Combining Combinatorics
and
D-optimal Designs
Dean Sterling Hoskins
Arizona State University
DOE Conference
Coauthors: Charles Colbourn, Douglas Montgomery, ASU
Nankai University
Design of Experiments Conference
July 9-13, 2006
Covering Arrays

Covering Arrays CAλ(N; t; k; v)






There are k factors (array columns) with v levels (1..v)
Each row is a run out of N total runs
t is known as the strength
Every combination of t columns must include all factor-level combinations
These combinations occur at least λ times
CA arrays are not usually full rank (important for statistical analysis!)
k=3
N=8
V=2
V=2
All
combinations
occur at least
once
λ =1
Nankai University
Design of Experiments Conference
July 9-13, 2006
D-Optimal Designs (Screening Type)

D-Optimal Designs OD(model, k ,v)




There are k factors (run matrix columns) with v levels (1..v)
The model determines how many factors and interactions we can
estimate (ME, ME+2FI, ME+2FI+3FI, etc.)
The model determines how many runs are necessary
Strength t is not a consideration but maximum orthogonality between runs is
k=3
Runs = 7
For ME+2FI
V=2
V=2
Max Tensile Strength = c1A1+c2A2+c3A3+ c11A1B1+…
Nankai University
Design of Experiments Conference
RESULTS
July 9-13, 2006
Hybrid Designs

Hybrid Designs




Best properties of covering arrays combined with best properties of DOptimal designs
Maximize covering and minimize losses in D-Efficiency in combined
designs
Designing an algorithm to produce such designs
Other discussions




Can D-Optimal algorithms generate covering arrays?
Can distance based algorithms generate covering arrays?
Do different D-Optimal algorithms perform better at generating covering
arrays?
Model complexity versus ability to generate covering arrays – is a
ME+2FI D-Optimal design as good as a ME+2FI+3FI design in
producing a 3-covering CA?
Nankai University
Design of Experiments Conference
July 9-13, 2006
Covering Arrays Generated by Optimal Designs

Results for Optimality Criterion versus Coverage




For 65 and 3421 designs
No optimality criterion with ME+2FI can produce 3-covering in a number
of runs given by best in literature covering arrays although can achieve
~90% coverage
Fedorov best followed closely by DETMAX
U and S distance criteria not generally as good
Nankai University
Design of Experiments Conference
July 9-13, 2006
Covering Arrays Generated by Optimal Designs

Results for Model Complexity versus Coverage

For 65 designs
ME+2FI best at producing 3-covering.

ME and ME+3FI do not produce good designs for coverage


Take Away

It is possible to design an algorithm to produce designs that are
covering and have good D-efficiencies at the same time
Nankai University
Design of Experiments Conference
July 9-13, 2006
Hybrid Design Generation

Coverage Ratio





Can ME+2FI D-Optimal designs incorporate 3 factor interaction
coverage?
Prospective Ratio (PR) = # runs to 3-cover / # runs for ME+2FI
Look at wide range of designs with factors that have the same number
of levels
PR < 1.0 means we can find
such a design
Take Away

Most designs that have factors with the same number of levels can 3cover and have an ME+2FI model at the same time
Nankai University
Design of Experiments Conference
July 9-13, 2006
Hybrid Design Generation

Hybrid Algorithm

Take a 3-covering CA and adds runs to produce an ME+2FI model

Utilizes greedy algorithm or ‘best in literature’ to generate covering
arrays
Utilizes AS-295 Fedorov algorithm developed by Miller (Fortran) to
enhance array until it meets ME+2FI model

Nankai University
Design of Experiments Conference
July 9-13, 2006
Hybrid Design Generation

Results
5 Designs looked at
3-covering achieved within boundary of ME+2FI model


D-efficiency loss of about 5% in each case
In most cases CA’s generated by greedy methods outperformed CA’s
that were ‘best’ in literature



Note

An interaction of 3 covering in a process is where these type of designs
will out class pure D-Optimal designs in estimating a process
Nankai University
Design of Experiments Conference
July 9-13, 2006
Overlapping Properties
D-Opt
D-Opt
CA t=3
?
ME+2FI
CA t=3
ME+2FI
ρ
Lower Defficiency
High Defficiency
B
A
CA t=3
D-Opt
‘Best’
ME+2FI
C
Classes of overlap between strength three covering and D-Optimal ME+2FI designs
Class A
Class B
Class C
There may or may not be High Defficiency D-Optimal designs that also
offer strength three covering properties.
We have only achieved cases in this
research with a 5% loss over the best
designs SAS produces. The questionable
overlap in this area would be small to
none for high D-efficiency designs
produced by Fedorov algorithms.
Lower D-efficiency D-Optimal designs in
essence can have any D-efficiency as
long as the rank of the X matrix is full.
There are examples of designs that are
both D-Optimal and strength three
covering shown in the hybrid section. As
the D-efficiency drops the overlap (ρ) in
this area becomes larger and larger.
Research has indicated that at least for
the CA’s studied, D-Optimal designs
while producing reasonable covering in a
ME+2FI number of runs do not come
close to minimum CA’s in literature
(minimum number of runs known).
Likewise for the designs studied, ‘best’
CA’s do not have a sufficient number of
runs to estimate a ME+2FI model.
Nankai University
Design of Experiments Conference
July 9-13, 2006
Questions?
Nankai University
Design of Experiments Conference
July 9-13, 2006