# DOE - Fractional Factorial Design(Confounding Pattern or...

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Fractional Factorial Design (Confounding Pattern or Alias Structure) Example 1: 2^(3 - 1) design with 3 = 12 (X3 = X1 * X2) or C = AB (textbook notation) Design layout (or design matrix) is given in page 335 of the DOE handouts. See page 334 for a graphic presentation of the fractional factorial design (also given in lecture notes on 08/28/02). 1. Multiply X3 to both sides of X3 = X1 * X2. 2. Left hand side (LHS) becomes X3 * X3, which is the same column of +1 and - 1 multiplied by itself. Thus, it becomes a column of all +1’s, i.e., X3 * X3 = I = (+1, +1, …, +1). 3. Right hand side (RHS) becomes X1 * X2 * X3. 4. We thus obtain a “ generating relationship ”: I = X1 * X2 * X3 (or I = 123 in short). 5. X3 = X1 * X2 (or 3 = 12) is called the “ generator ” of the design. 6. Because there are three letters in the RHS of the generating relationship, the resolution ” of the design is 3. 7. From statement #4, if one multiplies X1 to both sides of the generating relationship, the resulted relationship becomes X1 = (X1 * X1) * X2 * X3 = X2 * X3; that is 1 = 23 in short. 8. Similarly, one can obtain 2 = 13 and 3 = 12 relationships. 9. You will find that the column of X2*X3 (i.e., 23) of the design layout will be the same as the column X1 (verify it yourself). Similarly, X2 column is the same as X1*X3, and X3 column and the same as X1*X2. 10. In summary, the “ confounding pattern ” of the design in this example is I = 123, 1 = 23, 2 = 13, and 3 = 12. 11. Then, the “ Estimable Effects ” are Average (for the column of one’s), X1 (which is equal to X2*X3), X2 (= X1*X3) and X3 (= X1*X2), i.e., only 4 estimable effects from the experiment of 4 runs. 1

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Example 1: 2^(3 - 1) Design A 2^3 Full Factorial Design is split into two blocks of 4-run design by using the generator, +3 = 12 or - 3 = 12. 2 Goal for data collection using the DOE procedures: 1. Estimate effects from the variables. 2. Estimate variance of effects. 3. Build models for process characterization and optimization.
3 #5 #3 #8 #2 Effect: “normalized” difference of averages of data at the high and low levels of a variable, e.g., Estimate of X1-effect from data (#2, #3, #5, #8) = [ ave(#2 + #8) - ave(#3 + #5) ] / 2

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4 Estimate of X3-effect from data (#2, #3, #5, #8) = [ ave(#2 + #3) - ave(#5 + #8) ] / 2 Similarly, X2-effect and the overall average (intercept in the model) can be estimated. Note that the slope in the following model is equal to half of the effect. Y = β 0 + β 1 X1 + β 2 X2 + β 3 X3. The fractional factorial design selects a fraction of the full factorial design carefully and intelligently to facilitate the “best” estimates of effects and model coefficients.
A) 2^3 Full Factorial Design in 8 runs: Run X1 X2 X3 1 - 1 - 1 - 1 2 +1 - 1 - 1 3 - 1 +1 - 1 4 +1 +1 - 1 5 - 1 - 1 +1 6 +1 - 1 +1 7 - 1 +1 +1 8 +1 +1 +1 B) 2^(3 - 1) Half-Fractional Factorial Design in 4 runs: Use 3 = 12 as the generator. Run

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## This note was uploaded on 06/27/2011 for the course ISYE 3370 taught by Professor L during the Spring '11 term at Georgia Tech.

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DOE - Fractional Factorial Design(Confounding Pattern or...

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