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Design and analysis of experiments (2011)

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Video material (part 1)
Download video: Link (plays in Google Chrome) [937Mb, 98 mins]

Video timing

00:00 to 03:33 Announcements
03:33 to 09:42 Overview of designed experiments
09:42 to 16:30 Experiments with one variable
16:30 to 31:56 The importance of randomization
31:56 to 42:04 Changing one variable at a single time (COST) approach
42:04 to 93:58 Factorials: 2 variables at 2 levels
93:58 to 98:12 Overview of mini-project

Video material(part 2)
Download video: Link (plays in Google Chrome) [1.2 Gb, 126 mins]

Video timing

00:00 to 28:25 3-factor factorial example
28:25 to 34:22 Announcements and overview of previous class
34:22 to 61:38 Determining significance of effects, and refitting after removing any effects
61:38 to 67:32 Comparison of COST and full factorials
67:32 to 86:16 Blocking for disturbances
86:16 to 125:26 Fractional factorial: half fractions, generators and defining relationships

Video material (part 3)
Download video: Link (plays in Google Chrome) [1.4 Gb, 146 mins]

Video timing

00:00 to 39:20 Example: analysis of a fractional factorial
39:20 to 43:00 Overview of DOE projects
43:00 to 48:20 Saturated designs for screening
48:20 to 54:15 Foldover for de-aliasing
54:15 to 59:08 Projectivity of fractional designs
59:08 to 66:20 Sequential optimization: single variable case
66:20 to 128:50 Response surface optimization in many variables
128:50 to 131:26 Evolutionary operation
131:26 to 136:45 General approach for running experiments
136:45 to 143:05 Handling constraints and optimal designs
143:05 to 146:09 Mixture designs

Contents

[edit] Course notes

[edit] Projector overheads

Class date: 7 March: covered slides 1 to 12
9 March: covered slides 16 to 28
10 March: covered slides 29 to 42
16 March: covered slides 42, 46 to 50, 63 to 66
17 March: guest speaker
21 March: covered slides 67 to 76
23 March: covered 77 to 88, and 103-107
24 March: covered 108 to 119
I want my notes with:  

  pages per physical page

Use page frames?

[edit] Guest speaker overheads

Class date: 17 March
I want my notes with:  

  pages per physical page

Use page frames?

[edit] Audio recordings of 2011 classes

Date Material covered Audio file
07 March 2011 Single variable experiments, randomization, changing one single variable at a time (COST) approach Class 22
09 March 2011 Analysis of a 2 factor experiment: without interactions, and with interactions Class 23
10 March 2011 Factorial analysis using least squares; 3-factor experiment example (contains an integer variable) Class 24
16 March 2011 Assess if factors are significant or not; blocking and confounding Not available
17 March 2011 Guest speaker, Emily Nichols, on DOEs Class 26
21 March 2011 Fractional factorials; generators and defining relationships Class 27
23 March 2011 Saturated designs; response surface methods Class 28
24 March 2011 Response surface methods Class 29
28 March 2011 Dealing with incorrect experiments and constraints; wrapping up DOEs; take-home review. Class 30

Thanks to the various students responsible for recording and making these files available

[edit] 3-factor example

The data are from a plastics molding factory which must treat its waste before discharge. The \(y\) variable represents the average amount of pollutant discharged [lb per day], while the 3 factors that were varied were:

  • \(C\): the chemical compound added [A or B]
  • \(T\): the treatment temperature [72°F or 100°F]
  • \(S\): the stirring speed [200 rpm or 400 rpm]
  • \(y\): the amount of pollutant discharged [lb per day]
Experiment Order \(C\) \(T\) [°F] \(S\) [rpm] \(y\) [lb]
1 5 A 72 200 5
2 6 B 72 200 30
3 1 A 100 200 6
4 4 B 100 200 33
5 2 A 72 400 4
6 7 B 72 400 3
7 3 A 100 400 5
8 8 B 100 400 4

We showed the cube plot for this system in the class on 10 March. From the cube plot we could already see the main factors, and even the CS interaction was noticeable.

Next, use computer software to verify that

\[y = 11.25 + 6.25x_C + 0.75x_T -7.25x_S + 0.25 x_C x_T -6.75 x_C x_S -0.25 x_T x_S - 0.25 x_C x_T x_S\]

The \(\mathbf{X}\) matrix and \(\mathbf{y}\) vector used to calculate the least squares model:

\[\begin{split}\begin{bmatrix} 5\\30\\6\\33\\4\\3\\5\\4 \end{bmatrix} &=\begin{bmatrix} +1 & -1 & -1 & -1 & +1 & +1 & +1 & -1\\ +1 & +1 & -1 & -1 & -1 & -1 & +1 & +1\\ +1 & -1 & +1 & -1 & -1 & +1 & -1 & +1\\ +1 & +1 & +1 & -1 & +1 & -1 & -1 & -1\\ +1 & -1 & -1 & +1 & +1 & -1 & -1 & +1\\ +1 & +1 & -1 & +1 & -1 & +1 & -1 & -1\\ +1 & -1 & +1 & +1 & -1 & -1 & +1 & -1\\ +1 & +1 & +1 & +1 & +1 & +1 & +1 & +1\\\end{bmatrix}\begin{bmatrix} b_0 \\ b_C \\ b_T \\ b_{S} \\ b_{CT} \\ b_{CS} \\ b_{TS} \\ b_{CTS} \end{bmatrix} \\\mathbf{y} &= \mathbf{X} \mathbf{b}\end{split}\]

[edit] Visualizing a response surface

[edit] Some advice

You can easily create your own response surfaces and visualize them. Then practice on them before and during the take-home exam. Here are equations for the 4 examples given in class on Thursday:

Binary variables should be treated exactly like continuous variables in all respects during the optimization. The only difference is that it makes sense to visualize and use them at the two levels.

[edit] Plotting code

MATLAB code used in class on 23 March 2011 to do the first factorial:

X = [+1 +1 +1 +1; ...
     -1 +1 -1 +1; ...
     -1 -1 +1 +1; ...
     +1 -1 -1 +1]';
 
% Profit values recorded from the factorial experiments
profit = [193 310 468 571]';
% Coefficient order = [Intercept, b_T, b_S, b_{TS}]
b = inv(X'*X)*X'*profit
 
% What does the model surface look like?
[T, S] = meshgrid(-2:0.1:2, -2:0.1:2);
Y = b(1) + b(2).*T + b(3).*S + b(4).*T.*S;
 
subplot(1, 2, 1)
surf(T, S, Y)
xlabel('T')
ylabel('S')
grid('on')
 
subplot(1, 2, 2)
contour(T, S, Y)
xlabel('T')
ylabel('S')
grid('on')
axis equal
colorbar

MATLAB code used in class on 24 March 2011 to continue with the central composite design factorial:

% CCD design around point 6
% Put 4 factorial points first, then the center point
% then the 4 axial (star) points
a = sqrt(2);
X = [1 -1 -1; ...
     1 +1 -1; ...
     1 -1 +1; ...
     1 +1 +1; ...
     1  0  0; ...
     1  0 -a; ...
     1 +a  0; ...
     1  0  a; ...
     1 -a  0];
X(:,4) = X(:,2) .* X(:,3);  % TS = T x S
X(:,5) = X(:,2) .* X(:,2);  % TT = T x T
X(:,6) = X(:,3) .* X(:,3);  % SS = S x S
y = [694 725 620 642  688  720 700 610 663]';
b = inv(X'*X)*X'*y
[T,S] = meshgrid(-2:0.1:3, -3:0.1:2);
Y = b(1) + b(2).*T + b(3).*S + b(4).*T.*S + b(5).*T.*T + b(6).*S.*S;
 
subplot(1, 2, 1)
surf(T, S, Y)
xlabel('T')
ylabel('S')
grid('on')
 
subplot(1, 2, 2)
contour(T, S, Y)
xlabel('T')
ylabel('S')
grid('on')
axis equal
colorbar
 
% Predicted optimum: $737, actual value at this point: $737.8
T = +2.25;
S = -1.75;
y_hat_opt = b(1) + b(2).*T + b(3).*S + b(4).*T.*S + b(5).*T.*T + b(6).*S.*S
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