Control charts

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Control charts

• 2WS02 Industrial Statistics
• A. Di Bucchianico

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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)

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Statistically versus technically in control

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

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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

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Shewhart control chart

• graphical display of product characteristic which
  is important for product quality

Upper Control Limit
Centre Line
Lower Control Limit
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Control charts
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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?
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Meaning of control limits

• limits at 3 x standard deviation of plotted
  statistic
• basic example

UCL
LCL
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Example

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

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Individual versus mean
group means
individual observations
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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

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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

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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.

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Strategy 1

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

process mean
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Strategy 2

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

process mean
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Phase I (Initial study) in control (1)
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Phase I (Initial study) in control (2)
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Phase I (Initial Study) not in-control
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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)

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Control chart patterns (1)

• Cyclic pattern,
• three arrows with different weight

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Control chart patterns (2)

• Trend,
• course of pin

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Control chart patterns (3)

• Shifted mean,
• Adjusted height Dartec

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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

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Phase II Control to standard (1)
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Phase II Control to standard (2)
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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

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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?

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Performance of control charts

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

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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

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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

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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

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Metrics for run lengths

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

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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

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Geometric distribution
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Numerical values

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

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Scale in Statgraphics

• Are our calculations wrong???

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Sample size and run lengths

• increase of sample size corresponding control
  limits
• same in-control run length
• decrease of out-of-control run length

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Numerical values

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

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Runs rules and run lengths

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

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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

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CUSUM Chart

• plot cumulative sums of observation

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CUSUM tabular form

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

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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

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CUSUM V-mask form
UCL
CL
change point
LCL
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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

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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

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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

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CUSUM for variability

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

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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!

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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?).

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

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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)

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c-chart

• Poisson distribution meanc and variancec

Control Limits for Nonconformities
is average number of nonconformities in sample
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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
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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

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Moving Range Chart

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

individual measurements
moving range
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Example Viscosity of Aircraft Primer Paint
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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
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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