Example: worst case balance improved
Ulrike Grömping
based on joint work with Roberto Fontana
n=3 runs
m=2 factors with s1=2 and s2=3 levels \[
d=\left (\begin{smallmatrix}
0 & 2\\
0 & 0\\
1 & 1
\end{smallmatrix}\right )
\]
n x m factorial design d:
\[ \begin{split} \\ D=\left ( \begin{smallmatrix} 0 & 0\\ 0 & 1\\ 0 & 2\\ 1 & 0\\ 1 & 1\\ 1 & 2 \end{smallmatrix} \right ), \mathbf r = \left ( \begin{smallmatrix} 1\\ 0\\ 1\\ 0\\ 1\\ 0 \end{smallmatrix}\right ) \end{split} \]
Representation by N x 1 counting vector r:
Mj the N x df(j) model matrix of all j-factor interactions
Mℱ,j the n x df(j) sub matrix of Mj
n2Aj = 1nTMℱ,jMℱ,jT1n = rTMjMjTr
Thus: Aj can be minimized
\[
D=
\begin{pmatrix}
0\\
1\\
2
\end{pmatrix},
\mathbf r=
\begin{pmatrix}
r_1\\
r_2\\
r_3
\end{pmatrix}
\] r1, r2, r3 ≥ 0 (integers) with sum n = 10
Main effects model matrix M1 (3x2)
in normalized orthogonal coding:
\[
\mathbf M_1=
\begin{pmatrix}
-\sqrt{3/2} & -\sqrt{1/2}\\
0 & \sqrt 2 \\
\sqrt{3/2} & -\sqrt{1/2}
\end{pmatrix}
\]
a single qualitative 3-level factor
N = 3 (unreplicated full factorial)
n = 10 runs to be assigned
r1, r2, r3 ≥ 0 (integers) with sum n = 10
M1 model matrix in norm. orthog. coding
Minimize
102A1 = rTM1M1Tr = rTH1r with
\[ \mathbf H_1= \begin{pmatrix} 2 & -1 & -1\\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{pmatrix} \]
a single qualitative 3-level factor
N = 3 (unreplicated full factorial)
n = 10 runs to be assigned
r1, r2, r3 ≥ 0 (integers) with sum n = 10
M1 model matrix in norm. orthog. coding
add free variables y1 and y2 to the problem, with
\[
\mathbf y = \begin{pmatrix} y_1 \\ y_2
\end{pmatrix} = \mathbf M_1^\top \mathbf r
=
\frac{1}{\sqrt 2}\begin{pmatrix}
\sqrt 3 (r_3-r_1)\\
2r_2-r_1+r_3
\end{pmatrix}
\]
The objective function in terms of y: \[
\mathbf {rM}_1\mathbf M_1^\top \mathbf r = \mathbf{y^\top y}=y_1^2+y_2^2
\]
r1, r2, r3 ≥ 0 (integers) with sum n = 10
y=M1r
minimize quadratic objective
rTM1M1Tr = y12 + y22
quadratic \(\rightarrow\) linear:
add variable t1, constrained by t12 ≥ y12 + y22
minimize linear objective t1, subject to
minimize t1 subject to
r1, r2, r3 ≥ 0 (integrality not required),
y = M1Tr,
and
t12 ≥ y12 + y22
relaxed optimum:
r1 = r2 = r3 = 10/3
objective value: 0 (lower bound)
relaxed optimum: r1 = r2 = r3 = 10/3 objective value: 0 (lower bound)
branch 1 with r1 ≥ 4:
relaxed problem has integer optimum
r1 = 4, r2 = r3 = 3
incumbent
objective value: 2 (upper bound)
branch 2 with r1 ≤ 3
relaxed problem has non-integer optimum
r1 = 3, r2 = r3 = 3.5
objective value: 0.5 (lower bound)
branch 2 with r1 ≤ 3
branch 2.1 with r2 ≥ 4
relaxed problem has integer optimum
r1 = 3, r2 = 4, r3 = 3
objective value: 2
branch 2.2 with r2 ≤ 3
relaxed problem has integer optimum
r1 = r2 = 3, r3 = 4
objective value: 2
two branch solutions, objective values equal to those of previous incumbent
Thus: upper bound = lower bound, gap=0.
In problems of realistic size,
72 runs feasible
actual experiment (VSGFS in R package DoE.base):
constructed by column optimization from the OA(72, 243384161) provided by Kuhfeld (2009)
GWLP of the actual experiment: (A1, A2, A3, A4) = (0, 0, 0.451, 3.247)
Lower bound for A3 in a resolution III array: 2/27 = 0.074 < 0.451
our algorithm provides such an array
DoE.MIParray’s result strongly depends on the order of factor levels:
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