Determination of the UCL i , WCL i of the Optimal VSS Loss Function

where L* belongs to an out-of-control process.

The ANOS of the fixed parameters loss function chart is

ARL n ANOS

Fp

_ = ₀⋅ (41) ARL can refer to equation (21).

4.3 Determination of the UCL

_i

, WCL

_i

of the Optimal VSS Loss Function Chart

If the six design parameters (n₁,n₂,UCL₁,UCL₂,WCL₁,WCL₂) of the VSS loss function chart are not known. Then, this section provides the application technique to determine the optimal design parameters through the direct search approach. The objective function of the optimization is ATS which is the function of the six design parameters and subjects to (1)

α

₁ =

α

₂ =

α

₀, where

α

₀ =

P

(

L

≥

UCL

₀), (2) the range

The mathematical model can be expressed as MinimumATS = f(n₁,n₂,UCL₁,UCL₂,WCL₁,WCL₂)

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The procedures search the optimal design parameters (n₁,n₂,UCL₁,UCL₂,WCL₁,WCL₂) are described as follows.

Step1: Specify

n

_L,

n

_U,α₀,

p

,δ,

h

, R.

Step2: Searching available combinations (n₀, k₀) under theα . ₀

Step3: Searching n1 within feasible region of [

n

_L, n₀) and searching n2 within the feasible region of (

n ,

₀

n

_U] for minimizing ATS.

p can be determined by

₀ equation (36) when n0, n1 and n2 are known.

Step 4: k₁ can be obtained whenα and n₀ 1 are known, k₂ can be obtained whenα and ₀ n2 are known.

Step5: Determine UCL₁ by using equation (29) when n₁, k₁ and p are known.

Determine UCL2 by using equation (31) when n2, k2 and p are known.

Step6: Determine WCL₁ by using equation (30) when UCL₁ and

p are known.

₀ Determine WCL2 by using equation (32) when UCL2 and

p are known.

₀ Step7: Check ifn₁,n₂and h satisfy the constraint

AIR

≤ . Then, the design

R

parameters n₁*,n₂*,UCL₁*,UCL₂*,WCL₁*,WCL₂*can be determined under the minimum ATS.

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Figure11. Flow Chart of the Design of the Optimal VSS Loss Function Chart Check

n ,

₁

n and h to

₂

satisfyAIR≤ R

YES

NO

Not Available Using equation (17) to determine various

combinations of (n0, k0) under the specifiedα ₀

Using equation (29) to calculate UCL1 by known n1, k1 and p.

Using equation (31) to calculate UCL2 by known n2, k2 and p.

Find the optimal design (n₁*,n₂*,UCL₁*,UCL₂*,WCL₁*,WCL₂*)from all feasible solutions with minimal ATS

Input: Specify

R h p n

n

_L, _U,α₀, ,δ, ,

Determine k1 by knownα , p and n₀ 1

Determine k2 by knownα , p and n₀ 2

Searching

n

₁∈[

n

_L,

n

₀), ] , ( ₀

2

n n

U

n

∈

Using equation (36) to calculate

p

₀by known

n ,

₁

n

₂

Using equation (30) to calculate WCL1 by known UCL1 and

p .

₀ Using equation (32) to calculate WCL2 by known UCL2 and

p .

₀

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4.4 Performance Comparisons

Find the design parameters of the optimal VSS loss function chart by approach describe in Section 4.3. First, specify5≤ n≤200,_α0=0.00283(

ARL =355), h=1 and

₀ R=500. The in-control p =0.001, 0.01, 0.02, 0.05, 0.1, 0.3, 0.5 andδ=1.5, 2, 2.5.

Table11 and Tables12 show the optimal design

parametersn₁*,n₂*,UCL₁*,UCL₂*,WCL₁*,WCL₂*, percentage of saved ATS and percentage of saved ANOS compared to Fp loss function chart. The percentage of saved ATS is

% VSS_ 100

=

% ATS

Saved − ⋅

ATS Fp

ATS Optimal

ATS

Fp

(42)

The percentage of saved ANOS is

% VSS_ 100

=

% ANOS

Saved − ⋅

ANOS Fp

ANOS Optimal

ANOS

Fp

(43)

Table11 shows optimal design of the VSS loss function chart with p=0.001, 0.01 and 0.02. Table12 shows optimal design of the VSS loss function chart with p=0.05, 0.1, 0.3 and 0.5. Due to the trait of discrete distribution, binomial,α₁(

1 _ 0

1 ARL ) andα₂(

2 _ 0

1

ARL ) are not easy to be exactly the same. Thus, control

0 5

1 _

0 − ARL <

ARL

and

ARL

_{0_2} − ARL₀ <5of calculation, except p=0.001. When p=0.001, let

ARL

_{0_1}− ARL₀ <10 and

ARL

_{0_2} − ARL₀ <10 for existent results.

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Table11. Optimal VSS and Fp with R=500, p=0.001, 0.01, 0.02

p 0.001 0.01 0.02

δ 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5

n0 77 77 77 138 138 138 119 119 119

k0 11.61 11.61 11.61 6.85 6.85 6.85 6.00 6.00 6.00

n₁* 76 76 76 58 58 58 70 70 70

n₂* 78 78 78 185 185 185 174 174 174

WCL1* 1 1 1 1 1 1 4 4 4

UCL1* 1.08 1.08 1.08 9.96 9.96 9.96 27.37 27.37 27.37

WCL₂* 1 1 1 4 4 4 16 16 16

UCL₂* 4 4 4 49 49 49 100 100 100

VSS_ATS* 166.99 96.11 62.95 52.20 14.06 5.93 30.73 6.77 2.84 Fp_ATS 163.67 94.37 61.91 54.51 16.73 7.50 36.49 9.50 4.07 VSS_ANOS 12725.33 7330.20 4804.27 7699.83 2297.96 1011.32 4436.72 1082.63 469.05

Fp_ANOS 12602.90 7266.82 4767.00 7521.74 2309.02 1035.30 4342.62 1131.08 484.90 ARL0 359.24 359.24 359.24 357.37 357.37 357.37 354.76 354.76 354.76 ARL0_1 368.57 359.24 359.24 362.24 362.24 362.24 355.50 355.50 355.50 ARL₀_2 350.26 359.24 359.24 356.65 356.65 356.65 351.65 351.65 351.65 AIR 76.22 76.28 76.35 148.77 167.03 176.81 145.91 164.31 171.25 Saved ATS% -2.02 -1.84 -1.68 4.23 15.97 20.91 15.80 28.81 30.29 Saved ANOS% -0.97 -0.87 -0.78 -2.37 0.48 2.32 -2.17 4.28 3.27

In Table 11 and Table 12, the optimal VSS loss function chart can save more ATS than the Fp loss function chart, except p=0.001 and 0.3. The optimal VSS loss function chart can save at least 3.86% and at most 30.47% without considering p=0.001 and 0.3. When p is 0.001 or 0.3, the VSS loss function did not have better performance than the Fp loss function chart. Compared ANOS of the VSS and Fp loss function chart, VSS consumes more observations frequently. It is better to adopt the Fp loss function when p is too small or too large.

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Table12. Optimal VSS and Fp with R=500, p=0.05, 0.1, 0.3, 0.5

P 0.05 0.1 0.3 0.5

δ 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 1.5

n0 83 83 83 110 110 82 19 19 19 70

k0 5.22 5.22 5.22 4.24 4.24 4.47 4.44 4.44 4.44 3.28

n₁* 49 49 49 62 89 62 19 19 19 48

n₂* 95 95 95 154 154 110 33 33 33 89

WCL1* 4 4 4 49 121 64 144 144 144 625

UCL1* 55.48 55.48 55.48 172.61 302.55 172.61 133.29 133.29 133.29 1089.37 WCL₂* 16 16 16 256 324 169 324 324 324 2025 UCL₂* 144 144 144 729 729 441 324 324 324 3364 VSS_ATS* 19.93 4.23 1.89 4.85 1.26 1.07 11.48 2.05 1.08 1.01 Fp_ATS 21.87 4.87 2.14 6.98 1.58 1.18 11.48 2.05 1.08 1.06 VSS_ANOS 1823.08 397.44 178.99 710.04 190.68 116.78 218.07 38.95 20.60 90.25

Fp_ANOS 1815.00 404.58 177.80 767.65 174.15 96.87 218.07 38.95 20.60 73.86 ARL0 354.45 354.45 354.45 355.84 355.84 357.10 354.28 354.28 354.28 358.25 ARL0_1 357.25 357.25 357.25 355.71 351.26 355.71 354.28 354.28 354.28 362.63 ARL₀_2 357.77 357.77 357.77 353.30 353.30 355.84 355.48 355.48 355.48 360.69 AIR 91.74 94.46 94.93 150.05 153.85 109.99 20.34 28.84 32.95 89.00 Saved ATS% 8.86 13.25 11.62 30.47 20.51 9.68 0.00 0.00 0.00 3.86 Saved ANOS% -0.44 1.76 -0.67 7.50 -9.49 -20.56 -0.00 -0.00 0.00 -22.19

Figures 12 and 13 are the main effect plots of the optimal VSS loss function chart under various p andδ. We summarize the results from data analyses and plots as follows. (1) The average of ATS decreases when p increases, except p=0.3. (2) The average of WCL1 and WCL2 increase when p increases. The average of UCL1 and UCL₂ increase when p increases, except p=0.3. (3) The average of ATS decreases whenδincreases. (4) The average of UCL1, UCL2, WCL1 and WCL2 decrease when δincreases. (5) The VSS loss function chart outperforms Fp loss function chart, except p=0.001 and p=0.3.

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Figure12. Main Effect Plot of the Optimal VSS Loss Function Chart (p, R=500)

Figure13. Main Effect Plot of the Optimal VSS Loss Function Chart (δ, R=500)

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4.5 Example

A manager is concerned about the defect proportion of baked pies at store. From the analyses in Section 4.4, the optimal VSS loss function chart outperforms the Fp loss function chart when p is not extremely large of small. Hence, the manager

determines to construct an optimal VSS loss function chart for monitoring loss caused by the process proportion shift. The historical record shows the in-control proportion of defected pies, p, is 0.02. The store can accept the sample size within [5, 200], the sampling interval is set as 1 hour, and an AIR is less than 500 due to the process capacity. Considerα₀ =0.00282(ARL₀=355) and proportion scaleδ=2.5. Base on those information,

5 . 2 , 500 ,

00282 . 0 ,

1 , 200 ,

5 , 02 .

0 = = = ₀ = = =

=

n n h α R δ

p

_{L U} .

The manager uses the approach in describes Section 4.3 to determine the optimal design parameters of the VSS loss function chart as follows (see Table13). Use those design parameters, the optimal VSS loss function chart for the pie store can be established as Figure14.

Table13. Optimal Design of the VSS Loss Function Chart

p n k n1 n2 WCL1 UCL1 WCL2 UCL2 ARL0

0.02 119 6 70 174 4 27.37 16 100 354.76

UCL1=27.37

WCL1=4

UCL2=100

WCL2=16

I3 (Action region)

I₂ (Warning region) (n2=174) (n2=174)

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Bases on this plan, the store collectes 24 samples with h=1 hour and constructs an optimal VSS loss function chart to monitor the quality of pies. If the current sample point is located within the central regionI₁, the next sample should adopt

n1=70 as sample size. If the current sample point is located within the warning regionI₂, the next sample should adoptn₂=174 as sample size. If the current sample point is located outside the UCL, the occurred S.C. should be searched and removed from the process.

Table14 shows the sampling results using the optimal VSS loss function chart scheme. The first sample size was decided randomly by the probability

p =0.15 of

₀' using

n =70 and the probability (1-

₁

p )=0.85 of using

₀'

n =174. In this example, first

₂ sample usedn₁=70 as sample size with UCL₁=27.37 and WCL₁=4. The 1^st data point L=1 is located within WCL1=4, thus, the 2^nd sample should adopt 70 as sample size with UCL₁=27.37 and WCL₁=4. The 2^nd data point L=9 is located between WCL₁=4 and UCL1=27.37, thus, the 3^rd sample should adopt 174 as its sample size with UCL2

=100 and WCL₂=16. The 3^rd data point L=9 is located within WCL₂=16, thus, the 4^th sample should adopt 70 as its sample size. The other data points follow the same rule to determine the use of sample sizes, UCL_i and WCL_i.

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Table14. Sampling data and VSS Loss Function Chart

Sample ni X L Located

Region Sample ni X L Located

Region 1 random choose

n1=70 1 1 I1 13 n1=70 3 9 I2

2 n1=70 3 9 I2 14 n2=174 2 4 I1

3 n2=174 3 9 I1 15 n1=70 2 4 I2

4 n1=70 3 9 I2 16 n2=174 2 4 I1

5 n2=174 1 1 I1 17 n1=70 5 25 I2

6 n1=70 1 1 I1 18 n2=174 1 1 I1

7 n1=70 2 4 I2 19 n1=70 2 4 I2

8 n2=174 4 16 I2 20 n2=174 10 100 I3

9 n2=174 3 9 I1 21 random choose

n1=70 3 9 I2

10 n1=70 3 9 I2 22 n2=174 2 4 I1

11 n2=174 2 4 I1 23 n1=70 3 9 I2

12 n1=70 1 1 I1 24 n2=174 2 4 I1

Figures15 shows the constructed optimal VSS loss function chart. The point on the 20^th sample falls on action region, thus, the occurred S.C. should be searched and removed from the process. ATS of the optimal VSS loss function chart is 2.84 and ANOS is 469.05 after calculation.

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The manager wants to know how much the optimal VSS loss function chart can save for the pie store by comparing with Fp loss function chart. Figure9 shows the Fp loss function chart. The ATS of Fp loss function chart is 4.07 and the ANOS of Fp loss function chart is 484.9 after calculation.

Compared performance between the Fp and optimal VSS loss function chart, the latter saves around 30.29% ATS and 3.27 %ANOS (see Table15). The VSS loss function chart outperforms Fp loss function chart significantly and it can help store to monitor defect proportion of pies more effective. Thus, it is better to apply an optimal VSS loss function chart to control the loss and quality of pies.

Table15. Comparison of the Fp and VSS Loss Function Chart Chart ATS Saved ATS% ANOS Saved ANOS%

VSS 2.84

30.29 469.05

3.27

Fp 4.07 484.9

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在文檔中 適應性計數值損失函數管制圖之設計 - 政大學術集成 (頁 39-50)