lecture14 - http:/ocw.mit.edu _ MIT OpenCourseWare Spring...

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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 .
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2.830J/6.780J/ESD.63J 1 M anufacturing Control of Manufacturing Processes Subject 2.830/6.780/ESD.63 Spring 2008 Lecture #14 Aliasing and Higher Order Models April 3, 2008
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2.830J/6.780J/ESD.63J 2 M anufacturing Outline •L a s t T i m e – 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
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2.830J/6.780J/ESD.63J 3 M anufacturing 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: – We will not be able to find all 8 coefficients y = β 0 + β 1 x 1 + β 2 x 2 + β 12 x 1 x 2 + β 3 x 3 13 x 1 x 3 + β 23 x 2 x 3 + β 123 z x 1 x 2 x 3 + ε
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2.830J/6.780J/ESD.63J 4 M anufacturing 2 3-1 Experiment • Consider doing 4 experiments instead of 8; e.g.: 1 2 3 4 x 1 x 2 x 1 x 2 1 1 + 1 + 1 1 1 1 + 1 1 + 1 + 1 + 1 •Th is a 2 2 array • Could also be for 3 inputs if we define x 3 = x 1 x 2
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2.830J/6.780J/ESD.63J 5 M anufacturing 2 3-1 Experiment 1 2 3 4 x 1 x 2 x 3 1 1 + 1 + 1 1 1 1 + 1 1 + 1 + 1 + 1 But now we can only define 4 coefficients in the model: e.g.: ) y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 i.e. no interaction terms
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2.830J/6.780J/ESD.63J 6 M anufacturing 2 3-1 Experiment Or we could choose other terms: ) y = β 0 + β 1 x 1 + β 2 x 2 + β 13 x 1 x 3 or: ) y = β 0 + β 1 x 1 + β 12 x 1 x 2 + β 3 x 3 or:
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2.830J/6.780J/ESD.63J 7 M anufacturing Confounding / Aliasing • We actually have the following: • where the z variable represent sums of the various input terms, e.g. • where the specific choice of the experimental array determines what these sums are ) y = β 0 + β ' 1 z 1 + β ' 2 z 2 + β ' 3 z 3 z 1 = x 1 x 2 + x 3 z 2 = x 1 + x 2 x 3 L
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2.830J/6.780J/ESD.63J 8 M anufacturing Confounding / Aliasing Test I A B AB C AC BC ABC (1) 1 -1 1 1 1 a 1 1 - 1- 1 1 1 b 11- 1 - 11 ab 1 1 1 1 c 1 1 1 1 ac 1 1 1 1 b c1- 1 a b c 11111111 2 3 Array: (Our X matrix)
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2.830J/6.780J/ESD.63J 9 M anufacturing Confounding / Aliasing Consider upper half: Tes t I A B AB C AC BC ABC ( 1 ) 1- 1 - 11- 11 1 a 1 1 - 1 1 1 b 1 - ab 1 1 1 1 -1 -1 -1 -1 c 1 -1 -1 1 1 -1 -1 1 ac 1 1 -1 -1 1 1 -1 -1 b c1- 1 a b c 11111111 Look at columns for C - no change at all! or C = -I Also AC = -A and BC = -B, and ABC = -AB
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2.830J/6.780J/ESD.63J 10 M anufacturing Confounding / Aliasing Contrast A =[ -(1)+a-b+ab] Contrast AC =[ (1)-a+b-ab] AC is an alias of A Note that alias of A =A*(-C) Defining Relation I = -C Test I A B AB C AC BC ABC (1) 1 -1 -1 1 -1 1 1 -1 a 1 1 -1 -1 -1 -1 1 1 b 1- 11- 1 - 11 a b 1111 - 1 - 1 - 1 - 1 c 1 -1 -1 1 1 -1 -1 1 ac 1 1 -1 -1 1 1 -1 -1 b c1- 1 a b c 11111111
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2.830J/6.780J/ESD.63J 11 M anufacturing Choice of Design?
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lecture14 - http:/ocw.mit.edu _ MIT OpenCourseWare Spring...

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