Performance Comparison for VP Weighted Loss Control Charts

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6.5 Performance Comparison for VP Weighted Loss Control Charts

6.5.1 VP Weighted Loss Control Charts with Specified Variable Parameters

Consider the levels of the process parameters to be : γ₁=(0.05 ,0.1) ,

) 1 . 0 , 05 . 0

2 =(

γ , δ₁=(0.5,1.5), δ₂ =(1.5,2),δ₃ =(0 ,1), δ₄=(0.5,1.5) and δ₅ =(1.5 ,2), and the specified values of

t

_q,

n

_q,

α

_q are as follows : (t₁,t₂)=

(

(0.2,0.9),(0.4,0.6)

)

(

⁽¹¹^,⁸^,³^),⁽²⁵^,¹⁴^,⁴⁾

)

) , ,

(n₁ n₂ n₃ = and (α₁,α₃)=

(

(0.005,0.0025),(0.01,0.002)

)

. To investigate the effects of the parameters

( γ

₁,

γ

₂,

δ

₁ ,

δ

₂,

δ

₃,

δ

₄ ,

δ

₅ ,

t

n

α

) on AATS,

ANOS and the out-of-control average loss per unit product, and to compare the performances among specified VP WL₁ and WL₂ charts, FP WL₁ and WL₂ charts

and FP 2

X Z

Z − and 2

Z

− charts, we adopted 16 combinations of these

parameters using an orthogonal array table L16 (2¹⁰) (Table 6.7a). The out-of-control average loss per unit product for WL₁ and WL₂ charts and for FP 2

X Z

Z − and

Z

− charts can be expressed in equations (6.26) and (6.27). In order to simply the calculation, we let the two-step in-control means are zero and the variances are one.

2 4 2 2

5 2 2 3 1

1 2

2 1 2

1 WL ( S_X) (1 )( S_X S_X ) ( S_e) (1 )( S_e)

WL + =λ δ + −λ δ +δ +λ δ + −λ δ

2 4 2 2

5 2 2 3 1 1 2

1δ (1 λ )(δ δ ) λ δ (1 λ )δ

λ + − + + + −

= (6.26)

2 5 2 4 2 2 2 1 2 5 2 4 2 2 2

1 ) ( ) ( ) ( )

(

2 + + = δ + δ + δ + δ =δ +δ +δ +δ

= _X _S _e _S _X _X _e _e

S L L L L S S S S

L X e (6.27)

The calculated values of (

λ

₁^*,

λ

^*₂),

t ,

₃ α , ₂

( w

1⁽

α

q^,

n

q^),

k

1⁽

α

q^,

n

q^),

w

2⁽

α

q^,

n

q^),

)

) ,

2( q

n

k α

, AATS*, ANOS and the out-of-control average loss per unit product are illustrated in Tables 6.7b and 6.7c. The results indicate that specified VP WL₁ and WL2 charts used less time than FP WL₁ and WL₂ charts, and the AATS saving from 0.14% to 36.01%. Since the in-control process mean was always assumed equal

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to the target for FP 2

X Z

Z − and 2

e Z

Z − charts, under

δ

₃ =0 specified VP

WL1 and WL₂ chart had smaller AATS and the out-of-control average loss per unit product than FP 2

X Z

Z − and 2

e Z

Z − charts, and the AATS savings from 13.02%

to 47.11% and loss savings from 50% to 60.4%.

We also created main effect plots to investigate the effects of all the parameters on the amount of saved percentage in the AATS, ANOS and loss. There were two main results in those effect plots:

1. Figure 6.7a indicates parameters δ₁,δ₂,δ₄ and

δ

₅ are significant, when

5 4 2 1,

δ

decrease, specified VP WL₁ and WL₂ charts would save more AATS percentage than FP WL₁ and WL₂ charts. Figure 6.7b indicates parameters

δ

₁ ,

δ

₂,

δ

₄,

δ

₅ and

n are significant, when

_q δ ,₁ δ ,₂ δ ,₄

δ

₅ decrease, specified VP WL₁ and WL₂ charts would waste less ANOS percentage than FP WL1 and WL₂ charts under (

n

₁,

n

₂,

n

₃)=(11,8,3). However, parameter

δ

₃ is not significant for saved AATS and ANOS percentage.

2. Figures 6.8a and 6.8b indicate parameters

γ

₁,

δ

₁,

δ

₂,

δ

₄,

δ

₅,

t ,

n

_q and

α

_q are significant, when

γ

₁,

δ

₁,

δ

₂,

δ

₄,

δ

₅ decrease, specified VP WL₁ and WL₂ charts would save more AATS percentage and waste less ANOS percentage than FP

X Z

Z − and 2

e Z

Z − charts under (

t

₁,

t

₂)=(0.4,0.6), )(

n

₁,

n

₂,

n

₃)=(11,8,3

and (

α

₁,

α

₃)=(0.01,0.002). Figure 6.8c indicates parameters δ₁,δ₂,δ₄ and

δ

₅ are not significant for saved loss percentage.

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Table 6.7a Orthogonal array L16(2¹⁰) for 16 combinations of various parameters Combination of various parameters

Trt. γ₁ γ₂ δ₁ δ₂

δ

₃ δ₄

δ

₅ (t₁,t₂) (

n

₁,

n

₂,

n

₃) (

α

₁,

α

₃) 1 0.05 0.05 0.5 1.5 0 0.5 1.5 (0.2, 0.9) (25, 14, 4) (0.01, 0.002) 2 0.05 0.05 0.5 2 0 1.5 2 (0.4, 0.6) (11, 8, 3) (0.01, 0.002) 3 0.05 0.05 1.5 1.5 1 0.5 2 (0.4, 0.6) (25, 14, 4) (0.01, 0.002) 4 0.05 0.05 1.5 2 1 1.5 1.5 (0.2, 0.9) (11, 8, 3) (0.01, 0.002) 5 0.05 0.1 0.5 1.5 1 1.5 1.5 (0.4, 0.6) (25, 14, 4) (0.005, 0.0025) 6 0.05 0.1 0.5 2 1 0.5 2 (0.2, 0.9) (11, 8, 3) (0.005, 0.0025) 7 0.05 0.1 1.5 1.5 0 1.5 2 (0.2, 0.9) (25, 14, 4) (0.005, 0.0025) 8 0.05 0.1 1.5 2 0 0.5 1.5 (0.4, 0.6) (11, 8, 3) (0.005, 0.0025) 9 0.1 0.05 0.5 1.5 1 1.5 2 (0.2, 0.9) (11, 8, 3) (0.005, 0.0025) 10 0.1 0.05 0.5 2 1 0.5 1.5 (0.4, 0.6) (25, 14, 4) (0.005, 0.0025) 11 0.1 0.05 1.5 1.5 0 1.5 1.5 (0.4, 0.6) (11, 8, 3) (0.005, 0.0025) 12 0.1 0.05 1.5 2 0 0.5 2 (0.2, 0.9) (25, 14, 4) (0.005, 0.0025) 13 0.1 0.1 0.5 1.5 0 0.5 2 (0.4, 0.6) (11, 8, 3) (0.01, 0.002) 14 0.1 0.1 0.5 2 0 1.5 1.5 (0.2, 0.9) (25, 14, 4) (0.01, 0.002) 15 0.1 0.1 1.5 1.5 1 0.5 1.5 (0.2, 0.9) (11, 8, 3) (0.01, 0.002) 16 0.1 0.1 1.5 2 1 1.5 2 (0.4, 0.6) (25, 14, 4) (0.01, 0.002)

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Table 6.7b Optimum variable parameters for specified VP WL and ₁ WL charts (₂ ARL₀ =185) Optimum variable parameters for specified VP WL and ₁ WL charts ₂

Trt. (

λ

^*₁,

λ

^*₂) t3 α ₂

(

w1⁽α1^,n1^),k1⁽α1^,n1^),w2⁽α1^,n1^),k2⁽α1^,n1⁾

) (

w₁(α₂,n₂),k₁(α₂,n₂),w₂(α₂,n₂),k₂(α₂,n₂)

) (

w₁(α₃,n₃),k₁(α₃,n₃),w₂(α₃,n₃),k₂(α₃,n₃)

)

1 0.37 0.37 1.01 0.0092 0.59 0.71 0.59 0.71 0.69 0.88 0.69 0.88 1.25 2.26 1.25 2.26 2 0.40 0.34 1.25 0.0031 0.59 1.03 0.52 0.92 0.64 1.39 0.57 1.27 0.91 2.96 0.84 2.81 3 0.41 0.41 1.04 0.0092 1.47 1.73 0.64 0.77 1.65 2.05 0.75 0.95 2.48 4.11 1.31 2.36 4 0.49 0.25 1.11 0.0031 1.33 2.11 0.42 0.78 1.40 2.67 0.47 1.15 1.73 4.62 0.75 2.76 5 0.57 0.25 1.04 0.0045 1.43 1.73 0.43 0.56 1.59 2.04 0.53 0.76 2.32 3.71 1.07 2.06 6 0.64 0.41 1.11 0.0028 1.30 2.18 0.61 1.13 1.36 2.61 0.66 1.43 1.63 4.56 0.91 2.90 7 0.26 0.34 1.01 0.0045 0.44 0.57 0.55 0.70 0.54 0.77 0.65 0.89 1.08 2.06 1.20 2.13 8 0.34 0.37 1.25 0.0028 0.52 1.00 0.56 1.05 0.57 1.29 0.61 1.34 0.84 2.71 0.87 2.78 9 0.57 0.34 1.11 0.0028 1.32 2.20 0.52 1.00 1.38 2.62 0.57 1.29 1.68 4.45 0.84 2.71 10 0.64 0.37 1.04 0.0045 1.42 1.71 0.59 0.75 1.58 2.03 0.70 0.95 2.29 3.75 1.24 2.19 11 0.25 0.25 1.25 0.0028 0.42 0.86 0.42 0.86 0.47 1.17 0.47 1.17 0.75 2.66 0.75 2.66 12 0.34 0.41 1.01 0.0045 0.55 0.70 0.65 0.82 0.65 0.89 0.76 1.03 1.20 2.13 1.31 2.29 13 0.36 0.40 1.25 0.0031 0.55 0.95 0.59 1.03 0.60 1.31 0.64 1.39 0.86 2.85 0.91 2.96 14 0.41 0.26 1.01 0.0092 0.64 0.77 0.44 0.53 0.75 0.95 0.53 0.70 1.31 2.36 1.09 2.14 15 0.40 0.36 1.11 0.0031 1.35 2.22 0.55 0.95 1.43 2.83 0.60 1.31 1.78 4.88 0.86 2.85 16 0.51 0.35 1.04 0.0092 1.43 1.67 0.56 0.67 1.60 1.96 0.66 0.85 2.37 3.86 1.21 2.22

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Table 6.7c Performance comparison among specified VP WL₁ and WL₂ charts, FP WL₁ and WL₂ charts and FP 2 SX

X Z

Z − and 2

e Z

Z − charts (ARL₀ =185) Specified VP WL and ₁ WL charts ₂ FP WL and ₁ WL charts ₂ FP 2

X Z

Z − and 2

e Z

Z − charts

Trt. AATS* ANOS WL1+WL2 AATS ANOS Saved AATS(%)

Saved

ANOS(%) AATS ANOS L_S ^Saved AATS(%)

Saved ANOS(%)

Saved loss(%) 1 2.54 20.36 1.98 3.96 19.82 36.01 -2.73 4.80 23.98 5.00 47.11 15.08 60.40 2 1.02 7.73 4.60 1.12 5.61 9.03 -37.62 1.28 6.41 10.50 20.36 -20.49 56.24

3 1.00 8.82 6.40 1.16 5.79 13.89 -52.26

4 0.90 6.12 7.40 0.90 4.48 0.14 -36.62

5 1.30 11.42 4.50 1.68 8.42 22.99 -35.58

6 1.22 8.81 5.16 1.42 7.12 14.37 -23.74

7 0.88 6.69 5.10 0.89 4.43 0.32 -51.08 1.01 5.07 10.75 13.02 -31.83 52.60 8 1.92 15.91 3.84 2.75 13.75 30.07 -15.65 3.25 16.26 8.75 40.85 2.19 56.17

9 1.88 14.17 5.10 2.50 12.49 24.68 -13.44

10 1.55 13.53 4.36 2.11 10.57 26.71 -27.97

11 0.96 7.20 4.50 1.01 5.03 5.03 -43.02 1.19 5.94 9.00 19.56 -21.13 50.00 12 0.96 7.34 4.63 1.01 5.06 4.95 -45.06 1.15 5.75 10.50 16.29 -27.75 55.88 13 1.73 14.30 2.72 2.41 12.07 28.39 -18.48 2.81 14.04 6.75 38.45 -1.84 59.70 14 1.10 8.87 4.04 1.19 5.96 7.81 -48.99 1.38 6.91 8.75 20.57 -28.37 53.86

15 1.64 12.24 5.62 2.10 10.49 21.78 -16.65

16 0.79 6.34 7.97 0.80 4.01 1.15 -58.00

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average of saved AATS(%)

0.10 0.05 25

0.10

0.05 0.5 1.5 1.5 2.0

1 0 25

1.5

0.5 1.5 2.0 (0.2,0.9) (0.4,0.6)

(25,14,4) (11, 8, 3) 25

(0.01,0.002) (0.005,0.0025)

r1 r2 δ1 δ2

δ3 δ4 δ5 (t1,t2)

(n1,n2,n3) (α1,α3)

main effects plot for various parameters

average of saved ANOS(%)

0.10 0.05 -23

-33

-43

0.10

0.05 0.5 1.5 1.5 2.0

1 0 -23

-33

-43

1.5

0.5 1.5 2.0 (0.2,0.9) (0.4,0.6)

(25,14,4) (11, 8, 3) -23

-33

-43

(0.01,0.002) (0.005,0.0025)

r1 r2 δ1 δ2

δ3 δ4 δ5 (t1,t2)

(n1,n2,n3) (α1,α3)

main effects plot for various parameters

Figure 6.7a The main effects plots for saved AATS percentage Figure 6.7b The main effects plots for saved ANOS percentage for specified VP WL charts compare with FP WL charts for specified VP WL charts compare with FP WL charts

average of saved AATS(%)

0.10 0.05

0.10

0.05 0.5 1.5

2.0 1.5 35

1.5

0.5 1.5 2.0

(0.4,0.6) (0.2,0.9) 35

(25,14,4)

(11, 8, 3) (0.005,0.0025) (0.01,0.002)

r1 r2 δ1

δ2 δ4 δ5

(t1,t2) (n1,n2,n3) (α1,α3)

main effects plot for various parameters

average of saved ANOS(%)

0.10 0.05 0 -10

-20

0.10

0.05 0.5 1.5

2.0 1.5 0 -10

-20

1.5

0.5 1.5 2.0

(0.4,0.6) (0.2,0.9) 0 -10 -20

(25,14,4)

(11, 8, 3) (0.005,0.0025) (0.01,0.002)

r1 r2 δ1

δ2 δ4 δ5

(t1,t2) (n1,n2,n3) (α1,α3)

main effects plot for various parameters

Figure 6.8a The main effects plots for saved AATS percentage for specified Figure 6.8b The main effects plots for saved ANOS percentage for specified VP WL charts compare with FP ₂

X Z

Z − and 2

e Z

Z − charts VP WL charts compare with FP ₂

X Z

Z − and 2

e Z

Z − ^charts

average of saved loss(%)

1.5 0.5

2.0 1.5

1.5 0.5

2.0 1.5

δ1 δ2

δ4 δ5

main effects plot for various parameters

Figure 6.8c The main effects plots for saved loss percentage for specified VP WL charts compare with FP ₂

X Z

Z − ^and 2 Se

e Z

Z − ^charts

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6.5.2 VP Weighted Loss Control Charts with Optimum Variable Parameters

Sometimes, the variable parameters

t

_q ,

n

_q,

α

_q ,

w

_i(

α

_q,

n

_q),

k

_i(

α

_q,

n

_q),

i

=1,2 ,

3 , 2 ,

q

and the weighted factors

λ

₁ and

λ

₂ for VP WL₁ and WL₂ control charts cannot be specified. However, the optimal values for VP WL₁ and WL₂ charts need to be present. Here, the optimal variable parameters and weighted factors for the proposed charts are determined using an optimization technique, the OPTIM subroutine in the R program to minimize AATS under the following constraints. That is,

Objective function : Minimize AATS

Subject to :

0.1≤

t

₁<

t

₂<1<

t

₃≤2, 30 5

1≤

n

₃ < <

n

₂ <

n

₁≤ ,

, 1 . 0

α

₃ <

α

₂ <

α

₁< (6.28)

3 , 2 , 1 , ) , ( ) , ( 0 , 1 0

, 3 , 2 , 1 , ) , ( ) , ( 0 , 1 0

2 2

1 1

∞

q n

k n w

q n

k n w

q q q

α α

We considered 16 combinations of (γ₁,γ₂,δ₁,δ₂,

δ

₃,

δ

₄,

δ

₅) using orthogonal array table L16 (2⁷) (see Table 6.8a), and used the aforementioned optimization technique to determine the optimum variable parameters for minimum AATS. The optimum results are shown in Tables 6.8b and 6.8c. The results indicate that optimum VP

WL and

₁

WL charts use less time than FP

₂ WL₁ and WL₂ charts, and the AATS is saved from 8.3% to 49.98% . Under

δ

₃ =0, optimum VP WL₁ and WL₂ charts also save time and result in less loss than FP 2

X Z

Z − and

Z

− charts, and the AATS savings from 24.06% to 58.36% and loss saving from 50% to 62%.

We also created main effects plots for the saved AATS, ANOS and loss percentages versus various parameters within the optimized VP WL₁ and WL₂

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charts. There were two main results in those plots:

1. Figures 6.9a and 6.9b indicate that parameters δ₁,δ₂,δ₄ and

δ

₅ are significant, and optimum VP WL₁ and WL₂ charts would save more AATS percentage and waste less ANOS percentage than FP WL₁ and WL₂ charts when the two dependent process means and variances both have small shifts. However, parameters γ₁,γ₂ and

δ

₃ are not significant for saved AATS and ANOS percentage.

2. Figures 6.10a and 6.10b indicate that parameters γ ,₁ δ₁,δ₂,δ₄ and

δ

₅ are significant, when

γ

₁,

δ

₁,

δ

₂,

δ

₄,

δ

₅ decrease, and optimum VP WL₁ and WL₂ charts would save more AATS percentage and waste less ANOS percentage than

FP 2

X Z

Z − and 2

Z

− charts. Figure 6.10c indicates parameters

4 2 1,δ ,δ

δ and

δ

₅ are not significant for saved loss percentage.

Based on the above main effects plots, optimum VP WL₁ and WL₂ charts save more AATS percentage and waste less ANOS percentage when the means and variances of the two dependent process steps both have small shifts for

5 . 1 , 5 .

0 ≤δ₁ δ₄≤ and 0.5≤

δ

₂,

δ

₅ ≤1.5. To check the effects of the smaller

δ

_i, 2

. 1 , 2 .

0 ≤δ₁ δ₄ ≤ and 1.3≤

δ

₂,

δ

₅ ≤1.7, their optimum results are calculated and presented in Tables 6.9b and 6.9c. The results in Table 6.9d indicate that smaller δ ,1 δ ,₂ δ and₄

δ

₅ increased much amount of saved AATS, ANOS and loss percentages compared to FP WL₁ and WL₂ charts and FP 2

X Z

Z − and

Z

− charts.

From the results of the previously specified-parameter and optimum-parameter data analyses, we conclude that the performance of optimum VP WL₁ and WL₂ charts is better than specified VP WL₁ and WL₂ charts, and specified VP WL₁ and

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WL2 charts outperform FP WL₁ and WL₂ charts and FP 2

X Z

Z − and 2

Z

−

charts. In particular, when the two dependent process shifts are much smaller, VP WL1 and WL₂ charts would save more AATS and loss percentages and waste less ANOS percentages than the other charts.

Table 6.8a Orthogonal array L16(2⁷) for 16 combinations of various parameters Combination of various parameters

Trt. γ₁ γ₂ δ₁ δ₂

δ

₃ δ₄

δ

₅

1 0.05 0.05 0.5 1.5 0 0.5 1.5

2 0.05 0.05 0.5 2 0 1.5 2

3 0.05 0.05 1.5 1.5 1 0.5 2 4 0.05 0.05 1.5 2 1 1.5 1.5 5 0.05 0.1 0.5 1.5 1 1.5 1.5

6 0.05 0.1 0.5 2 1 0.5 2

7 0.05 0.1 1.5 1.5 0 1.5 2 8 0.05 0.1 1.5 2 0 0.5 1.5 9 0.1 0.05 0.5 1.5 1 1.5 2 10 0.1 0.05 0.5 2 1 0.5 1.5 11 0.1 0.05 1.5 1.5 0 1.5 1.5

12 0.1 0.05 1.5 2 0 0.5 2

13 0.1 0.1 0.5 1.5 0 0.5 2 14 0.1 0.1 0.5 2 0 1.5 1.5 15 0.1 0.1 1.5 1.5 1 0.5 1.5

16 0.1 0.1 1.5 2 1 1.5 2

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Table 6.8b Optimum variable parameters for optimum VP WL₁ and WL₂ charts (ARL₀ =185) Optimum variable parameters for optimum VP WL₁ and WL₂ charts

Trt. (

λ

^*₁,

λ

^*₂)

(

^, ^, ^, ⁽ ^, ^), ⁽ ^, ^), ⁽ ^, ^), ⁽ ^, 1^*⁾

)

* 1 2

* 1

* 1 2

* 1

* 1 1

* 1

* 1 1

* 1

1 n w n k n w n k n

t α α α α α

(

^, ^, ^, ⁽ ^, ^), ⁽ ^, ^), ⁽ ^, ^), ⁽ ^, ^*2⁾

)

* 2 2

* 2

* 2 2

* 2

* 2 1

* 2

* 2 1

* 2

2 n w n k n w n k n

t α α α α α

(

t₃^*,n₃^*,α₃^*,w₁(α₃^*,n₃^*),k₁(α₃^*,n^*₃),w₂(α₃^*,n₃^*),k₂(α₃^*,n^*₃)

)

1 0.35 0.35 0.1 18 0.0098 0.54 0.76 0.54 0.76 0.1 12 0.0077 0.60 0.93 0.60 0.93 1.24 3 0.0013 1.12 3.00 1.12 3.00 2 0.41 0.34 0.1 13 0.0046 0.73 1.06 0.63 0.93 0.1 11 0.0033 0.77 1.19 0.67 1.05 1.15 4 0.0026 1.18 2.28 1.08 2.12 3 0.40 0.41 0.1 13 0.0041 1.58 2.28 0.71 1.07 0.1 10 0.0032 1.68 2.58 0.77 1.25 1.18 4 0.0026 2.20 4.02 1.13 2.28 4 0.54 0.23 0.1 15 0.0035 1.51 2.06 0.45 0.73 0.1 11 0.0033 1.61 2.30 0.52 0.89 1.15 4 0.0026 2.15 3.71 0.93 2.05 5 0.59 0.23 0.1 15 0.0100 1.45 1.87 0.44 0.63 0.1 10 0.0078 1.58 2.17 0.52 0.84 1.17 4 0.0017 2.06 3.88 0.89 2.20 6 0.67 0.41 0.1 13 0.0046 1.49 2.08 0.71 1.06 0.1 10 0.0031 1.58 2.36 0.77 1.25 1.18 4 0.0026 2.02 3.78 1.13 2.28 7 0.24 0.34 0.1 15 0.0035 0.48 0.74 0.62 0.89 0.1 12 0.0034 0.53 0.85 0.66 1.00 1.13 4 0.0026 0.99 2.05 1.12 2.12 8 0.33 0.35 0.1 16 0.0100 0.53 0.76 0.56 0.79 0.1 12 0.0069 0.58 0.90 0.60 0.94 1.25 3 0.0015 1.09 2.91 1.12 2.94 9 0.58 0.34 0.1 18 0.0094 1.42 1.79 0.56 0.74 0.1 10 0.0091 1.58 2.14 0.67 0.98 1.17 4 0.0015 2.07 3.93 1.05 2.29 10 0.67 0.37 0.1 15 0.0098 1.45 1.86 0.62 0.85 0.1 10 0.0079 1.57 2.16 0.71 1.05 1.17 4 0.0017 2.03 3.98 1.08 2.31 11 0.22 0.22 0.1 14 0.0043 0.45 0.73 0.45 0.73 0.1 11 0.0033 0.50 0.89 0.50 0.89 1.15 4 0.0026 0.92 2.06 0.92 2.06 12 0.34 0.41 0.1 16 0.0035 0.59 0.86 0.69 0.99 0.1 11 0.0033 0.67 1.05 0.77 1.18 1.15 4 0.0026 1.08 2.12 1.19 2.28 13 0.37 0.41 0.1 15 0.0098 0.62 0.85 0.68 0.921 0.1 10 0.0089 0.71 1.03 0.77 1.11 1.17 4 0.0015 1.08 2.35 1.14 2.45 14 0.41 0.23 0.1 15 0.0033 0.69 1.03 0.44 0.73 0.1 10 0.0032 0.77 1.25 0.52 0.96 1.17 4 0.0026 1.14 2.28 0.89 2.05 15 0.40 0.37 0.1 17 0.0093 1.52 1.95 0.62 0.81 0.1 11 0.0096 1.67 2.23 0.70 0.98 1.14 4 0.0016 2.30 4.25 1.13 2.33 16 0.53 0.33 0.1 15 0.0035 1.56 2.07 0.61 0.87 0.1 13 0.0035 1.61 2.16 0.64 0.93 1.11 4 0.0026 2.28 3.72 1.14 2.11

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Table 6.8c Performance comparison among optimum VP WL₁ and WL₂ charts, FP WL₁ and WL₂ charts and FP 2 SX

X Z

Z − and 2

e Z

Z − charts (ARL₀ =185)

Optimum VP WL and ₁ WL charts ₂ FP WL and ₁ WL charts ₂ FP 2 SX

X Z

Z − and 2 Se

e Z

Z − charts

Trt. AATS* ANOS WL1+WL2 AATS ANOS Saved AATS(%)

Saved

ANOS(%) AATS ANOS L_S ^Saved AATS(%)

Saved ANOS(%)

Saved loss (%) 1 2.00 22.45 1.90 3.99 19.96 49.98 -12.49 4.80 23.98 5 58.36 6.37 62.00 2 0.87 7.90 4.63 1.12 5.61 22.02 -40.93 1.28 6.41 10.5 31.82 -23.23 55.88

3 0.87 8.11 6.44 1.16 5.79 24.93 -40.05

4 0.76 6.73 7.29 0.90 4.48 14.77 -50.18

5 1.13 11.18 4.50 1.70 8.48 33.34 -31.87

6 1.01 9.60 5.21 1.43 7.14 29.29 -34.44

7 0.77 6.78 5.10 0.89 4.43 13.08 -52.89 1.01 5.07 10.75 24.06 -33.58 52.60 8 1.60 17.35 3.78 2.77 13.83 42.05 -25.40 3.25 16.26 8.75 50.71 -6.67 56.83

9 1.48 14.65 5.10 2.51 12.53 41.11 -16.88

10 1.34 13.24 4.41 2.12 10.61 36.77 -24.77

11 0.81 7.58 4.50 1.02 5.08 20.16 -49.13 1.19 5.94 9 31.75 -27.48 50.00 12 0.83 7.34 4.63 1.01 5.06 17.77 -45.13 1.15 5.75 10.5 27.58 -27.81 55.88 13 1.47 14.63 2.78 2.41 12.04 38.81 -21.49 2.81 14.04 6.75 47.52 -4.19 58.85 14 0.90 8.43 4.04 1.19 5.97 24.38 -41.19 1.38 6.91 8.75 34.66 -22.00 53.86

15 1.34 13.29 5.64 2.09 10.45 36.03 -27.21

16 0.74 6.25 7.89 0.80 4.02 8.30 -55.56

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average of saved AATS(%)

0.10 0.05

0.10

0.05 0.5 1.5

2.0 1.5 40

0 0.5 1.5

2.0 1.5 40

γ1 γ2 δ1

δ2 δ3 δ4

δ5

main effects plot for various parameters

average of saved ANOS(%)

0.10 0.05 -25

-35

-45

0.10

0.05 0.5 1.5

2.0 1.5 -25

-35

-45

0 0.5 1.5

2.0 1.5 -25

-35

-45

γ1 γ2 δ1

δ2 δ3 δ4

δ5

main effects plot for various parameters

Figure 6.9a The main effects plots for saved AATS percentage Figure 6.9b The main effects plots for saved ANOS percentage for optimum VP WL charts compare with FP WL charts for optimum VP WL charts compare with FP WL charts

average of saved AATS(%)

0.10 0.05 50

0.10

0.05 0.5 1.5

2.0 1.5 50

1.5

0.5 1.5 2.0

γ1 γ2 δ1

δ2 δ4 δ5

main effects plot for various parameters

average of saved ANOS(%)

0.10 0.05 -10

-20

-30

0.10

0.05 0.5 1.5

2.0 1.5

-10

-20

-30

1.5

0.5 1.5 2.0

γ1 γ2 δ1

δ2 δ4 δ5

main effects plot for various parameters

Figure 6.10a The main effects plots for saved AATS percentage for optimum Figure 6.10b The main effects plots for saved ANOS percentage for optimum VP WL charts compare with FP ₂

X Z

Z − ^and 2

e Z

Z − charts VP WL charts compare with FP ₂

X Z

Z − ^and 2

e Z

Z − ^charts

average of saved loss(%)

1.5 0.5

2.0 1.5

1.5 0.5

2.0 1.5

δ1 δ2

δ4 δ5

main effects plot for various parameters

Figure 6.10c The main effects plots for saved loss percentage for optimum VP WL charts compare with FP ₂

X Z

Z − ^and 2

e Z

Z − ^charts

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Table 6.9a Orthogonal array L16(2⁷) for 16 combinations of various parameters with smaller δ₁,δ₂,δ₄ and

δ

₅

Combination of various parameters Trt. γ₁ γ₂ δ₁ δ₂

δ

3 δ₄

δ

1 0.05 0.05 0.2 1.3 0 0.2 1.3 2 0.05 0.05 0.2 1.7 0 1.2 1.7 3 0.05 0.05 1.2 1.3 1 0.2 1.7 4 0.05 0.05 1.2 1.7 1 1.2 1.3 5 0.05 0.1 0.2 1.3 1 1.2 1.3 6 0.05 0.1 0.2 1.7 1 0.2 1.7 7 0.05 0.1 1.2 1.3 0 1.2 1.7 8 0.05 0.1 1.2 1.7 0 0.2 1.3 9 0.1 0.05 0.2 1.3 1 1.2 1.7 10 0.1 0.05 0.2 1.7 1 0.2 1.3 11 0.1 0.05 1.2 1.3 0 1.2 1.3 12 0.1 0.05 1.2 1.7 0 0.2 1.7 13 0.1 0.1 0.2 1.3 0 0.2 1.7 14 0.1 0.1 0.2 1.7 0 1.2 1.3 15 0.1 0.1 1.2 1.3 1 0.2 1.3 16 0.1 0.1 1.2 1.7 1 1.2 1.7

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Table 6.9b Optimum variable parameters for optimum VP WL₁ and WL₂ charts with smaller δ₁,δ₂,δ₄ and

δ

₅ (ARL₀ =185) Optimum variable parameters for optimum VP WL₁ and WL₂ charts with smaller δ₁,δ₂,δ₄ and

δ

₅

Trt. (

λ

^*₁,

λ

^*₂)

(

^, ^, ^, ⁽ ^, ^), ⁽ ^, ^), ⁽ ^, ^), ⁽ ^, 1^*⁾

)

* 1 2

* 1

* 1 2

* 1

* 1 1

* 1

* 1 1

* 1

1 n w n k n w n k n

t α α α α α

(

^, ^, ^, ⁽ ^, ^), ⁽ ^, ^), ⁽ ^, ^), ⁽ ^, 2^*⁾

)

* 2 2

* 2

* 2 2

* 2

* 2 1

* 2

* 2 1

* 2

2 n w n k n w n k n

t α α α α α

(

^, ^, ^, ⁽ ^, ^), ⁽ ^, ^), ⁽ ^, ^), ⁽ ^, ^*3⁾

)

* 3 2

* 3

* 3 2

* 3

* 3 1

* 3

* 3 1

* 3

3 n w n k n w n k n

t α α α α α

1 0.38 0.38 0.1 34 0.0100 0.56 0.66 0.557 0.66 0.1 22 0.0096 0.61 0.75 0.61 0.75 1.10 3 0.0019 1.49 2.92 1.49 2.92 2 0.43 0.31 0.1 14 0.0100 0.72 0.98 0.55 0.77 0.1 10 0.0073 0.80 1.18 0.63 0.95 1.17 4 0.0018 1.17 2.46 0.99 2.20 3 0.34 0.43 0.1 15 0.0088 1.52 2.10 0.69 0.97 0.1 9 0.0069 1.70 2.59 0.80 1.25 1.21 4 0.0017 2.18 4.45 1.11 2.48 4 0.51 0.18 0.1 13 0.0039 1.53 2.16 0.40 0.74 0.1 10 0.0032 1.62 2.42 0.46 0.92 1.18 4 0.0026 2.09 3.75 0.83 2.08 5 0.63 0.19 0.1 28 0.0100 1.34 1.59 0.32 0.42 0.1 11 0.0100 1.58 2.04 0.46 0.72 1.14 4 0.0016 2.15 3.94 0.92 2.25 6 0.73 0.43 0.1 13 0.0092 1.44 1.97 0.71 1.02 0.1 9 0.0072 1.55 2.30 0.79 1.24 1.21 4 0.0016 1.93 4.16 1.11 2.49 7 0.18 0.32 0.1 11 0.0054 0.43 0.79 0.62 0.96 0.1 10 0.0031 0.46 0.93 0.64 1.08 1.18 4 0.0026 0.83 2.08 1.00 2.09 8 0.27 0.35 0.1 35 0.0099 0.40 0.49 0.50 0.61 0.1 24 0.0098 0.44 0.56 0.54 0.68 1.14 2 0.0016 1.69 4.19 1.79 4.35 9 0.64 0.31 0.1 35 0.0097 1.32 1.52 0.46 0.55 0.1 20 0.0097 1.43 1.73 0.52 0.66 1.12 3 0.0018 2.49 4.77 1.34 2.81 10 0.71 0.4 0.1 22 0.0100 1.40 1.69 0.63 0.785 0.1 12 0.0093 1.57 2.01 0.75 1.00 1.12 4 0.0018 2.19 4.05 1.23 2.37 11 0.18 0.18 0.1 11 0.0049 0.41 0.80 0.41 0.80 0.1 9 0.0047 0.46 0.94 0.46 0.94 1.22 4 0.0022 0.77 2.14 0.77 2.14 12 0.32 0.42 0.1 15 0.0099 0.56 0.76 0.69 0.94 0.1 10 0.0057 0.64 1.00 0.78 1.19 1.17 4 0.0021 1.01 2.16 1.15 2.38 13 0.39 0.4 0.1 32 0.0100 0.55 0.69 0.57 0.702 0.1 15 0.0097 0.65 0.89 0.67 0.90 1.17 3 0.0014 1.32 3.08 1.33 3.11 14 0.42 0.17 0.1 15 0.0100 0.69 0.94 0.36 0.56 0.1 10 0.0073 0.78 1.16 0.44 0.80 1.17 4 0.0018 1.15 2.43 0.83 2.23 15 0.31 0.38 0.1 33 0.0100 1.42 1.70 0.56 0.66 0.1 20 0.0096 1.56 1.94 0.62 0.78 1.12 3 0.0018 2.86 5.37 1.44 2.94 16 0.50 0.30 0.1 14 0.0033 1.51 2.15 0.55 0.86 0.1 10 0.0032 1.62 2.42 0.62 1.04 1.17 4 0.0026 2.10 3.76 0.98 2.07

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Table 6.9c Performance comparison among optimum VP WL and ₁ WL charts with smaller ₂ δ₁,δ₂,δ₄ and

δ

₅, FP WL and ₁ WL charts and ₂

FP 2

X Z

Z − and 2

e Z

Z − charts (ARL₀ =185)

Optimum VP WL₁ and WL₂ charts FP WL and ₁ WL charts ₂ FP 2

X Z

Z − and 2

e Z

Z − charts

Trt. AATS* ANOS WL1+WL2 AATS ANOS Saved AATS(%)

Saved

ANOS(%) AATS ANOS L_S ^Saved AATS(%)

Saved ANOS(%)

Saved loss (%) 1 5.60 56.54 1.33 12.27 61.36 54.35 7.86 14.41 72.04 3.46 61.12 21.52 61.45 2 1.24 12.32 3.16 2.00 9.98 37.88 -23.44 2.37 11.84 7.26 47.60 -4.12 56.54 3 1.23 12.87 5.03 2.10 10.51 41.69 -22.41

4 0.95 9.44 5.33 1.42 7.10 33.21 -32.91 5 2.26 21.48 3.09 3.49 17.43 35.17 -23.22 6 1.59 16.14 3.76 2.85 14.23 44.04 -13.44

7 0.96 9.27 3.39 1.39 6.93 30.59 -33.62 1.66 8.32 7.46 42.18 -11.31 54.57 8 3.75 40.85 2.45 6.76 33.79 44.47 -20.89 7.47 37.36 6.06 49.78 -9.33 59.59 9 3.67 37.69 3.49 6.39 31.95 42.55 -17.93

10 2.85 26.31 3.17 4.71 23.56 39.58 -11.69

11 1.05 10.74 2.97 1.72 8.62 39.17 -24.63 2.11 10.56 6.26 50.37 -1.67 52.56 12 1.12 10.98 3.14 1.68 8.42 33.36 -30.48 1.99 9.93 7.26 43.52 -10.59 56.74 13 3.07 32.64 1.86 5.29 26.47 41.98 -23.30 6.02 30.10 4.66 48.98 -8.42 60.01 14 1.28 13.35 2.72 2.16 10.81 40.60 -23.54 2.57 12.87 6.06 50.14 -3.71 55.12 15 2.72 30.52 4.53 4.22 21.10 35.58 -44.64

16 0.88 8.21 5.74 1.16 5.81 23.80 -41.33

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Table 6.9d Performance comparison between optimum VP WL₁ and WL₂ charts with smaller δ₁,δ₂,δ₄ and

δ

₅ and optimum VP WL₁ and WL₂ charts Compare with FP WL and ₁ WL charts 2 Compare with FP 2

X Z

Z − and 2

e Z

Z − charts

Trt. Saved AATS(%)*

Saved AATS(%)

Saved ANOS(%)*

Saved ANOS(%)

Saved AATS(%)*

Saved AATS(%)

Saved ANOS(%)*

Saved ANOS(%)

Saved loss (%)*

Saved loss (%) 1 54.35 49.98 7.86 -12.49 61.12 58.36 21.52 6.37 61.45 62.00 2 37.88 22.02 -23.44 -40.93 47.60 31.82 -4.12 -23.23 56.54 55.88

3 41.69 24.93 -22.41 -40.05

4 33.21 14.77 -32.91 -50.18

5 35.17 33.34 -23.22 -31.87

6 44.04 29.29 -13.44 -34.44

7 30.59 13.08 -33.62 -52.89 42.18 24.06 -11.31 -33.58 54.57 52.60 8 44.47 42.05 -20.89 -25.40 49.78 50.71 -9.33 -6.67 59.59 56.83

9 42.55 41.11 -17.93 -16.88

10 39.58 36.77 -11.69 -24.77

11 39.17 20.16 -24.63 -49.13 50.37 31.75 -1.67 -27.48 52.56 50.00 12 33.36 17.77 -30.48 -45.13 43.52 27.58 -10.59 -27.81 56.74 55.88 13 41.98 38.81 -23.30 -21.49 48.98 47.52 -8.42 -4.19 60.01 58.85 14 40.60 24.38 -23.54 -41.19 50.14 34.66 -3.71 -22.00 55.12 53.86

15 35.58 36.03 -44.64 -27.21

16 23.80 8.30 -41.33 -55.56

Note * means the save percentage of optimum VP WL and ₁ WL₂ charts with smaller δ₁,δ₂,δ₄ and

δ

₅ compare with other charts

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7. CONCLUSIONS

We propose a single VSI, VSSI and VP weighted loss control charts to effectively monitor the target and variance simultaneously on a single process step and two dependent process steps. The performance of the adaptive weighted loss control charts are measured using AATS, ANOS and the out-of-control average loss per unit product. A Markov chain approach is applied to calculate the AATS and ANOS. Following numerical analysis, the specified VP weighted loss control charts have smaller AATS and larger ANOS than FP weighted loss charts and FP 2

X Z

Z −

and 2

e Z

Z − charts and smaller loss than FP 2

X Z

Z − and 2

e Z

Z − charts.

Furthermore, the optimum VP weighted loss control charts outperform the specified VP weighted loss control charts, and thus, is recommended when users cannot specify the variable parameters.

We find when the shift scales for the two dependent process are smaller, the adaptive weighted loss control charts can save more AATS and loss percentage and waste less ANOS percentage than the other charts. Compare with FP WL charts, the saved AATS percentage range for optimum VSI WL charts is 4.15 to 37.92; for optimum VSSI WL charts is 8.37 to 48.78; for optimum VP WL charts is 8.3 to 49.98. Compare with FP 2

X Z

Z − and 2

e Z

Z − charts, the saved AATS and loss percentage range for optimum VSI WL charts is 12.41 to 48.69 and 42.16 to 59.6;

for optimum VSSI WL charts is 24.09 to 57.37 and 50 to 62; for optimum VP WL charts is 24.06 to 58.36 and 50 to 62. The optimum VP WL charts can save more AATS percentage than optimum VSI and VSSI charts. When the shifts are much smaller, optimum VP WL

charts can save more AATS and loss percentage, and

waste less ANOS percentage.

Numerical examples are used to illustrate the application of adaptive weighted loss charts. The results indicated that the detection ability of adaptive weighted loss charts is better than FP WL charts and FP X−S_X and e−S_e charts. However, if the weighted loss control chart for outgoing quality chart is misused in the second step, it may increase the false alarm rate and lead to over adjustment of a process.

Since the better performance of the adaptive weighted loss control charts, the convenience of using a single chart for each step and the relative ease of explaining their use to process monitors, adaptive weighted loss control charts are recommended

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for use in industry and science. However, using the weighted loss control charts can not tell if the mean and/or variance are out-of-control. To solve the problem, we need to check if the mean and/or variance of the out-of-control statistic are significantly different from the in-control mean and/or variance. Future work in this field may develop important extensions for weighted loss charts, such as EWMA, CUSUM, multivariate and nonparametric regression charts.

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REFERENCES

[1] Amin, R. W. and Miller, R. W. (1993), “A robustness study of

X Charts with

variable sampling intervals,” Journal of Quality Technology 25, 36-44.

[2] Amin, R. W., Wolff, H., Besenfelder, W. and Baxley, R. JR. (1999), “EWMA control charts for the smallest and largest observations,” Journal of Quality

Technology 31, 189-206.

[3] Chen, G., Cheng, S. W. and Xie, H. (2001), “Monitoring process mean and variability with one EWMA chart,” Journal of Quality Technology 33(2), 223-233.

[4] Chengular, I. N., Arnold, J. C. and Reynolds, M. R., JR. (1989), “Variable sampling intervals for multiparameter Shewhart charts, ” Communications in

Statistics – Theory and Methods 18, 1769–1792.

[5] Cinlar, E. (1975), Introduction to stochastic process. Englewood Cliffs, NJ:Prentice-Hall.

[6] Constable, G. K., Cleary, M. J., Tickel, C. and Zhang, G. X. (1988), “Use of Cause-Selecting Charts in the Auto Industry,” ASQC Quality Congress

Transactions. American Society for Quality Control, 597-602.

[7] Costa, A. F. B. (1994), “ X Charts with variable sample size,” Journal of Quality

Technology 26(3), 155-163.

[8] Costa, A. F. B. (1997), “ X Charts with variable sample size and sampling intervals,” Journal of Quality Technology 29(2), 197-204.

[9] Costa, A. F. B. (1998), “Joint

X and R Charts with variable parameters,” IIE Transactions 30(4), 505-514

[10] Costa, A. F. B. (1999a), “Joint X and R Charts with variable sample size and sampling intervals,” Journal of Quality Technology 31, 387-397.

[11] Costa, A. F. B. (1999b), “ X Charts with variable parameters,” Journal of

Quality Technology 31, 408-416.

[12] Cyrus, D. (1997), Statistical Aspects of Quality Control Academic Press, London.

[13] Daudin, J. J. (1992), “Double sampling

X Charts,” Journal of Quality Technology 24, 78-87.

[14] Grabov, P. and Ingman, D. (1996), “Adaptive control limits for bivariate process monitoring,” Journal of Quality Technology 28, 320-330.

[15] Mandel, B. J. (1969), “The Regression Control Chart,” Journal of Quality

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Technology 1, 1-9.

[16] Penrose, K., Nelson, A. and Fisher, A. (1985), “Generalized body composition prediction equation for men using simple measurement techniques” Medicine and

Science in Sports and Exercise,17(2), 189.

[17] Prabhu, S. S., Montgomery, D. C. and Runger, G. G. (1994), “ A combine dadaptive sample size and sampling interval

X control scheme,” Journal of Quality Technology, 26(3), 164–176.

[18] Prabhu, S. S., Runger, G. C. and Keats, J. B. (1993), “An adaptive sample size

X chart,” International Journal of Production Research, 31,2895–2909.

[19] Reynolds, M. R., JR. (1996), “Variable-sampling-interval control charts with Sampling at Fixed Times,” IIE Transactions 28, 497-510.

[20] Reynolds, M. R., JR. and Glosh, B. K. (1981), “Designing control charts for means and variances,” ASQC Quality Congress Transactions, American Society

for Quality Control ,San Francisco ,400-407.

[21] Reynolds, M. R., JR., Amin, R. W., Arnold, J. C. and Nachlas, J. A (1988), “ X charts with variable sampling intervals,” Technometrics 30(2), 181-192.

[22] Reynolds, M. R., JR. and Arnold, J. C., (1989), “ Optimal one-sided Shewhart control chart with variable sampling intervals,” Sequential Analysis 8, 51-77.

[23] Reynolds, M. R., JR., Arnold, J. C. and Baik (1996), “ Variable sampling interval

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在文檔中適應性加權損失管制圖之研究 - 政大學術集成 (頁 80-99)

Performance Comparison for VP Weighted Loss Control Charts

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6.5 Performance Comparison for VP Weighted Loss Control Charts

6.5.1 VP Weighted Loss Control Charts with Specified Variable Parameters

t

n

α

(

)

(

)

(

)

( γ

γ

δ

δ

δ

δ

δ

t

n

α

) on AATS,

Z

Z

Z

Z

λ

λ

t ,

( w

α

n

k

α

n

w

α

n

)

n

k α

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δ

δ

δ

δ

δ

δ

δ

δ

δ

δ

n are significant, when

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立政治大學

立政治大學

立政治大學

立政治大學

立政治大學

立政治大學

立政治大學