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Control charts

- 2WS02 Industrial Statistics
- A. Di Bucchianico

Goals of this lecture

- Further discussion of control charts
- variable charts
- Shewhart charts
- rational subgrouping
- runs rules
- performance
- CUSUM charts
- EWMA charts
- attribute charts (c, p and np charts)
- special charts (tool wear charts, short-run

charts)

Statistically versus technically in control

- Statistically in control
- stable over time /
- predictable
- Technically in control
- within specifications

Statistically in control vs technically in control

- statistically controlled process
- inhibits only natural random fluctuations (common

causes) - is stable
- is predictable
- may yield products out of specification
- technically controlled process
- presently yields products within specification
- need not be stable nor predictable

Shewhart control chart

- graphical display of product characteristic which

is important for product quality

Upper Control Limit

Centre Line

Lower Control Limit

Control charts

Basic principles

- take samples and compute statistic
- if statistic falls above UCL or below LCL, then

out-of-control signal

how to choose control limits?

Meaning of control limits

- limits at 3 x standard deviation of plotted

statistic - basic example

UCL

LCL

Example

- diameters of piston rings
- process mean 74 mm
- process standard deviation 0.01 mm
- measurements via repeated samples of 5 rings

yields

Individual versus mean

group means

individual observations

Range chart

- need to monitor both mean and variance
- traditionally use range to monitor variance
- chart may also be based on S or S2
- for normal distribution
- E R d2 E S (Hartleys constant)
- tables exist
- preferred practice
- first check range chart for violations of control

limits - then check mean chart

Design control chart

- sample size
- larger sample size leads to faster detection
- setting control limits
- time between samples
- sample frequently few items or
- sample infrequently many items?
- choice of measurement

Rational subgroups

- how must samples be chosen?
- choose sample size frequency such that if a

special cause occurs - between-subgroup variation is maximal
- within-subgroup variation is minimal.

Strategy 1

- leads to accurate estimate of ?
- maximises between-subgroup variation
- minimises within-subgroup variation

process mean

Strategy 2

- detects contrary to strategy 1 also temporary

changes of process mean

process mean

Phase I (Initial study) in control (1)

Phase I (Initial study) in control (2)

Phase I (Initial Study) not in-control

Trial versus control

- if process needs to be started and no relevant

historic data is available, then estimate µ and ?

or R from data (trial or initial study) - if points fall outside the control limits, then

possibly revise control limits after inspection.

Look for patterns! - if relevant historical data on µ and ? or R are

available, then use these data (control to

standard)

Control chart patterns (1)

- Cyclic pattern,
- three arrows with different weight

Control chart patterns (2)

- Trend,
- course of pin

Control chart patterns (3)

- Shifted mean,
- Adjusted height Dartec

Control chart patterns (4)

- A pattern can give explanation of the cause
- Cyclic ? different arrows, different weight
- Trend ? course of pin
- Shifted mean ? adjusted height Dartec
- Assumption a cause can be verified by a pattern
- The feather of one arrow is damaged ? outliers

below

Phase II Control to standard (1)

Phase II Control to standard (2)

Runs and zone rules

- if observations fall within control limits, then

process may still be statistically

out-of-control - patterns (runs, cyclic behaviour) may indicate

special causes - observations do not fill up space between control

limits - extra rules to speed up detection of special

causes - Western Electric Handbook rules
- 1 point outside 3?-limits
- 2 out of 3 consecutive points outside 2 ? -limits
- 4 out of 5 consecutive points outside 1 ? -limits
- 8 consecutive points on one side of centre line
- too many rules leads to too high false alarm rate

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Warning limits

- crossing 3 ? -limits yields alarm
- sometimes warning limits by adding 2 ? -limits

no alarm but collecting extra information by - adjustment time between taking samples and/or
- adjustment sample size
- warning limits increase detection performance of

control chart

Detection meter stick production

- mean 1000 mm, standard deviation 0.2 mm
- mean shifts from 1000 mm to 0.3 mm?
- how long does it take before control chart

signals?

Performance of control charts

- expressed in terms of time to alarm (run length)
- two types
- in-control run length
- out-of-control run length

Statistical control and control charts

- statistical control observations
- are normally distributed with mean ? and

variance ?2 - are independent
- out of (statistical) control
- change in probability distribution
- observation within control limits
- process is considered to be in control
- observation beyond control limits
- process is considered to be out-of-control

In-control run length

- process is in statistical control
- small probability that process will go beyond 3 ?

limits (in spite of being in control) - false

alarm! - run length is first time that process goes beyond

3 ? limits - compare with type I error

Out-of-control run length

- process is not in statistical control
- increased probability that process will go beyond

3 ? limits (in spite of being in control) - true

alarm! - run length is first time that process goes beyond

3 sigma limits - until control charts signals, we make type II

errors

Metrics for run lengths

- run lengths are random variables
- ARL Average Run Length
- SRL Standard Deviation of Run Length

Run lengths for Shewhart Xbar- chart

- in-control p 0.0027

- time to alarm follows geometric distribution
- mean 1/p 370.4
- standard deviation (?(1-p))/p 369.9

Geometric distribution

Numerical values

- Shewhart chart for mean (n1)
- single shift of mean

Scale in Statgraphics

- Are our calculations wrong???

Sample size and run lengths

- increase of sample size corresponding control

limits - same in-control run length
- decrease of out-of-control run length

Numerical values

- Shewhart chart for mean (n5)
- single change of standard deviation (? - c?)

Runs rules and run lengths

- in-control run length decreases (why?)
- out-of-control run length decreases (why?)

Performance Shewhart chart

- in-control run length OK
- out-of-control run length
- OK for shifts 2 standard deviation group

average - Bad for shifts average
- extra run tests
- decrease in-control length
- decrease out-of-control length

CUSUM Chart

- plot cumulative sums of observation

CUSUM tabular form

- assume
- data normally distributed with known ?
- individual observations

Choice K and H

- K is reference value (allowance, slack value)
- C measures cumulative upward deviations of µ0K
- C- measures cumulative downward deviations of

µ0-K - for fast detection of change process mean µ1
- K½ µ0- µ1
- H5? is good choice

CUSUM V-mask form

UCL

CL

change point

LCL

Drawbacks V-mask

- only for two-sided schemes
- headstart cannot be implemented
- range of arms V-mask unclear
- interpretation parameters (angle, ...) not well

determined

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Rational subgroups and CUSUM

- extension to samples
- replace ? by ?/?n
- contrary to Shewhart chart , CUSUM works best

with individuals

Combination

- CUSUM charts appropriate for small shifts (
- CUSUM charts are inferior to Shewhart charts for

large shifts(1.5?) - use both charts simultaneously with 3.5?

control limits for Shewhart chart

Headstart (Fast Initial Response)

- increase detection power by restart process
- esp. useful when process mean at restart is not

equal at target value - set C0 and C-0 to non-zero value (often H/2 )
- if process equals target value µ0 is, then CUSUMs

quickly return to 0 - if process mean does not equal target value µ0,

then faster alarm

CUSUM for variability

- define Yi (Xi-µ0)/ ? (standardise)
- define Vi (?Yi-0.822)/0.349
- CUSUMs for variability are

Exponentially Weighted Moving Average chart

- good alternative for Shewhart charts in case of

small shifts of mean - performs almost as good as CUSUM
- mostly used for individual observations (like

CUSUM) - is rather insensitive to non-normality

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Why control charts for attribute data

- to view process/product across several

characteristics - for characteristics that are logically defined on

a classification scale of measure - N.B. Use variable charts whenever possible!

Control charts for attributes

Attributes are characteristics which have a

countable number of possible outcomes.

- Three widely used control charts for attributes
- p-chart fraction non-conforming items
- c-chart number of non-conforming items
- u-chart number of non-conforming items per unit
- For attributes one chart only suffices (why?).

p-chart

Number of nonconforming products is binomially

distributed

sample fraction of nonconforming

mean

variance

p-chart

average of sample fractions

Fraction Nonconforming Control Chart

Assumptions for p chart

- item is defect or not defect (conforming or

non-conforming) - each experiment consists of n repeated

trials/units - probability p of non-conformance is constant
- trials are independent of each other

c-chart

- Counts the number of non-conformities in sample.
- Each non-conforming item contains at least one

non-conformity (cf. p chart). - Each sample must have comparable opportunities

for non-conformities - Based on Poisson distribution
- Prob( nonconf. k)

c-chart

- Poisson distribution meanc and variancec

Control Limits for Nonconformities

is average number of nonconformities in sample

u-chart

- monitors number of non-conformities per unit.

- n is number of inspected units per sample
- c is total number of non-conformities

Control Chart for Average Number of

Non-conformities Per Unit

Moving Range Chart

- use when sample size is 1
- indication of spread moving range
- Situations
- automated inspection of all units
- low production rate
- expensive measurements
- repeated measurements differ only because of

laboratory error

Moving Range Chart

- calculation of moving range
- d2, D3 and D4 are constants depending number of

observations

individual measurements

moving range

Example Viscosity of Aircraft Primer Paint

Viscosity of Aircraft Primer Paint

- since a moving range is calculated of n2

observations, d21.128, D30 and D43.267

CC for individuals

CC for moving range

Viscosity of Aircraft Primer Paint

X

MR

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Tool wear chart

- known trend is removed (regression)
- trend is allowed until maximum
- slanted control limits

USL

UCL

reset

LCL

LSL

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Pitfalls

- bad measurement system
- bad subgrouping
- autocorrelation
- wrong quality characteristic
- pattern analysis on individuals/moving range
- too many run tests
- too low detection power (ARL)
- control chart is not appropriate tool (small

ppms, incidents, ...) - confuse standard deviation of mean with individual