Statistical Process Control

Control charts

Quality

•Quality is an old concept•Artisan’s or craftsmen’s work characterised by qualities such as strength, beauty or finish.•In mass production, reproducibility is a big issue•Particularly dimensions of component parts•Quality was obtained through complete inspection•In 1931, Walter Shewhart published ‘Economic Control of Quality of Manufactured Product ‘•Foundation of modern Statistical Process Control (SPC)

Process variability

• Basketball player shoots 100 FT/day• Typical sequence 84/100, 67/100, 77/100• All processes have this type of variability• Process variation has two components:a) Natural variation, common cause or

system variation (fluctuates around long run %)

b) Special cause or assignable cause variation (e.g. player has hand injury)

Common causes

• Deming – 85-94% of problems are common cause• System is the responsibility of management

•Inappropriate procedures. •Poor design. •Poor maintenance of machines. •Lack of clearly defined standard operating procedures. •Poor working conditions,e.g. lighting, noise, dirt, temperature, ventilation. •Machines not suited to the job. •Substandard raw materials.

•Measurement error. •Vibration in industrial processes. •Ambient temperature and humidity. •Insufficient training. •Normal wear and tear. •Variability in settings. •Computer response time

Special causes

• Workforce responsibility

• Can be addressed in the short term

•Operator absent. •Poor adjustment of equipment. •Operator falls asleep. •Faulty controllers. •Machine malfunction. •Computer crashes. •Poor batch of raw material. •Power surges.

Basic idea of control charting

• Draw samples from process of interest at a sequence of time points

• Choose a statistic, such as sample mean or sample proportion to describe state of process

• Values of statistic plotted against time

Example: Monitoring thermostat production

• 4 thermostats per hour sampled from assembly line and tested

• Test temperature = 75.• Past experience shows thermostat response

varies from test temp with s.d. = 0.5.• How should monitoring be done?

Control chart format

• Center line = aim of process = • Control limits = range of normal operation• Control limits decided by common cause

variability ()

Identifying special causes

•Limits chosen to make excursions rare (e.g. 1 in 1000)•Many other ways in which process can be out of control

Setting control limits(, known)•e.g. Monitoring hourly averages

997.033

n

Xn

P

n

NX t

2

,~

n

n1

2 3 4

UCL = n = 75 +3 x 0.5/ 4 = 75.75LCL = n = 75 - 3 x 0.5/ 4 = 74.25

Estimating process parameters

• Need to know common cause variability (for limits)• Special causes will disturb estimate of variability• Observations close in time are likely to differ in only

common cause variation• Choose a rational subgroup of observations

12 3 4

Within subgroup variation

• Common cause variation () = variability within subgroups

• Estimate from within subgroup standard deviation (S)

• E(S) = an

n 2 3 4 5 6 7 8 9 10 an .780 .886 .921 .940 .952 .959 .965 .969 .973

•If we have k subgroups, we pool s.d.’s to get overall estimate:

s.d. process of estimate unbiased ,/ˆ

1

1

n

k

ii

aS

Sk

S

Between subgroup variation

• Variability across (between) subgroups is given by changes in subgroup means

• Changes in subgroup mean are measured relative to process mean ( if known)

linecenter as k

1x use unknown, If

1

k

iix

•A point lies outside control limits when its between subgroup variation is large. •We conclude that this is due to a special cause

Example: Monitoring measurement process

• Testing Measurement method for chemical assay (% solids in chemical)

• Take a sample. Split into 3 parts. Make 3 separate measurements.

• 17 different samples.All from same source.

• What is being measured?

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

8.8 4.3 5.8 7.5 8.2 5.1 7.5 6.6 6.3 4.9 6.0 7.6 7.0 5.8 7.5 5.0 7.5

9.2 3.3 6.1 6.2 7.3 7.0 7.7 7.9 7.3 4.7 6.2 7.3 6.5 7.1 5.9 6.1 6.3

9.1 3.8 7.6 6.0 7.2 6.9 7.6 5.7 7.3 5.4 6.8 6.2 5.7 5.2 6.0 5.7 7.4

Data from 17 samples

Measurement process for chemical assays

9.0 3.1 6.5 6.6 7.6 6.3 7.6 6.7 7.0 5.0 6.3 7.0 6.4 6.0 6.5 5.6 7.

Average  

0.21 0.50 0.96 0.81 0.55 1.07 0.10 1.11 0.58 0.36 0.42 0.74 0.66 0.97 0.90 0.56 0.67

Standard Deviation (s)

0.40 1.00 1.80 1.50 1.00 1.90 0.20 2.20 1.00 0.70 0.80 1.40 1.30 1.90 1.60 1.10 1.20

Range (R)

Calculation of control limits

• First estimate process mean: 51.6x

• Then estimate process s.d.:

74.0886.0/66.0/ˆ naS

22.53/74.0351.63/ˆ3 LCL

80.73/74.0351.63/ˆ3 UCL

x

x

Sometimes range in used : E(R) = bn

Control chart for measurement data

•Solid evidence process in not in control

Common types of special causes

•Point out of limits

• Run of points above or below centerline

Need to takeimmediate action

Common types of special causes

•run of points going up or down

• Repetitive patterns observed

Use of control charts• Charts can be done manually or automatically• Crucial: What happens next?• Control charts don't improve processes, people do • Incidents need to be recorded in real time • Need to track down root cause and remove it• Don’t confuse specification limits with control charts• Specifications – what is desirable• Control limits – what is currently possible• Specifications pertain to individual items• Control limits pertain to averages

Effectiveness of control charts• Two types of mistakes possible• Type I error: Stopping the process when it is in

control.• In American or Japanese system:

3Xor 3XP error) I P(Type ttnn

003.0997.01331

n

Xn

P

In British system:

002.0998.0109.309.31

n

Xn

P

Type II errors = 1- Power

• Not stopping the process when it is out of control (e.g. process mean has shifted)

•

n

Xn

P 33 00

nZnP 33

0.25 0.5 0.75 1.0 1.5 2.0 2.5 3.0

when n = 4 0.99 0.97 0.93 0.84 0.5 0.15 0.06 0.001

Average Run Length (ARL)

• ARL = (Avg.) time needed to identify an out-of-limits signal

limits) control outside is ( sXPp limits control outside is for which sfirst sXY

pYEpGeomY /1)();(~

If process is in control, p =

26.384026.0

111)E( ARL

pY

ARL for out of control process

• If process is out of control,e.g.

• p = 1- Y) = 1/p, e.g.

0.25 0.5 0.75 1.0 1.5 2.0 2.5 3.0

when n = 4 0.99 0.97 0.93 0.84 0.5 0.15 0.06 0.001

ARL 156 44 15 6 2 1.19 1.07 1.001

•Chart takes very long to detect is shift is small•CUSUM charts have shorter ARL

Full SPC cycle

• Identify special causes of variation• Bring process into statistical control• Reduce common cause effects through process

improvement (narrow control limits towards centerline)

Control Charts for Process Variation

• X bar charts look to control the central tendency of a process

• Equally important to control the process variation

• Many practitioners first control variation and then location

• Variation measured by S (historically R)

Sampling distribution of s2

• What is it’s sampling distribution ?

variancesample theis )(1

1

1

22

n

ii xx

ns

then ),N( i.i.d are If 2iX 21

2 ~ iX

212

2

~)1(

n

sn

then)1,0N( i.i.d are If iX

etc. ~ 22

22

21 XX

)N(0, i.id. Y ,... 2i

21

22

21

2 nYYYs

•Sums of squares of i.i.d normals are chi-squared with as many d.f. as there are terms.

Density of Z

X

f(x)

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4

Density of Z^2

X

f(x)

0 1 2 3 4

01

23

4

Chisquared densities

X

f(x)

0 20 40 60 80 100

0.0

0.0

50

.10

0.1

5

0 1 2 3 4 5

01

00

20

03

00

samplevars

Sampling Distribution of sample variances , X~N(0,1), n=5

Control limits for S chart

• E(S2) = 2 (S2 is an unbiased estimator of 2)

• V(S) = E(S2) – (E(S))2 = 2 – (an2

2( ) 1S nV S a

2

2

LCL = 3 1 /

UCL = 3 1 /

n n

n n

s s a a

s s a a

variancesample theis )(1

1

1

22

n

ii xx

nS

Normal approx:

S chart for Measurement data