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Constrained Optimization

Constrained Optimization. 3-8 Continuous &/or Discrete. Linear + Cross-products (interactions). Good predictions of effects and interactions. 2-Level Factorial (+ Center Points). Relative Importance of Three Stages of Experimentation.

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Constrained Optimization

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  1. Constrained Optimization 3-8 Continuous &/or Discrete Linear + Cross-products (interactions) Good predictions of effects and interactions 2-Level Factorial (+ Center Points)

  2. Relative Importance of Three Stages of Experimentation U-Unconstrained Optimization (Response Surfaces Chapter 10) S U C C-Constrained Optimization (Main effects and interactions) S-Screening Experiments (What Factors are important)

  3. A Poor Solution is to Use One-at-a-Time Experiments Run A B C D E F G H 1 - - - - - - - - 2 + - - - - - - - 3 - + - - - - - - 4 - - + - - - - - 5 - - - + - - - - 6 - - - - + - - - 7 - - - - - + - - 8 - - - - - - + - 9 - - - - - - - +

  4. Fractional Factorial Experiments • Method for Strategically Picking a Subset of a 2k Design • Used for Screening purposes • Has much Higher Power for detecting Effects through hidden replication • Can be used to estimate some interactions and limited optimization

  5. Half-Fraction of 23 I = ABC A = C I = C

  6. Paradigms That Justify Use of Fractional Factorials • •

  7. Hierarchical Ordering Principle Venus – Moon – Jupiter align Jupiter Mars Venus Crescent Moon ▪Although its possible that three planets may align with the moon, its more often that two planets will align with moon than three ▪Likewise though three factor interactions and higher order interactions are possible, its more likely that large effects will be main effects or two factor interactions

  8. Creating a half fraction design in SAS

  9. I = ABCDE

  10. Would these conclusions have been reached using one at a time experimentation?

  11. · In a one half fraction of a 2k experiment every effect that could be estimated was confounded with one other effect, thus one half the effects had to be assumed negligible in order to interpret or explain the results · In a one quarter fraction of a 2k experiment every effect that can be estimated is confounded with three other effects, thus three quarters of the effects must be assumed negligible in order to interpret or explain the results · In a one eighth fraction of a 2k experiment every effect that can be estimated is confounded with seven other effects, thus seven eights of the effects must be assumed negligible in order to interpret or explain the results, etc.

  12. Creating a 2k-p Design 1. Create a full two-level factorial in k-p factors 2. Add each of the remaining p factors by assigning them to a column of signs for an interaction among the first k-p columns

  13. These are the generators

  14. the generators the defining relation the generalized interaction

  15. Confounding Pattern or Alias Structure Defining Relation

  16. 26-3 design base design in 6-3 = 3Factors A, B, C

  17. The three factor generalized interaction is The defining relation is

  18. Select Design… New Two-Level Design ► Define Variables… ► Add>

  19. Example ¼ Fraction of 26 One possible set of generators is: Resulting in the following Alias Structure

  20. Another possible set of generators is: Resulting in the following Alias Structure

  21. Resolution as a criteria for choosing generators R, the resolution, is the length of the shortest word in the defining relation. Resolution III – main effects confounded with two-factor interactions Resolution IV – main effects confounded with three-factor interactions, and two factor interactions confounded with other two-factor interactions Resolution V – main effects confounded with four-factor interactions, two-factor interactions confounded with three-factor interactions. In this case if you are willing to assume three factor interactions and higher are negligible, you can estimate all main effects and two factor interactions Higher Resolution means main effects are confounded with higher order interactions

  22. Minimum Aberration as a criteria for choosing generators d1 F = ABCD, G = ABCE I = ABCDF = ABCEG = DEFG d2 F = ABC, G = ADE I = ABCF = ADEG = BCDEFG Which is better? Word length pattern: length 3 length 4 d1(0, 1, 2) length 5 d2(0, 2, 0, 1)

  23. Symbolically: (A3, A4, A5, …) Aris number of words of length r

  24. Number of clear Effects as a criteria for choosing generators An effect is defined to be clear if none of its aliases are main effects or two factor interactions See Example64.sas

  25. acid treatment Only 56% Eucalyptus used in Brazilian forests Hemicellulose hydrolyzate Edible Biomass rich in essential amino acids Paecilomyces variolii Fermentation

  26. Generators for minimum aberration

  27. BH

  28. EG Maximum appears to be with Ammonium Sulfate and Sodium Phosphate both at 2 g/L

  29. CH

  30. BEG

  31. Recap 8 Factors would require 28 = 256 for full factorial 16 + 8 = 24 resulted in plausible interpretation and identification of optimal results

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