. However, when

b  500

and



₃



1 the FSI-AL chart performs poorer than the WV

X

and R charts. The FSI-AL chart seem to have poorest performance among all of the other charts in small



₁ and



₂ when

 500



b

and



₃



0.

The EWMA-AL chart seems to have better performance than the FSI-AL chart when



₁ and



₂ are small. However, when (

b

,



₃)



(



500,1) or (500,0) the EWMA-AL chart performs poorer than the FSI-AL chart. The EWMA-AL chart performs at least as good as all of other FSI charts when

b  0

no matter



₁ and



2 are small or large. The EWMA-AL chart performs better than WV

X

and R charts and SC

X

and R charts for small



₁ and



₂ under



₃



1 and

b   500

, 0 or 500. The EWMA-AL chart performs at least as good as than WV and SC methods in detecting moderate to large shift of



₁ and



₂ when



₃



0,1 and

b

be either

 500

or 0.

The WV

X

and R charts is comparable to the SC

X

and R charts. However, the ATS0 of the WV

X

and R charts is much higher than 185, implies a much higher false alarm rate. On the other hand, the ATS0 of the SC

X

and R charts is lower than 185. The SC

X

and R charts has a better in-control performance than the WV

X

and R charts.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

Table 5-9 The performance comparison among the eight charts

b

3 ^{-500 0 500}

Small shift 0

Opt.VSI-ML>Opt.VSI-AL

>EWMA-ML=WV=SC

>FSI-ML>EWMA-AL>FSI-AL

Opt.VSI-ML>Opt.VSI-AL

>EWMA-AL>FSI-AL>WV>SC

>EWMA-ML>FSI-ML

Opr.VSI-AL>Opt.VSI-ML

>FSI-AL>WV>EWMA-AL>SC

>EWMA-ML>FSI-ML

1

Opt.VSI-ML>Opt.VSI-AL

>FSI-AL>EWMA-AL>WV>SC

>FSI-ML>EWMA-ML

Opt.VSI-ML>Opt.VSI-AL

>EWMA-AL>FSI-AL=WV=SC

>EWMA-ML>FSI-ML

Opt.VSI-AL>Opt.VSI-ML

>EWMA-AL>WV>FSI-AL>SC

>EWMA-ML>FSI-ML

Moderate to large shift 0

Opt.VSI-ML>Opt.VSI-AL

>EWMA-ML=EWMA-AL

=FSI.ML=FSI-AL=WV=SC

Opt.VSI-ML>Opt.VSI-AL

>EWMA-AL=FSI-AL=WV

=SC>EWMA-ML>FSI-ML

Opt.VSI-ML>Opt.VSI-AL

>FSI-AL=WV>SC>EWMA-AL

>EWMA-ML>FSI-ML

1

Opt.VSI-ML>Opt.VSI-AL

>FSI-AL>EWMA-AL>FSI-ML

>WV>SC>EWMA-ML

Opt.VSI-ML>Opt.VSI-AL

>EWMA-AL>FSI-AL>WV>SC

>EWMA-ML>FSI-ML

>EWMA-ML>FSI-ML

在文檔中 偏斜常態分配下損失管制圖之設計 - 政大學術集成 (頁 69-108)