Constrained Optimization 3 8 Continuous or Discrete Linear

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 U U-Unconstrained Optimization (Response Surfaces Chapter 10) C S C-Constrained Optimization (Main effects and interactions) S-Screening Experiments (What Factors are important)

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

Fractional Factorial Experiments • Method for Strategically Picking a Subset of a 2 k 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

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

Paradigms That Justify Use of Fractional Factorials • •

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

Creating a half fraction design in SAS

I = ABCDE

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

· In a one half fraction of a 2 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 2 k 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 k · · In a one eighth fraction of a 2 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. k

Creating a 2 k-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

These are the generators

the generators the generalized interaction the defining relation

Defining Relation Confounding Pattern or Alias Structure

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

The three factor generalized interaction is The defining relation is

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

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

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

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

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

Symbolically: (A 3, A 4, A 5, …) Ar is number of words of length r

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 Example 64. sas

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

Generators for minimum aberration

BH

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

CH

BEG

Recap 8 Factors would require 28 = 256 for full factorial 16 + 8 = 24 resulted in plausible interpretation and identification of optimal results Label Factor Optimal Setting B Rice Bran 30. 0 g/L E Ammonium Sulfate 2. 0 g/L G Sodium Phosphate 0. 0 g/L

Reverse signs of coded factor levels for Factor B

Example +

Creating Design Augmented by Foldover in SAS Data Step ADX

Augmenting a resolution IV by mirror image or foldover does not break strings of confounded two factor interactions , H=ABD Augment by design with signs reversed on Factor A only

Augment by design Reversing signs on A

Example: High concentration of arsenic reported in ground water in countries such as Bangladesh, Chile, India, Poland, Nepal … causing people to be prone to various forms of cancer

Simple IOCS filters have been used in Bangladesh and Nepal to remove arsenic from ground water

Simple household filters are effective raw water iron oxide coated sand pourous membrane purified water

IOCS Coating solution made of ferric nitrate and sodium hydroxide with NAOH added to control p. H. Age Coating Solution Pour over clean sand yes Mix Dry Filter Spiked Water Sample repeat Mix Coating Solution no Ramakrishna et. al. (2006) conducted experiments to optimize The coating process.

What can be done to separate AD+CF

AD CF 17 18 19 20 3 3 + + 0 0 0 0 + + + + + + + No longer orthogonal Fit model Y=A B F AD CF by regression

Exchange Algorithm for maximizing det(X’X) Candidate x’s -1, -. 5, 0, . 5, 1 Step 1 replace 0 with -1 Step 3 replace. 5 with 1 Step 2 replace -. 5 with -1

Choose additional runs to maximize the

●Plackett-Burman Designs are Resolution III, but there is no defining relation ●Main Effects are confounded with two-factor interactions, but rather than being completely confounded with a few two-factor interactions, they are partially confounded with many two-factor interactions Alias Matrix shows the alias structure

Example

Implications of Partial Confounding 1. We can use Alias matrix to determine what two-factor interactions are confounded with large unassigned effects 2. Models involving main effects and some partially confounded can be fit by regression since X‘X matrix is not saingular

Creating a Plackett-Burman Design in SAS

Read in the data and merge it with the design created earlier Fit the model and output the parameter estimates

Create interactions and do all subsets regression

OA(12, 31, 24) Run 1 2 3 4 5 1 0 0 0 2 0 0 1 3 0 1 1 4 0 1 1 1 0 5 1 0 0 1 1 6 1 0 1 1 0 7 1 1 0 0 1 8 1 1 1 0 0 9 2 0 0 10 2 0 1 11 2 1 0 0 0 12 2 1 1

Data Similar to Experiment with Teaching Methods in Chapter 2

Dummy variables represent effect of chair style