MIT OpenCourseWare ______http://ocw.mit.edu

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

For information about citing these materials or our Terms of Use, visit: ______http://ocw.mit.edu/terms. Control of Manufacturing Processes Subject 2.830/6.780/ESD.63 Spring 2008 Lecture #9

Advanced and Multivariate SPC

March 6, 2008

Manufacturing 1 Agenda • Conventional Control Charts – Xbar and S

• Alternative Control Charts – Moving average –EWMA – CUSUM

• Multivariate SPC

Manufacturing 2 Xbar Chart Process Model: x ~ N(5,1), n = 9

6.0 UCL=6.000

5.5

5.0 µ0=5.000 Mean of x 4.5

4.0 LCL=4.000

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

• Is process in control?

Manufacturing 3 Run Data (n=9 sample size)

11

9 8 7

5 Avg=4.91 4 Run Chart of x 3

1 0 0 9 18 27 36 45 54 63 72 81 90 99 Run • Is process in control?

Manufacturing 4 S Chart

3.0

2.5

2.0 UCL=1.707 1.5

1.0 µ0=0.969 Standard Deviation of x 0.5 LCL=0.232 0.0 1 2 3 4 5 6 7 8 9 10 11 12 13 Sample

• Is process in control?

Manufacturing 5 Alternative Charts: Running Averages

• More averages/Data • Can use run data alone and average for S only • Can use to improve resolution of mean shift 1 j + n x = x Rj ∑ i Running Average n measurements n i = j j +n at sample j 2 1 2 SRj = ∑ (xi − x Rj ) Running Variance n − 1 i = j

Manufacturing 6 Simplest Case: Moving Average • Pick window size (e.g., w = 9)

9 8 7 6 UCL 5 µ0=5.00 4 LCL

Moving Avg x of 3 2 1 8 16 24 32 40 48 56 64 72 80 88 96 Sample Manufacturing 7 General Case: Weighted Averages

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

• 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?

Manufacturing 8 Consider an Exponential Weighted Average

0.25 Define a weighting function 0.2 i Wt − i = r(1 − r)

0.15 Exponential Weights 0.1

0.05

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Manufacturing 9 Exponentially Weighted Moving Average: (EWMA)

Ai = rxi + (1 − r)Ai −1 Recursive EWMA time 2 ⎛ σ x ⎞ ⎛ r ⎞ 2t σ = ⎜ ⎟ 1 − ()1− r A ⎝ n ⎠ ⎝ 2 − r⎠ [] 2 σ ⎛ r ⎞ σ = x A n ⎝ 2 − r⎠ UCL, LCL = x ± 3σ A for large t

Manufacturing 10 Effect of r on σ multiplier

0.9

0.8 plot of (r/(2-r)) vs. r 0.7

0.6

0.5

0.4 wider control limits 0.3

0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 r

Manufacturing 11 SO WHAT? • The variance will be less than with xbar,

σ x ⎛ r ⎞ ⎛ r ⎞ σA = = σ x n ⎝ 2 − r⎠ ⎝ 2 − r⎠ • 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

Manufacturing 12 EWMA vs. Xbar

1

0.9 r=0.3

0.8 Δμ = 0.5 σ

0.7

xbar 0.6 EWMA UCL EWMA 0.5 LCL EWMA grand mean UCL 0.4 LCL

0.3

0.2

0.1

0 0 50 100 150 200 250 300

Manufacturing 13 Mean Shift Sensitivity EWMA and Xbar comparison 1.2 xbar

1 EWMA

UCL 0.8 EWMA LCL EWMA

Grand 0.6 3/6/03 Mean

UCL 0.4 LCL Mean shift = .5 σ 0.2 n=5 r=0.1 0 1 5 9 13172125293337414549 Manufacturing 14 Effect of r

1

0.9 xbar

0.8 EWMA

0.7 UCL EWMA

0.6 LCL EWMA

0.5 Grand Mean

0.4 UCL

0.3 LCL

0.2

0.1 r=0.3 0 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

Manufacturing 15 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

Manufacturing 16 Another Approach: Cumulative Sums

• Add up deviations from mean – A Discrete Time Integrator j

C j = ∑(xi − x) i=1

• 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

Manufacturing 17 Mean Shift Sensitivity: CUSUM

8

7 t

6 Ci = ∑ (xi − x ) i =1

5

4 Mean shift = 1σ 3 Slope cause by

2 mean shift Δμ

1

0 1 3 5 7 9 31 33 35 37 39 41 43 45 47 49 11 13 15 17 19 21 23 25 27 29 -1

Manufacturing 18 Control Limits for CUSUM

• Significance of Slope Changes? – Detecting Mean Shifts • Use of v-mask – Slope Test with Deadband 2 ⎛ 1 − β ⎞ d = ln⎜ ⎟ δ ⎝ α ⎠ Upper decision line Δx δ = σ θ x −1 ⎛ Δx ⎞ θ = tan ⎜ ⎟ ⎝ 2k ⎠ d where Lower decision line k = horizontal scale factor for plot

Manufacturing 19 8 Use of Mask

7 θ=tan-1(Δμ/2k) 6 k=4:1; Δμ=0.25 (1σ)

5 tan(θ) = 0.5 as plotted

4

3

2

1

0 1 3 5 7 9 31 33 35 37 39 41 43 45 47 49 11 13 15 17 19 21 23 25 27 29

-1

Manufacturing 20 An Alternative • Define the Normalized Statistic X − μ Which has an Z = i x i σ expected mean of x 0 and variance of 1 • And the CUSUM statistic t Z Which has an ∑ i expected mean of S = i =1 i t 0 and variance of 1

Chart with Centerline =0 and Limits = ±3

Manufacturing 21 Example for Mean Shift = 1σ

5 Normalized CUSUM

4

3

2

1

0 1 3 5 7 9 31 33 35 37 39 41 43 45 47 11 13 15 17 19 21 23 25 27 29 Mean Shift = 1 σ

-1

Manufacturing 22 Tabular CUSUM

• Create Threshold Variables:

+ + Ci = max[0, xi − (μ0 + K ) + Ci −1 ] Accumulates deviations − − Ci = max[0,(μ0 − K ) − xi + Ci −1 ] from the mean K= threshold or slack value for accumulation Δμ K = Δμ = mean shift to detect typical 2 H : alarm level (typically 5σ)

Manufacturing 23 Threshold Plot

6 μ 0.495

5 σ 0.170 k=δμ/2 0.049

4 h=5σ 0.848

C+ 3 C-

C+ C- H threshold

2

1

0 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

Manufacturing 24 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!

Manufacturing 25 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

Manufacturing 26 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?

Manufacturing 27 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!

Manufacturing 28 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.

Manufacturing 29 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

Manufacturing 30 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?

Manufacturing 31 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

Manufacturing 32 LFM Application

RobRob YorkYork ‘95‘95 “Distributed“Distributed GagingGaging MethodologiesMethodologies forfor VariationVariation ReductionReduction inin andand AutomotiveAutomotive BodyBody Shop”Shop”

Body in White “Indicator”

Manufacturing 33 Manufacturing 34 Independent Random Variables

4 4

2

3

0 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

2 -2

-4

1 -6 x1

0 -8 x2 -4 -3 -2 -1 0 1 2 3 4

-1

4 -2

3

2 -3 1

0 1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829303132333435363738394041424344454647484950515253 -4 -1

-2 x -3 1 -4 x2 -5 Proper Limits?

Manufacturing 35 Correlated Random Variables

4

4 3

2

1 3

0 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

-1 2

-2

-3 x 1 1

-4

0 x2 -4 -3 -2 -1 0 1 2 3 4

4 -1

3

2 -2

1

0 -3 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

-1

-2 -4 x -3 2

-4 x1 Proper Limits?

Manufacturing 36 Outliers?

4

3

4 2

1

3 0 1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829303132333435363738394041424344454647484950515253

-1

2 -2

-3

-4 1

0 x2 -4 -3 -2 -1 0 1 2 3 4

4 -1 3

2

-2 1

0 1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829303132333435363738394041424344454647484950515253

-3 -1

-2

-4 -3

-4

x1

Manufacturing 37 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

Manufacturing 38 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)

Manufacturing 39 Multivariate Elements • Given a vector of measurements

• We can define vector of means: where p = # parameters • and covariance matrix:

variance

covariance

Manufacturing 40 Joint Probability Distributions

• Single Variable Normal Distribution

squared standardized distance from mean • Multivariable Normal Distribution

squared standardized distance from mean

Manufacturing 41 Sample Statistics • For a set of samples of the vector x

• Sample Mean

• Sample Covariance

Manufacturing 42 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)

Manufacturing 43 Control Chart for χ2? • Assume an acceptable probability of Type I errors (upper α) 2 • UCL = χ α, p –where p = order of the system

2 • If process means are µ1 and µ2 then χ 0 < UCL

Manufacturing 44 Univariate vs. χ2 Chart

Joint control region for x1 and x2

UCLx1

x1

LCLx1

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

x2 LCLx UCLx 1 2 2 2 UCL = xa.2 2 3 4 5 6 7 x2 0 8 9 10 11 12 13 10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 14 Sample number 15 16

Figure by MIT OpenCourseWare. 45 Manufacturing Montgomery, ed. 5e Multivariate Chart with No Prior Statistics: T2

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

• Where x is the vector of the averages for each variable over all measurements • S is the matrix of sample covariance over all data

Manufacturing 46 Similarity of T2 and t2

vs.

Manufacturing 47 Distribution for T2

– Given by a scaled F distribution

α is type I error probability p and (mn–m –p + 1) are d.o.f. for the F distribution n is the size of a given sample m is the number of samples taken p is the number of outputs

Manufacturing 48 F - Distribution

F ν = 8,ν = 16 ν1 ,ν2 1 2

Significance level α

Tabulated Values Fα , p,(mn− m − p+1)

Manufacturing 49 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

Manufacturing 50 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

Manufacturing 51 Data Set

Output x1 Output x2 Sample Subgroup n=4 R1 Subgroup n=4 1 808278857 19 22 20 20 2 757884819 24 21 18 21 3 838684874 19 24 21 22 4 798480835 18 20 17 16 5 828178868 23 21 18 22 6 868485873 21 20 23 21 7 848882856 19 23 19 22 8 768478828 22 17 19 18 9 858885873 18 16 20 16 10 80 78 81 83 5 18 19 20 18 11 86 84 85 86 2 23 20 24 22 12 81 81 83 82 2 22 21 23 21 13 81 86 82 79 7 16 18 20 19 14 75 78 82 80 7 22 21 23 22 15 77 84 78 85 8 22 19 21 18 16 86 82 84 84 4 19 23 18 22 17 84 85 78 79 7 17 22 18 19 18 82 86 79 83 7 20 19 23 21 19 79 88 85 83 9 21 23 20 18 20 80 84 82 85 5 18 22 19 20

Mean Vector Covariance Matrix

82.46 7.51 -0.35 20.18 -0.35 3.29

Manufacturing 52 Cross Plot of Data

X2 bar

23.00 23.00

22.00 22.00 21.00

21.00 20.00 X2 bar

19.00 20.00 18.00

19.00 17.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

18.00

X1 bar

17.00 88.00 86.00 78.00 80.00 82.00 84.00 86.00 88.00 84.00

82.00 X1 bar 80.00

78.00

76.00

74.00 1234567891011121314151617181920

Manufacturing 53 T2 Chart

18

16

14

12

10 T2 UCL 8

6 α = 0.0054 4 (Similar to ±3σ limits on 2 two xbar charts combined)

0 1 2 3 4 5 6 7 8 9 1011121314151617181920

Manufacturing 54 Individual Xbar Charts

88.00

86.00

84.00

82.00 X1 bar UCL X1 80.00 LCL X1

78.00

76.00

74.00 1 2 3 4 5 6 7 8 91011121314151617181920

24.00

23.00

22.00

21.00 X2 bar UCL X2 20.00 LCL X2

19.00

18.00

17.00 1234567891011121314151617181920

Manufacturing 55 Finding Cause of Alarms • With only one variable to plot, which variable(s) caused an alarm? • Montgomery – Compute T2 2 th – Compute T (i) where the i variable is not included – Define the relative contribution of each variable as 2 2 • di = T - T (i)

Manufacturing 56 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

Manufacturing 57 Principal Component Analysis

• x1 and x2 are highly correlated in the z1 direction

• 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

Manufacturing 58 Principal Component Analysis

• Finding the cij that define the principal components: –Find Σ covariance matrix for data x – Let eigenvalues of Σ be th – Then constants cij are the elements of the i 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

Manufacturing 59 Extension to EWMA and CUSUM

• Define a vector EWMA

Zi = rxi + (1 + r)Zi −1 • And for the control chart plot where T 2 = Z T Σ −1Z i i Zi i r Σ = [1 − (1 − r)2 ]Σ Zi i 2 − r

Manufacturing 60 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

Manufacturing 61