1©2007 Jan Rohlén, Varians AB

Lektion 3

2007-11-14

Chapter 5

Control Charts for Variables

Sources of variation

• chance, random causes– Random variation

– White noise. Background variation

– In control, stable process

• Systematic, assignable causes– There is a reason

– Out of control

– Process not stable

• The purpose of SPC is to detect and eliminate systematic sources of variation.

Control of variables

• Measurable characteristics of a product.

• Control both average and spread!

-5 0 50

0.1

0.2

0.3

0.4

L U -5 0 50

0.1

0.2

0.3

0.4

L U -5 0 50

0.1

0.2

0.3

0.4

L U

0

0

µ µ

σ σ

=

=0

0

µ µ

σ σ

>

=

0

0

µ µ

σ σ

=

>

2©2007 Jan Rohlén, Varians AB

Control charts

• Allways two diagrams

Control limits

( , )X N µ σ∼

( , )X Nn

σµ∼

UCL

CL

LCL

Time, sampling number2

2

UCL zn

LCL zn

α

α

σµ

σµ

= +

= −

and are known.µ σ

2

3 Popular limits

3.09 Probability limitszα

=

We do neither know µ nor σ!

• X-R-diagram

– No need of computers

– R more intuitive

– Vaste of information

• X-S-diagram

– Computer (calculator) is needed.

– More efficient.

– Standard deviation not very easy to understand.

3©2007 Jan Rohlén, Varians AB

X-R-diagram

1 2ˆ mx x x

xm

µ+ + +

= =…

,1 ,2 ,

1 2

max( ) min( ), 1,...,

i i i n

i

m

i ij ij

x x xx

n

R R RR

m

R x x j n

+ + + =

+ + +

=

= − =

…

…

2

ˆR

dσ =

sampling groups

size of sampling group

m

n

2 2 and are found in

appendix page 725

A d2

2

ˆ 33 R A R

n d n

σ= =

2

2

ˆˆ 3

ˆ

ˆˆ 3

Control limits for -diagram

UCL x A Rn

CL x

LCL x A Rn

x

σµ

µ

σµ

= + = +

= =

= − = −

4

3

ˆ3

ˆ3

Control limits for -diagram

R

R

UCL R D R

CL R

LCL R D R

R

σ

σ

= + =

=

= − =

2 3 4, and are found in

appendix page 725

A D D

Control limits x-R

Exempel:

Steel springs with stiffness 75N/mm. 25 sample groups, each of size 4.

Figure 5.6 shows no evidence against normality.

4©2007 Jan Rohlén, Varians AB

75.1065

0.9440

x

R

=

=

Histogram of spring stiffness

Spring stiffness –ex.

2

2

75.11 0.729 0.9 75.79

75.11

7

44

75.11 0.729 0.944 4.42

Control limits for -diagram

UCL x A R

CL x

LCL x A R

x

= + = + ⋅ =

= =

= − = − ⋅ =

4

3

2.282 0. 2.15

0.944

0

944

ˆ3 0 0.944

Control limits for -diagram

R

UCL D R

CL R

LCL R D R

R

σ

= = ⋅ =

= =

= − = = ⋅ =

5©2007 Jan Rohlén, Varians AB

Is the spring process capable?

75 2

77

75

73

k N mm

USL N mm

T N mm

LSL N mm

= ±

=

= =

Toleransgränser

The spread of spring stiffness

2

The standard deviation is estimated by:

0.944ˆ 0.46

2.059

RN mm

dσ = = =

( ) ( )-5

The percentage defective is estimated by:

p ( 73) ( 77)

73- 75.11 77 - 75.11 1

0.46 0.46

4.59 1 4.11

2.21 10

P x P x= < + >

= Φ + − Φ

= Φ − + − Φ

= ⋅

Process capability

Process Capability Ratio:

6

77 73ˆ 1.48ˆ6 6 0.46

p

p

USL LSLC

USL LSLC

σ

σ

−=

− −= = =

⋅

Process "natural"

tolerance limits:

3

3

UNTL

LNTL

µ σ

µ σ

= +

= −

(More about process capability in ch. 7)

6©2007 Jan Rohlén, Varians AB

Control limits, tolerance limits and

natural tolerance limits

Obs!

There is no connection between

control limits (natural tolerance limits)

and tolerance limits!

Rational sampling groups

diagram controls average (Between group variation)

diagram controls the variation within the group. (Within group variation)

x

R

−

−

time

dim

Estimate from

within group variation!

σ

A comment on R-method

• R=max – min

– Easy, but inefficient for large sample group sizes!

Relative efficiency

2 1.000

3 0.992

4 0.975

5 0.955

6 0.930

10 0.850

n

Not so good to detect small

shifts in process average.

7©2007 Jan Rohlén, Varians AB

R- and other limits

• Probability limits

– α=0.002 � 99.9% och 0.01%

– Välj k=Zα/2=3.09 istället för 3 (medevärdediag.)

– Välj D0.001σ och D0.999σ som R-gränser.

• Standard values

– Warning! Do you really know you proces so well?

Interpretation of diagrams

• Cyclic patterns

• Mixes

• Change in process average

• Trends

• Stratification (to small withingroup variation)

0 20 40 60 80 100 1204

5

6

7

8Dygnsvariation

Timmar

pH

3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.50

20

40

60

80

pH

Ex: 24 hour variation

Suggestion: Use a

process model that

incoorporates the

daily variation.

8©2007 Jan Rohlén, Varians AB

Ex: Cavity variation

0 10 20 30 40 50 60 70 800

5

10

15

-2 0 2 4 6 8 10 12 140

2

4

6

8

Suggestion: Study every

cavity individually.

Ex: Variation in raw material.

0 5 10 15 20 25 30 35 402

4

6

8

10

12

2 3 4 5 6 7 8 9 10 11 120

5

10

15

20

25

Suggestion: Try to decrease

the variation, alternatively

to make the product more

robust.

Ex: Worn tool.

0 10 20 30 40 50 60 705.5

6

6.5

7

7.5

5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 7 7.2 7.40

5

10

15

20

Truncated distribution.

Under estimated

capability

Suggestion: Special SPC

and capability.

9©2007 Jan Rohlén, Varians AB

X-R-diagram and

not normal data

• Avereage diagram rather stable!

– Central limit theorem (thank!)

– If n>4 then it is normally OK. (depends on process)

• The spread diagram is sensitive!

– Unsymmetrical

Ex. Distance to hole center

o

y

x

Normal N(0,σ)

Norm

al N

(0,σ

)

a

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

2 2a x y= +

Rayleigh R(σ)

OC-curve and ARL

-4 -3 -2 -1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Skift i medelvärdet

β a

ccepta

nssannolik

het

OC-kurva för Shewhartdiagram

n=4

n=10

n=40

( ) ( )0( | )

P LCL x UCL k

L k n L k n

β µ µ σ= ≤ ≤ = +

= Φ − − Φ − −

( )1

1

11

1

r

r

ARL rβ ββ

∞−

=

= − =−

∑

1 β−

( )1β β−

( )2 1β β−

Sample 1

Sample 2

Sample 3

10©2007 Jan Rohlén, Varians AB

OC-kurva för R

R-diagram not effective

for small changes in

standard deviation.

Control charts for x and s

• Estimate σ with sample standard deviation.

• Computational aid is needed.

• More effective than R for large sample sizes.

• Variable sample sizes possible.

The Diagram

,1 ,2 ,

1 2

2

,

1

1 2

1( )

1

i i i n

i

m

n

i i j i

j

m

x x xx

n

x x xx

m

s x xn

s s ss

m

=

+ + +=

+ + +=

= −−

+ + +=

∑

…

…

…

3

4

3

4

2

4 4

4

2

4 3

4

-diagram

3

3

-diagram

3 1

3 1

x

sUCL x x A s

c n

CL x

sLCL x x A s

c n

s

sUCL s c B s

c

CL s

sLCL s c B s

c

= + = +

=

= − = −

= + − =

=

= − − =

12

4

4 3 4 3

2 2

11

2

See page 725 for constants

, , ,

n

cnn

c B B A

Γ

= −− Γ

11©2007 Jan Rohlén, Varians AB

x-s diagram for spring stiffness-

example

3

3

4

3

-diagram

75.11 1.628 0.423

-diagram

2.266 0.423

0 0.423

75.79

75.11

74.42

0.96

0.43

0

x

UCL x A s

CL x

LCL x A s

s

UCL B s

CL s

LCL B s

= + = + ⋅ =

= = = − =

= = ⋅ =

= = = = ⋅ =

0 5 10 15 20 2574

74.5

75

75.5

76

Sty

vhet

x-diagram

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1s-diagram

Shewhart when n=1

• Automtized measurement. No sample groups.

• Very low production speed.

• Bulk production. Only measurement error

present.

• Many measurements on the same product.

Variation between barrellsAlcohol (%)

Batchnummer

12©2007 Jan Rohlén, Varians AB

Design of diagram

1

1 2

i i i

m

MR x x

MR MR MRMR

m

−= −

+ + +=

…

2

2

4

-diagram

3

3

MR-diagram

0

x

MRUCL x

d

CL x

MRLCL x

d

UCL D MR

CL MR

LCL

= +

= = −

=

= =

Exempel MR

Problem with Moving Range

• Very sensitive for not normal data

• ARL large for small shifts.

• Alternatives CUMSUM or EWMA

• Montgomery warns!