Table 4 The ARL results of Bayesian Max-EWMA control chart as a result of a shift in Weibull parameters W(1, 1.5) with \(\gamma\)= 0.25 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

2.47 (1.17)

2.75 (0.80)

2.78 (0.70)

2.83 (0.64)

2.88 (0.58)

2.92 (0.53)

2.97 (0.49)

3.00 (0.42)

3.03 (0.40)

3.05 (0.36)

3.05 (0.31)

3.01 (0.27)

5

1.88 (0.77)

2.11 (0.47)

2.13 (0.39)

2.13 (0.36)

2.12 (0.33)

2.12 (0.32)

2.12 (0.32)

2.11 (0.31)

2.13 (0.33)

2.11 (0.32)

2.09 (0.29)

2.03 (0.19)

7

1.60 (0.61)

1.84 (0.42)

1.90 (0.33)

1.94 (0.25)

1.97 (0.17)

1.99 (0.11)

1.99 (0.06)

1.99 (0.05)

1.99 (0.03)

1.99 (0.03)

1.99 (0.05)

1.98 (0.11)

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

1.96 (1.01)

5.70 (2.94)

6.25 (3.06)

6.72 (3.00)

7.27 (3.10)

7.61 (3.09)

7.84 (3.01)

7.95 (2.98)

7.68 (2.81)

7.24 (2.59)

5.62 (1.82)

4.54 (1.23)

5

1.46 (0.61)

3.97 (1.69)

4.15 (1.56)

4.34 (1.48)

4.52 (1.42)

4.55 (1.25)

4.57 (1.18)

4.44 (1.09)

4.20 (1.00)

3.88 (0.90)

3.09 (0.65)

2.65 (0.53)

7

1.24 (0.45)

3.19 (1.17)

3.28 (1.07)

3.41 (1.00)

3.52 (0.93)

3.52 (0.80)

3.49 (0.77)

3.36 (0.71)

3.16 (0.65)

2.92 (0.62)

2.36 (0.49)

2.06 (0.25)

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

1.49 (0.69)

13.20 (10.09)

23.99 (20.21)

45.10 (39.36)

59.07 (52.32)

49.75 (42.61)

32.81 (27.26)

22.31 (16.65)

15.80 (10.53)

12.26 (7.33)

7.34 (3.14)

5.43 (1.76)

5

1.18 (0.41)

8.48 (5.46)

15.52 (11.75)

22.41 (17.77)

23.30 (17.46)

16.24 (10.92)

10.63 (6.13)

7.90 (3.76)

6.19 (2.49)

5.20 (1.77)

3.66 (0.92)

3.02 (0.62)

7

1.07 (0.26)

1.06 (0.25)

11.24 (7.46)

14.63 (10.05)

13.77 (8.68)

9.61 (5.19)

6.55 (2.84)

5.14 (1.87)

4.23 (1.28)

3.68 (1.00)

2.76 (0.59)

2.29 (0.46)

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

1.31 (0.56)

12.59 (9.60)

62.01 (57.77)

370.68 (367.77)

342.12 (338.03)

141.43 (133.84)

66.40 (60.28)

38.90 (32.79)

26.38 (20.91)

22.65 (17.18)

8.89 (4.28)

6.40 (2.44)

5

1.09 (0.29)

7.57 (4.99)

39.35 (35.73)

370.09 (368.89)

178.01 (172.69)

47.26 (41.65)

20.19 (14.80)

12.40 (7.76)

8.85 (4.72)

7.15 (3.18)

4.15 (1.19)

3.37 (0.77)

7

1.02 (0.15)

5.58 (3.17)

28.08 (24.05)

370.89 (366.23)

103.09 (96.58)

24.88 (19.76)

11.43 (6.92)

7.46 (3.57)

5.69 (2.26)

4.75 (1.61)

3.05 (0.71)

2.56 (0.55)

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

1.32 (0.56)

7.08 (4.74)

17.28 (13.83)

31.99 (27.51)

40.31 (35.12)

35.71 (30.64)

42.15 (36.38)

36.27 (29.78)

28.51 (22.04)

23.20 (17.39)

11.22 (6.17)

7.63 (3.39)

5

1.04 (0.21)

4.58 (2.49)

10.78 (7.52)

20.34 (16.36)

22.16 (17.43)

21.14 (16.01)

17.36 (12.27)

13.67 (8.39)

10.44 (5.59)

8.70 (4.17)

4.93 (1.60)

3.77 (0.98)

7

1.01 (0.10)

3.59 (1.71)

8.08 (5.08)

14.32 (10.43)

15.03 (10.69)

13.55 (8.81)

11.07 (6.33)

8.55 (4.09)

6.75 (2.75)

5.71 (2.06)

3.54 (0.92)

2.83 (0.62)

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

1.15 (0.39)

4.44 ( 2.58)

7.41 (4.76)

9.51 (6.17)

9.74 (6.22)

9.69 (5.88)

9.36 (5.20)

9.04 (4.82)

8.59 (4.23)

8.17 (3.69)

6.86 (2.36)

6.01 ( 1.67)

5

1.06 ( 0.24)

3.08 ( 1.48)

5.16 ( 2.83)

6.34 ( 3.55)

6.24 ( 3.27)

6.04 ( 2.88)

5.75 ( 2.54)

5.52 ( 2.23)

5.17 ( 1.85)

4.93 ( 1.66)

4.33 ( 1.17)

3.62 ( 0.73)

7

1.00 ( 0.08)

4.39 ( 2.15)

4.04 ( 1.92)

4.95 ( 2.47)

4.79 ( 2.18)

4.58 ( 1.89)

4.86 ( 1.91)

4.19 ( 1.41)

3.98 ( 1.21)

3.77 ( 1.01)

3.24 ( 0.65)

2.35 ( 0.48)

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

1.10 ( 0.31)

2.44 (1.25)

3.50 (1.79)

3.84 (1.84)

3.81 (1.69)

2.26 (0.65)

3.61 (1.34)

3.49 (1.17)

3.39 (1.09)

3.34 ( 0.96)

3.11 (0.70)

2.98 (0.55)

5

1.01 ( 0.12)

1.79 ( 0.77)

2.58 ( 1.09)

2.81 ( 1.12)

2.80 ( 1.05)

2.72 ( 0.91)

2.63 ( 0.79)

2.55 ( 0.69)

2.49 ( 0.63)

2.45 ( 0.58)

2.09 ( 0.29)

2.15 ( 0.36)

7

1.00 ( 0.04)

1.52 ( 0.60)

2.18 ( 0.83)

2.06 ( 0.71)

2.34 ( 0.76)

2.26 ( 0.66)

2.21 ( 0.58)

2.14 ( 0.49)

2.10 ( 0.43)

2.08 ( 0.39)

2.00 ( 0.22)

1.99 ( 0.11)

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

1.05 (0.23)

1.68 (0.80)

2.35 (1.08)

2.58 (1.10)

2.57 (0.96)

2.52 (0.88)

2.44 (0.76)

2.40 (0.69)

2.34 (0.61)

2.32 (0.56)

2.18 (0.42)

2.10 ( 0.31)

5

1.00 ( 0.07)

1.32 ( 0.52)

2.04 (1.09)

1.99 ( 0.75)

1.98 ( 0.67)

1.94 ( 0.60)

1.91 ( 0.54)

1.89 ( 0.49)

1.91 ( 0.31)

1.86 ( 0.41)

1.85 ( 0.35)

1.86 ( 0.34)

7

1.00 ( 0.03)

1.15 ( 0.36)

1.55 ( 0.60)

1.68 (0.61)

1.66 ( 0.56)

1.66 ( 0.53)

1.62 ( 0.51)

1.60 ( 0.50)

1.57 ( 0.49)

1.57 ( 0.49)

1.47 ( 0.49)

1.38 ( 0.48)

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

1.05 (0.23)

1.36 (0.58)

1.78 (0.82)

2.01 (0.83)

2.01 (0.75)

2.00 ( 0.67)

1.95 (0.61)

1.94 (0.55)

1.91 (0.50)

1.90 (0.44)

1.88 (0.35)

1.89 (0.31)

5

1.00 ( 0.09)

1.12 ( 0.34)

1.39 ( 0.55)

1.57 ( 0.60)

1.58 ( 0.58)

1.55 ( 0.54)

1.52 ( 0.52)

1.50 ( 0.51)

1.46 ( 0.50)

1.48 ( 0.56)

1.45 ( 0.54)

1.26 ( 0.44)

7

1.00 ( 0.02)

1.03 ( 0.19)

1.21 ( 0.42)

1.41 ( 0.53)

1.59 ( 0.62)

1.67 ( 0.65)

1.72 ( 0.67)

1.21 ( 0.41)

1.17 ( 0.38)

1.14 ( 0.35)

1.05 ( 0.23)

1.01 ( 0.11)

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

1.04 (0.20)

1.13 (0.36)

1.27 ( 0.51)

1.41 (0.59)

1.48 (0.59)

1.46 (0.55)

1.44 (0.52)

1.41 (0.50)

1.39 (0.49)

1.36 (0.48)

1.27 (0.44)

1.18 (0.38)

5

1.00 ( 0.06)

1.02 ( 0.16)

1.07 ( 0.27)

1.17 ( 0.38)

1.19 ( 0.40)

1.15 ( 0.36)

1.12 ( 0.33)

1.09 ( 0.29)

1.07 ( 0.25)

1.05 ( 0.22)

1.01 ( 0.10)

1.00 ( 0.04)

7

1.00 ( 0.01)

1.00 ( 0.06)

1.02 ( 0.14)

1.06 ( 0.25)

1.07 ( 0.26)

1.04 ( 0.21)

1.02 ( 0.16)

1.01 ( 0.11)

1.00 ( 0.08)

1.00 ( 0.08)

1.00 ( 0.01)

1.0 (0.0)