IOE366-Ch11-Multifactor Studies

IOE366-Ch11-Multifactor Studies - Ch 11 Multifactor...

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Ch 11 - Multifactor Experiments 11.1 Two Factor ANOVA with K ij = 1 11.2 Two Factor ANOVA with K ij > 1 11.3 Three Factor ANOVA 11.4 2 p Factorial Experiments 1 Design of Multifactor Experiments Experiment (Signal) Design (Noise) Factorial Randomization Full Fractional Nested Complete Blocking 2 Engineering Experiments ± Reduce time to design/develop new roducts & processes products & processes ± Improve performance of existing processes prove eliability nd ± Improve reliability and performance of products ± Achieve product & process robustness ± Evaluation of materials, design alternatives, setting component & system tolerances, etc. 3 One-Variable-At-A-Time Experiments 4 Experiment #1: Study Effects of Reaction Time on Yield (Reaction Temperature held fixed at 225 o C)
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ollow- n Experiment Follow on Experiment 5 Experiment #2: Study Effects of Reaction Temperature on Yield (Reaction Time held fixed at 130 minutes) EAKNESS OF WEAKNESS OF ONE-VARIABLE-AT-A-TIME EXPERIMENTS Experiment #1: Study Effects of Reaction ime on Yield Experiment #2: Study Effects of eaction Temperature on Yield 6 Time on Yield (Reaction Temperature held fixed at 225 o C) Reaction Temperature on Yield (Reaction Time held fixed at 130 minutes) eakness of One t ime Expts Weakness of One-at-a-Time Expts 1. Invariably arrive at local ptima (or worse) optima (or worse) 2. Hawthorne Studies 3. Response Surface Designs (or EVOP) offer efficient advantages g 7 Interaction effects 8
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Interaction effects 9 11.2 Two Factor Layout ( K > 1) Factor B μ 11 μ 12 μ 1J A 1 B 1 B 2 B J μ 21 μ 22 μ 2J A 2 ctor A A Fa c 10 μ I2 I μ I1 μ IJ Two Factor Model Model : X k = μ + ε k i = 1, …,I ijk ij ijk j = 1 , , J k = 1 , , K ε ijk ~ N (0, σ 2 ) Decompose the model into the following: 11 X ijk = μ + α i + β j + αβ ij + ε ijk Model Hypotheses 01 2 : ... 0 I H α αα == 2 : ... 0 J ββ β 1 1 2 : ... 0 IJ αβ 1 : at least one parameter 0 12
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Estimation of Parameters l ... x μ = l l .. ... () i xx α = m . . ... j β = l m . .. . . ... ij xxx αβ −− =+ ++ + 13 . μμ = = Breakdown of SSTR . ... IJK k ∑∑∑ 2 = .. ... 2 + . . ... 2 + . .. . . ... + 2 14 NOVA (Fixed Effects) ANOVA (Fixed Effects) Source df MS E(MS) Between Trt IJ – 1 MSTR A I – 1 MSA σ 2 + JK 2 1 I SB 2 2 B J 1 MSB σ + IK 1 J B )( ) SAB 2 JI n ∑∑ AB (I – 1)(J – 1) MSAB σ 2 + (1 ) ) IJ Within Trt IJ(K – 1) MSE σ 2 15 Total IJK – 1 Example #1: Quality Example w Interaction Shift 1 Shift 2 Shift 3 Supplier 1 71.43 102.48 80.78 8 83 8 51 2 86 78.83 88.51 82.86 83.58 86.15 80.46 75.65 91.17 78.18 5 82 7 31 3 04 75.82 97.31 63.04 Supplier 2 77.53 93.72 84.81 84.00 83.01 78.80 3.43 7.24 1.47 83.43 87.24 71.47 82.06 88.97 72.67 78.69 87.69 82.82 6.19 2.34 3.01 16 Supplier 3 86.19 82.34 83.01 81.36 93.13 77.11 78.00 77.68 79.13
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General Linear Model: Quality versus Supplier, Shift Factor Type Levels Values Supplier fixed 3 Supplier 1 Supplier 2 Supplier 3 Shift fixed 3 Shift 1 Shift 2 Shift 3 Analysis of Variance for Quality, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P upplier 2 19 60 19 60 9 80 0 33 0 720 Supplier 19.60 19.60 9.80 0.33 0.720 Shift 2 704.07 704.07 352.04 11.93 0.000 Supplier*Shift 4 430.75 430.75 107.69 3.65 0.014 Error 36 1062.05 1062.05 29.50 otal 44 2216 47 17 Total 2216.47 Main Effects Plot - Data Means for Quality Shift Supplier 86 84 82 Quality 80 8 S h i f t 3 2 1 u p l e r 78 18 Interaction Plot - Data Means for Quality Supplier 1 3 Supplier Supplier 2
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IOE366-Ch11-Multifactor Studies - Ch 11 Multifactor...

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