Data Description and Determining the Optimum Producer Inspection and

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37 CHAPTER 5. DETERMINING THE BEST λ OF THE ECONOMIC

EWMA

X-bar

CONTROL CHART UNDER DIFFERENT

SHIFT SCALES IN THE MEAN AND VARIANCE

5.1 Data Description and Determining the Optimum Producer Inspection and the Design Parameters of the Economic EWMA

_X-bar

Control Chart

In this section, we use routine “rgamma” to simulate one type of in-control gamma distribution and five types of out-of-control gamma distributions.

We first simulate 25 samples of size 4 data from the in-control gamma distribution with a=25 and b=0.2, with a mean and variance of 5 and 1, respectively.

Table 5-1. In-control Data with Gamma (a=25, b=0.2)

No. simulation data with n=4

X

No. simulation data with n=4

X

1 6.283 6.397 4.894 5.691 5.816 14 5.849 6.036 5.879 3.814 5.394 2 4.071 5.513 4.908 5.222 4.929 15 6.359 4.882 4.89 2.939 4.767 3 5.222 4.037 5.591 5.076 4.982 16 6.288 3.448 6.242 2.928 4.727 4 2.54 5.11 3.753 5.155 4.14 17 5.595 5.352 4.419 4.756 5.03 5 6.439 4.634 5.628 4.692 5.348 18 5.124 4.054 5.542 4.358 4.77 6 5.2 5.276 3.601 2.725 4.201 19 5.816 4.549 5.241 5.435 5.26 7 5.265 4.934 5.253 3.719 4.793 20 4.593 4.515 5.584 5.111 4.951 8 3.954 5.004 5.628 6.364 5.237 21 6.918 5.952 5.245 4.799 5.728 9 4.04 3.106 5.334 4.119 4.15 22 4.881 4.955 6.066 4.761 5.166 10 4.102 7.045 4.045 5.118 5.077 23 4.525 5.358 5.983 3.144 4.753 11 4.317 4.943 5.044 4.384 4.672 24 5.442 4.729 5.419 5.011 5.15 12 3 5.595 5.794 4.064 4.613 25 7.023 5.363 5.197 3.921 5.376

13 4.992 5.155 7.028 5.743 5.729

X  ₄ _. ₉₉

As a producer, we do not know the parameters of the gamma distribution; thus, we used the MLE method, which maximizes cumulative products of p.d.f for given data to estimate

aˆ

and

bˆ of the simulation data.

With 100 data in Table 5-1, we estimate the parameters of in-control data, and obtain

a ˆ

 24 . 349

b

ˆ_I



0.205, Mean

 a

ˆ_I

 b

ˆ_I



4.99

︿

, and

Var  a ˆ

 b ˆ

_I²

 1 . 023

︿

‧

subject to 0.5≦h≦8. If the producer decides to inspect, we maximize EAP (Equation 4-7) to determine the optimum h* and ω*, subject to 0.5≦h≦8 and 2≦ω.

I. To compare the profit model with different λ, we adopt moderate shifts in the mean and variance of out-of-control gamma data.

Let a=26 and b=0.25, that is, the out-of-control mean and variance are 6.5 and 1.625, respectively, which means δ1=1 and δ2=0.05. We simulate 15 samples of size 4 data from the out-of-control gamma distribution with a=26 and b=0.25, as follows:

Table 5-2. Out-of-control Data with Gamma (a=26, b=0.25)

No. simulation data with n=4

X

With 60 data in Table 5-2, we estimate the parameters of out-of-control data, and obtain

a ˆ

 25 . 268

b

ˆ_O



0.265 ,

ˆ 0 . 919

‧

Table 5-3. The Optimum Results Comparison of Profit Model with Different λ

Inspection Without With

λ

1 0.6 0.05 1 0.6 0.05 According to Table 5-3, with and without inspection, h*, EWMAX-bar chart, and

ARL

1 are the same at each λ. However, with inspection, we increased the profit per unit time as follows:

(1) If λ=0.05, we increase 54.9% profit per unit time when we have an inspection.

(2) If λ=0.6, we increase 42.5% profit per unit time when we have an inspection.

(3) If λ=1, we increase 44.2% profit per unit time when we have an inspection.

If we use the economic EWMAX-bar chart with λ=0.6, we have the largest EAP*

and smallest ARL1 for the moderate shifts in the mean and variance. Therefore, we suggest that the producer takes inspection with USL*=8.66, use the economic EWMAX-bar chart with λ=0.6 and take four samples every 0.5 unit time.

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40

To find the detection ability for the three types of EWMAX-bar chart, we plot the in-control and out-of-control statistics on them.

Figure 5-1. The Economic EWMAX-bar Chart (λ=1) with In-control Data

For EWMAX-bar chart with λ=1, Figure 5-1 shows that no points are out of limits for in-control samples.

Figure 5-2. The Economic EWMAX-bar Chart (λ=1) with Out-of-control Data Plots the out-of-control statistics on the EWMAX-bar Chart with λ=1, Figure 5-2 shows that No. 6, 8, 10, 12, and 15 are out of limits; the first true alarm is on No. 6.

Figure 5-3. The Economic EWMA_X-bar Chart (λ=0.6) with In-control Data For EWMA_X-bar chart with λ=0.6, Figure 5-3 shows that no points are out of limits for in-control samples.

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Figure 5-4. The Economic EWMAX-bar Chart (λ=0.6) with Out-of-control Data Plots the out-of-control statistics on the EWMAX-bar Chart with λ=0.6, Figure 5-4 shows that No. 3 to No. 15 are out of limits; the first true alarm is on No. 3.

Figure 5-5. The Economic EWMAX-bar Chart (λ=0.05) with In-control Data For EWMAX-bar chart with λ=0.05, Figure 5-5 shows that no points are out of limits for in-control samples.

Figure 5-6. The Economic EWMA_X-bar Chart (λ=0.05) with Out-of-control Data Plots the out-of-control statistics on the EWMA_X-bar Chart with λ=0.05, Figure 5-6 shows that No. 4 to No. 15 are out of limits; the first true alarm is on No. 4.

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42

II. To compare the profit model with different λ, we adopt small shifts in the mean and variance of out-of-control gamma data.

Let a=28 and b=0.21, that is, the out-of-control mean and variance are 5.88 and 1.2348, respectively, which means δ1=3 and δ2=0.01. We simulate 15 samples of size 4 data from the out-of-control gamma distribution with a=28 and b=0.21, as follows:

Table 5-4. Out-of-control Data with Gamma (a=28, b=0.21)

No. simulation data with n=4

X

1 6.156 5.224 7.595 6.145 6.28

2 5.026 4.593 8.429 6.001 6.012

3 5.205 5.889 4.731 3.281 4.776

4 4.828 5.989 4.256 4.911 4.996

5 7.144 7.129 5.382 6.513 6.542

6 5.664 5.578 6.608 6.595 6.111

7 7.939 4.867 5.966 6.424 6.299

8 5.524 5.661 7.156 6.023 6.091

9 5.271 5.992 5.396 7.1 5.94

10 7.212 5.952 7.16 5.058 6.345

11 7.683 5.009 6.831 5.237 6.19

12 6.226 4.889 4.416 5.096 5.157

13 5.478 4.99 5.898 6.196 5.641

14 6.961 6.639 4.215 6.964 6.195

15 5.797 7.595 7.614 7.204 7.053

975 .

 5 X

With 60 data in Table 5-4, we estimate the parameters of out-of-control data, and obtain

a ˆ

 30 . 883

b

ˆ_O



0.193 ,

ˆ 6 . 534





ˆ 0 . 012

 



, Mean^︿



5.975 , and

156 . 1

Var^︿



. Hence, we have a 0.974 mean shift scale and a 1.063 s.d. shift scale.

‧

same as the X-bar probability chart. We compare the optimum results of profit model with λ=1, 0.5, and 0.05, as follows:

Table 5-5. The Optimum Results Comparison of Profit Model with Different λ

Inspection Without With

λ

1 0.5 0.05 1 0.5 0.05 According to Table 5-5, with and without inspection, h*, EWMAX-bar chart, and

ARL

1 are the same at each λ. However, with inspection, we increased the profit per unit time as follows:

(1) If λ=0.05, we increase 1.66% profit per unit time when we have an inspection.

(2) If λ=0.5, we increase 1.6% profit per unit time when we have an inspection.

(3) If λ=1, we increase 1.85% profit per unit time when we have an inspection.

If we use the economic EWMAX-bar chart with λ=0.5, we have the largest EAP*

and smallest ARL1 for the small shifts in the mean and variance. Therefore, we suggest that the producer takes inspection with USL*=8.66, use the economic EWMAX-bar chart with λ=0.5 and take four samples every 0.5 unit time.

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44

To find the detection ability for the three types of EWMAX-bar chart, we plot the in-control and out-of-control statistics on them.

Figure 5-7. The Economic EWMAX-bar Chart (λ=1) with In-control Data

For EWMAX-bar chart with λ=1, Figure 5-7 shows that no points are out of limits for in-control samples.

Figure 5-8. The Economic EWMAX-bar Chart (λ=1) with Out-of-control Data Plots the out-of-control statistics on the EWMAX-bar Chart with λ=1, Figure 5-8 shows that No. 15 is out of limits; the first true alarm is on No. 15.

Figure 5-9. The Economic EWMA_X-bar Chart (λ=0.5) with In-control Data For EWMA_X-bar chart with λ=0.5, Figure 5-9 shows that no points are out of limits for in-control samples.

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Figure 5-10. The Economic EWMAX-bar Chart (λ=0.5) with Out-of-control Data Plots the out-of-control statistics on the EWMAX-bar Chart with λ=0.5, Figure 5-10 shows that No. 6 to No. 11, 14, and 15 are out of limits; the first true alarm is on No. 6.

Figure 5-11. The Economic EWMAX-bar Chart (λ=0.05) with In-control Data For EWMAX-bar chart with λ=0.05, Figure 5-11 shows that no points are out of limits for in-control samples.

Figure 5-12. The Economic EWMA_X-bar Chart (λ=0.05) with Out-of-control Data Plots the out-of-control statistics on the EWMA_X-bar Chart with λ=0.05, Figure 5-12 shows that No. 7 to No. 15 are out of limits; the first true alarm is on No. 7.

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III. To compare the profit model with different λ, we adopt only moderate shifts in the mean of out-of-control gamma data.

Let a=42.45 and b=0.154, that is, the out-of-control mean and variance are 6.5 and 1, respectively, which meansδ1=17.25 and δ2=-0.046. We simulate 15 samples of size 4 data from the out-of-control gamma distribution with a=42.45 and b=0.154, as follows:

Table 5-6. Out-of-control Data with Gamma (a=42.45, b=0.154)

No. simulation data with n=4

X

1 5.211 6.206 8.77 6.833 6.755

2 6.232 6.273 7.556 6.781 6.71

3 7.759 6.392 4.876 6.144 6.293

4 8.373 7.25 6.74 5.986 7.087

5 5.014 7.998 5.377 5.547 5.984

6 7.65 5.385 5.822 6.622 6.37

7 7.911 6.213 10.067 6.721 7.728

8 7.416 5.252 6.553 6.646 6.467

9 6.259 7.502 5.828 7.118 6.677

10 4.747 5.193 7.432 6.134 5.876

11 5.911 7.173 6.807 6.404 6.574

12 7.335 6.388 6.383 6.643 6.687

13 6.699 8.889 5.93 5.889 6.852

14 6.41 4.786 5.894 7.576 6.167

15 6.871 6.969 7.435 5.815 6.773

6 .

 6 X

With 60 data in Table 5-6, we estimate the parameters of out-of-control data, and obtain

a ˆ

 41 . 331

b

ˆ_O



0.16 ,

ˆ 16 . 983





ˆ 0 . 045

 



, Mean^︿



6.6 , and

05 . 1

Var^︿



. Hence, we have a 1.603 mean shift scale and a 1.017 s.d. shift scale.

‧

the X-bar probability chart. We compare the optimum results of profit model with λ=1, 0.6, and 0.05, as follows:

Table 5-7. The Optimum Results Comparison of Profit Model with Different λ

Inspection Without With

λ

1 0.6 0.05 1 0.6 0.05

first true alarm on which sample According to Table 5-7, with and without inspection, h*, EWMAX-bar chart, and

ARL

1 are the same at each λ. However, with inspection, we increased the profit per unit time as follows:

(1) If λ=0.05, we increase 8.6% profit per unit time when we have an inspection.

(2) If λ=0.6, we increase 7.4% profit per unit time when we have an inspection.

(3) If λ=1, we increase 7.7% profit per unit time when we have an inspection.

If we use the economic EWMAX-bar chart with λ=0.6, we have the largest EAP*

and smallest ARL1 for the only moderate shifts in the mean. Therefore, we suggest that the producer takes inspection with USL*=8.66, use the economic EWMAX-bar

chart with λ=0.6 and take four samples every 0.5 unit time.

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To find the detection ability for the three types of EWMAX-bar chart, we plot the in-control and out-of-control statistics on them.

Figure 5-13. The Economic EWMAX-bar Chart (λ=1) with In-control Data For EWMAX-bar chart with λ=1, Figure 5-13 shows that no points are out of limits for in-control samples.

Figure 5-14. The Economic EWMAX-bar Chart (λ=1) with Out-of-control Data Plots the out-of-control statistics on the EWMAX-bar Chart with λ=1, Figure 5-14 shows that No. 1, 2, 4, 7, 9, 12, 13, and 15 are out of limits; the first true alarm is on No. 1.

Figure 5-15. The Economic EWMA_X-bar Chart (λ=0.6) with In-control Data For EWMA_X-bar chart with λ=0.6, Figure 5-15 shows that no points are out of limits for in-control samples.

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Figure 5-16. The Economic EWMAX-bar Chart (λ=0.6) with Out-of-control Data Plots the out-of-control statistics on the EWMAX-bar Chart with λ=0.6, Figure 5-16 shows that No. 2 to No. 15 are out of limits; the first true alarm is on No. 2.

Figure 5-17. The Economic EWMAX-bar Chart (λ=0.05) with In-control Data For EWMAX-bar chart with λ=0.05, Figure 5-17 shows that no points are out of limits for in-control samples.

Figure 5-18. The Economic EWMA_X-bar Chart (λ=0.05) with Out-of-control Data Plots the out-of-control statistics on the EWMA_X-bar Chart with λ=0.05, Figure 5-18 shows that No. 3 to No. 15 are out of limits; the first true alarm is on No. 3.

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IV. To compare the profit model with different λ, we adopt only moderate shifts in the variance of out-of-control gamma data.

Let a=15.385 and b=0.325, that is, the out-of-control mean and variance are 5 and 1.625, respectively, which means δ1=-9.615 and δ2=0.125. We simulate 15 samples of size 4 data from the out-of-control gamma distribution with a=15.385 and b=0.325, as follows:

Table 5-8. Out-of-control Data with Gamma (a=15.385, b=0.325)

No. simulation data with n=4

X

1 5.783 4.26 7.003 4.471 5.379

2 5.485 4.479 7.158 3.814 5.234

3 4.143 7.228 7.417 4.916 5.926

4 6.487 5.547 2.772 4.822 4.907

5 5.967 4.88 5.348 3.423 4.904

6 4.063 3.395 5.573 2.929 3.99

7 5.252 4.555 6.222 4.037 5.016

8 3.769 4.488 2.513 4.25 3.755

9 4.227 7.095 4.826 5.706 5.463

10 4.821 4.497 6.032 4.128 4.87

11 5.136 4.276 7.218 5.864 5.624

12 4.27 4.185 7.084 5.688 5.307

13 6.958 3.505 6.282 4.767 5.378

14 4.593 3.912 4.769 7.98 5.313

15 5.699 7.59 4.129 5.879 5.824

126 .

 5 X

With 60 data in Table 5-8, we estimate the parameters of out-of-control data, and obtain

a ˆ

 15 . 608

b

ˆ_O



0.328,

ˆ 8 . 741

 



ˆ 0 . 123





, Mean^︿



5.126 , and

684 . 1

Var^︿



. Hence, we have a 0.126 mean shift scale and a 1.281 s.d. shift scale.

‧

the X-bar probability chart. We compare the optimum results of profit model with λ=1, 0.3, and 0.05,, as follows:

Table 5-9. The Optimum Results Comparison of Profit Model with Different λ

Inspection Without With

λ

1 0.3 0.05 1 0.3 0.05

first true alarm on which sample

According to Table 5-9, with and without inspection, h*, EWMA_X-bar chart, and

ARL

₁ are the same at each λ. However, with inspection, we increased the profit per unit time as follows:

(1) If λ=0.05, we increase 1.5% profit per unit time when we have an inspection.

(2) If λ=0.3, we increase 1.4% profit per unit time when we have an inspection.

(3) If λ=1, we increase 1.4% profit per unit time when we have an inspection.

If we use the economic EWMA_X-bar chart with λ=0.3, we have the largest EAP*

and smallest ARL₁ for the only moderate shifts in the variance. Therefore, we suggest that the producer takes inspection with USL*=8.66, use the economic EWMA_X-bar chart with λ=0.3 and take four samples every 0.5 unit time.

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52

To find the detection ability for the three types of EWMAX-bar chart, we plot the in-control and out-of-control statistics on them.

Figure 5-19. The Economic EWMAX-bar Chart (λ=1) with In-control Data For EWMAX-bar chart with λ=1, Figure 5-19 shows that no points are out of limits for in-control samples.

Figure 5-20. The Economic EWMAX-bar Chart (λ=1) with Out-of-control Data Plots the out-of-control statistics on the EWMAX-bar Chart with λ=1, Figure 5-20 shows that no points are out of limits; it has no true alarm.

Figure 5-21. The Economic EWMA_X-bar Chart (λ=0.3) with In-control Data For EWMA_X-bar chart with λ=0.3, Figure 5-21 shows that no points are out of limits for in-control samples.

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Figure 5-22. The Economic EWMAX-bar Chart (λ=0.3) with Out-of-control Data Plots the out-of-control statistics on the EWMAX-bar Chart with λ=0.3, Figure 5-22 shows that no points are out of limits; it has no true alarm.

Figure 5-23. The Economic EWMAX-bar Chart (λ=0.05) with In-control Data For EWMAX-bar chart with λ=0.05, Figure 5-23 shows that no points are out of limits for in-control samples.

Figure 5-24. The Economic EWMA_X-bar Chart (λ=0.05) with Out-of-control Data Plots the out-of-control statistics on the EWMA_X-bar Chart with λ=0.05, Figure 5-24 shows that no points are out of limits; it has no true alarm.

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V. To compare the profit model with different λ, we adopt only small shifts in the variance of out-of-control gamma data.

Table 5-10. Out-of-control Data with Gamma (a=15.385, b=0.325)

No. simulation data with n=4

X

1 3.461 5.507 6.388 3.679 4.759

2 3.152 3.912 3.297 4.57 3.733

3 4.175 3.619 3.699 5.961 4.363

4 5.62 5.381 6.398 3.62 5.255

5 4.68 3.814 5.188 5.141 4.706

6 3.949 5.531 6.154 5.891 5.381

7 5.054 4.684 6.065 5.079 5.22

8 6.155 3.538 3.748 4.434 4.469

9 7.259 5.163 5.59 5.38 5.848

10 6.283 6.908 5.931 6.023 6.287

11 4.941 4.361 6.621 3.688 4.903

12 4.891 5.098 5.064 4.271 4.831

13 4.771 5.773 7.21 4.154 5.477

14 4.146 6.678 4.142 4.289 4.814

15 5.892 5.977 5.689 4.045 5.401

03 .

 5 X

With 60 data in Table 5-10, we estimate the parameters of out-of-control data, and obtain

a ˆ

 22 . 087

b

ˆ_O



0.228,

ˆ 2 . 261

 



ˆ 0 . 023





, Mean^︿



5.03, and

145 . 1

Var^︿



. Hence, we have a 0.039 mean shift scale and a 1.058 s.d. shift scale.

‧

same as the X-bar probability chart. We compare the optimum results of profit model with λ=1, 0.2, and 0.05, as follows:

Table 5-11. The Optimum Results Comparison of Profit Model with Different λ

Inspection Without With

λ

1 0.2 0.05 1 0.2 0.05

first true alarm on which sample

According to Table 5-11, with and without inspection, h*, EWMA_X-bar chart, and

ARL

₁ are the same at each λ. However, with inspection, we increased the profit per unit time as follows:

(1) If λ=0.05, we increase 0.5% profit per unit time when we have an inspection.

(2) If λ=0.2, we increase 0.5% profit per unit time when we have an inspection.

(3) If λ=1, we increase 0.5% profit per unit time when we have an inspection.

If we use the economic EWMA_X-bar chart with λ=0.2, we have the largest EAP*

and smallest ARL₁ for the only small shifts in the variance. Therefore, we suggest that the producer takes inspection with USL*=8.66, use the economic EWMA_X-bar chart with λ=0.2 and take four samples every 0.5 unit time.

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To find the detection ability for the three types of EWMAX-bar chart, we plot the in-control and out-of-control statistics on them.

Figure 5-25. The Economic EWMAX-bar Chart (λ=1) with In-control Data For EWMAX-bar chart with λ=1, Figure 5-25 shows that no points are out of limits for in-control samples.

Figure 5-26. The Economic EWMAX-bar Chart (λ=1) with Out-of-control Data Plots the out-of-control statistics on the EWMAX-bar Chart with λ=1, Figure 5-26 shows that no points are out of limits; it has no true alarm.

Figure 5-27. The Economic EWMA_X-bar Chart (λ=0.2) with In-control Data For EWMA_X-bar chart with λ=0.2, Figure 5-27 shows that no points are out of limits for in-control samples.

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Figure 5-28. The Economic EWMAX-bar Chart (λ=0.2) with Out-of-control Data Plots the out-of-control statistics on the EWMAX-bar Chart with λ=0.2, Figure 5-28 shows that no points are out of limits; it has no true alarm.

Figure 5-29. The Economic EWMAX-bar Chart (λ=0.05) with In-control Data For EWMAX-bar chart with λ=0.05, Figure 5-29 shows that no points are out of limits for in-control samples.

Figure 5-30. The Economic EWMA_X-bar Chart (λ=0.05) with Out-of-control Data Plots the out-of-control statistics on the EWMA_X-bar Chart with λ=0.05, Figure 5-30 shows that no points are out of limits; it has no true alarm.

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VI. Comparing the profit model with different λ using an example of service time data.

We consider a quality variable with an exponential distribution, and take the service time data from Yang et al. (2012). The service time is an important quality characteristic for a bank branch in Taiwan. To measure the efficiency in the service system of a bank branch, the sampling service times (in minutes) are measured from 10 counters every 2days for 30days; that is, 15 samples of size n=10 are taken from an in-control service system. These data have been analyzed and have a right-skewed distribution, as shown in Table 5-12.

Table 5-12. In-control Service Time Data.

No. In-control service data with n=10

X

1 0.88 0.78 5.06 5.45 2.93 6.11 11.59 1.2 0.89 3.21 3.81 2 3.82 13.4 5.16 3.2 32.27 3.68 3.14 1.58 2.72 7.71 7.67 3 1.4 3.89 10.88 30.85 0.54 8.4 5.1 2.63 9.17 3.94 7.68 4 16.8 8.77 8.36 3.55 7.76 1.81 1.11 5.91 8.26 7.19 6.95 5 0.24 9.57 0.66 1.15 2.34 0.57 8.94 5.54 11.69 6.58 4.73 6 4.21 8.73 11.44 2.89 19.49 1.2 8.01 6.19 7.48 0.07 6.97 7 15.08 7.43 4.31 6.14 10.37 2.33 1.97 1.08 4.27 14.08 6.71 8 13.89 0.3 3.21 11.32 9.9 4.39 10.5 1.7 10.74 1.46 6.74 9 0.03 12.76 2.41 7.41 1.67 3.7 4.31 2.45 3.57 3.33 4.16 10 12.89 17.96 2.78 3.21 1.12 12.61 4.23 6.18 2.33 6.92 7.02 11 7.71 1.05 1.11 0.22 3.53 0.81 0.41 3.73 0.08 2.55 2.12 12 5.81 6.29 3.46 2.66 4.02 10.95 1.59 5.58 0.55 4.1 4.50 13 2.89 1.61 1.3 2.58 18.65 10.77 18.23 3.13 3.38 6.34 6.89 14 1.36 1.92 0.12 11.08 8.85 3.99 4.32 1.71 1.77 1.94 3.71 15 21.52 0.63 8.54 3.37 6.94 3.44 3.37 6.37 1.28 12.83 6.83

766 .

 5 X

Since the 150 in-control data follows exponential distribution, we estimate the parameters of the exponential distribution, and obtain

b

ˆ_I



5.766, Mean^︿



5.766, and Var^︿



33.244. We use the routine “ks.test” to test in-control data with Kolmogorov-Smirnov test method and have a p-value = 0.6714; therefore, we do not reject the data drawn from the exponential distribution with bI=5.766.

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The new data set of service times from a new automatic service system of the bank branch, 10 new samples of size 10, were collected and listed in Table 5-13.

The out-of-control service time data are as follows:

Table 5-13. Out-of -control Service Time Data.

No. Out-of-control service data with n=10

X

1 3.54 0.01 1.33 7.27 5.52 0.09 1.84 1.04 2.91 0.63 2.42 2 0.86 1.61 1.15 0.96 0.54 3.05 4.11 0.63 2.37 0.05 1.53 3 1.45 0.19 4.18 0.18 0.02 0.7 0.8 0.97 3.6 2.94 1.50 4 1.37 0.14 1.54 1.58 0.45 6.01 4.59 1.74 3.92 4.82 2.62 5 3 2.46 0.06 1.8 3.25 2.13 2.22 1.37 2.13 0.25 1.87 6 1.59 3.88 0.39 0.54 1.58 1.7 0.68 1.25 6.83 0.31 1.88 7 5.01 1.85 3.1 1 0.09 1.16 2.69 2.79 1.84 2.62 2.22 8 4.96 0.55 1.43 4.12 4.06 1.42 1.43 0.86 0.67 0.13 1.96 9 1.08 0.65 0.91 0.88 2.02 2.88 1.76 2.87 1.97 0.62 1.56 10 4.56 0.44 5.61 2.79 1.73 2.46 0.53 1.73 7.02 2.13 2.90

045 .

 2 X

With 100 data in Table 5-13, we estimate the parameters of out-of-control data, and obtain

b

ˆ_O



2.045,

ˆ 3 . 72

 



, Mean^︿



2.045, and Var.^︿



4.184. Hence, we have a -0.645 small mean shift scale and a 0.355 small s.d. shift scale. During calculation, the out-of-control mean and variance are smaller than the in-control mean and variance.

We use the routine “ks.test” to test out-of-control data and we have a

p-value=0.4182; therefore, we do not reject the data drawn from the exponential

distribution with bO=2.045.

We let aI =1, n=10, θ=0.01, e=0.05, D=20, T=250, s0=5, s1=0.1, W=500, kc=10,

A=600, IC=0.1, P

C=300, PU=150, and R=200. If the producer decides not to inspect,

we maximize EAP (Equation 3-12) to determine the optimum h*, subject to 0.5≦h≦8. If the producer decides to inspect, we maximize EAP (Equation 4-7) to

determine the optimum h* and ω*, subject to 0.5≦h≦8 and 2≦ω.

‧

Table 5-14. The Optimum Results Comparison of Profit Model with Different λ

Inspection Without With

λ

1 0.4 0.1 1 0.4 0.1

EAP*

-42444.27 -43125.13 -36886.39 -5577.36 -5973.12 -2346.79

P

_W 292.532 292.532 292.532 - - -

ARL

₁ 2.69

2.55

3.9 2.69

2.55

3.9

UCL

12.786 8.972 6.98 12.786 8.972 6.98

LCL

1.778 3.531 4.718 1.778 3.531 4.718

first true alarm on which sample According to Table 5-14, with and without inspection, h*, EWMAX-bar chart, and

ARL

1 are the same at each λ. However, with inspection, we increased the profit per unit time as follows:

(1) If λ=0.1, we increase 93.6% profit per unit time when we have an inspection.

(2) If λ=0.4, we increase 86.1% profit per unit time when we have an inspection.

(3) If λ=1, we increase 86.9% profit per unit time when we have an inspection.

If we use the economic EWMAX-bar chart with λ=0.1, we have the largest EAP*, but largest ARL1. If we use the economic EWMAX-bar chart with λ=0.4, we have the smallest EAP*, but smallest ARL1. To maximize EAP* for small shifts in the mean and variance we suggest that the producer takes inspection with USL*=17.297, use the economic EWMAX-bar chart with λ=0.1 and take 10 samples every 8 unit time.

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To find the detection ability for the three types of EWMAX-bar chart, we plot the in-control and out-of-control statistics on them.

Figure 5-31. The Economic EWMAX-bar Chart (λ=1) with In-control Data For EWMAX-bar chart with λ=1, Figure 5-31 shows that no points are out of limits for in-control samples.

Figure 5-32. The Economic EWMAX-bar Chart (λ=1) with Out-of-control Data Plots the out-of-control statistics on the EWMAX-bar Chart with λ=1, Figure 5-32 shows that No. 2, 3, and 9 are out of limits; the first true alarm is on No. 2.

Figure 5-33. The Economic EWMA_X-bar Chart (λ=0.4) with In-control Data For EWMA_X-bar chart with λ=0.4, Figure 5-33 shows that no points are out of

在文檔中最大利潤下規格上限與EWMA管制圖之設計 - 政大學術集成 (頁 47-73)

Data Description and Determining the Optimum Producer Inspection and

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CHAPTER 5. DETERMINING THE BEST λ OF THE ECONOMIC

EWMA

CONTROL CHART UNDER DIFFERENT

SHIFT SCALES IN THE MEAN AND VARIANCE

5.1 Data Description and Determining the Optimum Producer Inspection and the Design Parameters of the Economic EWMA

Control Chart

X

X

X  4 . 99

aˆ

bˆ of the simulation data.

a ˆ

 24 . 349

b



 a

 b



Var  a ˆ

 b ˆ

 1 . 023

‧

X

a ˆ

 25 . 268

b



ˆ 0 . 919

‧

λ

ARL

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X

975 .

 5 X

a ˆ

 30 . 883

b



ˆ 6 . 534





ˆ 0 . 012

 







‧

λ

ARL

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立政治大學

X  ₄ _. ₉₉

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