Specified VSI Loss Function Chart for binomial data

國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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)

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

(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 proportion scaleδ=2.5. Bases on those information,

5 . 2 , 500 ,

355 ,

2 ,

200 ,

5 , 02 .

0 = = = ₀ = = =

=

n n h ARL R δ

p

_{L U U} .

The manager uses the approach describes in Section 3.4 to determine the optimal design parameters of the VSI loss function chart as follows (see Table6). Use those

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

design parameters, the optimal VSI loss function chart for the pie store can be established as Figure7.

Table6. Optimal Design of the VSI Loss Function Chart

p n h₁ h₂ UCL WCL ARL0

0.02 119 1.6 0.2 36 9 354.76

Figure7. Optimal VSI Loss Function Chart

Bases on this plan, the store collects 24 samples with n=119 and constructs an optimal VSI 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 h₁=1.6 as sampling interval. If the current sample point is located within the warning regionI₂, the next sample should adopt h₂=0.2 as sampling interval. If the current sample point is located outside the UCL, the occurred S.C. should be searched and removed from the process.

Table7 shows the sampling results using the optimal VSI loss function chart scheme. The first sampling interval was decided randomly by the probability

0'

p =0.13 of using h =1.6 and the probability (1-

₁

p )=0.87 of using

₀'

h =0.2. In this

₂

WCL=9

LCL=0

UCL=36

I2 (Warning region)

I3 (Action region)

I1 (Central region)

(h2=0.2)

(h1=1.6)

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

9 and 36, thus, the 3^rd sample should adopt 0.2 as its sampling interval. After 1.6 hours from 2^nd sample, the 3^rd was taken. The 3^rd data point L=9 is located between 9 and 36, thus, the 4^th sample should still adopt 0.2 as its sampling interval. After 0.2 hour from the 3^rd sample, the 4^th sample was taken. The other data points follow the same rule to determine the use of sampling intervals.

Table7. Sampling data and VSI Loss Function Chart

Sample hi X L Located

Region Sample hi X L Located

Region

1 random

choose h1=1.6 1 1 I1 13 h1=1.6 3 9 I2

2 h1=1.6 3 9 I2 14 h2=0.2 2 4 I1

3 h2=0.2 3 9 I2 15 h1=1.6 2 4 I1

4 h2=0.2 3 9 I2 16 h1=1.6 2 4 I1

5 h2=0.2 1 1 I1 17 h1=1.6 5 25 I2

6 h1=1.6 1 1 I1 18 h2=0.2 1 1 I1

7 h1=1.6 2 4 I1 19 h1=1.6 2 4 I1

8 h1=1.6 4 16 I2 20 h1=1.6 10 100 I3

9 h2=0.2 3 9 I2 21 random choose

h1=1.6 3 9 I2

10 h2=0.2 3 9 I2 22 h2=0.2 2 4 I1

11 h2=0.2 2 4 I1 23 h1=1.6 3 9 I2

12 h1=1.6 1 1 I1 24 2 4 I1

Figures8 shows the constructed optimal VSI 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 VSI loss function chart is 1.27 after calculation.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

Figure8. Optimal VSI Loss Function Chart (red points adopt h2; blue points adopt h1)

The manager wants to know how much the optimal VSI loss function chart can save for the pie store by comparing with Fp loss function chart. Table8 shows parameters of the Fp loss function chart. Bases on the setting, Fp loss function chart was built as Figure9. The point on the 20^th sample was out of the UCL and the occurred S.C. should be searched. ATS of the Fp loss function chart is 4.07 after calculation.

Table8. Fp Loss Function Chart

p n h UCL WCL ARL0

0.02 119 1 36 0 354.76

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

Compared performance between the Fp and optimal VSI loss function chart, the latter saves around 68.8% ATS (see Table9). The VSI loss function chart outperforms the 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 VSI loss function chart to control the loss and quality of pies.

Table9. Comparison of the Fp and VSI Loss Function Chart Chart ATS Saved ATS%

VSI 1.27

68.8

Fp 4.07

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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)

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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)

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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)

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

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

Table14 shows the sampling results using the optimal VSS loss function chart

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 22-0)