2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008
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1 MIT OpenCourseWare J / 6.780J / ESD.63J Control of Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit:
2 Control of Processes Subject 2.830/6.780/ESD.63 Spring 2008 Lecture #14 Aliasing and Higher Order Models April 3, J/6.780J/ESD.63J 1
3 Outline Last Time Full Factorial Models Experimental Design Blocks and Confounding Single Replicate Designs Today Fractional Factorial Designs Aliasing Patterns Implications for Model Construction Process Optimization using DOE 2.830J/6.780J/ESD.63J 2
4 Fractional Factorial Experiments What if we do less than full factorial 2 k? Example: run < 2 3 experiments for 3 inputs From regression model for 3 inputs: y = β 0 + β 1 + β 2 + β 12 + β 3 x 3 +β 13 x 3 + β 23 x 3 + β 123z x 3 + ε We will not be able to find all 8 coefficients 2.830J/6.780J/ESD.63J 3
5 2 3-1 Experiment Consider doing 4 experiments instead of 8; e.g.: This is a 2 2 array Could also be for 3 inputs if we define x 3 = 2.830J/6.780J/ESD.63J 4
6 2 3-1 Experiment x But now we can only define 4 coefficients in the model: e.g.: ) y = β 0 + β 1 + β 2 + β 3 x 3 i.e. no interaction terms 2.830J/6.780J/ESD.63J 5
7 2 3-1 Experiment Or we could choose other terms: ) y = β 0 + β 1 + β 2 + β 13 x 3 or: ) y = β 0 + β 1 + β 12 + β 3 x 3 or: 2.830J/6.780J/ESD.63J 6
8 Confounding / Aliasing We actually have the following: ) y = β + β ' z + β ' z + β ' z where the z variable represent sums of the various input terms, e.g. z 1 = + x 3 z 2 = + x 3 L where the specific choice of the experimental array determines what these sums are 2.830J/6.780J/ESD.63J 7
9 Confounding / Aliasing 2 3 Array: (Our X matrix) Test I A B AB C AC BC ABC (1) a b ab c ac bc abc J/6.780J/ESD.63J 8
10 Consider upper half: Confounding / Aliasing Test I A B AB C AC BC ABC (1) a b ab c ac bc abc Look at columns for C - no change at all! or C = -I Also AC = -A and BC = -B, and ABC = -AB 2.830J/6.780J/ESD.63J 9
11 Confounding / Aliasing Test I A B AB C AC BC ABC (1) a b ab c ac bc abc Contrast A =[ -(1)+a-b+ab] Contrast AC =[ (1)-a+b-ab] Defining Relation I = -C AC is an alias of A Note that alias of A =A*(-C) 2.830J/6.780J/ESD.63J 10
12 Aliases Choice of Design? Must have one of the pair assumed negligible ( sparsity of effects ) Balance/Orthogonality Sufficient excitation of inputs Enable short-cut estimation of model effects and model coefficients 2.830J/6.780J/ESD.63J 11
13 Balance and Orthogonality Test I A B AB C AC BC ABC (1) a b ab c ac bc abc Note: All columns have equal number of + and - signs (Balance) Sum of product of any two columns = 0 (Orthogonality) -All combinations occur the same number of times 2.830J/6.780J/ESD.63J 12
14 Balance/Orthogonality in Test I A B C AB AC BC ABC a b c ab ac bc abc A and B are balanced; C is not A, B and C are orthogonal 2.830J/6.780J/ESD.63J 13
15 Better Subset Balanced/Orthogonal Test I A B C AB AC BC ABC a b c ab ac bc abc With this array: - balance for A, B, C - all but ABC are orthogonal - defining relation I=ABC e.g. aliases of A: A*ABC=A*I A*A = I BC aliased with A Aliases: A BC B AC C AB I ABC 2.830J/6.780J/ESD.63J 14
16 Resolution III Design Resolution No main aliases with other main effects Main - interaction aliases Resolution IV No alias between main effects and 2 factor effects, but others exist Resolution V No main and no 2 factor aliases J/6.780J/ESD.63J 15
17 With this array: - balance for A, B, C - all but A B C are orthogonal - defining relation I=ABC Design Resolution Test I A B C AB AC BC ABC a b c ab ac bc abc III e.g. aliases of A: A*ABC=A*I A*A = I BC aliased with A Main effects aliased with interactions only Aliases: A BC B AC C AB I ABC 2.830J/6.780J/ESD.63J 16
18 p = 1 1/2 fraction p = 2 1/4 fraction p 1/2 p Smaller Fraction 2 k-p 2.830J/6.780J/ESD.63J 17
19 A Different Fraction Consider I = AC Test I A B AB C AC BC ABC (1) a b ab c ac bc abc J/6.780J/ESD.63J 18
20 A Different Fraction Consider I = AC Test I A B AB C AC BC ABC (1) b ac abc Balance? Orthogonality? I=AC Aliases A with C B with ABC AB with BC 2.830J/6.780J/ESD.63J 19
21 How Decide What Aliasing To Choose? Prior knowledge of process Rules of thumb Sparsity of effects Hierarchy of effects Inheritance of effects 2.830J/6.780J/ESD.63J 20
22 Sparsity of Effects An experimenter may list a large number of effects for consideration A small number of effects usually explain the majority of the variance A B C D Courtesy of Prof. Dan Frey 2.830J/6.780J/ESD.63J 21
23 Main effects are usually more important than twofactor interactions Hierarchy Two-way interactions are usually more important than three-factor interactions And so on A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD Courtesy of Prof. Dan Frey 2.830J/6.780J/ESD.63J 22
24 Two-factor interactions are most likely when both participating factors (parents?) are strong Inheritance Two-way interactions are least likely when neither parent is strong A B C AB AC BC D AD BD CD And so on ABC ACD ABD BCD ABCD Courtesy of Prof. Dan Frey 2.830J/6.780J/ESD.63J 23
25 Resolution III Design Resolution No main aliases with other main effects Main - interaction aliases Resolution IV No alias between main effects and 2 factor effects, but others exist Resolution V III I = ABC No main and no 2 factor aliases IV I = ABCD V I = ABCDE 2.830J/6.780J/ESD.63J 24
26 2 4-2 A B C D Four Main Effects Four tests? Suppose we want to alias A with BCD and ABC What are the defining relations? 2.830J/6.780J/ESD.63J 25
27 2 4-2 Suppose we want to alias A with BCD and ABC I A B AB C AC BC ABC D AD BD CD ABD ACD BCD ABCD a b ab c ac bc abc d ad bd cd abd acd bcd abcd A BCD = I Run only (1), bc, ad and abcd A ABC = BC = I 2.830J/6.780J/ESD.63J 26
28 2 4-2 Suppose we want to alias A with BCD and ABC I A B AB C AC BC ABC D AD BD CD ABD ACD BCD ABCD (1) bc ad abcd A BCD = I A ABC = BC = I ( NB AD = I also) Aliases? Defining Relations I=BC I=AD I=ABCD A - ABC B -C C -ABD D -ABC A - D B - ABD C - ACD D - BCD A - BCD B - ACD 2.830J/6.780J/ESD.63J 27
29 Outline Fractional Factorial Designs Aliasing Patterns Implications for Model Construction Process Optimization using DOE 2.830J/6.780J/ESD.63J 28
30 Consider Higher Order Model y = β 0 + β 1 + β 21 2 Quadratic Model Now we need all 3 tests y = x x + SS Residuals =SS within y J/6.780J/ESD.63J 29
31 General Quadratic Equation η m = β 0 + k β i i =1 x im Problem k β 2i i =1 k k i =1 im + β ij j =1 j <i x im x j m + h.o.t. + ε m ŷ = β 0 + β 1 + β 2 + β 12 + β β β β β How many levels for each input? 2.830J/6.780J/ESD.63J 30
32 Quadratic Solution Same as before with matrix equation: η=xβ+ε β 0 η L β 1 ε 1 η L β 2 ε 2 η 2 M = L M M M M M M O β 11 β 22 + ε 2 M η N L β 12 M ε N 2.830J/6.780J/ESD.63J 31
33 Experimental Design for Quadratic: Full factorial 3 k Three levels per test Central Composite Design adding to 2x2 design Partial Factorials and Aliases 2.830J/6.780J/ESD.63J 32
34 Consider a Quadratic Model w/interaction Includes linear terms, quadratic terms and all first and second-order interactions =3 k N k No Interactions Full Model J/6.780J/ESD.63J 33
35 3 2 Full Factorial Quadratic Model ŷ = β 0 +β 1 +β 2 +β 12 +β β β β β (1) A B AB A2 B2 A2B B2A A2B2 y y y y y y y y y J/6.780J/ESD.63J 34
36 Which Partial Fraction? ŷ = β 0 + β 1 + β 2 + β 12 + β β 22 2 (1) A B AB A2 B2 A2B B2A A2B2 y y y y y y y y y J/6.780J/ESD.63J 35
37 Which Partial Fraction? (1) A B AB A2 B2 A21 B21 AB22 y y y y y y y y y J/6.780J/ESD.63J 36
38 Which Partial Fraction? (1) A B AB A2 B2 A21 B21 AB22 y y y y y y y y y (1) A B AB A2 B2 A21 B21 AB22 y y y y y y y y y J/6.780J/ESD.63J 37
39 Which Partial Fraction? (1) A B AB A2 B2 A21 B21 AB22 y y y y y y y y y (1) A B AB A2 B2 y y y y y y J/6.780J/ESD.63J 38
40 Quadratic Solution ŷ = β 0 + β 1 + β 2 + β β β 12 y β 0 y 2 y 3 y 4 y 5 = β 1 β 2 β 11 β 22 y = Xβ β = X 1 y y β J/6.780J/ESD.63J 39
41 A Quadratic Surface y = J/6.780J/ESD.63J 40
42 A Standard 3 2 Full Factorial Design Test x1 X J/6.780J/ESD.63J 41
43 Central Composite Design Consider the case: First Experiment is 2 2 with 4 tests Model is shown to have poor fit High SS Quad for intermediate point Decide to go to Quadratic Not Sure of Shape of Surface 2.830J/6.780J/ESD.63J 42
44 Central Composite Design +1 = Add 5 additional points: One at center One equidistant from center along each axis J/6.780J/ESD.63J 43
45 Central Composite Test x1 X original tests additional tests 2.830J/6.780J/ESD.63J 44
46 Outline Fractional Factorial Designs Aliasing Patterns Implications for Model Construction Process Optimization using DOE 2.830J/6.780J/ESD.63J 45
47 Process Optimization Create an Objective Function J Minimize or Maximize max J x min J x J=J(factors) ; J(x); J(α) Adjust J via factors with constraints, such as J/6.780J/ESD.63J 46
48 Methods for Optimization Analytical Solutions y/ x = 0 Gradient Searches Hill climbing (steepest ascent/descent) Local min or max problem Excel solver given a convex function 2.830J/6.780J/ESD.63J 47
49 Basic Optimization Problem y 0 = J 0 y (or J) x 0 x 2.830J/6.780J/ESD.63J 48
50 3D Problem 5 z J/6.780J/ESD.63J 49
51 Analytical y y(x) x = 0 x Need Accurate y(x) Analytical Model Dense x increments in Experiment Difficult with Sparse Experiments Easy to missing optimum 2.830J/6.780J/ESD.63J 50
52 Sparse Data Procedure y β 1 x - x + Linear models with small increments Move along desired gradient Near zero slope change to quadratic model x 2.830J/6.780J/ESD.63J 51
53 Extension to 3D J/6.780J/ESD.63J 52
54 Linear Model Gradient Following S21 S S19 S18 S y S16 S15 S14 S13 y S12 S11 S10 S9 S8 1.0 S7 S6 0.5 S5 0.0 S13 S16 S19 S4 S3 S2 S7 S10 x1 S1 S1 S4 y ˆ = β + β x + β x + β x x J/6.780J/ESD.63J 53
55 Steepest Descent y ˆ = β + β x + β x + β x x g x1 g = y = β 1 + β 12 = y = β 2 + β 12 g 1 g 2 G S21 S20 S19 S18 S17 S16 S15 S14 S13 y S12 S11 S10 S9 S8 S7 S6 S5 S4 Make changes in and along G x1 S3 S2 S1 Δ = g g x2 Δ 2.830J/6.780J/ESD.63J 54
56 Experimental Optimization WHY NOT JUST PICK BEST POINT? Why not optimize on-line? Skip the Modeling Step! Adaptive Methods Learn how best to model as you go. e.g. Adaptive OFACT 2.830J/6.780J/ESD.63J 55
57 Evolutionary Operation EVOP Pick best y i y 3 y 0 y 2 y y 1 4 ±δ ±δ Re-center process Do again J/6.780J/ESD.63J 56
58 Confirming Experiments Checking Intermediate points Y X Rechecking -1the optimum -1 X2 Data only at corners Test at interior point Evaluate error Consider Central Composite? 2.830J/6.780J/ESD.63J 57
59 A Procedure for DOE/Optimization Study Physics of Process Define Important Inputs Intuition about model Limits on inputs Define Optimization Penalty Function J=f(x) max J x min J x For us, x = u or α 2.830J/6.780J/ESD.63J 58
60 Procedure Identify model (linear, quadratic, terms to include) Define inputs and ranges Identify noise parameters to vary if possible (Δα s) Perform Experiment Appropriate order randomization blocking against nuisance or confounding effects 2.830J/6.780J/ESD.63J 59
61 Solve for ß s Apply ANOVA Data significant? Terms significant? Lack of Fit Significant? Procedure Drop Insignificant Terms Add Higher Order Terms as needed 2.830J/6.780J/ESD.63J 60
62 Search for Optimum Analytically Piecewise Continuously Procedure 2.830J/6.780J/ESD.63J 61
63 Find Optimum value x* Perform Confirming experiment Test Model at x* Procedure Evaluate error with respect to model Test hypothesis that y(x*) = ˆ y (x*) 2.830J/6.780J/ESD.63J 62
64 If hypothesis fails Procedure Consider new ranges for inputs Consider higher order model as needed Boundary may be optimum! 2.830J/6.780J/ESD.63J 63
65 Summary Fractional Factorial Designs Aliasing Patterns Implications for Model Construction Process Optimization using DOE 2.830J/6.780J/ESD.63J 64
2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 2.830J / 6.780J / ESD.63J Control of Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
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