/k Control charts 2WS02 Industrial Statistics A. Di Bucchianico

/k 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)

/k Statistically versus technically in control “Statistically in control” stable over time / predictable “Technically in control” within specifications

/k 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

/k Shewhart control chart graphical display of product characteristic which is important for product quality Upper Control Limit Centre Line Lower Control Limit

/k Control charts

/k 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?

/k Meaning of control limits limits at 3 x standard deviation of plotted statistic basic example: UCL LCL

/k Example diameters of piston rings process mean: 74 mm process standard deviation: 0.01 mm measurements via repeated samples of 5 rings yields:

/k Individual versus mean group means individual observations

/k Range chart need to monitor both mean and variance traditionally use range to monitor variance chart may also be based on S or S 2 for normal distribution: –E R = d 2 E S (Hartley’s constant) –tables exist preferred practice: –first check range chart for violations of control limits –then check mean chart

/k 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

/k 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. between subgroup variation within subgroup variation

/k Strategy 1 leads to accurate estimate of  maximises between-subgroup variation minimises within-subgroup variation process mean

/k Strategy 2 detects contrary to strategy 1 also temporary changes of process mean process mean

/k Phase I (Initial study): in control (1)

/k Phase I (Initial study): in control (2)

/k Phase I (Initial Study): not in-control

/k 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)

/k Control chart patterns (1) Cyclic pattern, three arrows with different weight

/k Control chart patterns (2) Trend, course of pin

/k Control chart patterns (3) Shifted mean, Adjusted height Dartec

/k 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

/k Phase II: Control to standard (1)

/k Phase II: Control to standard (2)

/k 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

/k 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?

/k Performance of control charts expressed in terms of time to alarm (run length) two types: –in-control run length –out-of-control run length

/k 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

/k 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

/k 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

/k Metrics for run lengths run lengths are random variables –ARL = Average Run Length –SRL = Standard Deviation of Run Length

/k Run lengths for Shewhart Xbar- chart in-control: p = UCL LCL time to alarm follows geometric distribution: – mean 1/p = – standard deviation: (  (1-p))/p = 369.9

/k Geometric distribution

/k Numerical values Shewhart chart for mean (n=1) single shift of mean:

/k Scale in Statgraphics Are our calculations wrong???

/k Sample size and run lengths increase of sample size + corresponding control limits: –same in-control run length –decrease of out-of-control run length

/k Numerical values Shewhart chart for mean (n=5) single change of standard deviation (  -> c  ) c P(|Xbar|>3  ARLSRL

/k Runs rules and run lengths in-control run length: decreases (why?) out-of-control run length: decreases (why?)

/k Performance Shewhart chart in-control run length OK out-of-control run length –OK for shifts > 2 standard deviation group average –Bad for shifts < 2 standard deviation group average extra run tests –decrease in-control length –decrease out-of-control length

/k CUSUM Chart plot cumulative sums of observation  change point

/k CUSUM tabular form assume –data normally distributed with known  –individual observations

/k Choice K and H K is reference value (allowance, slack value) C + measures cumulative upward deviations of µ 0 +K C - measures cumulative downward deviations of µ 0 -K for fast detection of change process mean µ 1 : –K=½ |µ 0- µ 1 | H=5  is good choice

/k CUSUM V-mask form UCL LCL CL change point

/k Drawbacks V-mask only for two-sided schemes headstart cannot be implementedheadstart 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

/k Combination CUSUM charts appropriate for small shifts (<1.5  ) CUSUM charts are inferior to Shewhart charts for large shifts(>1.5  ) use both charts simultaneously with ±3.5  control limits for Shewhart chart

/k Headstart (Fast Initial Response) increase detection power by restart process esp. useful when process mean at restart is not equal at target value set C + 0 and C - 0 to non-zero value (often H/2 ) if process equals target value µ 0 is, then CUSUMs quickly return to 0 if process mean does not equal target value µ 0, then faster alarm

/k CUSUM for variability define Y i = (X i -µ 0 )/  (standardise) define V i = (  |Y i |-0.822)/0.349 CUSUMs for variability are:

/k 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!

/k Control charts for attributes 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?). Attributes are characteristics which have a countable number of possible outcomes.

/k p-chart Number of nonconforming products is binomially distributed sample fraction of nonconforming: mean:variance

/k p-chart average of sample fractions: Fraction Nonconforming Control Chart:

/k 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

/k 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

/k c-chart Poisson distribution: mean=c and variance=c Control Limits for Nonconformities: is average number of nonconformities in sample

/k 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:

/k 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

/k Moving Range Chart calculation of moving range: d 2, D 3 and D 4 are constants depending number of observations individual measurements moving range

/k Example: Viscosity of Aircraft Primer Paint BatchViscosityMR BatchViscosityMR

/k Viscosity of Aircraft Primer Paint since a moving range is calculated of n=2 observations, d 2 =1.128, D 3 =0 and D 4 =3.267 CC for individuals CC for moving range

/k 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 LSL USL LCL UCL reset

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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