4 5 6 7 8 9 10 11 12 13 14 15 Subgroup No.

1 2 3 Date 12/26

Control chart (Ex 3-2)

Time 8:50 Measurement X1 X2 X3 35 40 32 X4 37 Average X-bar Range R Comment 11:30 46 37 36 41 1:45 3:45 34 69 40 64 34 68 36 59 6.65

0.1

New, temporary operator 4:20 8:35 38 42 34 41 44 43 40 34 12/27 12/28 9:00 9:40 1:30 2:50 8:30 1:35 2:25 2:35 44 33 48 47 38 37 40 38 41 41 44 43 41 37 38 39 41 38 47 36 39 41 47 45 46 36 45 42 38 37 35 42 3:35 50 42 43 45

1

Control chart (Ex 3-2)

20 21 22 23 24 Subgroup No.

16 17 18 19 25 Date 12/29 12/30 Time 8:25 9:25 11:00 2:35 3:15 9:35 10:20 11:35 2:00 4:25 Measurement X1 33 X2 35 X3 29 X4 39 Average X-bar 41 38 40 44 29 28 34 58

6.42

35 56 38 41 55 40 37 45 45 38 48 37

6.51

Range R

0.3

0.11

Comment Damaged oil line Bad material 39 42 43 39 42 39 36 38 35 39 35 43 40 36 38 44

Sum R=2.19

Sum X-bar =160.25

2

Control chart (Ex 3-2)

3

Revised Central Lines

X

new

 

X



X

d d

and R

new

 

where g X

d

R

d d



discarded subgroup averages



number of discarded subgroups



discarded subgroup ranges



d d

4

Standard Values

X

0 

X

new

R

0 

R

new

and

 0 

R

0

d

2

UCL X LCL X



X

0 

A

 0 

X

0 

A

 0

UCL R



D

2  0

LCL R



D

1  0 5

6

State of Control

7

State of Control

When a point (subgroup value) falls outside its control limits, the process is out of control.

Out of control means a change in the process due to a special cause. A process can also be considered out of control even when the points fall inside the 3ơ limits Besterfield Quality Control 8th Ed 8

State of Control

 It is not natural for seven or more consecutive points to be above or below the central line.

 Also when 10 out of 11 points or 12 out of 14 points are located on one side of the central line, it is unnatural.

 Six points in a row are steadily increasing or decreasing indicate an out of control situation Besterfield Quality Control 8th Ed 9

Patterns in Control Charts

Besterfield Quality Control 8th Ed 10

Patterns in Control Charts

Figure 5-13 Simplified rule for out-of-control pattern Besterfield Quality Control 8th Ed 11

Out-of-Control Condition

1. Change or jump in level.

2. Trend or steady change in level 3. Recurring cycles 4. Two populations (also called mixture) 5. Mistakes Besterfield Quality Control 8th Ed 12

Out-of-Control Patterns

Change or jump in level Trend or steady change in level Recurring cycles Two populations Besterfield Quality Control 8th Ed 13

14

Individual values VS Averages

Comparison of individual values compared to averages 15

Individual values VS Averages

Calculations of the average for both the individual values and for the subgroup averages are the same.

However the sample standard deviation is different.



x

 

n

where   population

x

  population standard deviation standard deviation of subgroup of individual averages values n  subgroup size if we assume normality, then the population standard deviation

s

 can be estimated standard deviation from : of samples,  ˆ 

s c

4 c 4  factor (in Table B) for computing if

n

 20

, c

4  4

(n

 1

) (

4

n

 3

)

 ˆ 16

Central Limit Theorem

If the population from which samples are taken is not normal, the distribution of sample averages will tend toward normality provided that the sample size,

n

, is at least 4. This tendency gets better and better as the sample size gets larger. 17

Central Limit Theorem

Illustration of central limit theorem 18

Central Limit Theorem

Dice illustration of central limit theorem 19

Control Limits & Specifications

Figure 5-21 Relationship of limits, specifications, and distributions 20

Control Limits & Specifications

 The control limits are established as a function of the average (ค่า control limits เป็นค่าที่ค านวน จากค่าเฉลี่ยที่ได้จากการเก็บข้อมูล)  Specifications are the permissible variation in the size of the part and are, therefore, for individual values ของข้อมูลแต่ละตัว) (ค่า Specifications เป็นค่าที่ ก าหนดขึ้นเพื่อใช ้เป็นขอบเขตของการกระจายส าหรับค่า  The specifications or tolerance limits are established by design engineers or customers to meet a particular function 21

Process Capability & Tolerance

 The process spread will be referred to as the process capability and is equal to 6 σ  The difference between specifications is called the tolerance  When the tolerance is established by the design engineer without regard to the spread of the process, undesirable situations can result 22

Process Capability & Tolerance

Three situations are possible:  Case I: When the process capability is less than the tolerance 6 σUSL-LSL 23

Process Capability & Tolerance

Case I: When the process capability is less than the tolerance 6 σ

Process Capability & Tolerance

Case II: When the process capability is less than the tolerance 6 σ=USL-LSL Case I 6σ=USL-LSL 25

Process Capability & Tolerance

Case III: When the process capability is less than the tolerance 6 σ>USL-LSL Case I 6σ>USL-LSL 26

Capability Index

Process capability and specifications or tolerance are combined to form the

capability index, C p .

C p



USL

 6  0

LSL

where

C p

 capability index

USL



LSL

 tolerance or specificat ion 6  0  process capability  0  standard deviation of averages of subgroup after the averages due to designable causes are discarded 27

Capability Index

The capability index does not measure process performance in terms of the nominal or target value. This measure is accomplished by C pk .

C pk



Min

{(

USL



X

3 

o

), (

X



LSL

) where

C p k

USL  capability index LSL  tolerance or specificat ion 28

Capability Index

1. The C p value does not change as the process center changes 2. C p = C pk when the process is centered 3. C pk is always equal to or less than C p 4. A C pk = 1 (and C p = 1.33) is a de facto standard. It indicates that the process is producing product that conforms to specifications 5. A C pk < 1 indicates that the process is producing product that does not conform to specifications 29

Capability Index

6. A C p < 1 indicates that the process is not capable 7. A C pk = 0 indicates the average is equal to one of the specification limits 8. A negative C pk value indicates that the average is outside the specifications 30

C

pk

Measures

C pk = negative number C pk = zero C pk = between 0 and 1 C pk = 1 (and C p = 1) C pk > 1

31

1-How to estimate Process Capability

This following method of calculating the process capability

assumes that the process is stable or in statistical control

:  Take 25 (g) subgroups of size 4 for a total of 100 measurements   0 

R d

2   Calculate the range,

R

, for each subgroup

R

ΣR/g Calculate the estimate of the population standard deviation using: 

o



R d

2  Process capability will equal 6 σ 0 32

2-How to estimate Process Capability

The process capability can also be obtained by using the standard deviation:  Take 25 (g) subgroups of size 4 for a total of 100 measurements  Calculate the sample standard deviation,

s

, for each subgroup   Calculate the estimate of the population standard

s

Σs/g deviation 

o



s c

4  Process capability will equal 6 σ o 33

Capability Index (EX. 3-3)

1.

2.

3.

4.

A new process is started, and the sum of the sample standard deviations for 25 groups of size 4 is 105. Approximate the process capability.

Determine the capability index before ( σ o = 0.038

) and after ( 0.03) improvement using specification of 6.40 ± 0.15.

σ o = What is the C pk value after improvement for Question 1 when the process center is 6.40? When the process center is 6.30?

A new process is started, and the sum of the sample standard deviations for 25 subgroups of size 4 is 750. If the specifications are 700 ± 80, what is the process capability index? What action would you recommend?

34

Additional control charts

1. Standard deviation chart (or s chart) • This chart is nearly the same X-bar and R chart. However, for subgroup sizes ≥ 10, an s chart is more accurate than an R Chart 2. Moving average and Moving range chart • This chart is used to combine a number of individual values and plot them on the chart. This technique is quite common in the chemical industry, where only one reading (datum) is possible at a time ( Can chemical engineering students give an example?) 3. Exponential Weighted Moving-Average (EWMA) chart • The EWMA chart gives the greatest weight to the most recent data and less weight to all previous data. It primary advantage is the ability to detect small shifts in the process average; however, it does not react as quickly to large shifts as the X-bar chart.

35

S Control Chart

For subgroup sizes ≥10, an s chart is more accurate than an R Chart. Trial control limits are given by:

x

 

i g

 1

x g s

 

g i

 1

s

,

g UCL x



x



A

3

s UCL s LCL x



x



A

3

s LCL s



B

4

s



B

3

s where

A 3 , B 3 , B 4  factors for the control limits found in Table B 36

Revised Limits for

x

0

s

0  

UCL x LCL x x new s new

   

S control chart



g x

 

g d x d



g s



g d s d

,  0 

s

0

c

4

x

0

x

0  

A

 0

UCL s A

 0

LCL s

 

B

6  0

B

5  0

where s d

 discarded subgroup averages c 4 , A, B 5 , B 6  factors found in Table B 37

Subgroup No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Date 12/26 12/27 12/28

S chart (Ex. 3-4)

Measurement Average S.D.

Time X1 X2 6.40

X3 6.32

X4 6.37

X-bar 6.35

S 0.034

8:50 6.35

6.37

6.36

6.41

6.46

0.045

11:30 6.46

6.40

6.34

6.36

6.34

0.028

1:45 3:45 6.34

6.69

6.64

6.34

6.68

6.44

6.59

6.40

6.69

6.38

0.045

0.042

4:20 6.38

6.41

6.43

6.34

6.42

0.041

8:35 6.42

6.41

6.41

6.46

6.44

0.024

9:00 6.44

6.41

6.38

6.36

6.33

0.034

9:40 6.33

6.44

6.47

6.45

6.48

0.018

1:30 6.48

6.43

6.36

6.42

6.47

0.045

2:50 6.47

6.41

6.39

6.38

6.38

0.014

8:30 6.38

6.37

6.41

6.37

6.37

0.020

1:35 6.37

6.38

6.47

6.35

6.40

0.051

2:25 2:35 6.40

6.38

6.39

6.45

6.42

6.38

0.032

3:35 6.50

0.036

6.50

6.42

6.43

6.45

Comment New, temporary operator

38

S chart (Ex. 3-4)

22 23 24 25 18 19 20 21 Subgroup No.

16 17 Date 12/29 12/30 Time 8:25 9:25 11:00 2:35 3:15 9:35 10:20 11:35 2:00 4:25 Measurement X1 X2 X3 6.33

6.41

6.35

6.4

6.29

6.29

6.38

6.35

6.56

6.38

6.44

6.41

6.55

6.4

6.28

6.37

6.45

6.45

6.39

6.42

6.43

6.39

6.42

6.39

6.36

6.38

6.35

6.39

6.35

6.43

X4 6.39

6.34

6.58

6.38

6.48

6.37

6.4

6.36

6.38

6.44

Average X-bar 6.34

6.36

6.42

6.37

6.51

6.40

6.39

6.39

6.38

6.41

Range S 0.042

0.056

0.125

0.025

0.054

0.036

0.029

0.024

0.036

0.029

Comment Damaged oil line Bad material

39

Moving average and Moving range chart

This chart is used to combine a number of individual values and plot them on the chart. This technique is quite common in the chemical industry, where only one reading (datum) is possible at a time Value 35 26 28 32 36 .

.

.

.

Three-period moving sum 35+26+28 =89 86 96 .

.

.

.

X-bar (35+26+28)/3 =29.6

28.6

32 .

.

.

.

S x-bar = R 35-26 = 9 6 S R = .

.

8 .

.

40

Exponential Weighted Moving Average (EWMA) chart

The EWMA is defined by the equation : where

V t V t

 

X t

 ( 1   )

V t

 1  the EWMA of the most recent plotted point

V t

 1   the EWMA of the previous plotted point  the weight given to the subgroup average or individual value

X t

 the subgroup average or individual value 41

Exponential Weighted Moving Average (EWMA) chart

The value of lamda,  , should be between 0.05

and 0.25, with lower valu es giving a better ability to detect smaller shifts.

Values of 0.08, 0.10, and 0.15

work well In order to start the sequential calculatio ns,

V t-

1 , is equal to

X

.

UCL



X



A

2

R

 ( 2   )

LCL



X



A

2

R

 ( 2   ) 42

Exponential Weighted Moving-Average (EWMA) chart (EX. 3-5) 43

Exponential Weighted Moving-Average (EWMA) chart 44

Quiz 3-1

Control charts for X-bar and R are to be established on a certain dimension part, measured in millimeters. Data were collected in subgroup sizes of 6 and are given below. Determine the trial central and control limits. Assume assignable causes and revise the central line and limits.

Subgroup Number 1 2 3 4 5 6 7 8 9 10 11 12 13 X-bar 20.35

20.40

20.36

20.65

20.20

20.40

20.43

20.37

20.48

20.42

20.39

20.38

20.4

R 0.34

0.36

0.32

0.36

0.36

0.35

0.31

0.34

0.30

0.37

0.29

0.30

0.33

Subgroup Number 14 15 16 17 18 19 20 21 22 23 24 25 X-bar 20.41

20.45

20.34

20.36

20.42

20.50

20.31

20.39

20.39

20.40

20.41

20.40

R 0.36

0.34

0.36

0.37

0.73

0.38

0.35

0.38

0.33

0.32

0.34

0.30

45

Quiz 3-2

Control charts for X-bar and s are to be established on the Brinell hardness of hardened tool steel in kilograms per square millimeter. Data for subgroup sizes of 8 are shown below. Determine the trail central lime and control limits for the X-bar and s charts. Assume that the out-of-control points have assignable causes. Calculate the revised limits and central line.

Subgroup Number 1 2 3 4 5 6 7 8 9 10 11 12 13 X-bar 540 534 545 561 576 523 571 547 584 552 541 545 546 S.D.

26 23 24 27 25 50 29 29 23 24 28 25 26 Subgroup Number 14 15 16 17 18 19 20 21 22 23 24 25 X-bar 551 522 579 549 508 569 574 563 561 548 556 553 S.D.

24 29 26 28 23 22 28 33 23 25 27 23 46

Quiz 3-3

Use data in Quiz 5-2 to establish EWMA chart, using  = 0.1

Subgroup Number 1 2 3 4 5 6 7 8 9 10 11 12 13 X-bar 20.35

20.40

20.36

20.65

20.20

20.40

20.43

20.37

20.48

20.42

20.39

20.38

20.4

R 0.34

0.36

0.32

0.36

0.36

0.35

0.31

0.34

0.30

0.37

0.29

0.30

0.33

Subgroup Number 14 15 16 17 18 19 20 21 22 23 24 25 X-bar 20.41

20.45

20.34

20.36

20.42

20.50

20.31

20.39

20.39

20.40

20.41

20.40

R 0.36

0.34

0.36

0.37

0.73

0.38

0.35

0.38

0.33

0.32

0.34

0.30

47