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4. Design of (DOE) (The 2k Factorial Designs)

Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University

Complex Sys. Des. 1 Example: Golfing

— How to improve my score in Golfing? — Practice!!!

— Other than that? — Type of driver used (oversized or regular sized) — Type of ball (2 piece or 3 piece) — Walking or riding cart — Drinking water or beer — Etc…

What combination of the factors is the best for me?

Complex Sys. Des. 2 How to find my best condition?

— One-factor-at-a time strategy

Any Problem??

Complex Sys. Des. 3 Interaction effect between the factors

— Interaction effect between type of driver and beverage

Complex Sys. Des. 4 Factorial

Two factors with 2 level for each factor

Complex Sys. Des. 5 Factorial Design of Experiments

Three factors

Four factors

Any Problem?? Complex Sys. Des. 6 Fractional Factorial Design of Experiments

16 experiments -> 8 experiments

Question for the semester How to effectively reduce the number of experiments? How to analyze the results of experiments?

Complex Sys. Des. 7 Introduction to 2k Factorial Designs

— Special case of the general factorial design; k factors, all at two levels — The two levels are usually called low and high (they could be either quantitative or qualitative) — Very widely used in industrial experimentation — Form a basic “building block” for other very useful experimental designs — Useful for factor screening

Complex Sys. Des. 8 Chemical Process Example

A = reactant concentration, B = catalyst amount, y = recovery

Complex Sys. Des. 9 The Simplest Case: The 22

“-” and “+” denote the low and high levels of a factor, respectively — Low and high are arbitrary terms — Geometrically, the four runs form the corners of a square — Factors can be quantitative or qualitative, although their treatment in the final model will be different

Complex Sys. Des. 10 Notation of the 2k Designs

— A special notation is used to represent the runs. In general, a run is represented by a series of lower case letters. If a letter is present, then the corresponding factor is set at the high level in that run; if it is absent, the factor is run at its low level. For example, run a indicates that factor A is at the high level and factor B is at the low level. The run with both factors at the low level is represented by (1). — This notation is used throughout the 2k design series. For example, the run in a 24 with A and C at the high level and B and D at the low level is denoted by ac.

Complex Sys. Des. 11 Estimation of Factor Effects

A= y+ - y - The letters (1), a, b, and ab also AA represent the totals of all n ab+ a b + (1) = - observations taken at these 2n 2 n design points. 1 =2n [ab + a - b - (1)]

B= y+ - y - BB ab+ b a + (1) = - 2n 2 n 1 =2n [ab + b - a - (1)] ab+(1) a + b AB = - 2n 2 n 1 =2n [ab + (1) - a - b ] Orthogonal Design Complex Sys. Des. 12 Contrasts in the 22

— Recall contrasts a C= y + y - y - y A ABABABAB+ + + - - + - - C = å ci y i. i=1 1 =[ab + a - b - (1)] n — Effect = /2 — Sum of Square of Contrasts 2 æa ö2 é1 ù ç c y ÷ ê[ab+ a - b - (1)] ú 2 çå i i. ÷ ën û [ab+ a - b - (1)] èi=1 ø SSA = = SSc = 1 4n 1 a (4) c 2 n n å i i=1 (Contrast)2 = 4 / n

Complex Sys. Des. 13 Sum of Squares of the 22 Designs

— The analysis of is [a + ab - b - (1)]2 SS = completed by computing A 4n the total sum of squares SST (with 4n-1 degrees of [b + ab - a - (1)]2 SS = freedom) as usual, and B 4n obtaining the error sum of squares SSE [with 4(n-1) [ab + (1) - a - b]2 SS = degrees of freedom] by AB 4n subtraction.

Complex Sys. Des. 14 ANOVA of the Chemical Processing

The F-test for the “model” source is testing the significance of the overall model; that is, is either A, B, or AB or some combination of these effects important?

Complex Sys. Des. 15 Regression Model

— Regression model for 2k Designs

y=bo + b1 x 1 + b 2 x 2 + b 3 x 1 x 2 + e

— Where x1 is coded variable of Factor A and x2 is coded variable of Factor B — Low lever = -1 and High level = +1 — Relationship between natural and coded variables

AAA-(+ + - ) / 2 x = 1 (AA+- - )/ 2

Complex Sys. Des. 16 Regression Model for Chemical Processing

— Since interaction effect is very small, the regression model employed is y=bo + b1 x 1 + b 2 x 2 + e

— where x1 is coded variable of the reactant concentration and x2 is coded variable of the amount of catalyst

Conc-( Conchigh + Conc low ) / 2 x1 = Catalyst -1.5 (Conchigh- Conc low )/ 2 x = 2 0.5 Conc-(25 + 15) / 2 Conc - 20 = = (25- 15) / 2 5

Complex Sys. Des. 17 Regression Model for Chemical Processing

— Estimating b 0 ,, b 1 b 2 of the regression model, using least square method — We will return to least square method in response surface method — Regression model with coded factors is æ ö æ ö ç8.33÷ ç- 5.00 ÷ yˆ =27.5 +ç÷ x1 + ç ÷ x 2 èç2÷ ø èç 2 ÷ø ˆ ˆ — where 27.5 is grand average of all observation, b 1 , b 2 is one-half of the corresponding factor effect estimates — Regression model with uncoded factors

æ8.33 öæConc--- 20 ö æ 5.00 öæ Catalyst 1.5 ö yˆ =27.5 +ç÷ ç ÷ + ç ÷ ç ÷ èç2÷ øè ç 5 ÷ø èç 2 ÷øè ç 0.5 ÷ø =18.33 + 0.8333Conc - 5.00 Catalyst

Complex Sys. Des. 18 Residual Analysis of Chemical Processing

— Residual e =y - yˆ æ8.33 ö æ- 5.00 ö e =28 - 25.835 yˆ =27.5 +ç÷ ( - 1) + ç ÷ ( - 1) For example 1 èç2÷ ø èç 2 ÷ø

Complex Sys. Des. 19 Review of Analysis Procedure — Estimate factor effects — Main effects, interaction effects — Formulate model 2 — 2 design example y=bo + b1 x 1 + b 2 x 2 + b 3 x 1 x 2 + e — Statistical testing (ANOVA) — Refine the model

— Chemical processing example y=bo + b1 x 1 + b 2 x 2 + e — Regression model estimation ˆ ˆ ˆ ˆ — By Least Square Method y=bo + b1 x 1 + b 2 x 2 — Analyze residuals (graphical) — Normal probability plot of residuals — Interpret results

Complex Sys. Des. 20 The 23 Factorial Design

Complex Sys. Des. 21 Factor Effect of the 23 Designs

— 3 factors, each at two levels — 8 factor-level combinations — 3 main effects: A,B,C — 3 two-factor interactions: AB, AC,BC — 1 three-factor interaction: ABC

Complex Sys. Des. 22 Factor Effect of the 23 Designs

— Main effect of A 1 A=[ a + ab + ac + abc -(1) - b - c - bc] 4n — Main effect of B 1 B=[ b + ab + bc + abc -(1) - a - c - ac] 4n — Main effect of C

1 C=[ c + ac + bc + abc -(1) - a - b - ab] 4n

Complex Sys. Des. 23 Factor Effect of the 23 Designs

— Interaction effect of AB 1 AB=é AB()() C + AB C ù 2 ëêhigh low ûú where 1 1 AB( C )= [ ab + (1)] - [ a + b ] low 2n 2 n 1 1 AB()[][] C= abc + c - ac + bc high 2n 2 n Therefore 1 AB=[ ab + (1) + abc + c - b - a - bc - ac ] 4n — The same approach can be applied for the interaction effect of BC and AC

Complex Sys. Des. 24 Factor Effect of the 23 Designs

— Interaction effect of ABC is defined as the average difference between the AB interaction at the two different level of C

1 ABC=[ AB( C high ) - AB ( C low )] 2 1ïìé 1 1 ù é 1 1 ù =íïê(abc + c ) - ( ac + ab ) ú - ê( ab + (1)) -( a + b) ú 2îïëê 2n 2 n ûú ëê 2n 2 n ûú 1 = [abc - bc - ac + c - ab + b + a -(1)] 4n — How to memorize the sign of coefficients?

Complex Sys. Des. 25 Factor Effect of the 23 Designs

Complex Sys. Des. 26 Properties of the Table

— Except for column I, every column has an equal number of + and – signs — The sum of the product of signs in any two columns is zero — Multiplying any column by I leaves that column unchanged (identity element) — The product of any two columns yields a column in the table:

A´ B = AB AB´ BC = AB2 C = AC

— Orthogonal design — is an important property shared by all factorial designs

Complex Sys. Des. 27 Effects, Sum of Squares, and Contrast

— The 23 Designs — Effect = Contrast/4 — Sum of squares = n(Contrast)2/8 — Contrast for factor A 1 Contrast=[ a + ab + ac + abc -(1) - b - c - bc] A n — Main effect of factor A 1 A= Contrast/ 4 =[ a + ab + ac + abc - (1) - b - c - bc] A 4n — Sum of Square of factor A

1 2 SS= n( Contrast )2 / 8 =[ a + ab + ac + abc - (1) - b - c - bc] AA 8n

Complex Sys. Des. 28 Plasma Etching Process

— A 23 factorial design was used to develop a nitride etch process on a single-wafer plasma etching tool. The design factors are the gap between the

electrodes, the gas flow (C2F6 is used as the reactant gas), and the RF power applied to the cathode. Each factor is run at two levels, and the design is replicated twice. The response variable is the etch rate for silicon nitride (Å/m)

A = gap, B = Flow, C = Power, y = Etch Rate

Complex Sys. Des. 29 Plasma Etching Process

Gap Gas flow Power

Plasma Etching Etch rate Wafer Process

Complex Sys. Des. 30 ANOVA Summary – Full Model

Important effects by A, C, AC,

Complex Sys. Des. 31 The Regression Model with Reduced Factors

Complex Sys. Des. 32 The Regression Model with Reduced Factors

Complex Sys. Des. 33 Cube Plot of Ranges

What do the large ranges when gap and power are at the high level tell you?

Complex Sys. Des. 34 The General 2k Factorial Design

Contrast Effect = 2k-1 n() Contrast 2 SS = 2k

Complex Sys. Des. 35 Unreplicated 2k Factorial Designs — These are 2k factorial designs with one observation at each corner of the “cube” — An unreplicated 2k factorial design is also sometimes called a “single replicate” of the 2k — These designs are very widely used — Risks…if there is only one observation at each corner, is there a chance of unusual response observations spoiling the results?

Complex Sys. Des. 36 Spacing of Factor Levels in the Unreplicated 2k Factorial Designs

If the factors are spaced too closely, it increases the chances that the noise will overwhelm the signal in the More aggressive spacing is usually best

Complex Sys. Des. 37 Unreplicated 2k Factorial Designs — Lack of causes potential problems in statistical testing — Replication admits an estimate of “pure error” (a better phrase is an internal estimate of error) — With no replication, fitting the full model results in zero degrees of freedom for error — Potential solutions to this problem — Pooling high-order interactions to estimate error — Normal probability plotting of effects (Daniels, 1959)

Complex Sys. Des. 38 Example of an Unreplicated 2k Design

— A chemical product is produced in a pressure vessel. A factorial is carried out in the pilot plant to study the factors thought to influence the filtration rate of this product . — The factors are A = temperature, B = pressure, C = mole ratio, D= stirring rate — A 24 factorial was used to investigate the effects of four factors on the filtration rate of a resin — Experiment was performed in a pilot plant

Complex Sys. Des. 39 The Resin Plant Experiment

Complex Sys. Des. 40 Contrast Constants for the 24 Design

Complex Sys. Des. 41 Estimates of the Effects

Complex Sys. Des. 42 ANOVA Summary for the Model as a 23 in Factors A, C, and D

Complex Sys. Des. 43 The Regression Model

Complex Sys. Des. 44 Experiments with the larger number of factors

— The system is usually dominated by the main effects and low-order interactions. Higher interactions are usually negligible. — When the number of factors is larger than 3 or 4, a common practice is to run only a single replicate design and then pool the higher order interactions as an estimate of error. — Normal probability plot of the effects may be useful — If none of the effects is significant, then the estimates will behave like a random sample drawn from a normal distribution with zero , and the plotted effects will lie approximately along a straight line. — Those effects that do not plot on the line are significant factors.

Complex Sys. Des. 45