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1. Introduction

Control charts are powerful statistical tools for monitoring production quality.

The technology industry has widely applied the Shewhart control charts for detecting process variations. A Shewhart control chart involves three design parameters, including sample size n, sampling interval h and control limit factor k. From a statistical viewpoint of a control chart design, a special cause (S.C.) is detected when the process operation is no longer stable. The determination of design parameters of a control chart depends on the required false alarm rate and power.

However, from a producer viewpoint, the company may not only be concerned about the speed of detecting an unstable process, but also focuses on the cost (loss) produced from processes. Duncan (1956) first developed the economic control chart several decades ago, based on the cost-control concept, which considered several cost parameters in the model. The economic model and minimal cost occurring within the production cycle determine the n, h, and k of a control chart. Many researchers have followed Duncan’s work, and been reviewed in the literature by Montgomery (1980) and Vance (1983). Nevertheless, the complicated nature of the economic model is a fact that cannot be ignored. Woodall (1986) considered weaknesses of the economic control chart to include “awkward time interval, difficulties in calculation and

imprecise process parameters.” Although computer technology has improved many of the calculation tasks, estimating the cost parameters still hinders using the model.

Taguchi (1984) commented on quality loss as “the loss to society caused by the product after it is shipped out.” Researchers have applied the concept of using loss

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out of economic model, and use Taguchi loss function to establish a new chart directly. In contrast with economic charts, the Taguchi loss function provides a different approach to control cost (loss). The Taguchi loss function presents loss through the difference between target value and measured value. The concept of obtaining quality and loss information at the same time is what this article wants to catch.

Researchers have recently demonstrated that adaptive control charts can help companies detect a S.C. more rapid than the fixed parameters control charts (Fp control charts). Studies have widely investigated adopting the variable sampling interval (VSI), variable sample size (VSS), variable sample size and sampling interval (VSSI) and variable parameters (VP), especially for variable data. Reynolds et al. (1988) proposed the first VSI

X charts which outperformed the Fp X charts.

Many researchers extended the ideas to improve detective ability. Tagaras (1998) published a reviewed paper of recent adaptive control charts. According to it, Rendtel (1987, 1990) proposed VSS and VSI charts for attribute data with a CUSUM-scheme.

Vaughan (1993) published a VSI np chart by the Bayesian model, Calabrese (1995), Porteus and Angelus (1997) proposed adaptive control charts for attribute data by the Bayesian model. However, compared to the prosperous researches of variable charts with attribute charts, the latter still leave much to explore. In recent researches, from 1999 to 2009, Costa (2001) proposed the VSS np chart with the Markov chain method, Luo and Wu (2002) developed one optimal algorithm for the VSS and VSI np charts, Costa (2003) developed a general method for the VP with c, np, u, and p charts, Luo and Wu (2004) worked on the VSSI np chart by an optimal algorithm that used attribute data on an adaptive control chart. The attribute control chart includes several attractive characteristics, such as easy implementation and measurement time saving, not to mention that variable data is sometimes undesirable because of process

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restrictions or technological limits. Therefore, this study is particularly interested in binomial data with an adaptive scheme.

This study establishes a new type of control chart, bases on the Taguchi loss function, and abbreviated here as a loss function chart. The purposes of this new control chart are monitoring the cost variation (loss) and maintaining a certain level of process quality. The chart controls cost in real-time and ensures production quality at the same time. The current study also develops an adaptive scheme. Section 2 introduces the distribution of the loss function for binomial data. Section 3 displays the design of the VSI loss function chart for binomial data. Section 4 shows the design of the VSS loss function chart. Section 5 demonstrates the design of the VP loss function chart. Data analyses and example are given and Section 6 is the conclusion and future study.

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2. Distribution of the Loss Function for binomial data

A loss function chart is a new control chart, bases on the statistics of a Taguchi loss function. This method is an easy and efficient way to monitor process costs.

From an economic viewpoint, the complicated economic model can estimate and control process loss. However, practical applications need to eliminate calculation obstacles for easy-usage, making the loss function chart a good alternative. Loss functions are used to describe quality cost (loss) while the product quality

characteristic is far from a specified target (T). An appropriate type of loss function is important when using the loss function chart. The choice of loss function may depend on the industry and process. Although there are various forms of loss functions, this study focuses on the quadratic loss function.

Let random variable X be the number of nonconforming units found within each sampling when the process is in control and it follows a binomial distribution with sample size n, fraction nonconforming p. The chosen quadratic loss function L as a loss function describes the cost while the product quality characteristic is far from the specified target. The quadratic loss function L is

)2

(

X T

L

= − (1) If the process is out-of-control, the random variable X turns to follow a binomial distribution with sample size n, fraction nonconformingδ , where theδ is a p proportion scale of the process,

p 1<δ <1

　 . The target value T is set as zero (T=0) in this article because the best target value of found nonconforming units ought to be zero.

Thus, the loss function can express as

L

=( X)². Equations (2) and (3) show the results ofE(L)andVar(L). Appendix refers to the derivation ofE(L)andVar(L).

2 2 2 2]

[ )

(L E X np np n p

E = = − + (2)

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4 3 2

4 3 3 3 4 2 3 2 2

2 16 10 4 4 7 12 6

6 )

(L n p n p n p n p n p np np np np

Var = − + + − + − + − (3)

From equations (2) and (3), the mean and variance of L are the function of n and p.

Thus, the mean and variance of L are close to zero when p is extremely small.

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3. Design of the VSI Loss Function Chart for binomial data

3.1 Construction of the VSI Loss Function Chart for binomial data

The fixed parameters (Fp) loss function chart is built with the upper control limit (UCL), central line (CL) and lower control limit (LCL). If the statistic L falls within the UCL and the LCL, the process is still in control. If the L falls outside the UCL, the process is out of control and requires search action. Take L as the statistic of a loss function chart and its UCL, CL, and LCL can be expressed as follows.

) ( )

(L k Var L E

UCL= + ⋅ (4) )

(L E

CL= (5)

=0

LCL (6) The LCL is set as zero because cost cannot be negative. Nevertheless, UCL is the function of n and p. When p is extremely small, UCL is close to zero. Thus, the loss function chart is not appropriate to be used for extremely small p.

The VSI loss function chart is a control chart with adaptive sampling intervals. A VSI loss function chart is built with a UCL, a WCL (warning control limit), and an LCL (see Fig. 1). The LCL is set as zero in the model because the nonconforming fraction is the degree of deterioration. Thus, LCL=0 is the best level of quality loss.

Figure1. VSI Loss Function Chart

WCL

0

UCL

CL

I2 (Warning region)

I1 (Central region)

I3 (Action region)

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A VSI loss function chart can be expressed as follows. See equations (7)-(10).

UCL=E(L)+k⋅ Var(L) (7)

^{WCL=E(L)+w⋅ Var(L)} (8) ^CL=E(L) (9)

=0

LCL

(10) where k is control limit factor and w is warning limit factor, 0<

w

<

k

<∞. Refer to equations (2)-(3), the UCL and WCL can be expressed mathematically as equations (11)-(12).

4 3 2 4

3 3 3 4 2 3 2 2 2 2

2

2 n p k 6n p 16n p 10n p 4n p 4n p np 7np 12np 6np

np np

UCL= − + + ⋅ − + + − + − + − (11)

4 3 2 4

3 3 3 4 2 3 2 2 2 2

2

2 n p w 6n p 16n p 10n p 4n p 4n p np 7np 12np 6np

np np

WCL= − + + ⋅ − + + − + − + − (12)

In a VSI loss function chart, WCL is a key to shorten the time of detecting the process shift. A VSI loss function chart has a long sampling interval h1 and a short sampling interval h2. WCL is the guard to decide the use of h1 or h2 between the samples.

When using the VSI loss function chart, two different sampling intervals, h1 and h2, are adopted. Users have to decide on a long sampling interval and a short sampling interval, where h1 >h2. If the data point is plotted on the central region (I1), use the long interval h1 as the next sampling interval. If the data point is plotted on the warning region (I2), use the short interval h2 as the next sampling interval. If the data point is plotted on the action region (I3), find the S.C. and repair the process.

For comparing the VSI loss function chart with the Fp loss function chart under the same standard, the average sampling interval needs to be the same when the process is

‧

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<∞, and

p is the probability

₀

From equation (14), WCL can be expressed in terms of )

Take the inverse function of both sides.



^{WCL −1}



^{=FX−1[p0⋅FX(}



^UCL−1



^)].

Then, the WCL can be determined approximately as equation (15).

 

^{0 2}

1[ ( 1) ] 1]

[ − ⋅ +

≈ F ⁻ F UCL p

WCL _{X X} (15)

where _{ }x is the smallest integer not less than the corresponding elements of x.

The value of

p can be calculated through equation (13) and expressed as

₀

2

The UCL can be determined when n, p, k are given by using equation (11). The value of n and k are decided by in-control average run length (use

ARL ). The

₀

ARL

₀ can be expressed as equation (17). ARL₀ is a function of n, p and k.

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where^{a = [np−np2+n2p2+k⋅ 6n2p2−16n2p3+10n2p4+4n3p3−4n3p4+np−7np2+12np3−6np4}

3.2 Performance Measurement

The control chart performance can be measured by many indexes, such as average run length (ARL) and average time to signal (ATS), whereATS =h⋅ARL. When the process is out of control, the smaller ARL or ATS means better detective ability of the control chart. When the process is in control, the larger ARL (see equation (17)) or ATS can result in fewer false alarms and costs.

This study applies the Markov chain method to calculate ATS (Prabhu et al.

(1993, 1995)) of the VSI loss function chart. Calculating the ARL and ATS of the VSI loss function chart includes two assumptions. First, the loss function chart assumes that only one S.C. may occur during the process. Second, the process is out-of-control at the beginning of the process. The ATS is calculated under the zero-state mode. Due to the assumptions, the process has two transient states and one absorbing state of the Markov chain approach (see Table1).

Table1. State Definition of the VSI Loss Function Chart for binomial data State S.C. occur Location of the VSI Loss Function Chart

1 Yes I1

2 Yes I2

3 Yes I3

DenoteP_i,jas a transition probability from the previous state i to the current state j, i=1, 2, 3, j=1, 2, 3. The transition probabilities are as follows:

‧

From the elementary properties of the Markov chain, the average time to signal is

h

matrix of order 2, Q is a 2 by 2 transition probability matrix,

h

is the vector of the next sampling interval for state 1 and state 2.

b

=[

p

₀^' ,1-

p

₀^' ],

h

= respectively, where

p is the probability of being at state 1 at the

₀^'

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 ATS of the fixed parameters loss function chart is

Fp

_

ATS

=

h

₀⋅

ARL

‧

3.3 Determination of the UCL, WCL of the Specified VSI Loss Function Chart

This research establishes a design of the VSI loss function chart with a specified sampling interval. The design parameters, (

n

,

WCL

,

k

), can be determined by

minimizing ATS under specified

h and subjects to (1) a specified

_i

ARL , (2) the range

₀ of sample size,2≤

n

_L ≤

n

≤

n

_U <∞, (3) the range of average inspection rate probabilities for the out-of-control process. Since _



The mathematical model can be expressed as Minimum ATS = f(n,WCL,k)

Subject to

(1)

ARL

₀

(2)2≤

n

_L ≤

n

≤

n

_U <∞

(3)0<

AIR

≤

R

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The procedures search the optimal design parameters (n, WCL, k) are described as follows.

Step1: Specify

n

_L,

n

_U,

ARL

₀,

p

,δ,

h

₀,

h

₁,

h

₂, R.

Step2:

p can be determined by equation (16) when

₀

h and

₁

h are specified.

₂ Step3: Searching available combinations (n, k) under the

ARL .

₀

Step4: Determine UCL by using equation (11) when n, k and p are known.

Step5: Determine WCL by using by equation (15) when UCL and

p are known.

₀ Step6: Check ifh₁,h₂and n satisfy the constraint

AIR

≤ . Then, the design

R

parameters

n

*,

k

*,

WCL

*can be determined under the minimum ATS.

3.4 Determination of the UCL, WCL of the Optimal VSI Loss Function Chart

If the five design parameters (n,h₁,h₂,WCL,k) of the VSI 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 five design parameters and subjects to (1) a specified

ARL , (2) the range of sample size,

₀ 2≤

n

_L ≤

n

≤

n

_U <∞, (3) the range of sampling interval, 0<

h

₂ <

h

₀ <

h

₁ ≤

h

_U, (4) the range of AIR,

R AIR

≤

<

0 .

The mathematical model can be expressed as MinimumATS = f(n,h₁,h₂WCL,k)

Subject to

(1)

ARL

₀

(2)2≤

n

_L ≤

n

≤

n

_U <∞ (3)0<

h

₂ <

h

₀ <

h

₁ ≤

h

_U (4)0<

AIR

≤

R

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The procedures search the optimal design parameters (n,h₁,h₂WCL,k) are described as follows.

Step1: Specify

n

_L,

n

_U,

ARL

₀,

p

,δ,

h

₀,

h

_U,

R

.

Step2: Searching available combinations (n, k) under the

ARL .

₀

Step3: Determine UCL by using equation (11) when n, k and p are known.

Step4: Searching

h within the feasible region of (

₁

h ,

₀

h

_U] and searching

h within the

₂ feasible region of (0,

h ) for minimizing ATS.

₀

p can be determined by

₀ equation (16) whenh₁andh₂are known.

Step5: Determine WCL by using equation (15) when UCL and

p are known.

₀ Step6: Check ifh₁,h₂and n satisfy the constraint

AIR

≤ . Then, the design

R

parameters n*,k*,h₁*,h₂*,WCL*can be determined under the minimum ATS.

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Figure2. Flow Chart of the Design of the Optimal VSI Loss Function Chart Check n,h₁andh₂to

satisfy

AIR

≤

R

Using equation (17) to determine various combinations of (n, k) under the specified

ARL

₀

Using equation (15) to calculate WCL by knownUCL ,p₀

Find the optimal design (

n *, k *, h

₁

*, h

₂

*, WCL *

) from all feasible solutions with minimal ATS

Input: Specify

0

0, , ,

, , ,

,

n p h h R ARL

n

_{L U} δ _U

Using equation (11) to determine UCL

Let

h

₁ ∈(

h

₀,

h

_U] Let

h

₂∈(0,

h

₀)

Using equation (16) to calculate

p

₀by

known

h ,

₁

h

₂

Not Available NO

YES

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3.5 Performance Comparisons

3.5.1 Specified VSI Loss Function Chart for binomial data

Find the design parameters of the specified VSI loss function chart by approach describe in Section 3.3. First, specify5≤ n≤200,

ARL =355,

₀ h₁=1.8, h₂=0.1,

h0=1 and R=500. Let the in-control p = 0.01, 0.02, 0.05, 0.1, 0.2, 0.3, 0.5 andδ=1.5, 2, 2.5. Table2 shows the design parameters (n*, k*, WCL*) and percentage of saved ATS by comparing with Fp loss function chart. The percentage of saved ATS is

% VSI 100

= S

% ATS

Saved − ⋅

ATS Fp

ATS pecified

ATS

Fp

(22)

Table2. ATS of the VSI and Fp under various p with specified h1=1.8, h2=0.1, R=500 p δ n* k* WCL* VSI_ATS* Fp_ATS ARL0 AIR Saved ATS%

0.01

1.5 NA NA NA NA NA NA NA NA

2 NA NA NA NA NA NA NA NA

2.5 NA NA NA NA NA NA NA NA

0.02

1.5 NA NA NA NA NA NA NA NA

2 NA NA NA NA NA NA NA NA

2.5 NA NA NA NA NA NA NA NA

0.05

1.5 49 5.90 9 20.72 35.27 357.25 490.00 41.27 2 49 5.90 9 2.95 8.94 357.25 490.00 67.02 2.5 49 5.90 9 0.78 3.78 357.25 490.00 79.35

0.1

1.5 NA NA NA NA NA NA NA NA

2 NA NA NA NA NA NA NA NA

2.5 NA NA NA NA NA NA NA NA

0.2

1.5 NA NA NA NA NA NA NA NA

2 NA NA NA NA NA NA NA NA

2.5 NA NA NA NA NA NA NA NA

0.3

1.5 33 4.05 121 1.29 5.66 355.48 330.00 77.13 2 19 4.44 49 0.28 2.05 354.28 190.00 86.16 2.5 19 4.44 49 0.11 1.08 354.28 190.00 89.82

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From Table2, the specified VSI loss function chart saves much ATS than the Fp chart when p=0.05 and 0.3. When p=0.05, the VSI loss function chart saves at least 41.27% on ATS and saves at most 79.35%. When p=0.3, the VSI loss function chart saves at least 77.13% on ATS and saves at most 89.82%.

Table3. ATS of the VSI and Fp under various p with specified h1=1.8, h2=0.1, R=700 P δ n* k* WCL* VSI_ATS* Fp_ATS ARL0 AIR Saved ATS%

0.01

1.5 NA NA NA NA NA NA NA NA

2 NA NA NA NA NA NA NA NA

2.5 NA NA NA NA NA NA NA NA

0.02

1.5 70 6.79 4 40.25 53.66 355.50 700.00 24.99 2 70 6.79 4 8.29 16.34 355.50 700.00 49.28 2.5 70 6.79 4 2.58 7.29 355.50 700.00 64.54

0.05

1.5 49 5.90 9 20.72 35.27 357.25 490.00 41.27 2 49 5.90 9 2.95 8.94 357.25 490.00 67.02 2.5 49 5.90 9 0.78 3.78 357.25 490.00 79.35

0.1

1.5 62 4.72 49 5.37 13.71 355.71 620 60.81 2 62 4.72 49 0.46 2.83 355.71 620 83.75 2.5 62 4.72 49 0.16 1.40 355.71 620 88.72

0.2

1.5 65 4.01 196 0.99 4.84 358.16 650 79.64 2 65 4.01 196 0.13 1.23 358.16 650 89.55 2.5 54 4.13 144 0.15 1.39 358.27 540 88.92

0.3

1.5 33 4.05 121 1.29 5.66 355.48 330.00 77.13 2 19 4.44 49 0.28 2.05 354.28 190.00 86.16 2.5 19 4.44 49 0.11 1.08 354.28 190.00 89.82 0.5 1.5 70 3.28 1296 0.11 1.06 358.25 700 89.98

*NA is not available

However, the parameter design set cannot be found when p=0.01, 0.02, 0.1, 0.2 and 0.5 because the restriction of AIR is too small (R=500). In Table3, the range of R is enlarged to 700 and the design parameters results are existent, except=0.01. When R=700, the VSI loss function chart saves at least 24.99% ATS and at most 89.98%

ATS compared with the Fp loss function chart. The performance of the specified VSI

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loss function chart outperforms the Fp loss function chart with R=700.

We summarize results from data analyses as follows. (1) The value of k becomes smaller when p is increasing, except p=0.2, 0.3. (2) The bigger theδis, the smaller the ATS is. (3) The larger the p is, the smaller the ATS is. (4) The specified VSI loss function chart has better performance than the Fp loss function chart. (5) It is better to set R=700 rather than 500 of the specified VSI loss function chart.

3.5.2 Optimal VSI Loss Function Chart for binomial data

Find the design parameters of the optimal VSI loss function chart by approach describe in Section 3.4. First, specify5≤ n≤200,

ARL =355,

₀ 0<

h

₂ <

h

₀ =1<

h

₁ ≤2, and R=500. The in-control p = 0.01, 0.02, 0.05, 0.1, 0.2, 0.3, 0.5 andδ=1.5, 2, 2.5.

Table4 shows the optimal design parameters (n*, k*, h1*, h2*, WCL*) and percentage of saved ATS compared to Fp loss function chart. The percentage of saved ATS is

% VSI_ 100

=

% ATS

Saved − ⋅

ATS Fp

ATS Optimal

ATS

Fp

(23)

In Table4, the optimal VSI loss function chart saves ATS at least 37.09% and at most 89.97% ATS compared to the Fp loss function chart under various p.

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Table4. ATS of the Optimal VSI and Fp under various p with R=500

p δ n* k* h1* h2* WCL* VSI_ATS* Fp_ATS ARL0 AIR Saved ATS%

0.01

1.5 185 6.39 2 0.2 4 28.02 44.54 356.65 298.63 37.09 2 185 6.39 2 0.2 4 5.30 12.54 356.65 457.13 57.76 2.5 185 6.39 1.7 0.1 4 8.36 16.73 357.37 290.03 50.03

0.02

1.5 119 6.00 1.7 0.1 9 21.89 36.49 354.76 203.04 40.02 2 119 6.00 1.7 0.1 9 3.35 9.50 354.76 365.55 64.79 2.5 119 6.00 1.6 0.2 9 1.27 4.07 354.76 419.68 68.93

0.05

1.5 187 4.40 2 0.2 100 3.64 9.09 356.15 493.94 59.93 2 83 5.22 1.6 0.1 25 1.16 4.87 354.45 396.18 76.22 2.5 49 5.90 1.8 0.1 9 0.78 3.78 357.25 274.75 79.35

0.1

1.5 169 3.95 2 0.3 289 1.44 4.09 354.84 497.36 64.73 2 62 4.72 1.7 0.1 49 0.45 2.83 355.71 449.19 84.12 2.5 62 4.72 1.6 0.2 49 0.29 1.40 355.71 305.42 78.95

0.2

1.5 176 3.52 2 0.4 1156 0.62 1.55 356.76 439.16 59.78 2 176 3.52 2 0.4 1156 0.40 1.00 356.76 440.00 60.00 2.5 176 3.52 2 0.4 1156 0.40 1.00 356.76 440.00 60.00

0.3

1.5 93 3.53 1.9 0.2 784 0.34 1.64 355.74 460.42 79.49 2 19 4.44 2 0.1 36 0.23 2.05 354.28 179.54 88.86 2.5 19 4.44 2 0.1 36 0.11 1.08 354.28 189.96 89.97 0.5 1.5 70 3.28 2 0.2 1225 0.21 1.06 358.25 349.99 79.99

Figure3. Main Effect Plot of the Optimal VSI Loss Function Chart (p, R=500)

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Figure4. Main Effect Plot of the Optimal VSI Loss Function Chart (δ, R=500)

To investigate the effects of various p andδon optimal design parameters (n*, k*, h1*, h2*) and ATS, the main effect plots are illustrated in Figures 3 and 4. We summarize the results from data analyses and plots as follows. (1)The average of optimal sample size n becomes small when p increases, except p=0.2 and 0.5, the average of optimal sample size n also becomes small whenδincreases. (2)The average ATS decreases when p orδincreases. (3)The average optimal k decreases when p is getting large, except p=0.2, the average optimal k increases whenδis getting large. (4)The distance between h₁ and h₂ is a fixed constant under various p andδ. If user has known the length of h1 or h2, the other one sampling interval can be obtained by the fixed distance between long and short sampling intervals.

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Table5. ATS of the Optimal VSI and Fp under various p with R=400 p δ n* k* h1* h2* WCL* VSI_ATS* Fp_ATS ARL0 AIR Saved

ATS%

0.01

1.5 185 6.39 2 0.2 4 28.02 44.54 356.65 298.63 37.09 2 185 6.39 1.9 0.3 4 6.24 12.54 356.65 383.85 50.23 2.5 138 6.85 1.9 0.1 4 2.90 7.50 357.37 396.31 61.36

0.02

1.5 119 6.00 1.7 0.1 9 21.89 36.49 354.76 203.04 40.02 2 119 6.00 1.7 0.1 9 3.35 9.50 354.76 365.55 64.79 2.5 119 6.00 1.9 0.2 9 1.36 4.07 354.76 394.76 66.56

0.05

1.5 187 4.40 2 0.3 100 4.45 9.09 356.15 399.01 51.05 2 83 5.22 1.6 0.1 25 1.16 4.87 354.45 396.18 76.22 2.5 49 5.90 1.8 0.1 9 0.78 3.78 357.25 274.75 79.35

0.1

1.5 110 4.24 2 0.2 121 2.10 6.98 355.84 383.88 69.90 2 82 4.47 1.7 0.2 81 0.48 2.09 357.10 381.03 77.09 2.5 62 4.72 1.6 0.2 49 0.29 1.40 355.71 305.42 78.95

0.2

1.5 176 3.52 1.9 0.5 1156 0.78 1.55 356.76 351.53 49.81 2 176 3.52 1.9 0.5 1156 0.50 1.00 356.76 352.00 50.00 2.5 176 3.52 1.9 0.5 1156 0.50 1.00 356.76 352.00 50.00

0.3

1.5 93 3.53 1.1 0.2 1156 0.48 1.64 355.74 394.10 70.79 2 19 4.44 2 0.1 36 0.23 2.05 354.28 179.54 88.86 2.5 19 4.44 2 0.1 36 0.11 1.08 354.28 189.96 89.97 0.5 1.5 70 3.28 2 0.2 1225 0.21 1.06 358.25 349.99 79.99

Although the optimal VSI loss function chart saves more detective time than the Fp loss function chart, the AIR and n are quite big. The bigger AIR or n means cost consuming. Thus, R may reduce to 400 (see Table5) for saving loss. Compared the optimal design parameters among Table4 (R=500) and Table5 (R=400), (p=0.01,δ

=2.5, R=400) and (p=0.1,δ=1.5, R=400) save more sample size than design

parameters with R=500, n from 185 decreases to 138 and from 169 decreases to 110.

Only (p=0.1, δ=2) with R=400 increases sample size from 62 to 82. The other sets stay the same parameters results. The AIR are smaller averagely when R=400. Use R=400 is a good alternative to save cost, although we may sustain slightly increase of ATS.

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Figure5. Main Effect Plot of the Optimal VSI Loss Function Chart (p, R=400)

Figure6. Main Effect Plot of the Optimal VSI Loss Function Chart (δ, R=400)

Figures 5 and 6 are the main effect plots of the optimal VSI loss function chart with R=400 under various p andδ. We summarize the results from main effect plots

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(3) The average optimal k decreases when p becomes large, except p=0.3, the average optimal k increases whenδbecomes large. (4) The distance between h1 and h2 is a fixed constant under various p andδ.

From data analyses in Section 3.5, we summarize that (1) the optimal VSI loss function and the specified VSI outperform the Fp loss function chart, (2) the ATS of the optimal VSI loss function chart is smaller than the specified VSI loss function chart (see Table2 and Table4), (3) it is important to give proper R when using specified VSI loss function chart, or the design parameters results may not be

available, (4) R=400 is an good alternative to reduce loss when using the optimal VSI loss function chart, (5) n decreases when p andδincreases. Consequently, it is

recommended to adopt an optimal VSI loss function chart for detecting a S.C. rapidly.

3.6 Example

A manager is concerned about the defect proportion of baked pies at store. From the analyses in Section 3.5, the optimal VSI loss function chart outperforms the specified VSI loss function chart. Hence, the manager determines to construct an optimal VSI 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, and an AIR is less than 500 due to process capacity. ConsiderARL₀=355 and

The store can accept the sample size within [5, 200], the maximum sampling interval is 2, and an AIR is less than 500 due to process capacity. ConsiderARL₀=355 and

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