Title: Control charts
1Control charts
- 2WS02 Industrial Statistics
- A. Di Bucchianico
2Goals 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)
3Statistically versus technically in control
- Statistically in control
- stable over time /
- predictable
- Technically in control
- within specifications
4Statistically 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
5Shewhart control chart
- graphical display of product characteristic which
is important for product quality
Upper Control Limit
Centre Line
Lower Control Limit
6Control charts
7Basic principles
- take samples and compute statistic
- if statistic falls above UCL or below LCL, then
out-of-control signal
how to choose control limits?
8Meaning of control limits
- limits at 3 x standard deviation of plotted
statistic - basic example
UCL
LCL
9Example
- diameters of piston rings
- process mean 74 mm
- process standard deviation 0.01 mm
- measurements via repeated samples of 5 rings
yields
10Individual versus mean
group means
individual observations
11Range 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
12Design 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
13Rational 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.
14Strategy 1
- leads to accurate estimate of ?
- maximises between-subgroup variation
- minimises within-subgroup variation
process mean
15Strategy 2
- detects contrary to strategy 1 also temporary
changes of process mean
process mean
16Phase I (Initial study) in control (1)
17Phase I (Initial study) in control (2)
18Phase I (Initial Study) not in-control
19Trial 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)
20Control chart patterns (1)
- Cyclic pattern,
- three arrows with different weight
21Control chart patterns (2)
22Control chart patterns (3)
- Shifted mean,
- Adjusted height Dartec
23Control 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
24Phase II Control to standard (1)
25Phase II Control to standard (2)
26Runs 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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28Warning 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
29Detection 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?
30Performance of control charts
- expressed in terms of time to alarm (run length)
- two types
- in-control run length
- out-of-control run length
31Statistical 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
32In-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
33Out-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
34Metrics for run lengths
- run lengths are random variables
- ARL Average Run Length
- SRL Standard Deviation of Run Length
35Run lengths for Shewhart Xbar- chart
- time to alarm follows geometric distribution
- mean 1/p 370.4
- standard deviation (?(1-p))/p 369.9
36Geometric distribution
37Numerical values
- Shewhart chart for mean (n1)
- single shift of mean
38Scale in Statgraphics
- Are our calculations wrong???
39Sample size and run lengths
- increase of sample size corresponding control
limits - same in-control run length
- decrease of out-of-control run length
40Numerical values
- Shewhart chart for mean (n5)
- single change of standard deviation (? - c?)
41Runs rules and run lengths
- in-control run length decreases (why?)
- out-of-control run length decreases (why?)
42Performance 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
43CUSUM Chart
- plot cumulative sums of observation
44CUSUM tabular form
- assume
- data normally distributed with known ?
- individual observations
45Choice 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
46CUSUM V-mask form
UCL
CL
change point
LCL
47Drawbacks 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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49Rational subgroups and CUSUM
- extension to samples
- replace ? by ?/?n
- contrary to Shewhart chart , CUSUM works best
with individuals
50Combination
- 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
51Headstart (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
52CUSUM for variability
- define Yi (Xi-µ0)/ ? (standardise)
- define Vi (?Yi-0.822)/0.349
- CUSUMs for variability are
53Exponentially 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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56Why 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!
57Control 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?).
58p-chart
Number of nonconforming products is binomially
distributed
sample fraction of nonconforming
mean
variance
59p-chart
average of sample fractions
Fraction Nonconforming Control Chart
60Assumptions 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
61c-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)
62c-chart
- Poisson distribution meanc and variancec
Control Limits for Nonconformities
is average number of nonconformities in sample
63u-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
64Moving 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
65Moving Range Chart
- calculation of moving range
- d2, D3 and D4 are constants depending number of
observations
individual measurements
moving range
66Example Viscosity of Aircraft Primer Paint
67Viscosity 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
68Viscosity of Aircraft Primer Paint
X
MR
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70Tool wear chart
- known trend is removed (regression)
- trend is allowed until maximum
- slanted control limits
USL
UCL
reset
LCL
LSL
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72Pitfalls
- 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