M Full-Factorial Experiment 16888 E77 Specify levels for each factor Evaluate outputs at every combination of values n factors complete but expensive In observations / levels E Factor A B Al B1 2 factors, 3 levels each A1 B2 Al B3 In=32 =9 expts B1 4 factors 3 levels each 6 B3 7 A3 B1 n= 34=81 expts A3 B2 A3 B3 o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
6 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Full -Factorial Experiment Factorial Experiment • Specify levels for each factor • Evaluate outputs at every combination of values n factors – complete but expensive! l levels l n observations Expt Factor No. A B 1 A1 B1 2 A1 B2 3 A1 B3 4 A2 B1 5 A2 B2 6 A2 B3 7 A3 B1 8 A3 B2 9 A3 B3 2 factors, 3 levels each: l n = 3 2 = 9 expts 4 factors, 3 levels each: l n = 3 4 = 81 expts
Mlesd Fractional Factorial Experiments 16888 E77 Due to the combinatorial explosion, We cannot usually perform a full factorial experiment So instead we consider just some of the possible combinations · Questions: How many experiments do need? Which combination of levels should i choose? Need to balance experimental cost with design space coverage C Massachusetts Institute of Technology - Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
7 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Fractional Factorial Experiments Fractional Factorial Experiments • Due to the combinatorial explosion, we cannot usually perform a full factorial experiment • So instead we consider just some of the possible combinations • Questions: – How many experiments do I need? – Which combination of levels should I choose? • Need to balance experimental cost with design space coverage
Mlesd Fractional Factorial Design 16888 E77 Initially, it may be useful to look at a large number of factors superficially rather than a small number of factors in detail: 1112 1112:113:143… 21:122 213122::23:24…… VS 311325:133:1343 many levels n15n2 many factors 8 C Massachusetts Institute of Technology - Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
8 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Fractional Factorial Design Fractional Factorial Design Initially, it may be useful to look at a large number of factors superficially rather than a small number of factors in detail: 1 11 12 2 21 22 1 2 , , , n nn f ll f ll f ll # # 1 11 12 13 14 2 21 22 23 24 3 31 32 33 34 ,,,, , ,, , , , ,, , , f llll f ll ll f ll ll ! ! ! vs. many levels many factors
Mlesd DoE Techniques Overview 16888 ES077 TECHNIQUE COMMENT EXPENSE ( levels, n=# factors Full factorial Evaluates all possible in -grows design designs exponentially with number of factors Orthogonal arrays Dont always seem to Moderate-depends work -interactions? on which array One at a time Order of factors?1+n(l-1)-cheap Latin hypercubes Not reproducible, 1-cheap poor coverage if divisions are large Parameter studyCaptures no 1+n(l-1)-cheap interactions o Massachusetts Institute of Technology-Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
9 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics DoE Techniques Overview Techniques Overview TECHNIQUE COMMENT EXPENSE (l=# levels, n=# factors) Full factorial design Evaluates all possible designs. ln - grows exponentially with number of factors Orthogonal arrays Don’t always seem to work - interactions? Moderate – depends on which array One at a time Order of factors? 1+n(l-1) - cheap Latin hypercubes Not reproducible, poor coverage if divisions are large. l - cheap Parameter study Captures no interactions. 1+n(l-1) - cheap
Mest Parameter Stud 16888 E77 Specify levels for each factor Change one factor at a time, all others at base level Consider each factor at every level Expt Factor No n factors A D 1+n(-1) B1 D1 evaluations B1 C1 D1 /levels A3 B1 C1 D1 B2 C1 D1 4 factors 3 levels each 12345678q Al B3 D1 Al B1 D1 1+n(1-1) Al B1 C3 A1 B1 D2 1+4(3-1)=9 expts B1 Baseline: Al. B1. C1. D1 o Massachusetts Institute of Technology -Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
10 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Parameter Study Parameter Study • Specify levels for each factor • Change one factor at a time, all others at base level • Consider each factor at every level Expt Factor No. AB C D C1 D1 D1 D1 D1 D1 D1 D1 D2 D3 C1 C1 C1 C1 C2 C3 C1 C1 4 A1 B2 5 A1 B3 6 A1 B1 7 A1 B1 8 A1 B1 1 A1 B1 2 A2 B1 3 A3 B1 9 A1 B1 n factors 1+ n ( l-1) evaluations l levels 4 factors, 3 levels each: 1+ n ( l-1) = 1+4(3-1) = 9 expts Baseline : A1, B1, C1, D1