Construction of the VSS Loss Function Chart for binomial data…

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4. Design of the VSS Loss Function Chart for binomial data

4.1 Construction of the VSS Loss Function Chart for binomial data

As Section 3.1 describe, the Fp loss function chart is built with UCL, CL and LCL. The UCL, CL, and LCL of Fp loss function chart can be expressed as equations (4)-(6).

The VSS loss function chart is a control chart with adaptive sample sizes. A VSS loss function chart is built with two control limit UCL₁,UCL₂, two warning limit

2 1,WCL

WCL and a LCL s Fig.10). The LCL is set as zero because fraction

nonconforming is the degree of deterioration. Thus, LCL=0 is the best level of quality loss.

Figure10. VSS Loss Function Chart

A VSS loss function chart can be expressed as follows. See equations (24)-(28).

) ( )

( _{1 1 1}

1

E L k Var L

UCL

= + (24)

) ( )

( _{1 1 1}

1 E L w Var L

WCL = + (25)

) ( )

( _{2 2 2}

2

E L k Var L

UCL

= + (26)

0

I3 (Action region) UCL₁

WCL₁

UCL₂

I₂ (Warning region)

I1 (Central region)

WCL₂

‧

mathematically as equations (29)-(32).

4 of detecting process shift. A VSS loss function chart has a large sample size n2 and a small sample size n₁. WCL₁ and WCL₂ are the guard to decide the use of n₁ or n₂ between samples.

When using VSS loss function chart, two different sample size, n₁ and n₂, are adopted. Users have to decide on a large sample size n2 and a small sample size n1, where n₂ >n₁. If the data point is plotted on the central region (I₁), use the small sample size n₁,UCL₁and WCL₁of the next sample. If the data point is plotted on the warning region (I2), use the large sample size n2,UCL₂and WCL₂of the next sample. If the data point is plotted on the action region (I3), find the S.C. and repair the process.

For comparing the VSS loss function chart with the Fp loss function chart under the same standard, the average sample size needs to be the same when the process is in control. The equation (33) needs to be satisfied.

‧

The average sample size of the VSS loss function chart is the same as fixed sample size of the Fp loss function chart, wheren₀is fixed sample size of the Fp loss function chart, 0<

n

₁ <

n

₀ <

n

₂ <∞, and

p is the probability of being in central region (I

₀ 1)

From equation (34), WCL₁ can be expressed in terms of )

Take the inverse function of both sides.



¹



^[ 0 ⁽¹ 1^)]

The value of

p can be calculated through equation (33) and expressed as

₀

2

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4 0 3 0 2 0 0 3 4 0 3 3 0 2 4 0 2 3 0 2 2 0 0 2 2 0 2 0 0

0 n p n p n p k [6n p 16n p 10n p 4n p 4n p n p 7n p 12n p 6n p

UCL = − + + − + + − + − + − (37)

where k₀ is the control limit factor of the Fp loss function chart.

The UCL0 can be determined when n0, p, k0 are given by using equation (37). The UCL₁ can be determined when n₁, p, k₁ are given by using equation (29). The UCL₂ can be determined when n2, p, k2 are given by using equation (31).The value of (n0, k₀), (n₁, k₁) and (n₂, k₂) are decided by

ARL (see equation (17)).

₀

4.2 Performance Measurement

Performance of the VSS loss function chart can be measured by ARL, ATS and ANOS (average number of observations to signal), where

ANOS

=

n

⋅

ARL

. When the process is out of control, the smaller ARL, ATS or ANOS means better detective ability of the control chart. When the process is in control, the larger ARL can result in fewer false alarms and costs.

The Markov chain method is applied to calculate those performance

measurements. There are two assumptions of calculating ARL, ATS and ANOS of the VSS loss function chart. First, the loss function chart assumes only one S.C. may occur during the process. Second, process is out-of-control at the beginning of the process starts. ATS and ANOS are calculated under the zero-state mode. Due to the assumptions, the process has two transient states and one absorbing state of Markov chain approach (see Table10).

Table10. State Definition of the VSS Loss Function Chart for binomial data State S.C. occur Location of the VSS loss function Chart

1 Yes I₁₁

2 Yes I12

3 Yes I13

‧

Transition probabilities are as the following:

 

From the elementary properties of the Markov chain, the ATS and ANOS are h matrix of order 2, Q is a 2 by 2 transition probability matrix, n is the vector of the next sample size for state 1 and state 2, h is the vector of the next sampling interval for state 1 and state .

b

=[

p

₀^' ,1-

p

₀^' ],

n

^' =[

n

₁,n₂],

h

^' =[1,1] and

probability of being at state 1 at the beginning of the process when the process is out-of-control.1−

p

₀^' is the probability of being state 2 at the beginning of the process when the process is out-of-control.

‧

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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5. Design of the VP Loss Function Chart for binomial data

5.1 Construction of the VP Loss Function Chart for binomial data

The VP loss function chart is a control chart with variable sampling intervals (h₁, h₂), variable sample sizes (n₁, n₂) and variable control limit factors (k₁, k₂) to detect the process. A VP loss function chart is built with two control limitUCL₁,UCL₂, two warning limitWCL₁,WCL₂and a LCL (see Fig.16). The LCL is set as zero because fraction nonconforming is the degree of deterioration. Thus, LCL=0 is the best level of quality loss.

Figure16. VP Loss Function Chart

A VP loss function chart can be expressed as follows. See equations (44)-(48).

) ( )

( _{1 1 1}

1 E L k Var L

UCL = + (44)

) ( )

( _{1 1 1}

1 E L w Var L

WCL = + (45) )

( )

( _{2 2 2}

2 E L k Var L

UCL = + (46)

) ( )

( _{2 2 2}

2 E L w Var L

WCL = + (47)

LCL=0 (48)

1

0<WCL1<UCL

,

0<

WCL

2 <

UCL

2

0

I₃ (Action region) UCL₁

WCL₁

UCL₂

I2 (Warning region)

I₁ (Central region)

WCL₂

‧

detecting process shift. A VP loss function chart has a large sample size n2, a small sample size n₁, a long sampling interval h₁ and a short sampling interval h₂. WCL₁ and WCL2 are the guard to decide the use of (n1, h1) or (n2, h2) among samples. When using the VP loss function chart, two different sample sizes, n₁ and n₂, and two

different sampling intervals, h1and h2, are adopted. Users have to decide on (n1, h1) or (n₂, h₂), where n₂ >n₁, h₁ >h₂. If the data point is plotted on the central region (I₁), use small sample size n₁, long sampling interval h₁,UCL₁andWCL₁of the next sample. If the data point is plotted on the warning region (I₂), use large sample size n₂, short sampling interval h₂,UCL₂andWCL₂of the next sample. If the data point is plotted on the action region (I₃), find the S.C. and repair the process.

For comparing the VP loss function chart with the Fp loss function chart under the same standard, the average sampling interval, the average sample size and the average false alarm rate need to be the same when the process is in control. The equation (53) and equations (55)-(56) need to be satisfied.

0 The average sampling interval of the VP loss function chart is the same as the fixed

‧

sampling interval of the Fp loss function chart, where h₀ is the fixed sampling interval of the Fp loss function chart, 0<

h

₂ <

h

₀ <

h

₁<∞.

p is the probability of being in

₀ central region (I₁) when the process is in control and can be defined as

) The average sample size of the VP loss function chart is the same as fixed sample size of the Fp loss function chart, wheren₀is fixed sample size of the Fp loss function chart, 0<

n

₁ <

n

₀ <

n

₂ <∞. The average false alarm rate of the VP loss function chart is the same as false alarm rate of the Fp loss function chart, where α₀ =

P

(

L

≥

UCL

₀),α₁ =

P

(

L

≥

UCL

₁),

)

( ₂

2 =P L≥UCL

α . The UCL0 is the upper control limit of the Fp loss function chart (see equation (37)).

From equation (54), WCL1 can be expressed in terms of

1

Take the inverse function of both sides.



^{WCL1 −1}



^{=FX−1[p0⋅(1−α1)]}

Then, the WCL₁ can be determined approximately as equation (57).

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2 1

2 0

0 n n

n p n

−

= − (58)

By using similar derivation, WCL₂ ≈[F_X⁻¹(p₀(1−α₂))+1]².

α

₂ can be calculated through equation (56) and expressed as

1 ₀

0 1 0

2

p

p

−

=

α

−

α

α

(59)

The UCL0 can be determined when n0, p, k0 are given by using equation (37).

The UCL₁ can be determined when n₁, p, k₁ are given by using equation (49). The UCL2 can be determined when n2, p, k2 are given by using equation (51).The value of (n₀, k₀) are decided byα , the value of (n₀ 1, k₁) are decided byα and the value of (n₁ 2, k2) are decided byα . ₂

5.2 Performance Measurement

Performance of the VP loss function chart can be measured by ARL, ATS and ANOS. When the process is out of control, the smaller ARL, ATS or ANOS means better detective ability of the control chart. When the process is in control, the larger ARL can result in fewer false alarms and costs.

The Markov chain method is applied to calculate those performance

measurements. There are two assumptions of calculating ARL, ATS and ANOS of the VP loss function chart. First, the loss function chart assumes only one S.C. may occur during the process. Second, process is out-of-control at the beginning of the process starts. ATS and ANOS are calculated under the zero-state mode. Due to the

assumptions, the process has two transient states and one absorbing state of Markov chain approach (see Table16).

‧

Table16. State Definition of the VP Loss Function Chart for binomial data State S.C. occur Location of the VP loss function Chart

1 Yes I₁₁ or I₂₁

2 Yes I12 or I22

3 Yes I13 or I23

Transition probabilities are as the following:

 

From the elementary properties of the Markov chain, the ATS and ANOS are h matrix of order 2, Q is a 2 by 2 transition probability matrix, n is the vector of the next sample size for state 1 and state 2, h is the vector of the next sampling interval for state 1 and state 2.b=[p₀^' ,1- p₀^' ],

n

^' =[

n

₁,n₂],

h

^' =[

h

₁,h₂]and





= P_1,1(n₁,UCL₁,WCL₁), P_1,2(n₁,UCL₁,WCL₁)

respectively, where

p is the

^'

‧

process when the process is out-of-control.

'

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

The ATS and ANOS of the fixed parameters loss function chart are

ARL

ARL can refer to equation (21).

5.3 Determination of the UCL, WCL of the Optimal VP Loss Function Chart

If the eight design parameters (n₁,n₂,h₁,h₂,UCL₁,UCL₂,WCL₁,WCL₂) of the VP 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 eight design parameters and subjects to (1) a specified

α

₀, (2) the range of sample size,2≤

n

_L ≤

n

₁<

n

₀ <

n

₂ ≤

n

_U <∞, (3) the range of sampling interval,

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The mathematical model can be expressed as

MinimumATS = f(n₁,n₂,h₁,h₂,UCL₁,UCL₂,WCL₁,WCL₂) Subject to

(1)

ARL

₀ =1/α₀

(2)2≤

n

_L ≤

n

₁ <

n

₀ <

n

₂ ≤

n

_U <∞ (3)0<

h

₂ <

h

₀ <

h

₁ ≤

h

_U

(4)

α

_L <

α

₁ <

α

₀ <

α

₂ <

α

_U (5)0<

WCL

_i <

UCL

_i,i=1,2 (6)

0 < AIR ≤ R

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The procedures search the optimal design parameters

(n₁,n₂,h₁,h₂,UCL₁,UCL₂,WCL₁,WCL₂) are described as follows.

Step1: Specify

n

_L,

n

_U,α₀,

p

,δ,

h

_U,

h

₀,α_L,α_U,

R

.

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

Step3: Determine UCL0 by using equation (37) when n0, k0 and p are known.

Step4: 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 (58) when n₀, n₁ and n₂ are known.

Step 5: Searching k1 under

α

_L <

α

₁ <

α

₀and known n1 for minimizing ATS.

Step 6: Determine UCL₁ by using equation (49) when n₁, k₁ and p are known.

Determine WCL1 by using equation (50) when UCL1 and

p are known.

₀ Step 7: Determineα by using equation (59) when₂ α ,₁ α and₀

p are known. k

₀ 2 can be

obtained whenα , p and n₂ 2 are known.

Step8: Determine UCL₂ by using equation (51) when n₂, k₂ and p are known.

Determine WCL2 by using equation (52) when UCL2 and

p are known.

₀ Step9: Searching

h within the feasible region of (

₁

h ,

₀

h

_U] for minimizing ATS.

h can

₂

be determined by equation (53) when

h and

₁

p are known.

₀

Step10: Check ifn₁,n₂ h₁andh₂satisfy the constraintAIR≤R. Then, the design parametersn₁*,n₂*,h₁*,h₂*,UCL₁*,UCL₂*,WCL₁*,WCL₂*can be determined under the minimum ATS.

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Use equation (17) to determine various combinations of (

n

₀, k₀) under the specifiedα ₀

Use equation (49) to calculate UCL1

by known n1, k1 and p

Input: Specify

U L U

U

L

n p h h R

n

, , ,δ, ₀, , ,α₀,α ,α

Use equation (37) to determine

UCL by known n

₀ 0, k0 and p

Use equation (50) to calculate WCL1 by known

p₀, UCL₁ Search

n ,

₁

n ,

₂

k to minimize ATS, and satisfy

₁

) ,

[ ₀

1

n n

n

∈ _L ,

n

₂ ∈(

n

₀,

n

_U]and

α

_L <

α

₁ <

α

₀

Use equation (58) to determine

p by

₀ known n₁ and n₂

Use equation (59) to determineα by ₂ knownα ,₁ α and₀

p

₀

Use equation(52) to calculate WCL₂ by known

p0, UCL2

Determine k₂ by knownα , p and n₂ 2

Find the optimal design(n*,n *,h*,h *,UCL*,UCL *,WCL*,WCL *) from all feasible Use equation (51) to calculate UCL2 by

known n₂, k₂ and p

Search

h to

₁ minimize ATS and

h

₁ ∈(

h

₀,

h

_U]

Use equation (53) to determineh₂by knownh₁and

p

₀

Checkn₁,n₂,h₁andh₂to satisfyAIR≤R

YES

NO

Not Available

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5.4 Performance Comparisons

Find the design parameters of the optimal VP loss function chart by approach describe in Section 5.3. First, specify5≤ n≤200,α₀=0.00283(

ARL =355),

₀

αU=0.005(

ARL =200),

₀ α_L=0.00167(

ARL =600), the maximum sampling interval

₀ hU=2,h₀=1 and R=500. The in-control p = 0.001, 0.01, 0.02, 0.05, 0.1, 0.2, 0.3, 0.5 andδ=1.5, 2, 2.5. Table17 and Table18 show the optimal design parameters

*

*,

*,

*,

*,

*,

*,

*, _{2 1 2 1 2 1 2}

1 n h h UCL UCL WCL WCL

n , percentage of saved ATS and

percentage of saved ANOS compared to Fp loss function chart. The percentage of saved ATS is

% VP_ 100

=

% ATS

Saved − ⋅

ATS Fp

ATS Optimal

ATS

Fp

The percentage of saved ANOS is

% VP_ 100

=

% ANOS

Saved − ⋅

ANOS Fp

ANOS Optimal

ANOS

Fp

Table17 shows optimal design of the VP loss function chart with p=0.001, 0.01, 0.02 and 0.05. Table18 shows optimal design of the VP loss function chart with p=

Table17 shows optimal design of the VP loss function chart with p=0.001, 0.01, 0.02 and 0.05. Table18 shows optimal design of the VP loss function chart with p=

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