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... i

Abstract... ii

Contents ... iv

List of Tables... v

List of Figures... vii

1. Introduction... 1

2. Process Mean Shift Investigation for Non-normal Process... 3

2.1. Weibull Process ... 3

2.1.1. The Weibull Distribution ... 4

2.1.2. Calculating the Average Run Length of Weibull Process under Different

Magnitude of shift by Simulation... 8

2.1.3. The Modified Mean Adjustments for Weibull Process... 9

2.2. Log-normal Process ... 12

2.2.1. The Log-normal Distribution... 12

2.2.2. Calculating the Average Run Length of Log-normal Process under

Different Magnitude of shift by Simulation ... 15

2.2.3. The Modified Mean Adjustments for Lognormal Process ... 16

2.3. Gamma Process... 18

2.3.1. The Gamma Distribution ... 18

2.3.2. Calculating the Average Run Length of Gamma Process under Different

Magnitude of shift by Simulation... 21

3. Process Mean Shift Investigation for Gamma Process... 22

3.1. Statistical Properties of Gamma Distribution... 22

3.2. The Detection Power of Gamma Process under the Bothe’ Adjustments ... 23

3.3. The Modified Mean Adjustments for Gamma Process ... 24

3.4. The Modified Estimator of Process Capability C

pk

... 27

3.4.1. C in the Non-Normal Case ... 27

pk

3.4.2. Adjustment of

C ... 28

pk

4. Application... 29

5. Conclusion ... 32

References... 33

(7)

Table 1. Values of skewness and kurtosis for various weibull distributions...6

Table 2.

AS value for several subgroup sizes n and various values under

50

weibull distribution(shfit to right)...10

Table 3.

AS value for several subgroup sizes n and various values under

50

weibull distribution(shfit to left)...11

Table 4. Values of skewness and kurtosis for various Log-normal distributions.13

Table 5.

AS value for several subgroup sizes n and various values under

50

log-normal distribution ...17

Table 6. Values of skewness and kurtosis for various gamma distributions. ...19

Table 7. Detection power of various gamma distributions ...23

Table 8.

AS value for several subgroup sizes n and various

50

values...25

Table 9. The 100 observations are collected of the historical data. ...32

Table 10. Index values and the corresponding bounds on NCPPM for a Normal

process ...35

Table 11. Average run length of weibull with +0.5

σ

mean shift ...36

Table 12. Average run length of weibull with +1

σ

mean shift...37

Table 13. Average run length of weibull with +1.5

σ

mean shift...38

Table 14. Average run length of weibull with +2

σ

mean shift...39

Table 15. Average run length of weibull with +2.5

σ

mean shift...40

Table 16. Average run length of weibull with +3

σ

mean shift...41

Table 17. Average run length of weibull with -0.5

σ

mean shift...42

Table 18. Average run length of weibull with -1

σ

mean shift...43

Table 19. Average run length of weibull with -1.5

σ

mean shift...44

Table 20. Average run length of weibull with -2

σ

mean shift...45

Table 21. Average run length of weibull with -2.5

σ

mean shift...46

Table 22. Average run length of weibull with -3

σ

mean shift...47

Table 23. Average run length of Lognormal with +0.5

σ

mean shift ...48

Table 24. Average run length of Lognormal with +1

σ

mean shift ...49

Table 25. Average run length of Lognormal with +1.5

σ

mean shift ...50

Table 26. Average run length of Lognormal with +2

σ

mean shift ...51

Table 27. Average run length of lognormal with +2.5

σ

mean shift ...52

Table 28. Average run length of Lognormal with +3

σ

mean shift ...53

Table 29. Average run length of Log-normal with -0.5

σ

mean shift...55

Table 30. Average run length of Log-normal with -1

σ

mean shift...56

(8)

Table 34. Average run length of Log-normal with -3

σ

mean shift...60

Table 35. Average run length of Gamma with +0.5

σ

mean shift ...61

Table 36. Average run length of Gamma with +1

σ

mean shift ...62

Table 37. Average run length of Gamma with +1.5

σ

mean shift ...63

Table 38. Average run length of Gamma with +2

σ

mean shift ...64

Table 39. Average run length of Gamma with +2.5

σ

mean shift ...65

Table 40. Average run length of Gamma with +3

σ

mean shift ...66

Table 41. Average run length of Gamma with -0.5

σ

mean shift ...67

Table 42. Average run length of Gamma with -1

σ

mean shift to left...68

Table 43. Average run length of Gamma with -1.5

σ

mean shift ...69

Table 44. Average run length of Gamma with -2

σ

mean shift ...70

Table 45. Average run length of Gamma with -2.5

σ

mean shiftt ...71

(9)

Figure 1. Weibull distributions for selected values of the shape parameter k and

scale parameter

λ

= ...5

1 Figure 2. presents several weibull distributions along with a normal distribution

for the same mean and variance. Let k

= 0.5, 1, 2, 3, 4, and 5, while

fixing

λ

= ...7

1 Figure 3. Log-normal distributions for selected values of the shape parameter

s

and

µ

=0. ...13

Figure 4 presents several log-normal distributions along with a normal

distribution for the same mean and variance. Let ...14

Figure 5 presents several gamma distributions along with a normal distribution

for the same mean and variance. Let N= 0.5, 1, 2, 3, 4, and 5, while

fixing

θ

= . ...20

1 Figure 6 Probability density functions for Gamma distributions with different

sample sizes. ...22

Figure 7 show that

AS

₅₀

not be affected by changing

θ

values ...26

Figure 8. Power curve for subgroup sizes 3,4 and 5 when N =3. ...27

Figure 9. wire bonding process...30

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. J

C

Distribution

Skewness

Kurtosis

N(0,1)

W(1,0.5)

87.7200

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9.0000

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3.2451

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2.8803

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(20)

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0.297 0.449 0.519 0.549 0.562 0.566 0.574 0.577 0.581 0.586 0.593

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PDF for Lognormal (0,2) and Normal(7.4,2926.4))

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PDF for Lognormal (0,3/2) and Normal(3.08,80.53))

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PDF for Lognormal(0,1/2) and Normal(1.13,0.36))

& B3 4

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PDF for Gamma(0.5,1) and Normal(0.5,0.5))

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PDF for Gamma(1,1) and Normal(1,1))

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