麻省理工学院：《Robust System Design》Constructing Orthogonal arrays

C onstructing Orthogonal Arrays Robust System Design mit 人 16881

Constructing Orthogonal Arrays Robust System Design 16.881 MIT

Learning objectives Introduce explore orthogonality Study the standard oas Practice computing dof of an experiment Learn how to select a standard oa Introduce means to modify oas Consider studying interactions in Oas Robust System Design mit 人 16881

Learning Objectives • Introduce & explore orthogonality • Study the standard OAs • Practice computing DOF of an experiment • Learn how to select a standard OA • Introduce means to modify OAs • Consider studying interactions in OAs Robust System Design 16.881 MIT

What is orthogonality? Geometr Vector algebra x 0 Robust design Form contrasts for the columns(i) V1+W;p+V;3……+V 9 =0 Inner product of contrasts must be zero W Robust System Design mit 人 16881

What is orthogonality? • Geometry v v • Vector algebra x ⋅ y = 0 • Robust design – F o rm contrasts for the columns ( i) wi1 + wi2 + wi3 L + wi9 = 0 – Inner product of contrasts must be zero w ⋅w = 0 Robust System Design 16.881 MIT

Before Constructing an Array We must define Number of factors to be studied Number of levels for each factor 2 factor interactions to be studied Special difficulties in running experiments Robust System Design mit 人 16881

Before Constructing an Array We must define: • Number of factors to be studied • Number of levels for each factor • 2 factor interactions to be studied • Special difficulties in running experiments Robust System Design 16.881 MIT

Counting degrees of Freedom Grand mean Each control factor(e.g, A) ( of levels ofA-1) Each two factor interaction(e.g, AxB) (OF for A)x doF for B) Example --2 X37 Robust System Design mit 人 16881

Counting Degrees of Freedom • Grand mean –1 • Each control factor (e.g., A) – (# of levels of A -1) • Each two factor interaction (e.g., AxB) – (DOF for A)x(DOF for B) • E x a m p l e - - 2 1x37 Robust System Design 16.881 MIT

Breakdown of dof Ni n= Number of n values sS due to mean ( levels)-1 ( levels )-1 etc DOF for factor a factor B error Robust System Design mit 人 16881

Breakdown of DOF 16.881 Robust System Design MIT n 1 SS due to mean n-1 (# levels) -1 factor A n = Number of η values (# levels) -1 factor B DOF for error etc

DOF and Modeling equations · Additive model na,Bi, Ck, Di)=u+a,+b, +Ck+d, +e How many parameters are there? How many additional equations constrain the parameters? Robust System Design mit 人 16881

DOF and Modeling Equations • Additive model 0 η( Ai , B j , Ck , Di) = µ + ai + b j + c k + di + e • How many parameters are there? • How many additional equations constrain the parameters? Robust System Design 16.881 MIT

DOF -- Analogy with Rigid Body motion How many parameters define the position and orientation of a rigid body? How do we remove these dof? (X,Y, 2 Rotation Translation Robust System Design mit 人 16881

DOF -- Analogy with Rigid Body Motion • How many parameters define the position and orientation of a rigid body? • How do we remove these DOF? γ y z (X,Y,Z) y z α β x x Rotation Translation Robust System Design 16.881 MIT

Notation for Matrix Experiments L(3 Number of experiments Number of levels Number of factors (3-1)x4 Robust System Design mit 人 16881

Notation for Matrix Experiments Number of experiments L9 (3 4) Number of levels Number of factors 9=(3-1)x4+1 Robust System Design 16.881 MIT

Standard Orthogonal Arrays See table 7. 1 on Phadke page 152 Note: You can never use an array that has fewer rows than doF req'd Note: The number of factors of a given evel is a maximum You can put a factor with fewer columns into a column that has more levels But not fewer Robust System Design mit 人 16881

Standard Orthogonal Arrays • See table 7.1 on Phadke page 152 • Note: You can never use an array that has fewer rows than DOF req’d • Note: The number of factors of a given level is a maximum • You can put a factor with fewer columns into a column that has more levels – But NOT fewer! Robust System Design 16.881 MIT