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.
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Example 2A: 2^(6 - 2) resolution IV design with generators 5 = 123 and 6 = 234. 1. Design layout (matrix): 5 = 123, 6=234 Run X1 X2 X3 X4 X5 X6 12 35 1346 2456 1 - 1 - 1 - 1 - 1 - 1 - 1 +1 +1 +1 +1 2 +1 - 1 - 1 - 1 +1 - 1 - 1 - 1 - 1 - 1 3 - 1 +1 - 1 - 1 +1 +1 - 1 - 1 - 1 - 1 4 +1 +1 - 1 - 1 - 1 +1 +1 +1 +1 +1 5 - 1 - 1 +1 - 1 +1 +1 +1 +1 +1 +1 6 +1 - 1 +1 - 1 - 1 +1 - 1 - 1 - 1 - 1 7 - 1 +1 +1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 8 +1 +1 +1 - 1 +1 - 1 +1 +1 +1 +1 9 - 1 - 1 - 1 +1 - 1 +1 +1 +1 +1 +1 10 +1 - 1 - 1 +1 +1 +1 - 1 - 1 - 1 - 1 11 - 1 +1 - 1 +1 +1 - 1 - 1 - 1 - 1 - 1 12 +1 +1 - 1 +1 - 1 - 1 +1 +1 +1 +1 13 - 1 - 1 +1 +1 +1 - 1 +1 +1 +1 +1 14 +1 - 1 +1 +1 - 1 - 1 - 1 - 1 - 1 - 1 15 - 1 +1 +1 +1 - 1 +1 - 1 - 1 - 1 - 1 16 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 Note: 12 = 35 = 1346 = 2456 (confounded) 2. Generating Relationships: I = 1235 = 2346 = 1456, where the last relationship I = 1456 is from the multiplication of the first two relationships I = 1235 and I = 2346. Note that (23)*(23) = I. 3. Resolution in this design is IV = 4 (the minimum number of letters in the generating relationships is 4) . Note that resolution IV design in this example is “stronger” than the resolution III design in the first example. This will be explained after the discussion of confounding patterns. 4. Confounding Pattern (alias structure): (including the estimable effects)
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This note was uploaded on 06/27/2011 for the course ISYE 3039 taught by Professor Roshanv during the Spring '08 term at Georgia Tech.

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

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