Control of Manufacturing Processesdspace.mit.edu/bitstream/handle/1721.1/56569/2-830J... · 2019....

Control of Manufacturing Control of Manufacturing

ProcessesProcesses

Subject 2.830

Spring 2004Spring 2004

Lecture #10Lecture #10

Advanced SPCAdvanced SPC

Moving Average ApproachesMoving Average Approaches

March 9, 2004March 9, 2004

3/9/04 Lecture 101 © D.E. Hardt, all rights

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2

Beyond XbarBeyond Xbar

•• Good PointsGood Points

–– Simple and “transparent”Simple and “transparent”

–– Enforces AssumptionsEnforces Assumptions•• Normality (via Central Limit)Normality (via Central Limit)

•• Independent (via long sampling times)Independent (via long sampling times)

•• LimitationsLimitations

–– n>1 to get Xbar and Sn>1 to get Xbar and S

–– arlarl isis typciallytypcially largelarge•• Not very Not very sensitivbesensitivbe to small changesto small changes

–– Slow time responseSlow time response


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Beyond XbarBeyond Xbar

•• What if n=1?What if n=1?

–– Have a Lot of DataHave a Lot of Data

–– Want Fast Response to ChangesWant Fast Response to Changes

•• How to Compute Control Chart Statistics?How to Compute Control Chart Statistics?

–– Running Chart and Running Variance?Running Chart and Running Variance?

–– Running Average and Running Variance?Running Average and Running Variance?

–– Running Average with Forgetting FactorRunning Average with Forgetting Factor

•• How to Increase Sensitivity to Small, How to Increase Sensitivity to Small, Persistent Mean Shift?Persistent Mean Shift?

–– Integrate the ErrorIntegrate the Error


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Chart Design:Chart Design:

n=1 Designs n=1 Designs -- Running AveragesRunning Averages

•• Sensitivity: Ability to detect small Sensitivity: Ability to detect small

changes (e.g. mean shifts)changes (e.g. mean shifts)

•• Time Response: Ability to Catch Time Response: Ability to Catch

Changes QuicklyChanges Quickly

•• Noise Rejection: Reduction of Variance; Noise Rejection: Reduction of Variance;

increase S/N increase S/N


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5

Parallels with Linear Discrete Parallels with Linear Discrete

System DynamicsSystem Dynamics•• Dynamics of Sampled Data SystemsDynamics of Sampled Data Systems

Step1

z+0.5

Discrete

1

s+1

Continuous


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Parallels with Linear Discrete Parallels with Linear Discrete

System DynamicsSystem Dynamics•• Filtering of “Noisy” SignalsFiltering of “Noisy” Signals

Random

Number 0.5

z-0.5

Discrete

1

s+1

Continuous


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Xbar “Filtering”Xbar “Filtering”

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

Run Data

Xbar n=4


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FilteringFiltering

•• Reduced PeaksReduced Peaks

•• Hides intermediate dataHides intermediate data

•• Reduces the “frequency content” of the Reduces the “frequency content” of the

outputoutput


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Independence and Independence and

CorrelationCorrelation

•• Independence: Current output does not Independence: Current output does not

depend on priordepend on prior

•• Correlation: Measure of IndependenceCorrelation: Measure of Independence

–– e.g. auto correlation functione.g. auto correlation function

Rxx( ) E[x( t)x(t )]


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CorrelationCorrelation

Rxx( ) E[x( t)x(t )]

0

0.2

0.4

0.6

0.8

1

-4 -3 -2 -1 0 1 2 3 4

Tmin Tmax

For a linear 1st order system

T~ 1 sec:

For an uncorrelated

process


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Sampling: Frequency and Sampling: Frequency and

Distribution of SamplesDistribution of Samples

0

0.2

0.4

0.6

0.8

1

-4 -3 -2 -1 0 1 2 3 4

Tmin Tmax

0

0.2

0.4

0.6

0.8

1

-4 -3 -2 -1 0 1 2 3 4

Tmin Tmax

0

0.2

0.4

0.6

0.8

1

-4-3-2-101234

TmT

SAMPLE TIME


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Correlation and SamplingCorrelation and Sampling

Correlated

Samples

Uncorrelated

Samples

Correlation

Time (e.g.)

•Taking samples beyond correlation

time guarantees independence


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Sampling and AveragingSampling and Averaging

•• Sampling Frequency AffectsSampling Frequency Affects

–– Time ResponseTime Response

–– CorrelationCorrelation

•• AveragingAveraging

–– Filters DataFilters Data

–– Slows ResponseSlows Response


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Alternative ChartsAlternative Charts

Running AveragesRunning Averages

n measurements

at sample j

xRj

1

nxi

i j

j n

SR j

2 1

n 1(xi

i j

j n

xRj )2

Running Average

Running Variance

• More averages/Data

• Can use run data alone and

average for S only

• Can use to improve resolution

of mean shift


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Specific Case: Specific Case:

Weighted AveragesWeighted Averages

yj a1x j 1 a2x j 2 a3 xj 3 ...

•• How should we weight How should we weight

measurements??measurements??

–– All equally? (as with Running Average)All equally? (as with Running Average)

–– Based on how recent?Based on how recent?

•• e.g. Most recent are more relevant than e.g. Most recent are more relevant than

less recent?less recent?


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0

0.05

0.1

0.15

0.2

0.25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Consider an Exponential Consider an Exponential

Weighted AverageWeighted Average

Define a weighting function

Wt i r (1 r )i

Exponential Weights


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Exponentially Weighted Exponentially Weighted

Moving Average Statistic: Moving Average Statistic:

(EWMA):(EWMA):Ai rxi (1 r)Ai 1 Recursive EWMA

Ax

2

n

r

2 r1 1 r

2t

UCL, LCL x 3 Afor large tfor large t

Ax

2

n

r

2 r

time

Effect of r on Effect of r on multipliermultiplier

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

plot of (r/(2-r)) vs. r

wider control limits

r


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SO WHAT???SO WHAT???

•• The variance will be less than with xbar, The variance will be less than with xbar,

•• n=1 case is validn=1 case is valid

•• If r=1 we have “unfiltered” dataIf r=1 we have “unfiltered” data

–– Run data stays run dataRun data stays run data

–– Sequential averages remain Sequential averages remain

•• If r<<1 we get long weighting and long delaysIf r<<1 we get long weighting and long delays

–– “Stronger” filter; longer response time“Stronger” filter; longer response time

Ax

n

r

2 rx

r

2 r

Mean Shift SensitivityMean Shift Sensitivity

EWMA and Xbar comparisonEWMA and Xbar comparison

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

xbar

EWMA

UCL

EWMA

LCL

EWMA

Grand

Mean

UCL

LCL

3/6/03

Mean shift = 1 S

r=0.1

Effect of rEffect of r

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

xbar

EWMA

UCL

EWMA

LCL

EWMA

Grand

Mean

UCL

LCL

r=0.3


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Small Mean ShiftsSmall Mean Shifts

•• What if What if xx is small wrt is small wrt xx ??

•• But it is “persistent”But it is “persistent”

•• How could we detect?How could we detect?

–– arl for xbar would be too largearl for xbar would be too large


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Another Approach: Another Approach:

Cumulative SumsCumulative Sums•• Add up deviations from meanAdd up deviations from mean

–– A Discrete Time IntegratorA Discrete Time Integrator

•• Since E{Since E{xx-- }=0 this sum should stay }=0 this sum should stay

near zeronear zero

•• Any bias in Any bias in xx will show as a trendwill show as a trend

C j (xi

i 1

j

x)


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Mean Shift Sensitivity: Mean Shift Sensitivity:

CUSUMCUSUM

-1

0

1

2

3

4

5

6

7

81 3 5 7 9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

39

41

43

45

47

49

Mean shift = 1S

Slope due to

mean shift

Ci (xii 1

t

x )


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Control Limits for CUSUMControl Limits for CUSUM

•• Significance of Slope Changes?Significance of Slope Changes?

–– Detecting Mean ShiftsDetecting Mean Shifts

•• Use of vUse of v--maskmask

–– Slope Test with DeadbandSlope Test with Deadband

d

Upper decision line

Lower decision line

d2

ln1

x

x

tan1 x

2k

where

k scale factor for plot


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Use of MaskUse of Mask

-1

0

1

2

3

4

5

6

7

8

1 3 5 7 9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

39

41

43

45

47

49

=tan-1( /2k)

k=4:1; =0.25 (1 )

tan( ) = 0.5 as plotted


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An AlternativeAn Alternative

•• Define the Normalized StatisticDefine the Normalized Statistic

•• And the CUSUM statisticAnd the CUSUM statistic

Zi

Xi x

x

Si

Zii 1

t

t

Which has an

expected mean of 0

and variance of 1

Which has an

expected mean of 0

and variance of 1

Can Chart with Centerline =0 and Limits = ±3


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Example for Mean Shift = 1Example for Mean Shift = 1

-1

0

1

2

3

4

5

1 3 5 7 9

11

13

15

17

19

21

23

25

27

29

31

33

35

37

39

41

43

45

47

Normalized CUSUM

Mean Shift = 1


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Tabular CUSUMTabular CUSUM

•• Create Threshold Variables: Create Threshold Variables:

Ci max[0, xi ( 0 K) Ci 1 ]

Ci max[0,( 0 K) xi Ci 1 ]

K= threshold or slack value

K2

mean shift to detect

H : alarm level (typically 5


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Threshold PlotThreshold Plot

0.495

0.170

k= /2 0.049

h=5 0.848


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SummarySummary

•• Noisy Data Need Some FilteringNoisy Data Need Some Filtering

•• Sampling Strategy Can Guarantee Sampling Strategy Can Guarantee

IndependenceIndependence

•• Linear Discrete Filters have Been Linear Discrete Filters have Been

ProposedProposed

–– EWMAEWMA

–– Running IntegratorRunning Integrator

•• Choice Depends on Nature of Process Choice Depends on Nature of Process


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Summary of Part IISummary of Part II

•• Consider Process a Random ProcessConsider Process a Random Process

–– Can never predict precise valueCan never predict precise value

•• Model with Model with P(x)P(x) oror p(x)p(x)

–– AssumeAssume p(x,t) = p(x)p(x,t) = p(x)

•• Shewhart HypothesisShewhart Hypothesis

–– InIn-- control = purely random outputcontrol = purely random output•• Normal, independent stationaryNormal, independent stationary

•• “The best you can do!”“The best you can do!”

–– Not in Not in -- controlcontrol•• NonNon--random behaviorrandom behavior

•• Source can be found and eliminatedSource can be found and eliminated


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Summary of Part IISummary of Part II

•• P(x) normalP(x) normal

–– M and s only to defineM and s only to define

•• Find Sample StatisticsFind Sample Statistics

–– Xbar and SXbar and S

•• Plot SequentiallyPlot Sequentially

•• Look for unexpected behaviorLook for unexpected behavior

–– Confidence IntervalsConfidence Intervals

–– Mean Shift hypothesisMean Shift hypothesis

–– ……


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The SPC HypothesisThe SPC Hypothesis

...

...

In-Control

Not

In-Control

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Sample Number

ProcessY

p(y)

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