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Control charts

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k. Statistically in control vs technically in control. statistically ... k. Shewhart control chart ... k. Range chart. need to monitor both mean and variance ... – PowerPoint PPT presentation

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Title: Control charts


1
Control charts
  • 2WS02 Industrial Statistics
  • A. Di Bucchianico

2
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)

3
Statistically versus technically in control
  • Statistically in control
  • stable over time /
  • predictable
  • Technically in control
  • within specifications

4
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

5
Shewhart control chart
  • graphical display of product characteristic which
    is important for product quality

Upper Control Limit
Centre Line
Lower Control Limit
6
Control charts
7
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?
8
Meaning of control limits
  • limits at 3 x standard deviation of plotted
    statistic
  • basic example

UCL
LCL
9
Example
  • diameters of piston rings
  • process mean 74 mm
  • process standard deviation 0.01 mm
  • measurements via repeated samples of 5 rings
    yields

10
Individual versus mean
group means
individual observations
11
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

12
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

13
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.

14
Strategy 1
  • leads to accurate estimate of ?
  • maximises between-subgroup variation
  • minimises within-subgroup variation

process mean
15
Strategy 2
  • detects contrary to strategy 1 also temporary
    changes of process mean

process mean
16
Phase I (Initial study) in control (1)
17
Phase I (Initial study) in control (2)
18
Phase I (Initial Study) not in-control
19
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)

20
Control chart patterns (1)
  • Cyclic pattern,
  • three arrows with different weight

21
Control chart patterns (2)
  • Trend,
  • course of pin

22
Control chart patterns (3)
  • Shifted mean,
  • Adjusted height Dartec

23
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

24
Phase II Control to standard (1)
25
Phase II Control to standard (2)
26
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

27
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28
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

29
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?

30
Performance of control charts
  • expressed in terms of time to alarm (run length)
  • two types
  • in-control run length
  • out-of-control run length

31
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

32
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

33
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

34
Metrics for run lengths
  • run lengths are random variables
  • ARL Average Run Length
  • SRL Standard Deviation of Run Length

35
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

36
Geometric distribution
37
Numerical values
  • Shewhart chart for mean (n1)
  • single shift of mean

38
Scale in Statgraphics
  • Are our calculations wrong???

39
Sample size and run lengths
  • increase of sample size corresponding control
    limits
  • same in-control run length
  • decrease of out-of-control run length

40
Numerical values
  • Shewhart chart for mean (n5)
  • single change of standard deviation (? - c?)

41
Runs rules and run lengths
  • in-control run length decreases (why?)
  • out-of-control run length decreases (why?)

42
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

43
CUSUM Chart
  • plot cumulative sums of observation

44
CUSUM tabular form
  • assume
  • data normally distributed with known ?
  • individual observations

45
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

46
CUSUM V-mask form
UCL
CL
change point
LCL
47
Drawbacks V-mask
  • only for two-sided schemes
  • headstart cannot be implemented
  • range of arms V-mask unclear
  • interpretation parameters (angle, ...) not well
    determined

48
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49
Rational subgroups and CUSUM
  • extension to samples
  • replace ? by ?/?n
  • contrary to Shewhart chart , CUSUM works best
    with individuals

50
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

51
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

52
CUSUM for variability
  • define Yi (Xi-µ0)/ ? (standardise)
  • define Vi (?Yi-0.822)/0.349
  • CUSUMs for variability are

53
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

54
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55
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56
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!

57
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?).

58
p-chart
Number of nonconforming products is binomially
distributed
sample fraction of nonconforming
mean
variance
59
p-chart
average of sample fractions
Fraction Nonconforming Control Chart
60
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

61
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)

62
c-chart
  • Poisson distribution meanc and variancec

Control Limits for Nonconformities
is average number of nonconformities in sample
63
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
64
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

65
Moving Range Chart
  • calculation of moving range
  • d2, D3 and D4 are constants depending number of
    observations

individual measurements
moving range
66
Example Viscosity of Aircraft Primer Paint
67
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
68
Viscosity of Aircraft Primer Paint
X
MR
69
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70
Tool wear chart
  • known trend is removed (regression)
  • trend is allowed until maximum
  • slanted control limits

USL
UCL
reset
LCL
LSL
71
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72
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
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