lecture12 - http/ocw.mit.edu MIT OpenCourseWare 2.830J...

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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 #12 Full Factorial Models March 20, 2008
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2.830J/6.780J/ESD.63J 2 M anufacturing Outline • Modelling “Effects” from Multiple Inputs • ANOVA on Effects • Linear and Quadratic Models • Model Coefficient Calculation – Regression (General Approach) – Contrasts (for Factorial Designs)
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2.830J/6.780J/ESD.63J 3 M anufacturing What Is the Effect? • What is the relationship between Hold Time and Dimension? I.M. Hold Time Change 2.025 2.030 2.035 2.040 2.045 2.050 2.055 2.060 0.5 1 1.5 2 2.5 Hold Time Dimension
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2.830J/6.780J/ESD.63J 4 M anufacturing But Wait… There’s More! IM Run data 2.015 2.020 2.025 2.030 2.035 2.040 2.045 2.050 2.055 2.060 111111111111111222222222222222333333333333333444444444444444 Levels changes Velocity low high low high Hold time low low high high Do the two inputs interact? Dimension
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2.830J/6.780J/ESD.63J 5 M anufacturing Model Form • Linear • Quadratic • Exponential • General Polynomial? • Interactions What data needed to decide and/or estimate parameters of different model forms?
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2.830J/6.780J/ESD.63J 6 M anufacturing Multiple Input/Treatment Models • In general k inputs – If 2 levels for each ….. 2 k combinations – If 3 levels for each …… 3 k combinations • Why use more than one input? – More than one output – Change mean and variance • Process Robustness • Optimization of Quality Loss
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2.830J/6.780J/ESD.63J 7 M anufacturing linear term interaction term i = input index k = total number of inputs mean higher order terms (model form error) A General Linear Model for k inputs η=β 0 + β i i = 1 k x i + ij x i i = 1 k j = 1 j < i k x j + h . o . t . + ε residual error
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2.830J/6.780J/ESD.63J 8 M anufacturing Two Input Model η=β 0 + β 1 x 1 + β 2 x 2 + β 12 x 1 x 2 + h . o . t . + ε 4 coefficients to determine How many data points (factors, levels) are needed to uniquely identify?
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2.830J/6.780J/ESD.63J 9 M anufacturing Consider a One Input Case I.M. Hold Time Change 2.025 2.030 2.035 2.040 2.045 2.050 2.055 2.060 0.5 1 1.5 2 2.5 Hold Time Dimension
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2.830J/6.780J/ESD.63J 10 M anufacturing Linear One Input Example Linear model η=β 0 + β 1 x η = 1 2 X = 1 x 1 x + β = 0 1 ε = 0 (for means) Assume 2 levels x , x + = X + ε Since X is square and ε= = X 1 With 1 trial at each level we get:
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2.830J/6.780J/ESD.63J 11 M anufacturing Linear Model with Replicates • Line will no longer intersect specific points • What is “best fit? I.M. Hold Time Change 2.025 2.030 2.035 2.040 2.045 2.050 2.055 2.060 0.5 1 1.5 2 2.5 Hold Time Dimension
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2.830J/6.780J/ESD.63J 12 M anufacturing Minimum Error Line Fits • Define squared error for data for a given β o and 1 • Find o and 1 that lead to minimum of the sum of all e 2 • OR - Solve the matrix equation to get = ( X T X ) 1 X T η where X is a non-square matrix of all inputs for all replicates and η is the vector of all trial outputs e 2 = ( η− ( 0 − β 1 x )) 2
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2.830J/6.780J/ESD.63J 13 M anufacturing Aside η = X β + ε ε = η X squared error J = ε T = η X () T X ( ) The minimum value of J is then found by the vector partial derivative: J ∂β = 0 =− 2
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This note was uploaded on 09/24/2010 for the course MECHE 2.830J taught by Professor Davidhardt during the Spring '08 term at MIT.

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lecture12 - http/ocw.mit.edu MIT OpenCourseWare 2.830J...

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