Two Numerical Examples and the Results Comparison with the FSI

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Hence, the coefficient of the upper warning control limits (UWL) of the EWMAX-bar control chart is determined.

If EAP(n+1) is greater than EAP(n), then we choose EAP(n+1) to become EAP*.

Step5. With W1, let

ATS

₀

 p

^T_zs

 I  R

_I



^1

h





370

to solve W2 using the routine “uniroot” in the R program. Hence, lower warning control limits (LWL) of the EWMAX-bar control chart is determined.

The economic VSI EWMAX-bar control chart is then constructed.

Step6. Let n=n+1, 3≦n≦25. Proceed Step2.

The procedures to determine n, ω*, warning limits, control limits, and ATS1 of the economic VSI EWMAX-bar control chart with producer inspection is instead Step 4 of the previous procedure as follows:

Step4. With UCL and LCL, from

ATS

₀

 p

^T_zs

 I  R

_I



^1

h





370

, W2 is a function of W1. To determine optimum ω and W1, we use the routine

“DEoptim” of the R program to maximize EAP in Equation 6-7, subject to 2≦ω and 0≦W1≦L1 with specified parameters.

If we let h1=h2=h0, W1=0, and W2=0, then the VSI EWMAX-bar control chart becomes the FSI EWMAX-bar control chart.

6.5 Two Numerical Examples and the Results Comparison with the FSI EWMA

_X-bar

Control Chart

For the first numerical example, we use the procedures in Section 6.4 with the same specified parameters in Section 3.4 and choose λ=0.05, h₀=1, h₁=0.5, and h₂=2 to compare the optimum results of the VSI EWMA_X-bar control chart with the FSI EWMA_X-bar control chart, as follows:

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Table 6-1. Comparison of the optimum results of VSI and FSI EWMAX-bar control charts

Inspection Without With

Chart

FSI VSI FSI VSI

L

₁ 2.604 2.604 2.604 2.604

L

2 2.387 2.387 2.387 2.387

W

1 - 0.002 - 0.002

W

₂ - 0.929 - 0.929

n*

25 25 25 25

ω*

- - 2.311 2.311

USL*

8.66 8.66

Yield

0.965834 0.965834

EAP*

27428.1 27677.6 31998.4

32190 ATS

1 22.929 13.119 22.929

13.119 UCL*

3.2043 3.2043 3.2043 3.2043

UWL*

- 3.0002 - 3.0002

LWL*

- 2.9271 - 2.9271

LCL*

2.8127 2.8127 2.8127 2.8127

According to Table 6-1, n* and ω* are the same as in the model with inspection, but different FSI and VSI EWMAX-bar charts. We have the largest EAP* and the smallest ATS1 when we use the economic VSI EWMAX-bar chart with λ=0.05. With or without inspection, n*, EWMAX-bar chart, and ATS1 are the same. However, for EAP*, we increase the profit per unit time as follows:

(1) If we use the VSI EWMAX-bar control chart, we increase 16.3% profit per unit time when we have an inspection.

(2) If we use the FSI EWMAX-bar control chart, we increase 16.66% profit per unit time when we have an inspection.

(3) If we use the VSI EWMAX-bar chart, we increase 0.6% profit per unit time more than the FSI EWMAX-bar chart when the producer decides to inspect.

(4) If we use the VSI EWMAX-bar chart, we increase 0.91% profit per unit time more than the FSI EWMAX-bar chart when the producer decides not to inspect.

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Therefore, we suggest that the producer takes inspection with USL*=8.66, use the economic VSI EWMAX-bar chart with λ=0.05 and take 25 samples for having better performance. If the last sample point falls within the warning region, then take the next sample after 0.5unit time; if the last sample point falls within the center region, then take the next sample after 2 unit time. We then obtain 32190 profits per unit time.

For the second numerical example, we consider a special case of exponential distribution using the same service time data as that in the end of Section 5.1. We also use the same specified parameters and choose the same λ=0.1 and 0.4.

We compare the optimum results of the VSI EWMAX-bar control chart with the FSI EWMAX-bar control chart at λ=0.4, as follows:

Table 6-2. Comparison of the optimum results of the VSI and FSI EWMAX-bar charts at λ=0.4

Inspection Without With

Chart

FSI VSI FSI VSI

L

₁ 3.5169 3.5169 3.5169 3.5169

L

2 2.4510 2.4510 2.4510 2.4510

W

1 - 0.5403 - 0.5403

W

₂ - 0.2943 - 0.2943

ω*

- - 2 2

USL*

17.297 17.297

Yield

0.950213 0.950213

EAP*

-52723.27 -52546.4 -11560.23 -11457.41

ATS

₁ 2.55 2.778 2.55 2.778

UCL

8.972 8.972 8.972 8.972

UWL

- 6.258 - 6.258

LWL

- 5.497 - 5.497

LCL

3.531 3.531 3.531 3.531

first true alarm on which sample

No.2 (9 outliers)

No.2 (9 outliers)

No.2 (9 outliers)

No.2 (9 outliers) first true alarm on

which time 2 unit time 2.5 unit time 2 unit time 2.5 unit time

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According to Table 6-2, ω* is the same as in the model with inspection, but different FSI and VSI EWMAX-bar charts. We have the largest EAP*, but largest ATS1

when we use the economic VSI EWMAX-bar chart with λ=0.4. With or without inspection, ATS1 is the same. However, for EAP*, we increase the profit per unit time as follows:

(1) If we use the VSI EWMAX-bar control chart with λ=0.4, we increase 78.1%

profit per unit time when we have an inspection.

(2) If we use the FSI EWMAX-bar control chart with λ=0.4, we increase 86.1%

profit per unit time when we have an inspection.

(3) If we use the VSI EWMAX-bar chart, we increase 0.89% profit per unit time more than the FSI EWMAX-bar chart when the producer decides to inspect.

(4) If we use the VSI EWMAX-bar chart, we increase 0.34% profit per unit time more than the FSI EWMAX-bar chart when the producer decides not to inspect.

Therefore, for maximum EAP, we suggest that the producer takes inspection with USL*=17.297, use the economic VSI EWMAX-bar chart with λ=0.4 and take 10 samples for every0.5 or 2 unit time.

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

Figure 6-2. The Economic FSI EWMAX-bar Chart (λ=0.4) with In-control Data For FSI EWMAX-bar chart with λ=0.4, Figure 6-2 shows that no points are out of limits for in-control samples.

Figure 6-3. The Economic FSI EWMAX-bar Chart (λ=0.4) with Out-of-control Data Plots the out-of-control statistics on the FSI EWMAX-bar Chart with λ=0.4, Figure 6-3 shows that No. 2 to No. 10 are out of limits; the first true alarm is on No. 2.

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

Figure 6-4. The Economic VSI EWMAX-bar Chart (λ=0.4) with In-control Data According to the falling region of each in-control plotted point, we determine the sampling time as follows:

Table 6-3. Region and Sampling Time of Each In-control Statistic (λ=0.4)

No. X-bar EWMA Region

h

i

1 3.81 4.98 W.R. 2

2 7.67 6.06 C.R. 0.5

3 7.68 6.71 W.R. 2

4 6.95 6.80 W.R. 0.5

5 4.73 5.97 C.R. 0.5

6 6.97 6.37 W.R. 2

7 6.71 6.51 W.R. 0.5

8 6.74 6.60 W.R. 0.5

9 4.16 5.63 C.R. 0.5

10 7.02 6.18 C.R. 2

11 2.12 4.56 W.R. 2

12 4.50 4.54 W.R. 0.5

13 6.89 5.48 W.R. 0.5

14 3.71 4.77 W.R. 0.5

15 6.83 5.59 C.R. 0.5

For VSI EWMAX-bar chart with λ=0.4, Figure 6-4 and Table 6-3 show that no points are out of limits for in-control samples.

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Figure 6-5. The Economic VSI EWMAX-bar Chart (λ=0.4) with Out-of-control Data According to the falling region of each out-of-control plotted point, we determine the sampling time as follows:

Table 6-4. Region and Sampling Time of Each Out-of-control Statistic (λ=0.4)

No. X-bar EWMA Region

h

i

1 2.42 4.43 W.R. 2

2 1.53 3.27 A.R.* 0.5

3 1.50 2.56 A.R.* 0.5

4 2.62 2.58 A.R.* 2

5 1.87 2.30 A.R.* 2

6 1.88 2.13 A.R.* 0.5

7 2.22 2.16 A.R.* 0.5

8 1.96 2.08 A.R.* 0.5

9 1.56 1.88 A.R.* 2

10 2.90 2.29 A.R.* 2

If the plotted point falls inside the action region, then we randomly choose the sampling time.

Plots the out-of-control statistics on the EWMAX-bar Chart with λ=0.4, Figure 6-5 and Table 6-4 show that No. 2 to No. 10 out of limits; the first true alarm is on No. 2.

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We compare the optimum results of the VSI EWMAX-bar control chart with the FSI EWMAX-bar control chart at λ=0.1, as follows:

Table 6-5. Comparison of the optimum results of the VSI and FSI EWMAX-bar charts at λ=0.1

Inspection Without With

Chart

FSI VSI FSI VSI

L

₁ 2.9015 2.9015 2.9015 2.9015

L

2 2.5052 2.5052 2.5052 2.5052

W

1 - 0.2088 - 0.2088

W

₂ - 0.6048 - 0.6048

ω*

- - 2 2

USL*

17.297 17.297

Yield

0.950213 0.950213

EAP*

-51692.3 -51823.37 -10960.91 -11037.11

ATS

₁ 3.9 3.73 3.9 3.73

UCL

6.98 6.98 6.98 6.98

UWL

- 5.853 - 5.853

LWL

- 5.513 - 5.513

LCL

4.718 4.718 4.718 4.718

first true alarm on which sample

No.3 (8 outliers)

No.3 (8 outliers)

No.3 (8 outliers)

No.3 (8 outliers) first true alarm on

which time 3 unit time 3 unit time 3 unit time 3 unit time According to Table 6-5, ω* is the same in the model with inspection, but different FSI and VSI EWMAX-bar charts. We have the largest EAP*, but largest ATS1

when we use the economic FSI EWMAX-bar chart with λ=0.1. With or without inspection, ATS1 is the same. However, for EAP*, we increase the profit per unit time as follows:

(1) If we use the VSI EWMA_X-bar control chart with λ=0.1, we increase 78.8%

profit per unit time when we have an inspection.

(2) If we use the FSI EWMA_X-bar control chart with λ=0.1, we increase 93.6%

profit per unit time when we have an inspection.

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(3) If we use the FSI EWMAX-bar chart, we increase 0.7% profit per unit time more than the VSI EWMAX-bar chart when the producer decides to inspect.

(4) If we use the FSI EWMAX-bar chart, we increase 0.25% profit per unit time more than the VSI EWMAX-bar chart when the producer decides not to inspect.

Therefore, for maximum EAP, we suggest that the producer takes inspection with USL*=17.297, use the economic FSI EWMAX-bar chart with λ=0.1 and take 10 samples every0.5 or 2 unit time.

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

Figure 6-6. The Economic FSI EWMAX-bar Chart (λ=0.1) with In-control Data For FSI EWMAX-bar chart with λ=0.1, Figure 6-6 shows that no points are out of limits for in-control samples.

Figure 6-7. The Economic FSI EWMA_X-bar Chart (λ=0.1) with Out-of-control Data Plots the out-of-control statistics on the FSI EWMA_X-bar Chart with λ=0.1, Figure 6-7 shows that No. 3 to No. 10 are out of limits; the first true alarm is on No. 3.

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

Figure 6-8. The Economic VSI EWMAX-bar Chart (λ=0.1) with In-control Data According to the falling region of each in-control plotted point, we determine the sampling time as follows:

Table 6-6. Region and Sampling Time of Each In-control Statistic (λ=0.1)

No. X-bar EWMA Region

h

i

1 3.81 5.57 C.R. 2

2 7.67 5.78 C.R. 2

3 7.68 5.97 W.R. 2

4 6.95 6.07 W.R. 0.5

5 4.73 5.93 W.R. 0.5

6 6.97 6.04 W.R. 0.5

7 6.71 6.10 W.R. 0.5

8 6.74 6.17 W.R. 0.5

9 4.16 5.97 W.R. 0.5

10 7.02 6.07 W.R. 0.5

11 2.12 5.68 C.R. 0.5

12 4.50 5.56 C.R. 2

13 6.89 5.69 C.R. 2

14 3.71 5.49 W.R. 2

15 6.83 5.63 C.R. 0.5

For VSI EWMAX-bar chart with λ=0.1, Figure 6-8 and Table 6-6 show that no points are out of limits for in-control samples.

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Figure 6-9. The Economic VSI EWMAX-bar Chart (λ=0.1) with Out-of-control Data According to the falling region of each out-of-control plotted point, we determine the sampling time as follows:

Table 6-7. Region and Sampling Time of Each Out-of-control Statistic (λ=0.1)

No. X-bar EWMA Region

h

i

1 2.42 5.43 W.R. 2

2 1.53 5.04 W.R. 0.5

3 1.50 4.69 A.R.* 0.5

4 2.62 4.48 A.R.* 2

5 1.87 4.22 A.R.* 2

6 1.88 3.98 A.R.* 2

7 2.22 3.81 A.R.* 2

8 1.96 3.62 A.R.* 0.5

9 1.56 3.42 A.R.* 0.5

10 2.90 3.37 A.R.* 0.5

If the plotted point falls inside the action region, then we randomly choose the sampling time.

Plots the out-of-control statistics on the EWMAX-bar Chart with λ=0.1, Figure 6-9 and Table 6-7 show that No. 3 to No. 10 out of limits; the first true alarm is on No. 3.

In the second numerical example, the EAP* of FSI EWMA_X-bar chart with λ=0.1 in Table 6-5 is larger than that of VSI EWMA_X-bar chart with λ=0.4 in Table 6-2. The FSI EWMA_X-bar chart is slightly better than the VSI EWMA_X-bar chart when we fix

h

₀=1, h₁=0.5, and h₂=2.

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6.6 Sensitivity Analysis and the Optimum Results Comparison between the

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 79-90)