Construction of the VP Loss Function Chart for binomial data

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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=

0.1, 0.2, 0.3 and 0.5.

In Table17 and Table18, the optimal VP loss function chart can save more detective time than the Fp loss function chart, except p=0.001. The optimal VP loss function chart can save at least 24.18% and at most 89.67% without considering p=0.001.

When p is extremely small, the VP loss function did not save more time than the Fp loss function chart. It is better to adopt the Fp loss function when p is smaller than 0.001. Compared ANOS of the VP and Fp loss function chart, VP consumes more ANOS when p is bigger than 0.3.

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Table17. Optimal VP and Fp with R=500, p=0.001, 0.01, 0.02, 0.05

p 0.001 0.01 0.02 0.05

δ 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5

n0 77 77 77 138 138 138 119 119 119 187 83 83

k 11.61 11.61 11.61 6.85 6.85 6.85 6.00 6.00 6.00 4.43 5.22 5.22

n1* 76 76 76 51 51 85 67 88 86 172 68 67

n₂* 105 92 90 192 192 177 177 186 148 200 96 96

h1* 1.1 1.1 1.1 2 2 2 1.8 1.4 1.9 1.9 2 2

h2* 0.1 0.1 0.1 0.38 0.38 0.26 0.11 0.14 0.21 0.22 0.13 0.19

WCL₁* 4 4 1 1 1 1 4 9 4 81 16 16

UCL₁* 3.76 3.43 3.43 10.29 10.29 17.93 30.47 36.64 41.75 294.14 87.60 85.78 WCL2* 4 4 4 4.00 4.00 4.00 16.00 25.00 16.00 121.00 25.00 25.00 UCL2* 9 9 9 49.00 49.00 49.00 100.00 100.00 81.00 400.00 144.00 144.00 VP_ATS* 184.66 106.44 69.71 41.33 8.34 2.75 17.21 2.43 1.14 3.70 0.97 0.44

Fp_ATS 163.67 94.37 61.91 54.51 16.73 7.50 36.49 9.50 4.07 9.09 4.87 2.14 VP_ANOS 12758.29 7354.02 5663.77 7159.60 2099.45 1136.25 4238.70 915.92 465.37 1730.92 386.76 176.26

Fp_ANOS 12602.90 7266.82 4767.00 7521.74 2309.02 1035.30 4342.62 1131.08 484.90 1698.90 404.58 177.80 ARL0 359.24 359.24 359.24 357.37 357.37 357.37 354.76 354.76 354.76 356.15 354.45 354.45 ARL₀_1 368.57 368.57 368.57 581.94 581.94 591.50 444.04 505.25 576.75 385.28 504.85 565.73 ARL0_2 210.18 260.33 270.25 288.32 288.32 276.76 289.77 215.83 265.09 334.24 281.72 271.94

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Table18. Optimal VP and Fp with R=500, p=0.1, 0.2, 0.3, 0.5

p 0.1 0.2 0.3 0.5

δ 1.5 2 2.5 1.5 2 2.5 1.5 2 2.5 1.5

n0 110 62 62 176 176 176 93 19 19 70

k 4.24 4.72 4.72 3.52 3.52 3.52 3.53 4.44 4.44 3.28

n1* 93 47 47 97 97 97 47 12 12 53

n₂* 144 75 76 200 200 200 127 31 32 90

h1* 1.6 2 1.9 2 2 2 2 1.6 1.5 1.7

h2* 0.1 0.13 0.16 0.7 0.7 0.7 0.26 0.1 0.1 0.18

WCL₁* 121 25 25 289 289 289 196 25 25 784

UCL₁* 324.25 135.31 135.31 1018.87 1018.87 1018.87 572.31 65.63 65.63 1296.94 WCL2* 289.00 64.00 64.00 1369.00 1369.00 1369.00 1444.00 121.00 144.00 2116.00

UCL2* 625.00 256.00 256.00 3249.00 3249.00 3249.00 2809.00 289.00 324.00 3481.00

VP_ATS* 1.54 0.42 0.21 1.00 0.70 0.70 0.36 0.25 0.11 0.18

Fp_ATS 6.98 2.83 1.40 1.55 1.55 1.55 1.64 2.05 1.08 1.06

VP_ANOS 614.83 174.39 92.41 284.60 200.05 200.00 159.62 39.96 32.12 91.50 Fp_ANOS 767.65 175.56 86.65 273.37 273.37 273.37 152.59 38.95 20.60 73.86 ARL0 355.84 355.71 355.71 356.76 356.76 356.76 355.74 354.28 354.28 358.25 ARL₀_1 510.77 544.15 544.15 549.56 549.56 549.56 499.92 591.14 591.14 364.54 ARL0_2 221.48 273.60 268.82 322.39 322.39 322.39 293.24 210.02 203.13 351.12 AIR 494.04 498.55 472.62 285.66 285.66 285.66 487.81 301.73 319.99 500.00 Saved ATS% 77.97 85.10 84.94 35.60 54.92 54.93 78.31 87.57 89.67 82.52

Saved ANOS% 19.91 0.66 -6.65 -4.11 26.82 26.84 -4.61 -2.59 -55.97 -23.89

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Figures 18 and 19 are the main effect plots of the optimal VP 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. (2) The average of WCL1, WCL2 , UCL1 and UCL2 increases when p increases, except p=0.2. (3) The average of ATS decreases whenδincreases. (4) The average of UCL1, UCL2, WCL1 and WCL2 decreases whenδincreases. (5) The VP loss function chart significantly outperforms Fp loss function chart, except p=0.001.

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Figure19.Main Effect Plot of the Optimal VP Loss Function Chart (δ, R=500)

5.5 Example

A manager is concerned about the defect proportion of baked pies at store. From the analyses in Section 5.4, the optimal VP loss function chart outperforms the Fp loss

function chart when p is not extremely small. Hence, the manager determines to construct an optimal VP 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 maximum sampling interval is 2, the maximum false alarm rate_αU=0.005(

ARL =200), the minimum false alarm rate

₀

α =0.00167(

ARL =600), the fixed sampling interval h

0 is set as 1 hour, and an AIR is less

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than 500 due to process capacity. Consider_α0=0.00282 (ARL₀=355) and proportion scale δ=2.5. Based on those information,

5 . 2 , 500 ,

00282 . 0 ,

1 , 2 ,

200 ,

5 , 02 .

0 = = = ₀ = ₀ = = =

=

n n h h α R δ

p

_{L U U} ,

αU=0.005, α_L=0.00167.

The manager uses the approach describes in Section 5.3 to determine the optimal design parameters of the VP loss function chart as follows (see Table19). Use those design parameters, the optimal VP loss function chart for the pie store can be established as Figure20.

Table19. Optimal Design of the VP Loss Function Chart

p n0 n1 h1 WCL1 WCL2

0.02 119 86 1.9 4 16

ARL0 k₀ n₂ h₂ UCL₁ UCL₂

354.76 6 148 0.21 41.75 81

Figure20. Optimal VP Loss Function Chart

Bases on this plan, the store collectes 24 samples and constructs an optimal VP loss function chart to monitor the quality of pies. If the current sample point is located within the central region

I , the next sample should adopt (

₁

n =86,

₁

h =1.9) as next sample. If the

₁ current sample point is located within the warning region

I , the next sample should adopt

₂ (n₂=148, h₂=0.21) as next sample. If the current sample point is located outside the UCL,

UCL1=41.75

WCL1=4

UCL2=81

WCL2=16

LCL=0 LCL=0

(n2=148, h2=0.21) (n2=148, h2=0.21)

(n1=86, h1=1.9) (n1=86, h1=1.9)

I3 (Action region)

I₂ (Warning region)

I1 (Central region)

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(n₁=86,h₁=1.9) and the probability (1-

p )=0.92 of using(

₀' n₂=148,h₂=0.21). In this example, first sample used (

n =86,

₁

h =1.9) with UCL

₁ 1=41.75 and WCL1=4. The 1^st data point L=1 is located within WCL1=4, thus, the 2^nd sample should adopt (n₁=86, h₁=1.9) with UCL₁=41.75 and WCL₁=4. The 2^nd data point L=9 is located between WCL₁=4 and UCL1=41.75, thus, the 3^rd sample should adopt (

n =148,

₂

h =0.21) with UCL

₂ 2=81 and WCL₂=16. The 3^rd data point L=9 is located within WCL₂=16, thus, the 4^th sample should adopt (

n =86,

₁

h =1.9). The other data points follow the same rule to determine the use of

₁ sample sizes, sampling intervals, UCLi and WCLi.

Table20. Sampling data and VP Loss Function Chart Sample (ni,hi) X L Located

Region Sample (ni,hi) X L Located Region 1 random choose

(n1=86, h1=1.9) 1 1 I1 13 (n1=86, h1=1.9) 3 9 I2

2 (n1=86, h1=1.9) 3 9 I2 14 (n2=148, h2=0.21) 2 4 I1

3 (n2=148, h2=0.21) 3 9 I1 15 (n1=86, h1=1.9) 2 4 I2

4 (n1=86, h1=1.9) 3 9 I2 16 (n2=148, h2=0.21) 2 4 I1

5 (n2=148, h2=0.21) 1 1 I1 17 (n1=86, h1=1.9) 5 25 I2

6 (n1=86, h1=1.9) 1 1 I1 18 (n2=148, h2=0.21) 1 1 I1

7 (n1=86, h1=1.9) 2 4 I2 19 (n1=86, h1=1.9) 2 4 I2

8 (n2=148, h2=0.21) 4 16 I2 20 (n2=148, h2=0.21) 10 100 I3

9 (n2=148, h2=0.21) 3 9 I1 21 random choose

(n1=86, h1=1.9) 3 9 I2

10 (n1=86, h1=1.9) 3 9 I2 22 (n2=148, h2=0.21) 2 4 I1

11 (n2=148, h2=0.21) 2 4 I1 23 (n1=86, h1=1.9) 3 9 I2

12 (n1=86, h1=1.9) 1 1 I1 24 (n2=148, ) 2 4 I1

Figures21 shows the constructed optimal VP 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 VP loss function chart is 1.14 and ANOS is 465.37 after calculation.

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Figure21. Optimal VP Loss Function Chart (yellow points adpot n₁, h₁and right scale;

green points adpot n₂,h₂ and left scale)

The manager wants to know how much the optimal VP 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 VP loss function chart, the latter saves around 71.96% ATS and 4.03 %ANOS (see Table21). The VP 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 VP loss function chart to control the loss and quality of pies.

Table21. Comparison of the Fp and VP Loss Function Chart Chart ATS Saved ATS% ANOS Saved ANOS%

VP 1.14

71.96 465.37

4.03

Fp 4.07 484.9

Compared the performance among Fp, VSI, VSS and VP loss function chart via the example of Section 3.6, 4.5 and 5.5, the saved ATS% of the VSI is 68.8%, the saved ATS%

of the VSS is 30.29% and the saved ATS% of the VP is 71.96%. In this example, the VP

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6. Conclusions and Future Study

This article establishes a new type control chart, the loss function chart. Use the concept of the Taguchi loss function to build control charts. Control the process quality and reflect production loss at the same time. Compared the loss function chart with traditional np chart, the latter only reveals quality information. From producer viewpoint, control loss is more important under many circumstances. Thus, the loss function chart is better than the np chart under loss concern. This new chart is easier to operate than the economic chart, and covers cost information with a guaranteed quality standard. Through controlling ARL0 to minimize ATS, the loss function chart saves much production loss and gains better quality. The producer also monitors real-time cost variation of the process.

In this study, the adaptive schemes are applied. The current study proposes the

specified VSI loss function chart and the optimal VSI loss function chart. The optimal VSS and the optimal VP loss function chart are also demonstrated in this article. Comparing the performances among the specified VSI loss function chart, the optimal VSI loss function chart and the Fp loss function chart. The specified VSI loss function chart outperforms the Fp loss function chart. The optimal VSI loss function chart outperforms the specified VSI loss function chart. The range of saved time using the optimal VSI loss function chart is from the lowest 37.09% to the highest 89.97%. The VSS and VP loss function chart both can save more detective time than the Fp loss function chart. The range of saved time using the optimal VSS loss function chart is from the lowest 3.86% to the highest 30.47%.

The range of saved time using the optimal VP loss function chart is from the lowest

24.18% to the highest 89.67%. Numerical studies provide users optimal design of the VSI, VSS and VP loss function chart to use in real application. A example of the optimal VSI, VSS and VP loss function chart illustrate the construction processes and detective ability.

From the example, the saved ATS% of the VSI is 68.8%, the saved ATS% of the VSS is 30.29% and the saved ATS% of the VP is 71.96%. In this example, the VP loss function chart outperforms the VSI loss function chart, the VSI loss function chart outperforms the VSS loss function chart. Hence, the adaptive loss function chart is recommended for users due to better detective ability, especially the VSI and VP loss function chart.

Future studies of the loss function chart could consider using different loss functions to find the best model for each process. For accommodating complicated production processes, studies can be extended to multi-stage processes in the research or apply the loss function chart with multivariate or bivariate binomial data.

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References

[1] Alexander, S.M., Dillman, M.A., Usher, J.S. and Damodaran, B. (1995),“Economic Design of Control Charts using the Taguchi Loss Function,” Computers & Industrial

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[3] Serel, D.A. (2009),“Economic design of EWMA control charts based on loss function,” Mathematical and Computer Modelling 49, 745-759.

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X and S control charts using quadratic

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‧

E can be expressed as equation (1-1)

)

E

can be derived by the following processes.

Before deriving

E

(

X

⁴),

E

(

X

³)needs to be derived. Change position of items of the equation (1-3),

3 2

3) 3 ( ) 2 ( ) ( 1)( 2)

(

X E X E X n n n p

E

= − + − − . (1-4)

Put equation (1-1) into equation (1-4), then

3

‧

Change position of items of the equation (1-6),

4

Put equations (1-1) and (1-5) into equation (1-7), then

4

Put equations (1-1) and (1-8) into equation (1-2). Then,

Var (L )

can be expressed as

2

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