Table 3 The ARL results of Bayesian Max-EWMA control chart as a result of a shift in Weibull parameters W(1, 1.5) with \(\gamma\) = 0.10 and considering subgroup size as 3, 5, and 7 under LLF.

0.25

(µ, σ)

(−1.51, 2.04)

(−1.49, 0.89)

(−1.45, 0.72)

(−1.42, 0.61)

(−1.40, 0.53)

(−1.38, 0.47)

(−1.36, 0.42)

(−1.35, 0.38)

(−1.33, 0.35)

(−1.32, 0.32)

(−1.29, 0.25)

(−1.28, 0.20)

3

3.24 (1.24)

3.46 (0.75)

3.53 (0.66)

3.59 (0.60)

3.62 (0.56)

3.68 (0.53)

3.76 (0.49)

3.79 (0.47)

3.85 (0.42)

3.89 (0.37)

3.96 (0.24)

3.95 (0.23)

5

2.53 (0.80)

2.70 (0.56)

2.77 (0.50)

2.84 (0.43)

2.90 (0.36)

2.94 (0.29)

2.97 (0.22)

2.98 (0.17)

2.99 (0.13)

3.00 (0.10)

2.99 (0.09)

2.96 (0.17)

7

2.17 (0.64)

2.27 (0.45)

2.29 (0.45)

2.33 (0.47)

2.35 (0.47)

2.39 (0.48)

2.46 (0.49)

2.49 (0.49)

2.57 (0.49)

2.61 (0.48)

2.68 (0.46)

2.44 (0.49)

0.50

(µ, σ)

(−0.68, 3.16)

(−0.87, 1.11)

(−0.84, 0.90)

(−0.81, 0.76)

(−0.78, 0.66)

(−0.76, 0.58)

(−0.74, 0.53)

(−0.72, 0.48)

(−0.71, 0.44)

(−0.70, 0.40)

(−0.66, 0.31)

(−0.64, 0.25)

3

2.65 (1.14)

6.41 (2.40)

6.70 (2.17)

6.91 (1.96)

7.21 (1.82)

7.41 (1.67)

7.50 (1.57)

7.53 (1.50)

7.32 (1.40)

7.03 (1.36)

8.38 (1.89)

5.19 (0.91)

5

1.94 (0.71)

4.75 (1.50)

4.92 (1.31)

5.07 (1.22)

5.24 (1.09)

5.47 (1.47)

5.30 (0.90)

5.20 (0.86)

4.96 (0.82)

4.68 (0.78)

3.87 (0.61)

3.10 (0.36)

7

1.64 (0.57)

3.94 (1.11)

4.06 (0.98)

4.18 (0.88)

4.34 (0.83)

4.35 (0.72)

4.35 (0.67)

4.21 (0.63)

3.99 (0.62)

3.72 (0.59)

3.09 (0.39)

2.83 (0.37)

0.75

(µ, σ)

(0.01, 4.18)

(−0.42, 1.30)

(−0.39, 1.05)

(−0.36, 0.88)

(−0.34, 0.77)

(−0.32, 0.68)

(−0.29, 0.61)

(−0.28, 0.56)

(−0.26, 0.51)

(−0.25, 0.47)

(−0.21, 0.37)

(−0.18, 0.30)

3

1.99 (0.87)

13.27 (7.54)

19.93 (12.35)

24.68 (15.17)

25.30 (14.43)

21.44 (10.98)

16.99 (7.88)

13.86 (5.75)

11.28 (4.02)

9.75 (3.09)

7.15 (2.87)

5.87 (1.16)

5

1.47 (0.57)

9.32 (4.39)

13.42 (6.97)

15.43 (7.74)

14.67 (6.20)

11.85 (4.34)

9.16 (2.94)

7.61 (7.67)

6.46 (1.61)

5.73 (1.28)

4.42 (0.78)

3.79 (0.59)

7

1.25 (0.44)

7.47 (3.07)

10.52 (4.92)

11.77 (4.98)

9.65 (3.16)

8.69 (2.66)

6.80 (1.85)

5.72 (1.36)

4.95 (1.04)

4.43 (0.86)

3.50 (0.57)

3.04 (0.37)

1.0

(µ, σ)

(0.61, 5.00)

(−0.05, 1.47)

(−0.03, 1.18)

(0.00, 1.00)

(0.03, 0.87)

(0.05, 0.77)

(0.07, 0.69)

(0.09, 0.63)

(0.11, 0.58)

(0.12, 0.54)

(0.16, 0.41)

(0.19, 0.34)

3

1.66 (0.73)

13.61 (8.24)

54.88 (46.29)

369.33 (367.12)

173.35 (160.14)

57.52 (45.17)

29.07 (18.42)

19.61 (10.19)

15.23 (6.89)

12.76 (5.05)

8.05 (2.16)

6.53 (1.46)

5

1.27 (0.46)

8.47 (4.08)

31.78 (23.77)

370.89 (366.45)

79.18 (67.14)

24.21 (14.68)

13.64 (6.01)

10.17 (3.64)

8.21 (2.46)

7.13 (1.93)

4.89 (0.94)

4.14 (0.68)

7

1.11 (0.32)

6.52 (2.76)

23.51 (15.64)

369.87 (360.22)

49.61 (38.56)

15.99 (7.87)

9.67 (3.48)

7.42 (2.17)

6.17 (1.59)

5.39 (1.23)

3.83 (0.65)

3.28 (0.49)

1.25

(µ, σ)

(1.17, 5.69)

(0.28, 1.65)

(0.29, 1.31)

(0.32, 1.11)

(0.35, 0.96)

(0.37, 0.86)

(0.39, 0.77)

(0.41, 0.70)

(0.43, 0.64)

(0.45, 0.60)

(0.49, 0.46)

(0.52, 0.38)

3

1.47 (0.64)

8.56 (4.48)

17.12 (10.83)

25.07 (16.79)

25.43 (16.08)

23.33 (13.86)

20.32 (10.33)

17.52 (7.75)

15.06 (5.84)

13.33 (4.67)

10.08 (3.00)

7.28 (1.75)

5

1.17 (0.38)

5.67 (2.38)

11.37 (5.86)

16.89 (10.07)

16.33 (8.78)

14.76 (6.96)

12.62 (4.85)

10.71 (3.48)

9.05 (2.55)

8.12 (2.13)

5.55 (1.17)

4.55 (0.82)

7

1.05 (0.22)

4.50 (1.68)

8.83 (4.01)

13.31 (7.06)

12.58 (5.94)

11.15 (4.44)

9.70 (3.18)

8.19 (2.26)

6.99 (1.74)

6.22 (1.43)

4.33 (0.82)

3.60 (0.59)

1.50

(µ, σ)

(1.67, 6.26)

(0.58, 1.86)

(0.58, 1.44)

(0.61, 1.21)

(0.64, 1.05)

(0.66, 0.94)

(0.68, 0.84)

(0.70, 0.77)

(0.72, 0.71)

(0.74, 0.65)

(0.79, 0.51)

(0.82, 0.41)

3

1.36 (0.56)

5.68 (2.59)

8.58 (4.26)

9.80 (4.77)

9.54 (9.54)

9.19 (3.63)

8.77 (3.12)

8.38 (2.65)

8.09 (2.35)

7.73 (2.00)

6.91 (1.33)

6.36 (1.01)

5

1.11 (0.31)

3.98 (1.47)

6.23 (2.54)

7.05 (2.87)

6.80 (2.50)

6.56 (2.16)

6.29 (1.88)

6.02 (1.58)

5.81 (1.37)

5.54 (1.17)

5.20 (1.16)

4.47 (0.64)

7

1.02 (0.16)

3.30 (1.10)

5.08 (1.84)

5.78 (2.14)

5.57 (1.86)

5.34 (1.56)

5.13 (1.33)

4.93 (1.15)

4.75 (1.00)

4.56 (0.86)

4.07 (0.59)

3.66 (0.52)

2.00

(µ, σ)

(2.57, 7.16)

(1.16, 2.43)

(1.11, 1.70)

(1.13, 1.41)

(1.15, 1.23)

(1.18, 1.09)

(1.20, 0.98)

(1.23, 0.89)

(1.25, 0.82)

(1.26, 0.76)

(1.32, 0.59)

(1.36, 0.48)

3

1.24 (0.46)

2.20 (0.44)

4.51 (1.83)

4.68 (1.78)

4.63 (1.57)

4.49 (1.35)

4.38 (1.19)

4.25 (1.04)

4.14 (0.92)

4.12 (0.87)

3.88 (0.65)

3.74 (0.54)

5

1.05 (0.23)

2.41 (0.82)

3.40 (1.13)

3.61 (1.16)

3.55 (1.02)

3.43 (0.89)

3.35 (0.78)

3.25 (0.69)

3.19 (0.62)

3.17 (0.57)

3.02 (0.40)

2.96 (0.29)

7

1.01 (0.10)

1.62 (0.48)

2.10 (0.35)

2.44 (0.56)

2.79 (0.69)

3.15 (0.83)

3.51 ( 1.00)

3.50 (1.01)

4.20 (1.36)

4.53 (1.51)

5.64 (2.13)

2.46 (0.49)

2.50

(µ, σ)

(3.33, 7.82)

(1.74, 3.18)

(1.59, 2.05)

(1.59, 1.61)

(1.62, 1.39)

(1.64, 1.23)

(1.67, 1.11)

(1.69, 1.01)

(1.72, 0.93)

(1.73, 0.86)

(1.80, 0.67)

(1.84, 0.55)

3

1.18 (0.41)

2.29 (0.95)

3.14 (1.17)

3.31 ( 1.13)

3.27 (1.01)

3.20 (0.88)

3.12 (0.79)

3.08 (0.71)

3.02 (0.64)

2.98 (0.60)

2.87 (0.51)

2.84 ( 0.39)

5

1.05 (0.22)

1.75 (0.64)

2.45 ( 0.78)

2.58 ( 0.75)

2.55 (0.67)

2.50 (0.60)

2.43 (0.55)

2.39 (0.52)

2.34 (0.48)

2.31 (0.47)

2.17 (0.38)

2.08 (0.27)

7

1.00 (0.07)

1.17 (0.38)

2.09 (0.60)

2.24 (0.58)

2.65 (0.77)

2.72 (0.80)

2.91 (0.89)

3.08 (0.96)

3.19 ( 1.02)

2.04 (0.23)

2.00 (0.08)

2.00 (0.01)

3.00

(µ, σ)

(4.00, 8.33)

(2.35, 4.00)

(2.08, 2.58)

(2.03, 1.86)

(2.04, 1.55)

(2.07, 1.37)

(2.10, 1.24)

(2.12, 1.13)

(2.15, 1.04)

(2.17, 0.96)

(2.24, 0.75)

(2.29, 0.61)

3

1.14 (0.37)

1.76 (0.75)

2.40 (0.93)

2.63 (0.88)

2.63 (0.78)

2.58 (0.68)

2.51 (0.62)

2.48 (0.57)

2.43 (0.53)

2.39 (0.51)

2.27 (0.45)

2.17 (0.38)

5

1.02 ( 0.15)

1.36 (0.51)

1.88 (0.63)

2.22 (0.70)

2.48 (0.79)

2.33 (0.86)

2.06 (0.47)

2.03 (0.32)

2.01 (0.27)

2.01 (0.22)

1.99 (0.12)

1.99 (0.06)

7

1.00 (0.05)

1.17 (0.38)

1.63 (0.54)

1.83 (0.52)

1.86 (0.46)

1.85 (0.41)

1.84 (0.38)

1.85 (0.36)

1.85 (0.35)

1.85 (0.35)

1.86 (0.33)

1.99 ( 0.01)

4.00

(µ, σ)

(5.12, 9.07)

(3.61, 5.53)

(3.15, 3.99)

(2.93, 2.81)

(2.86, 2.07)

(2.86, 1.68)

(2.88, 1.48)

(2.91, 1.35)

(2.94, 1.24)

(2.96, 1.15)

(3.04, 0.89)

(3.10, 0.73)

3

1.10 (0.31)

1.33 (0.53)

1.59 (0.66)

1.88 (0.71)

1.98 (0.62)

1.98 (0.52)

1.97 (0.45)

1.96 ( 0.39)

1.95 ( 0.35)

1.94 (0.31)

1.96 (0.21)

1.97 (0.16)

5

1.01 (0.12)

1.09 (0.30)

1.27 (0.46)

1.48 (0.54)

1.59 (0.53)

1.59 (0.50)

1.58 (0.49)

1.58 (0.49)

1.55 (0.51)

1.53 (0.49)

1.48 (0.49)

1.42 (0.49)

7

1.00 (0.03)

1.02 (0.16)

1.12 (0.32)

1.28 ( 0.45)

1.34 (0.47)

1.32 (0.46)

1.27 (0.44)

1.23 (0.42)

1.21 (0.40)

1.17 (0.38)

1.07 (0.26)

1.02 (0.16)