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2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303)Spring 2008

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1Manufacturing

Control of Manufacturing Processes

Subject 2.830/6.780/ESD.63Spring 2008Lecture #9

Advanced and Multivariate SPC

March 6, 2008

2Manufacturing

Agenda• Conventional Control Charts

– Xbar and S

• Alternative Control Charts– Moving average– EWMA– CUSUM

• Multivariate SPC

3Manufacturing

Xbar ChartProcess Model: x ~ N(5,1), n = 9

• Is process in control?

4.0

4.5

5.0

5.5

6.0M

ean

of x

1 2 3 4 5 6 7 8 9 10 11 12 13Sample

µ0=5.000

LCL=4.000

UCL=6.000

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Run Data (n=9 sample size)

01

345

789

11R

un C

hart

of x

0 9 18 27 36 45 54 63 72 81 90 99Run

Avg=4.91

• Is process in control?

5Manufacturing

S Chart

0.0

0.5

1.0

1.5

2.0

2.5

3.0S

tand

ard

Dev

iatio

n of

x

1 2 3 4 5 6 7 8 9 10 11 12 13Sample

µ0=0.969

LCL=0.232

UCL=1.707

• Is process in control?

6Manufacturing

n measurementsat sample j

x R j =1n

xii = j

j + n

∑

SR j2 =

1n − 1

(xii = j

j +n

∑ − x Rj )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

Alternative Charts: Running Averages

7Manufacturing

Simplest Case: Moving Average• Pick window size (e.g., w = 9)

1

2

3

4

5

6

7

8

9

Mov

ing

Avg

of x

8 16 24 32 40 48 56 64 72 80 88 96Sample

µ0=5.00

LCL

UCL

8Manufacturing

yj = a1x j −1 + a2x j −2 + a3 xj −3 + ...

General Case: Weighted Averages

• How should we weight measurements?– All equally? (as with Moving Average)

– Based on how recent?• e.g. Most recent are more relevant than less

recent?

9Manufacturing

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 Weighted Average

Define a weighting function

Wt − i = r (1− r )i

Exponential Weights

10Manufacturing

Exponentially Weighted Moving Average: (EWMA)

Ai = rxi + (1 − r)Ai −1 Recursive EWMA

UCL, LCL = x ± 3σ A

σ A =σ x

2

n⎛ ⎝ ⎜

⎞ ⎠ ⎟

r2 − r

⎛ ⎝

⎞ ⎠ 1 − 1− r( )2 t[ ]

time

σA =σ x

2

nr

2 − r⎛ ⎝

⎞ ⎠

for large t

11Manufacturing

Effect of r on σ multiplier

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?• The variance will be less than with xbar,

• n=1 case is valid• If r=1 we have “unfiltered” data

– Run data stays run data– Sequential averages remain

• If r<<1 we get long weighting and long delays– “Stronger” filter; longer response time

σA =σ x

nr

2 − r⎛ ⎝

⎞ ⎠ = σ x

r2 − r

⎛ ⎝

⎞ ⎠

13Manufacturing

EWMA vs. Xbar

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 250 300

xbarEWMAUCL EWMALCL EWMAgrand meanUCLLCL

r=0.3

Δμ = 0.5 σ

14Manufacturing

Mean Shift SensitivityEWMA and Xbar comparison

0

0.2

0.4

0.6

0.8

1

1.2

1 5 9 13 17 21 25 29 33 37 41 45 49

xbar

EWMA

UCLEWMA

LCLEWMA

GrandMean

UCL

LCL

3/6/03

Mean shift = .5 σ

r=0.1n=5

15Manufacturing

Effect 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

UCLEWMA

LCLEWMA

GrandMean

UCL

LCL

r=0.3

16Manufacturing

Small Mean Shifts

• What if Δμx is small with respect to σx ?

• But it is “persistent”

• How could we detect?– ARL for xbar would be too large

17Manufacturing

Another Approach: Cumulative Sums

• Add up deviations from mean– A Discrete Time Integrator

• Since E{x-μ}=0 this sum should stay near zero when in control

• Any bias (mean shift) in x will show as a trend

C j = (xii=1

j

∑ − x)

18Manufacturing

Mean Shift Sensitivity: CUSUM

-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 = 1σSlope cause by mean shift Δμ

Ci = (xii =1

t

∑ − x )

19Manufacturing

Control Limits for CUSUM

• Significance of Slope Changes?– Detecting Mean Shifts

• Use of v-mask– Slope Test with Deadband

d

θ

Upper decision line

Lower decision line

d =2δ

ln1 − β

α⎛ ⎝ ⎜

⎞ ⎠ ⎟

δ = Δx σ x

θ = tan−1 Δx2k

⎛⎝⎜

⎞⎠⎟

where k = horizontal scale factor for plot

20Manufacturing

Use of Mask

-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

θ=tan-1(Δμ/2k)k=4:1; Δμ=0.25 (1σ)tan(θ) = 0.5 as plotted

21Manufacturing

An Alternative• Define the Normalized Statistic

• And the CUSUM statistic

Zi =Xi − μ x

σ x

Si =Zi

i =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

Chart with Centerline =0 and Limits = ±3

22Manufacturing

Example for Mean Shift = 1σ

-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 σ

23Manufacturing

Tabular CUSUM

• Create Threshold Variables:

Ci+ = max[0, xi − (μ0 + K ) + Ci −1

+ ]Ci

− = max[0,(μ0 − K ) − xi + Ci −1− ]

K= threshold or slack value for accumulation

K =Δμ2

Δμ = mean shift to detect

H : alarm level (typically 5σ)

Accumulates deviations from the mean

typical

24Manufacturing

Threshold Plot

0

1

2

3

4

5

6

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

C+

C- C+

C-H threshold

μ 0.495

σ 0.170

k=δμ/2 0.049

h=5σ 0.848

25Manufacturing

Alternative Charts Summary• Noisy data need some filtering• Sampling strategy can guarantee

independence• Linear discrete filters been proposed

– EWMA– Running Integrator

• Choice depends on nature of process• Noisy data need some filtering, BUT

– Should generally monitor variance too!

26Manufacturing

Motivation: Multivariate Process Control

• More than one output of concern– many univariate control charts– many false alarms if not designed properly– common mistake #1

• Outputs may be coupled– exhibit covariance– independent probability models may not be

appropriate– common mistake #2

27Manufacturing

Mistake #1 – Multiple Charts

• Multiple (independent) parameters being monitored at a process step– set control limits based on acceptable

α = Pr(false alarm)• E.g., α = 0.0027 (typical 3σ control limits), so 1/370 runs

will be a false alarm– Consider p separate control charts

• What is aggregate false alarm probability?

28Manufacturing

Mistake #1 – Multiple Tests for Significant Effects

• Multiple control charts are just a running hypothesis test – is process “in control” or has something statistically significant occurred (i.e., “unlikely to have occurred by chance”)?

• Same common mistake (testing for multiple significant effects and misinterpreting significance) applies to many uses of statistics – such as medical research!

29Manufacturing

The Economist (Feb. 22, 2007)

Text removed due to copyright restrictions. Please see The Economist, Science and Technology.

“Signs of the times.” February 22, 2007.

30Manufacturing

Approximate Corrections for Multiple (Independent) Charts

• Approach: fixed α’– Decide aggregate acceptable false alarm rate, α’– Set individual chart α to compensate

– Expand individual control chart limits to match

31Manufacturing

Mistake #2: Assuming Independent Parameters

• Performance related to many variables • Outputs are often interrelated

– e.g., two dimensions that make up a fit– thickness and strength– depth and width of a feature (e.g.. micro embossing)– multiple dimensions of body in white (BIW)– multiple characteristics on a wafer

• Why are independent charts deceiving?

32Manufacturing

Examples

• Body in White (BIW) assembly– Multiple individual dimensions measured– All could be OK and yet BIW be out of spec

• Injection molding part with multiple key dimensions

• Numerous critical dimensions on a semiconductor wafer or microfluidic chip

33Manufacturing

LFM Application

Rob York ‘95

“Distributed Gaging Methodologies for Variation Reduction in and Automotive Body Shop”

Rob York ‘95

“Distributed Gaging Methodologies for Variation Reduction in and Automotive Body Shop”

Body in White “Indicator”

34Manufacturing

35Manufacturing

Independent Random Variables

-8

-6

-4

-2

0

2

4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

-5

-4

-3

-2

-1

0

1

2

3

4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

-4

-3

-2

-1

0

1

2

3

4

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

x1

x1

x2

Proper Limits?x2

36Manufacturing

Correlated Random Variables

-4

-3

-2

-1

0

1

2

3

4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

-4

-3

-2

-1

0

1

2

3

4

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

-4

-3

-2

-1

0

1

2

3

4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

x2

x2

x1

x1

Proper Limits?

37Manufacturing

Outliers?

-4

-3

-2

-1

0

1

2

3

4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

-4

-3

-2

-1

0

1

2

3

4

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

-4

-3

-2

-1

0

1

2

3

4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

x2

x1

38Manufacturing

Multivariate Charts

• Create a single chart based on joint probability distribution– Using sample statistics: Hotelling T2

• Set limits to detect mean shift based on α• Find a way to back out the underlying

causes• EWMA and CUSUM extensions

– MEWMA and MCUSUM

39Manufacturing

Background• Joint Probability Distributions• Development of a single scale control chart

– Hotelling T2

• Causality Detection– Which characteristic likely caused a problem

• Reduction of Large Dimension Problems– Principal Component Analysis (PCA)

40Manufacturing

Multivariate Elements• Given a vector of measurements

• We can define vector of means:

• and covariance matrix:where p = # parameters

variance

covariance

41Manufacturing

Joint Probability Distributions

• Single Variable Normal Distribution

• Multivariable Normal Distribution

squared standardizeddistance from

mean

squared standardizeddistance from mean

42Manufacturing

Sample Statistics• For a set of samples of the vector x

• Sample Mean

• Sample Covariance

43Manufacturing

Chi-Squared Example - True Distributions Known, Two Variables

• If we know μ and Σ a priori:

will be distributed as

– sum of squares of two unit normals

• More generally, for p variables:

distributed as (and n = # samples)

44Manufacturing

Control Chart for χ2? • Assume an acceptable probability of Type I

errors (upper α)• UCL = χ2

α, p– where p = order of the system

• If process means are µ1 and µ2 then χ20 < UCL

45Manufacturing

Univariate vs. χ2 Chart

Montgomery, ed. 5e

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

LCLx1

UCLx1

x1

12345678910111213141516

LCLx2

UC

Lx2

x2

Joint control region for x1 and x2

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

UCL = xa.2

x02

2

Sample number

Figure by MIT OpenCourseWare.

46Manufacturing

Multivariate Chart with No Prior Statistics: T2

• If we must use data to get• Define a new statistic, Hotelling T2

• Where is the vector of the averages for each variable over all measurements

• S is the matrix of sample covariance over all data

x and S

x

47Manufacturing

Similarity of T2 and t2

vs.

48Manufacturing

Distribution for T2

– Given by a scaled F distribution

α is type I error probabilityp and (mn – m – p + 1) are d.o.f. for the F distributionn is the size of a given samplem is the number of samples takenp is the number of outputs

49Manufacturing

F - Distribution

Significance level α

Tabulated Values

Fν1 ,ν2ν1 = 8,ν2 = 16

Fα , p,(mn− m − p+1)

50Manufacturing

Phase I and II?• Phase I - Establishing Limits

• Phase II - Monitoring the Process

NB if m used in phase 1 is large then they are nearly the same

51Manufacturing

Example• Fiber production• Outputs are strength and weight• 20 samples of subgroups size 4

– m = 20, n = 4• Compare T2 result to individual control charts

52Manufacturing

Data SetOutput x1 Output x2

Sample Subgroup n=4 R1 Subgroup n=41 80 82 78 85 7 19 22 20 202 75 78 84 81 9 24 21 18 213 83 86 84 87 4 19 24 21 224 79 84 80 83 5 18 20 17 165 82 81 78 86 8 23 21 18 226 86 84 85 87 3 21 20 23 217 84 88 82 85 6 19 23 19 228 76 84 78 82 8 22 17 19 189 85 88 85 87 3 18 16 20 1610 80 78 81 83 5 18 19 20 1811 86 84 85 86 2 23 20 24 2212 81 81 83 82 2 22 21 23 2113 81 86 82 79 7 16 18 20 1914 75 78 82 80 7 22 21 23 2215 77 84 78 85 8 22 19 21 1816 86 82 84 84 4 19 23 18 2217 84 85 78 79 7 17 22 18 1918 82 86 79 83 7 20 19 23 2119 79 88 85 83 9 21 23 20 1820 80 84 82 85 5 18 22 19 20

Mean Vector Covariance Matrix

82.46 7.51 -0.3520.18 -0.35 3.29

53Manufacturing

Cross Plot of Data

17.00

18.00

19.00

20.00

21.00

22.00

23.00

78.00 80.00 82.00 84.00 86.00 88.00

X1 bar

74.00

76.00

78.00

80.00

82.00

84.00

86.00

88.00

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

X1 bar

X2 bar

17.00

18.00

19.00

20.00

21.00

22.00

23.00

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

X2 bar

54Manufacturing

T2 Chart

0

2

4

6

8

10

12

14

16

18

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

T2

UCL

α = 0.0054

(Similar to ±3σ limits ontwo xbar charts combined)

55Manufacturing

Individual Xbar Charts

17.00

18.00

19.00

20.00

21.00

22.00

23.00

24.00

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

X2 bar

UCL X2

LCL X2

74.00

76.00

78.00

80.00

82.00

84.00

86.00

88.00

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

X1 bar

UCL X1

LCL X1

56Manufacturing

Finding Cause of Alarms• With only one variable to plot, which

variable(s) caused an alarm?• Montgomery

– Compute T2

– Compute T2(i) where the i th variable is not included

– Define the relative contribution of each variable as• di = T2 - T2

(i)

57Manufacturing

Principal Component Analysis

• Some systems may have many measured variables p– Often, strong correlation among these variables:

actual degrees of freedom are fewer• Approach: reduce order of system to track only

q << p variables– where each z1 … zq is a linear combination of the

measured x1 … xp variables

58Manufacturing

Principal Component Analysis

• x1 and x2 are highly correlated in the z1direction

• Can define new axes zi in order of decreasing variance in the data

• The zi are independent• May choose to neglect dimensions with

only small contributions to total variance ⇒ dimension reduction

Truncate at q < p

59Manufacturing

Principal Component Analysis• Finding the cij that define the principal

components:– Find Σ covariance matrix for data x– Let eigenvalues of Σ be – Then constants cij are the elements of the ith

eigenvector associated with eigenvalue λis• Let C be the matrix whose columns are the eigenvectors• Then

where Λ is a p x p diagonal matrix whose diagonals are the eigenvalues

• Can find C efficiently by singular value decomposition (SVD)– The fraction of variability explained by the ith principal

components is

60Manufacturing

Extension to EWMA and CUSUM

• Define a vector EWMA

• And for the control chart plot

where

Zi = rxi + (1 + r)Zi −1

Ti2 = Zi

T ΣZi

−1Zi

ΣZi i =r

2 − r[1 − (1 − r)2 ]Σ

61Manufacturing

Conclusions

• Multivariate processes need multivariate statistical methods

• Complexity of approach mitigated by computer codes

• Requires understanding of underlying process to see if necessary– i.e. if there is correlation among the variables of

interest