Matrix Experiments Using Orthogonal Arrays 16 881 Robust System Design MIT Comments on HW 2 and Quiz 1 Questions on the Reading Quiz Brief Lecture Paper Helicopter Experiment 16 881 Robust System Design MIT Learning Objectives Introduce the concept of matrix experiments Define the balancing property and orthogonality Explain how to analyze data from matrix experiments Get some practice conducting a matrix experiment 16 881 Robust System Design MIT Static Parameter Design and the P Diagram Noise Factors Induce noise Product Process Signal Factor Hold constant for a static experiment 16 881 Response Optimize Control Factors Robust System Design Vary according to an experimental plan MIT Parameter Design Problem Define a set of control factors A B C Each factor has a set of discrete levels Some desired response A B C is to be maximized 16 881 Robust System Design MIT Full Factorial Approach Try all combinations of all levels of the factors A1B1C1 A1B1C2 If no experimental error it is guaranteed to find maximum If there is experimental error replications will allow increased certainty BUT experiments levels control factors 16 881 Robust System Design MIT Additive Model Assume each parameter affects the response independently of the others Ai B j Ck Di ai b j ck d i e This is similar to a Taylor series expansion f x y f xo yo 16 881 f x x xo x xo Robust System Design f y y yo h o t y yo MIT One Factor at a Time Expt No 1 2 3 4 5 6 7 8 9 16 881 Control Factors A B C D 2 1 3 2 2 2 2 2 2 2 2 2 1 3 2 2 2 2 2 2 2 2 2 1 3 2 2 Robust System Design 2 2 2 2 2 2 2 1 3 1 2 3 4 5 6 7 8 9 MIT Orthogonal Array Expt No 1 2 3 4 5 6 7 8 9 16 881 A 1 1 1 2 2 2 3 3 3 Control Factors B C 1 2 3 1 2 3 1 2 3 1 2 3 2 3 1 3 1 2 Robust System Design D 1 2 3 3 1 2 2 3 1 1 2 3 4 5 6 7 8 9 MIT Notation for Matrix Experiments L9 34 Number of experiments Number of levels Number of factors 9 3 1 x4 1 16 881 Robust System Design MIT Why is this efficient One factor at a time Estimated response at A3 is 3 a3 e3 Orthogonal array Estimated response at A3 is 3 a3 1 3 e7 e7 e7 Variance sums for independent errors Error variance 1 replication number 16 881 Robust System Design MIT Factor Effect Plots Factor Effects on the Mean Time sec 20 15 A1 A2 A3 B3 C1 C2 C3 D1 D2 D3 B1 10 5 A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3 Factor Effects on the Variance 25 20 15 10 5 0 A2 A1 A3 A1 A2 A3 16 881 B2 0 Time sec Which CF levels will you choose What is your scaling factor Robust System Design B1 B2 B3 B1 B2 B3 C1 C2 C1 C2 C3 D1 D2 C3 D3 D1 D2 D3 MIT Prediction Equation Time sec Factor Effects on the Variance 25 20 15 10 5 0 A2 A1 A1 A2 A3 A3 B1 B2 B3 B1 B2 B3 C1 C2 C1 C2 C3 D1 D2 C3 D3 D1 D2 D3 Ai B j Ck Di ai b j ck d i e 16 881 Robust System Design MIT Inducing Noise Expt No 1 2 3 4 5 6 7 8 9 16 881 Control Factors A B C D 1 1 1 2 2 2 3 3 3 1 2 3 1 2 3 1 2 1 1 2 3 2 3 1 3 1 2 1 2 3 2 1 2 3 3 1 1 2 3 4 5 6 7 8 9 Robust System Design Expt No 1 2 Noise Factor N 1 2 MIT Analysis of Variance ANOVA ANOVA helps to resolve the relative magnitude of the factor effects compared to the error variance Are the factor effects real or just noise I will cover it in Lecture 7 You may want to try the Mathcad resource center under the help menu 16 881 Robust System Design MIT