(1) ARL 1 Analysis of FP EWMA WL Chart and ARL 1 Comparison between FP EWMA Chart and FP WL Chart

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5.3 Data Analysis and ATS

Comparison among the FP EWMA, VSI EWMA and FP WL Charts

(1) ARL

Analysis of FP EWMA WL Chart and ARL

Comparison between FP EWMA Chart and FP WL Chart

Under



₀ 

0 

₀² 

1

, n₀ 

5

, h₀ 

1 

₀ 

0 . 0027

(or ARL₀= 370.37) and

6 .



0

a . Consider the levels of parameters,



₁ = (0.5, 1.2, 1.5),



₂ = (1.1, 1.5, 2),



3 = (0, 1, 2) and h₂= (0.1, 0.5, 0.9). The 27 combinations are arranged by L₂₇(3¹³).

Similar to Section 5.1, the values of L₁and L₂ under the combinations of λ= 0.05, 0.1, 0.2, 0.25, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and



₃= 0, 1, 2 are listed in Table 35.

Table 35. The Values of L1 and L2 under Various Values of λ and



₃

λ 0.05 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9

3 

0 

^L¹ ^2.74 ^3.15 ^3.41 ^3.61 ^3.78 ^3.92 ^4.16 ^4.35 ^4.50 ^4.62 ^4.71 ^4.76

L2 2.27 2.29 2.26 2.21 2.16 2.10 1.99 1.89 1.79 1.70 1.61 1.53

3 

1 

^L¹ ^2.68 ^3.05 ^3.28 ^3.45 ^3.59 ^3.71 ^3.90 ^4.06 ^4.18 ^4.28 ^4.35 ^4.39

L2 2.32 2.38 2.36 2.33 2.29 2.25 2.16 2.07 1.98 1.90 1.82 1.75

3 

2 

^L¹ ^2.63 ^2.95 ^3.14 ^3.27 ^3.38 ^3.47 ^3.61 ^3.73 ^3.82 ^3.89 ^3.95 ^3.98

L2 2.38 2.48 2.49 2.48 2.46 2.44 2.38 2.31 2.25 2.19 2.13 2.08

The ARL1s of the 27 combinations of



₁,



₂ and



₃ under λ = 0.05 are listed in Table 36.

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Table 36. ARL1 for FP EWMA WL and ATS1 Saved % Compared to FP WL Chart under λ = 0.05

No. ₁ ₂ ₃ L1 L2 UCL LCL EWMA

ARL1

FP ARL1

ARL1

saved % 1 0.5 1.1 0 2.74 2.27 0.87 0.52 25.50 109.54 76.72%

2 0.5 1.1 1 2.68 2.32 1.32 0.87 10.23 38.67 73.55%

3 0.5 1.1 2 2.63 2.38 2.63 1.96 8.92 29.36 69.61%

4 0.5 1.5 0 2.74 2.27 0.87 0.52 5.47 7.79 29.74%

5 0.5 1.5 1 2.68 2.32 1.32 0.87 4.81 5.90 18.43%

6 0.5 1.5 2 2.63 2.38 2.63 1.96 5.30 6.20 14.60%

7 0.5 2 0 2.74 2.27 0.87 0.52 2.71 2.36 -15.00%

8 0.5 2 1 2.68 2.32 1.32 0.87 2.81 2.31 -21.77%

9 0.5 2 2 2.63 2.38 2.63 1.96 3.36 2.66 -26.09%

10 1.2 1.1 0 2.74 2.27 0.87 0.52 7.02 21.67 67.59%

11 1.2 1.1 1 2.68 2.32 1.32 0.87 3.75 5.10 26.44%

12 1.2 1.1 2 2.63 2.38 2.63 1.96 3.44 3.60 4.34%

13 1.2 1.5 0 2.74 2.27 0.87 0.52 3.69 4.14 10.93%

14 1.2 1.5 1 2.68 2.32 1.32 0.87 2.83 2.42 -16.83%

15 1.2 1.5 2 2.63 2.38 2.63 1.96 2.86 2.21 -29.40%

16 1.2 2 0 2.74 2.27 0.87 0.52 2.30 1.88 -21.94%

17 1.2 2 1 2.68 2.32 1.32 0.87 2.11 1.59 -32.64%

18 1.2 2 2 2.63 2.38 2.63 1.96 2.30 1.63 -41.35%

19 1.5 1.1 0 2.74 2.27 0.87 0.52 4.79 9.86 51.49%

20 1.5 1.1 1 2.68 2.32 1.32 0.87 2.88 2.70 -6.44%

21 1.5 1.1 2 2.63 2.38 2.63 1.96 2.71 2.04 -32.89%

22 1.5 1.5 0 2.74 2.27 0.87 0.52 3.06 3.01 -1.56%

23 1.5 1.5 1 2.68 2.32 1.32 0.87 2.36 1.79 -31.27%

24 1.5 1.5 2 2.63 2.38 2.63 1.96 2.38 1.64 -45.17%

25 1.5 2 0 2.74 2.27 0.87 0.52 2.09 1.67 -25.16%

26 1.5 2 1 2.68 2.32 1.32 0.87 1.88 1.39 -35.25%

27 1.5 2 2 2.63 2.38 2.63 1.96 2.01 1.39 -44.75%

Note: ARL1 saved %:



FP ARL- 1FP EWMA ARL. - 1



FP ARL- 1%

To investigate the effect of



₁,



₂ and



₃ on the ARL₁ and ARL₁ saved %, the response table and response diagram show the results of the sensitivity analysis (Table

‧

From Table 36, Table 37, Figure 37 and Figure 38, we found a. ARL1 saved % from -45.17% to 76.72%.

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Table 38. ARL1 for FP EWMA WL and ARL1 Saved % Compared to FP WL Chart under λ=0.1

No. ₁ ₂ ₃ L1 L2 UCL LCL EWMA

ARL1

FP ARL1

ARL1

saved % 1 0.5 1.1 0 3.15 2.29 1.00 0.45 28.42 109.54 74.06%

2 0.5 1.1 1 3.05 2.38 1.48 0.77 9.73 38.67 74.84%

3 0.5 1.1 2 2.95 2.48 2.85 1.80 8.20 29.36 72.06%

4 0.5 1.5 0 3.15 2.29 1.00 0.45 4.93 7.79 36.63%

5 0.5 1.5 1 3.05 2.38 1.48 0.77 4.26 5.90 27.80%

6 0.5 1.5 2 2.95 2.48 2.85 1.80 4.64 6.20 25.19%

7 0.5 2 0 3.15 2.29 1.00 0.45 2.42 2.36 -2.42%

8 0.5 2 1 3.05 2.38 1.48 0.77 2.48 2.31 -7.23%

9 0.5 2 2 2.95 2.48 2.85 1.80 2.92 2.66 -9.53%

10 1.2 1.1 0 3.15 2.29 1.00 0.45 6.47 21.67 70.13%

11 1.2 1.1 1 3.05 2.38 1.48 0.77 3.27 5.10 35.85%

12 1.2 1.1 2 2.95 2.48 2.85 1.80 2.95 3.60 17.96%

13 1.2 1.5 0 3.15 2.29 1.00 0.45 3.27 4.14 20.99%

14 1.2 1.5 1 3.05 2.38 1.48 0.77 2.46 2.42 -1.90%

15 1.2 1.5 2 2.95 2.48 2.85 1.80 2.46 2.21 -11.35%

16 1.2 2 0 3.15 2.29 1.00 0.45 2.05 1.88 -8.66%

17 1.2 2 1 3.05 2.38 1.48 0.77 1.87 1.59 -17.07%

18 1.2 2 2 2.95 2.48 2.85 1.80 2.00 1.63 -22.83%

19 1.5 1.1 0 3.15 2.29 1.00 0.45 4.28 9.86 56.66%

20 1.5 1.1 1 3.05 2.38 1.48 0.77 2.49 2.70 7.81%

21 1.5 1.1 2 2.95 2.48 2.85 1.80 2.31 2.04 -13.55%

22 1.5 1.5 0 3.15 2.29 1.00 0.45 2.70 3.01 10.32%

23 1.5 1.5 1 3.05 2.38 1.48 0.77 2.05 1.79 -14.44%

24 1.5 1.5 2 2.95 2.48 2.85 1.80 2.04 1.64 -24.82%

25 1.5 2 0 3.15 2.29 1.00 0.45 1.87 1.67 -11.74%

26 1.5 2 1 3.05 2.38 1.48 0.77 1.67 1.39 -19.67%

27 1.5 2 2 2.95 2.48 2.85 1.80 1.75 1.39 -26.04%

Note: ARL1 saved %:



FP ARL- 1FP EWMA ARL. - 1



FP ARL- 1%

The response table and response diagram show the results of the sensitivity analysis (Table 39, Figure 39 and Figure 40).

‧

From Table 38, Table 39, Figure 39 and Figure 40, we found a. ARL1 saved % from -26.04% to 74.84%.

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Table 40. ARL1 for FP EWMA WL and ARL1 Saved % Compared to FP WL Chart under λ=0.15

No. ₁ ₂ ₃ L1 L2 UCL LCL EWMA

ARL1

FP ARL1

ARL1

saved % 1 0.5 1.1 0 3.41 2.26 1.11 0.40 32.85 109.54 70.02%

2 0.5 1.1 1 3.28 2.36 1.61 0.70 9.97 38.67 74.21%

3 0.5 1.1 2 3.14 2.49 3.03 1.68 8.15 29.36 72.24%

4 0.5 1.5 0 3.41 2.26 1.11 0.40 4.75 7.79 38.99%

5 0.5 1.5 1 3.28 2.36 1.61 0.70 4.04 5.90 31.60%

6 0.5 1.5 2 3.14 2.49 3.03 1.68 4.37 6.20 29.63%

7 0.5 2 0 3.41 2.26 1.11 0.40 2.28 2.36 3.31%

8 0.5 2 1 3.28 2.36 1.61 0.70 2.32 2.31 -0.48%

9 0.5 2 2 3.14 2.49 3.03 1.68 2.71 2.66 -1.76%

10 1.2 1.1 0 3.41 2.26 1.11 0.40 6.42 21.67 70.38%

11 1.2 1.1 1 3.28 2.36 1.61 0.70 3.06 5.10 39.96%

12 1.2 1.1 2 3.14 2.49 3.03 1.68 2.72 3.60 24.35%

13 1.2 1.5 0 3.41 2.26 1.11 0.40 3.10 4.14 25.24%

14 1.2 1.5 1 3.28 2.36 1.61 0.70 2.30 2.42 5.09%

15 1.2 1.5 2 3.14 2.49 3.03 1.68 2.27 2.21 -2.62%

16 1.2 2 0 3.41 2.26 1.11 0.40 1.93 1.88 -2.50%

17 1.2 2 1 3.28 2.36 1.61 0.70 1.75 1.59 -9.60%

18 1.2 2 2 3.14 2.49 3.03 1.68 1.85 1.63 -13.85%

19 1.5 1.1 0 3.41 2.26 1.11 0.40 4.10 9.86 58.46%

20 1.5 1.1 1 3.28 2.36 1.61 0.70 2.31 2.70 14.54%

21 1.5 1.1 2 3.14 2.49 3.03 1.68 2.12 2.04 -4.07%

22 1.5 1.5 0 3.41 2.26 1.11 0.40 2.54 3.01 15.63%

23 1.5 1.5 1 3.28 2.36 1.61 0.70 1.91 1.79 -6.30%

24 1.5 1.5 2 3.14 2.49 3.03 1.68 1.88 1.64 -14.85%

25 1.5 2 0 3.41 2.26 1.11 0.40 1.76 1.67 -5.45%

26 1.5 2 1 3.28 2.36 1.61 0.70 1.56 1.39 -12.28%

27 1.5 2 2 3.14 2.49 3.03 1.68 1.63 1.39 -16.91%

Note: ARL1 saved %:



FP ARL- 1FP EWMA ARL. - 1



FP ARL- 1%

The response table and response diagram show the results of the sensitivity analysis (Table 41, Figure 41 and Figure 42).

‧

From Table 40, Table 41, Figure 41 and Figure 42, we found a. ARL1 saved % from -16.91% to 74.21%.

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Table 42. ARL1 for FP EWMA WL and ARL1 Saved % Compared to FP WL Chart under λ=0.2

No. ₁ ₂ ₃ L1 L2 UCL LCL EWMA

ARL1

FP ARL1

ARL1

saved % 1 0.5 1.1 0 3.61 2.21 1.21 0.36 37.94 109.54 65.37%

2 0.5 1.1 1 3.45 2.33 1.73 0.64 10.58 38.67 72.65%

3 0.5 1.1 2 3.27 2.48 3.20 1.59 8.39 29.36 71.41%

4 0.5 1.5 0 3.61 2.21 1.21 0.36 4.70 7.79 39.58%

5 0.5 1.5 1 3.45 2.33 1.73 0.64 3.94 5.90 33.26%

6 0.5 1.5 2 3.27 2.48 3.20 1.59 4.24 6.20 31.74%

7 0.5 2 0 3.61 2.21 1.21 0.36 2.21 2.36 6.48%

8 0.5 2 1 3.45 2.33 1.73 0.64 2.23 2.31 3.42%

9 0.5 2 2 3.27 2.48 3.20 1.59 2.59 2.66 2.78%

10 1.2 1.1 0 3.61 2.21 1.21 0.36 6.59 21.67 69.60%

11 1.2 1.1 1 3.45 2.33 1.73 0.64 2.96 5.10 42.06%

12 1.2 1.1 2 3.27 2.48 3.20 1.59 2.59 3.60 27.97%

13 1.2 1.5 0 3.61 2.21 1.21 0.36 3.01 4.14 27.29%

14 1.2 1.5 1 3.45 2.33 1.73 0.64 2.20 2.42 9.10%

15 1.2 1.5 2 3.27 2.48 3.20 1.59 2.15 2.21 2.58%

16 1.2 2 0 3.61 2.21 1.21 0.36 1.86 1.88 1.01%

17 1.2 2 1 3.45 2.33 1.73 0.64 1.68 1.59 -5.21%

18 1.2 2 2 3.27 2.48 3.20 1.59 1.76 1.63 -8.37%

19 1.5 1.1 0 3.61 2.21 1.21 0.36 4.05 9.86 58.90%

20 1.5 1.1 1 3.45 2.33 1.73 0.64 2.20 2.70 18.46%

21 1.5 1.1 2 3.27 2.48 3.20 1.59 2.00 2.04 1.72%

22 1.5 1.5 0 3.61 2.21 1.21 0.36 2.46 3.01 18.45%

23 1.5 1.5 1 3.45 2.33 1.73 0.64 1.82 1.79 -1.45%

24 1.5 1.5 2 3.27 2.48 3.20 1.59 1.78 1.64 -8.68%

25 1.5 2 0 3.61 2.21 1.21 0.36 1.70 1.67 -1.74%

26 1.5 2 1 3.45 2.33 1.73 0.64 1.50 1.39 -7.82%

27 1.5 2 2 3.27 2.48 3.20 1.59 1.55 1.39 -11.37%

Note: ARL1 saved %:



FP ARL- 1FP EWMA ARL. - 1



FP ARL- 1%

The response table and response diagram show the results of the sensitivity analysis (Table 43, Figure 43 and Figure 44).

‧

From Table 42, Table 43, Figure 43 and Figure 44, we found a. ARL1 saved % from -11.37% to 72.65%.

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Table 44. ARL1 for FP EWMA WL and ARL1 Saved % Compared to FP WL Chart under λ=0.25

No. ₁ ₂ ₃ L1 L2 UCL LCL EWMA

ARL1

FP ARL1

ARL1

saved % 1 0.5 1.1 0 3.78 2.16 1.31 0.32 43.38 109.54 60.40%

2 0.5 1.1 1 3.59 2.29 1.85 0.59 11.43 38.67 70.44%

3 0.5 1.1 2 3.38 2.46 3.35 1.50 8.83 29.36 69.93%

4 0.5 1.5 0 3.78 2.16 1.31 0.32 4.73 7.79 39.20%

5 0.5 1.5 1 3.59 2.29 1.85 0.59 3.91 5.90 33.79%

6 0.5 1.5 2 3.38 2.46 3.35 1.50 4.18 6.20 32.62%

7 0.5 2 0 3.78 2.16 1.31 0.32 2.16 2.36 8.39%

8 0.5 2 1 3.59 2.29 1.85 0.59 2.18 2.31 5.84%

9 0.5 2 2 3.38 2.46 3.35 1.50 2.51 2.66 5.63%

10 1.2 1.1 0 3.78 2.16 1.31 0.32 6.90 21.67 68.14%

11 1.2 1.1 1 3.59 2.29 1.85 0.59 2.91 5.10 43.04%

12 1.2 1.1 2 3.38 2.46 3.35 1.50 2.51 3.60 30.19%

13 1.2 1.5 0 3.78 2.16 1.31 0.32 2.98 4.14 28.21%

14 1.2 1.5 1 3.59 2.29 1.85 0.59 2.14 2.42 11.62%

15 1.2 1.5 2 3.38 2.46 3.35 1.50 2.08 2.21 5.97%

16 1.2 2 0 3.78 2.16 1.31 0.32 1.82 1.88 3.24%

17 1.2 2 1 3.59 2.29 1.85 0.59 1.63 1.59 -2.32%

18 1.2 2 2 3.38 2.46 3.35 1.50 1.70 1.63 -4.74%

19 1.5 1.1 0 3.78 2.16 1.31 0.32 4.09 9.86 58.53%

20 1.5 1.1 1 3.59 2.29 1.85 0.59 2.14 2.70 20.90%

21 1.5 1.1 2 3.38 2.46 3.35 1.50 1.92 2.04 5.65%

22 1.5 1.5 0 3.78 2.16 1.31 0.32 2.41 3.01 20.01%

23 1.5 1.5 1 3.59 2.29 1.85 0.59 1.76 1.79 1.73%

24 1.5 1.5 2 3.38 2.46 3.35 1.50 1.71 1.64 -4.52%

25 1.5 2 0 3.78 2.16 1.31 0.32 1.66 1.67 0.60%

26 1.5 2 1 3.59 2.29 1.85 0.59 1.46 1.39 -4.88%

27 1.5 2 2 3.38 2.46 3.35 1.50 1.50 1.39 -7.70%

Note: ARL1 saved %:



FP ARL- 1FP EWMA ARL. - 1



FP ARL- 1%

The response table and response diagram show the results of the sensitivity analysis (Table 45, Figure 45 and Figure 46).

‧

From Table 44, Table 45, Figure 45 and Figure 46, we found a. ARL1 saved % from -7.70% to 70.44%.

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Table 46. ARL1 for FP EWMA WL and ARL1 saved % Compared to FP WL Chart under λ=0.3

No. ₁ ₂ ₃ L1 L2 UCL LCL EWMA

ARL1

FP ARL1

ARL1

saved % 1 0.5 1.1 0 3.92 2.10 1.40 0.29 48.99 109.54 55.27%

2 0.5 1.1 1 3.71 2.25 1.96 0.55 12.49 38.67 67.70%

3 0.5 1.1 2 3.47 2.44 3.50 1.42 9.42 29.36 67.91%

4 0.5 1.5 0 3.92 2.10 1.40 0.29 4.81 7.79 38.20%

5 0.5 1.5 1 3.71 2.25 1.96 0.55 3.92 5.90 33.58%

6 0.5 1.5 2 3.47 2.44 3.50 1.42 4.18 6.20 32.70%

7 0.5 2 0 3.92 2.10 1.40 0.29 2.14 2.36 9.49%

8 0.5 2 1 3.71 2.25 1.96 0.55 2.14 2.31 7.36%

9 0.5 2 2 3.47 2.44 3.50 1.42 2.46 2.66 7.51%

10 1.2 1.1 0 3.92 2.10 1.40 0.29 7.34 21.67 66.11%

11 1.2 1.1 1 3.71 2.25 1.96 0.55 2.89 5.10 43.30%

12 1.2 1.1 2 3.47 2.44 3.50 1.42 2.46 3.60 31.50%

13 1.2 1.5 0 3.92 2.10 1.40 0.29 2.97 4.14 28.35%

14 1.2 1.5 1 3.71 2.25 1.96 0.55 2.10 2.42 13.19%

15 1.2 1.5 2 3.47 2.44 3.50 1.42 2.03 2.21 8.28%

16 1.2 2 0 3.92 2.10 1.40 0.29 1.79 1.88 4.68%

17 1.2 2 1 3.71 2.25 1.96 0.55 1.60 1.59 -0.31%

18 1.2 2 2 3.47 2.44 3.50 1.42 1.66 1.63 -2.22%

19 1.5 1.1 0 3.92 2.10 1.40 0.29 4.19 9.86 57.57%

20 1.5 1.1 1 3.71 2.25 1.96 0.55 2.10 2.70 22.42%

21 1.5 1.1 2 3.47 2.44 3.50 1.42 1.87 2.04 8.39%

22 1.5 1.5 0 3.92 2.10 1.40 0.29 2.39 3.01 20.77%

23 1.5 1.5 1 3.71 2.25 1.96 0.55 1.72 1.79 3.90%

24 1.5 1.5 2 3.47 2.44 3.50 1.42 1.66 1.64 -1.53%

25 1.5 2 0 3.92 2.10 1.40 0.29 1.63 1.67 2.16%

26 1.5 2 1 3.71 2.25 1.96 0.55 1.43 1.39 -2.80%

27 1.5 2 2 3.47 2.44 3.50 1.42 1.46 1.39 -5.11%

Note: ARL1 saved %:



FP ARL- 1FP EWMA ARL. - 1



FP ARL- 1%

The response table and response diagram show the results of the sensitivity analysis (Table 47, Figure 47 and Figure 48).

‧

From Table 46, Table 47, Figure 47 and Figure 48, we found a. ARL1 saved % from -5.11% to 67.91%.

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Table 48. ARL1 for FP EWMA WL and ARL1 Saved % Compared to FP WL Chart under λ=0.4

No. ₁ ₂ ₃ L1 L2 UCL LCL EWMA

ARL1

FP ARL1

ARL1

saved % 1 0.5 1.1 0 4.16 1.99 1.59 0.24 60.32 109.54 44.93%

2 0.5 1.1 1 3.90 2.16 2.19 0.47 15.14 38.67 60.85%

3 0.5 1.1 2 3.61 2.38 3.80 1.28 11.02 29.36 62.47%

4 0.5 1.5 0 4.16 1.99 1.59 0.24 5.07 7.79 34.93%

5 0.5 1.5 1 3.90 2.16 2.19 0.47 4.03 5.90 31.72%

6 0.5 1.5 2 3.61 2.38 3.80 1.28 4.26 6.20 31.33%

7 0.5 2 0 4.16 1.99 1.59 0.24 2.12 2.36 10.34%

8 0.5 2 1 3.90 2.16 2.19 0.47 2.11 2.31 8.83%

9 0.5 2 2 3.61 2.38 3.80 1.28 2.41 2.66 9.38%

10 1.2 1.1 0 4.16 1.99 1.59 0.24 8.55 21.67 60.57%

11 1.2 1.1 1 3.90 2.16 2.19 0.47 2.95 5.10 42.26%

12 1.2 1.1 2 3.61 2.38 3.80 1.28 2.44 3.60 32.28%

13 1.2 1.5 0 4.16 1.99 1.59 0.24 3.02 4.14 27.22%

14 1.2 1.5 1 3.90 2.16 2.19 0.47 2.07 2.42 14.60%

15 1.2 1.5 2 3.61 2.38 3.80 1.28 1.97 2.21 10.85%

16 1.2 2 0 4.16 1.99 1.59 0.24 1.77 1.88 6.16%

17 1.2 2 1 3.90 2.16 2.19 0.47 1.56 1.59 2.07%

18 1.2 2 2 3.61 2.38 3.80 1.28 1.61 1.63 0.98%

19 1.5 1.1 0 4.16 1.99 1.59 0.24 4.52 9.86 54.20%

20 1.5 1.1 1 3.90 2.16 2.19 0.47 2.06 2.70 23.68%

21 1.5 1.1 2 3.61 2.38 3.80 1.28 1.80 2.04 11.78%

22 1.5 1.5 0 4.16 1.99 1.59 0.24 2.39 3.01 20.80%

23 1.5 1.5 1 3.90 2.16 2.19 0.47 1.68 1.79 6.41%

24 1.5 1.5 2 3.61 2.38 3.80 1.28 1.60 1.64 2.20%

25 1.5 2 0 4.16 1.99 1.59 0.24 1.60 1.67 3.89%

26 1.5 2 1 3.90 2.16 2.19 0.47 1.40 1.39 -0.29%

27 1.5 2 2 3.61 2.38 3.80 1.28 1.41 1.39 -1.73%

Note: ARL1 saved %:



FP ARL- 1FP EWMA ARL. - 1



FP ARL- 1%

The response table and response diagram show the results of the sensitivity analysis (Table 49, Figure 49 and Figure 50).

‧

From Table 48, Table 49, Figure 49 and Figure 50, we found a. ARL1 saved % from -1.73% to 62.47%.

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Table 50. ARL1 for FP EWMA WL and ARL1 Saved % Compared to FP WL Chart under λ=0.5

No. ₁ ₂ ₃ L1 L2 UCL LCL EWMA

ARL1

FP ARL1

ARL1

saved % 1 0.5 1.1 0 4.35 1.89 1.78 0.20 71.30 109.54 34.91%

2 0.5 1.1 1 4.06 2.07 2.41 0.40 18.39 38.67 52.45%

3 0.5 1.1 2 3.73 2.31 4.09 1.16 13.12 29.36 55.31%

4 0.5 1.5 0 4.35 1.89 1.78 0.20 5.42 7.79 30.44%

5 0.5 1.5 1 4.06 2.07 2.41 0.40 4.22 5.90 28.45%

6 0.5 1.5 2 3.73 2.31 4.09 1.16 4.44 6.20 28.47%

7 0.5 2 0 4.35 1.89 1.78 0.20 2.12 2.36 10.00%

8 0.5 2 1 4.06 2.07 2.41 0.40 2.10 2.31 9.00%

9 0.5 2 2 3.73 2.31 4.09 1.16 2.40 2.66 9.76%

10 1.2 1.1 0 4.35 1.89 1.78 0.20 10.14 21.67 53.22%

11 1.2 1.1 1 4.06 2.07 2.41 0.40 3.09 5.10 39.49%

12 1.2 1.1 2 3.73 2.31 4.09 1.16 2.47 3.60 31.28%

13 1.2 1.5 0 4.35 1.89 1.78 0.20 3.12 4.14 24.71%

14 1.2 1.5 1 4.06 2.07 2.41 0.40 2.07 2.42 14.47%

15 1.2 1.5 2 3.73 2.31 4.09 1.16 1.95 2.21 11.67%

16 1.2 2 0 4.35 1.89 1.78 0.20 1.76 1.88 6.43%

17 1.2 2 1 4.06 2.07 2.41 0.40 1.54 1.59 3.20%

18 1.2 2 2 3.73 2.31 4.09 1.16 1.58 1.63 2.58%

19 1.5 1.1 0 4.35 1.89 1.78 0.20 5.03 9.86 49.04%

20 1.5 1.1 1 4.06 2.07 2.41 0.40 2.07 2.70 23.27%

21 1.5 1.1 2 3.73 2.31 4.09 1.16 1.77 2.04 13.25%

22 1.5 1.5 0 4.35 1.89 1.78 0.20 2.43 3.01 19.44%

23 1.5 1.5 1 4.06 2.07 2.41 0.40 1.66 1.79 7.41%

24 1.5 1.5 2 3.73 2.31 4.09 1.16 1.57 1.64 4.16%

25 1.5 2 0 4.35 1.89 1.78 0.20 1.59 1.67 4.49%

26 1.5 2 1 4.06 2.07 2.41 0.40 1.38 1.39 1.08%

27 1.5 2 2 3.73 2.31 4.09 1.16 1.39 1.39 0.14%

Note: ARL1 saved %:



FP ARL- 1FP EWMA ARL. - 1



FP ARL- 1%

The response table and response diagram show the results of the sensitivity analysis (Table 51, Figure 51 and Figure 52).

‧

From Table 50, Table 51, Figure 51 and Figure 52, we found a. ARL1 saved % from 0.14% to 55.31%.

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Table 52. ARL₁ for FP EWMA WL and ARL₁ Saved % Compared to FP WL Chart under λ=0.6

No. ₁ ₂ ₃ L1 L2 UCL LCL EWMA

ARL1

FP ARL1

ARL1

saved % 1 0.5 1.1 0 4.50 1.79 1.98 0.17 81.53 109.54 25.57%

2 0.5 1.1 1 4.18 1.98 2.63 0.35 22.11 38.67 42.83%

3 0.5 1.1 2 3.82 2.25 4.38 1.05 15.70 29.36 46.54%

4 0.5 1.5 0 4.50 1.79 1.98 0.17 5.83 7.79 25.10%

5 0.5 1.5 1 4.18 1.98 2.63 0.35 4.48 5.90 24.13%

6 0.5 1.5 2 3.82 2.25 4.38 1.05 4.69 6.20 24.45%

7 0.5 2 0 4.50 1.79 1.98 0.17 2.15 2.36 8.90%

8 0.5 2 1 4.18 1.98 2.63 0.35 2.12 2.31 8.27%

9 0.5 2 2 3.82 2.25 4.38 1.05 2.42 2.66 9.08%

10 1.2 1.1 0 4.50 1.79 1.98 0.17 12.07 21.67 44.28%

11 1.2 1.1 1 4.18 1.98 2.63 0.35 3.31 5.10 35.08%

12 1.2 1.1 2 3.82 2.25 4.38 1.05 2.57 3.60 28.69%

13 1.2 1.5 0 4.50 1.79 1.98 0.17 3.27 4.14 21.16%

14 1.2 1.5 1 4.18 1.98 2.63 0.35 2.10 2.42 13.23%

15 1.2 1.5 2 3.82 2.25 4.38 1.05 1.96 2.21 11.26%

16 1.2 2 0 4.50 1.79 1.98 0.17 1.77 1.88 6.00%

17 1.2 2 1 4.18 1.98 2.63 0.35 1.54 1.59 3.52%

18 1.2 2 2 3.82 2.25 4.38 1.05 1.57 1.63 3.26%

19 1.5 1.1 0 4.50 1.79 1.98 0.17 5.71 9.86 42.14%

20 1.5 1.1 1 4.18 1.98 2.63 0.35 2.12 2.70 21.46%

21 1.5 1.1 2 3.82 2.25 4.38 1.05 1.77 2.04 13.30%

22 1.5 1.5 0 4.50 1.79 1.98 0.17 2.50 3.01 17.02%

23 1.5 1.5 1 4.18 1.98 2.63 0.35 1.66 1.79 7.36%

24 1.5 1.5 2 3.82 2.25 4.38 1.05 1.56 1.64 4.95%

25 1.5 2 0 4.50 1.79 1.98 0.17 1.60 1.67 4.43%

26 1.5 2 1 4.18 1.98 2.63 0.35 1.37 1.39 1.65%

27 1.5 2 2 3.82 2.25 4.38 1.05 1.37 1.39 1.15%

Note: ARL1 saved %:



FP ARL- 1FP EWMA ARL. - 1



FP ARL- 1%

The response table and response diagram show the results of the sensitivity analysis (Table 53, Figure 53 and Figure 54).

‧

From Table 52, Table 53, Figure 53 and Figure 54, we found a. ARL1 saved % from 1.15% to 46.54%.

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Table 54. ARL₁ for FP EWMA WL and ARL₁ Saved % Compared to FP WL Chart under λ=0.7

No. ₁ ₂ ₃ L1 L2 UCL LCL EWMA

ARL1

FP ARL1

ARL1

saved % 1 0.5 1.1 0 4.62 1.70 2.17 0.13 90.77 109.54 17.14%

2 0.5 1.1 1 4.28 1.90 2.86 0.29 26.15 38.67 32.36%

3 0.5 1.1 2 3.89 2.19 4.68 0.93 18.70 29.36 36.32%

4 0.5 1.5 0 4.62 1.70 2.17 0.13 6.29 7.79 19.18%

5 0.5 1.5 1 4.28 1.90 2.86 0.29 4.78 5.90 18.94%

6 0.5 1.5 2 3.89 2.19 4.68 0.93 5.00 6.20 19.47%

7 0.5 2 0 4.62 1.70 2.17 0.13 2.19 2.36 7.20%

8 0.5 2 1 4.28 1.90 2.86 0.29 2.15 2.31 6.88%

9 0.5 2 2 3.89 2.19 4.68 0.93 2.46 2.66 7.66%

10 1.2 1.1 0 4.62 1.70 2.17 0.13 14.29 21.67 34.07%

11 1.2 1.1 1 4.28 1.90 2.86 0.29 3.62 5.10 28.99%

12 1.2 1.1 2 3.89 2.19 4.68 0.93 2.72 3.60 24.44%

13 1.2 1.5 0 4.62 1.70 2.17 0.13 3.45 4.14 16.77%

14 1.2 1.5 1 4.28 1.90 2.86 0.29 2.15 2.42 11.08%

15 1.2 1.5 2 3.89 2.19 4.68 0.93 1.99 2.21 9.81%

16 1.2 2 0 4.62 1.70 2.17 0.13 1.79 1.88 5.05%

17 1.2 2 1 4.28 1.90 2.86 0.29 1.54 1.59 3.20%

18 1.2 2 2 3.89 2.19 4.68 0.93 1.57 1.63 3.20%

19 1.5 1.1 0 4.62 1.70 2.17 0.13 6.55 9.86 33.57%

20 1.5 1.1 1 4.28 1.90 2.86 0.29 2.21 2.70 18.31%

21 1.5 1.1 2 3.89 2.19 4.68 0.93 1.79 2.04 12.08%

22 1.5 1.5 0 4.62 1.70 2.17 0.13 2.60 3.01 13.77%

23 1.5 1.5 1 4.28 1.90 2.86 0.29 1.68 1.79 6.52%

24 1.5 1.5 2 3.89 2.19 4.68 0.93 1.56 1.64 4.83%

25 1.5 2 0 4.62 1.70 2.17 0.13 1.61 1.67 3.83%

26 1.5 2 1 4.28 1.90 2.86 0.29 1.37 1.39 1.79%

27 1.5 2 2 3.89 2.19 4.68 0.93 1.37 1.39 1.51%

Note: ARL1 saved %:



FP ARL- 1FP EWMA ARL. - 1



FP ARL- 1%

The response table and response diagram show the results of the sensitivity analysis (Table 55, Figure 55 and Figure 56).

‧

From Table 54, Table 55, Figure 55 and Figure 56, we found a. ARL1 saved % from 1.51% to 36.32%.

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Table 56. ARL₁ for FP EWMA WL and ARL₁ Saved % Compared to FP WL Chart under λ=0.8

No. ₁ ₂ ₃ L1 L2 UCL LCL EWMA

ARL1

FP ARL1

ARL1

saved % 1 0.5 1.1 0 4.71 1.61 2.37 0.10 98.84 109.54 9.77%

2 0.5 1.1 1 4.35 1.82 3.09 0.24 30.37 38.67 21.46%

3 0.5 1.1 2 3.95 2.13 4.99 0.82 22.05 29.36 24.90%

4 0.5 1.5 0 4.71 1.61 2.37 0.10 6.78 7.79 12.87%

5 0.5 1.5 1 4.35 1.82 3.09 0.24 5.13 5.90 13.08%

6 0.5 1.5 2 3.95 2.13 4.99 0.82 5.36 6.20 13.62%

7 0.5 2 0 4.71 1.61 2.37 0.10 2.24 2.36 5.13%

8 0.5 2 1 4.35 1.82 3.09 0.24 2.20 2.31 4.98%

9 0.5 2 2 3.95 2.13 4.99 0.82 2.51 2.66 5.63%

10 1.2 1.1 0 4.71 1.61 2.37 0.10 16.69 21.67 22.98%

11 1.2 1.1 1 4.35 1.82 3.09 0.24 4.03 5.10 21.11%

12 1.2 1.1 2 3.95 2.13 4.99 0.82 2.94 3.60 18.40%

13 1.2 1.5 0 4.71 1.61 2.37 0.10 3.66 4.14 11.68%

14 1.2 1.5 1 4.35 1.82 3.09 0.24 2.22 2.42 8.15%

15 1.2 1.5 2 3.95 2.13 4.99 0.82 2.05 2.21 7.46%

16 1.2 2 0 4.71 1.61 2.37 0.10 1.81 1.88 3.67%

17 1.2 2 1 4.35 1.82 3.09 0.24 1.55 1.59 2.51%

18 1.2 2 2 3.95 2.13 4.99 0.82 1.58 1.63 2.65%

19 1.5 1.1 0 4.71 1.61 2.37 0.10 7.55 9.86 23.47%

20 1.5 1.1 1 4.35 1.82 3.09 0.24 2.33 2.70 13.76%

21 1.5 1.1 2 3.95 2.13 4.99 0.82 1.84 2.04 9.52%

22 1.5 1.5 0 4.71 1.61 2.37 0.10 2.72 3.01 9.79%

23 1.5 1.5 1 4.35 1.82 3.09 0.24 1.71 1.79 4.96%

24 1.5 1.5 2 3.95 2.13 4.99 0.82 1.57 1.64 3.91%

25 1.5 2 0 4.71 1.61 2.37 0.10 1.62 1.67 2.88%

26 1.5 2 1 4.35 1.82 3.09 0.24 1.37 1.39 1.51%

27 1.5 2 2 3.95 2.13 4.99 0.82 1.37 1.39 1.37%

Note: ARL1 saved %:



FP ARL- 1FP EWMA ARL. - 1



FP ARL- 1%

The response table and response diagram show the results of the sensitivity analysis (Table 57, Figure 57 and Figure 58).

‧

From Table 56, Table 57, Figure 57 and Figure 58, we found a. ARL1 saved % from 1.37% to 24.90%.

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Table 58. ARL₁ for FP EWMA WL and ARL₁ Saved % Compared to FP WL Chart under λ=0.9

No. ₁ ₂ ₃ L1 L2 UCL LCL EWMA

ARL1

FP ARL1

ARL1

saved % 1 0.5 1.1 0 4.76 1.53 2.57 0.07 105.60 109.54 3.60%

2 0.5 1.1 1 4.39 1.75 3.33 0.18 34.60 38.67 10.52%

3 0.5 1.1 2 3.98 2.08 5.30 0.70 25.65 29.36 12.65%

4 0.5 1.5 0 4.76 1.53 2.57 0.07 7.29 7.79 6.40%

5 0.5 1.5 1 4.39 1.75 3.33 0.18 5.51 5.90 6.71%

6 0.5 1.5 2 3.98 2.08 5.30 0.70 5.77 6.20 7.08%

7 0.5 2 0 4.76 1.53 2.57 0.07 2.30 2.36 2.71%

8 0.5 2 1 4.39 1.75 3.33 0.18 2.25 2.31 2.68%

9 0.5 2 2 3.98 2.08 5.30 0.70 2.58 2.66 3.04%

10 1.2 1.1 0 4.76 1.53 2.57 0.07 19.19 21.67 11.46%

11 1.2 1.1 1 4.39 1.75 3.33 0.18 4.52 5.10 11.43%

12 1.2 1.1 2 3.98 2.08 5.30 0.70 3.23 3.60 10.34%

13 1.2 1.5 0 4.76 1.53 2.57 0.07 3.89 4.14 6.06%

14 1.2 1.5 1 4.39 1.75 3.33 0.18 2.31 2.42 4.43%

15 1.2 1.5 2 3.98 2.08 5.30 0.70 2.12 2.21 4.21%

16 1.2 2 0 4.76 1.53 2.57 0.07 1.85 1.88 1.97%

17 1.2 2 1 4.39 1.75 3.33 0.18 1.57 1.59 1.44%

18 1.2 2 2 3.98 2.08 5.30 0.70 1.60 1.63 1.54%

19 1.5 1.1 0 4.76 1.53 2.57 0.07 8.67 9.86 12.15%

20 1.5 1.1 1 4.39 1.75 3.33 0.18 2.50 2.70 7.70%

21 1.5 1.1 2 3.98 2.08 5.30 0.70 1.92 2.04 5.60%

22 1.5 1.5 0 4.76 1.53 2.57 0.07 2.86 3.01 5.18%

23 1.5 1.5 1 4.39 1.75 3.33 0.18 1.74 1.79 2.79%

24 1.5 1.5 2 3.98 2.08 5.30 0.70 1.60 1.64 2.32%

25 1.5 2 0 4.76 1.53 2.57 0.07 1.64 1.67 1.56%

26 1.5 2 1 4.39 1.75 3.33 0.18 1.38 1.39 0.86%

27 1.5 2 2 3.98 2.08 5.30 0.70 1.38 1.39 0.86%

Note: ARL1 saved %:



FP ARL- 1FP EWMA ARL. - 1



FP ARL- 1%

The response table and response diagram show the results of the sensitivity analysis (Table 59, Figure 59 and Figure 60).

‧

From Table 58, Table 59, Figure 59 and Figure 60, we found a. ARL1 saved % from 0.86% to 12.65%. scheme is more effective under small λ when the shifts of mean and/or variance are small so that the performance compare to FP WL chart is not great under large shifts of

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mean and/or variance.

For different values of λ in Tables 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56 and 58, the choice of the λ with maximal ARL₁ saved % for each combination of



₁,



₂ and



3 in L₂₇(3¹³) is listed in Table 62. That is, we may choose the λ for the given combination



₁,



₂ and



₃ from Table 60.

Table 60. The Optimal λ to Maximize the ARL1 Saved % Time

No. 1 ₂ ₃ λ^* ARL1 saved % No. 1 ₂ ₃ λ^* ARL1 saved %

1 0.5 1.1 0 0.05 76.72% 16 1.2 2 0 0.5 6.43%

2 0.5 1.1 1 0.1 74.84% 17 1.2 2 1 0.6 3.52%

3 0.5 1.1 2 0.15 72.24% 18 1.2 2 2 0.6 3.26%

4 0.5 1.5 0 0.2 39.58% 19 1.5 1.1 0 0.2 58.90%

5 0.5 1.5 1 0.25 33.79% 20 1.5 1.1 1 0.4 23.68%

6 0.5 1.5 2 0.3 32.70% 21 1.5 1.1 2 0.6 13.30%

7 0.5 2 0 0.4 10.34% 22 1.5 1.5 0 0.4 20.80%

8 0.5 2 1 0.5 9.00% 23 1.5 1.5 1 0.5 7.41%

9 0.5 2 2 0.5 9.76% 24 1.5 1.5 2 0.6 4.95%

10 1.2 1.1 0 0.15 70.38% 25 1.5 2 0 0.5 4.49%

11 1.2 1.1 1 0.3 43.30% 26 1.5 2 1 0.7 1.79%

12 1.2 1.1 2 0.4 32.28% 27 1.5 2 2 0.7 1.51%

13 1.2 1.5 0 0.3 28.35%

14 1.2 1.5 1 0.4 14.60%

15 1.2 1.5 2 0.5 11.67%

The results in Table 60 tell us that we have to choose the biggerλ when the values of

1, ₂ and/or ₃ are large. However, the ARL saved % of the combination #1 is 76.72% which is the maximum , that is, the FP EWMA WL chart has the best performance under λ=0.05 when the shifts of the mean and/or variance are much smaller.

(1) ARL 1 Analysis of FP EWMA WL Chart and ARL 1 Comparison between FP EWMA Chart and FP WL Chart

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5.3 Data Analysis and ATS

Comparison among the FP EWMA, VSI EWMA and FP WL Charts

(1) ARL

Analysis of FP EWMA WL Chart and ARL

Comparison between FP EWMA Chart and FP WL Chart

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