Chemical Engineering January 2019 - 55

active factors. The purpose here is
to quantify the effects of the latter
on the response in a more precise
and reliable way using linear models
(main effects and interactions).
Because the input variables are
changed simultaneously in a DOE,
possible synergistic and antagonistic
interactions between the input variables
can be detected; this is in contrast
to a so-called OVAT approach,
which involves changing only " one
variable at a time. " Equally important
to the detection of possible interactions
is the recognition of eventual
departure from linear relationships
between response and input variables
by including a center point in
the design. This is shown on the left
side of Figure 2.
The first-order design can detect
global curvature, but it cannot
separately estimate the quadratic
effects of each factor. At this stage
of experimentation, interpretation of
the results combined with process
and product expertise may allow
the identification of the direction of
steepest improvement toward an
optimum in the response, possibly
leading to another series of designed
experiments - this time closer to the
optimum - which can yield another
first-order model before proceeding
to the final stage. Finally, when it
comes to optimization of the manufacturing
process or product, a more
elaborate model will be needed to
describe the region around the optimal
response and to locate the
latter. A linear model will no longer
be sufficient. Instead, a quadratic
model will be used in a so-called response
surface method (RSM) to fit
the optimum.
Central composite design
Since we initially made use of a central
composite design (CCD) in our
case study discussed below, we will
shortly discuss this type of RSM first.
At the left side of Figure 2, a CCD is
depicted for three factors. The CCD
is very flexible as it can be set up in
a modular way: initially, one starts
with the execution of a two-level
factorial DOE, where each factor is
set at its low (-1) and high (+1) level
to verify the effects on the response
of interest. A center point, depicted
as (0,0,0) in the center of the 3-D
representation in Figure 2, may help
(-1, 1, 1)
(-1, -1, 1)
(-a, 0, 0)
X3
(0, 0, a)
(0, a, 0)
(1, 1, 1)
(1, -1, 1)
(-1, 1, -1)
(0, -a, 0)
(-1, -1, -1)
X1
(0, 0, -a) (1, -1, -1)
(0, 0, 0)
X2
(1, 1, -1)
(a, 0, 0)
(-1, -1, 0)
X3
(0, 0, 0)
(-1, -0, -1)
(0, 1, -1)
X1
(1, 1, -1)
FIGURE 2. Shown here is a central composite design (CCD; left side) and a definitive screening design
(DSD; right side) for three factors. For the CCD, a is the axial distance of the start points, which are included
in the experimental design to quantify quadratic (curvature) effects between the response and the
input factors
TABLE 1. DSD DESIGNS FOR 4 TO 6 FACTORS (ADAPTED FROM REF. 7)
N = 4
N = 5
run X1
1
2
3
4
5
6
7
8
9
-1
1
-1
1
-1
1
X2
1
-1
-1
1
1
-1
X3
-1
1
-1
1
1
-1
X4
-1
1
1
-1
-1
1
run X1 X2 X3 X4 X5
1
2
3
4
5
6
7
8
9
1
-1 0
1 -1
0 -1 -1 1
1
1 -1 0 1
-1 1
1
0 -1 -1
1 1
0 -1
1 -1 1 0
-1 1 -1 0
1
1 1
10 -1 -1 -1 -1
11
0 0
-1
1
1
-1
-1
1
1
-1
N = 6
run X1 X2 X3 X4 X5 X6
1
2
3
4
5
6
7
8
9
0 1 -1 -1 -1 -1
0 -1 1 1 1 1
1 0 -1 1 1 -1
-1 0 1 -1 -1 1
-1 -1 0 1 -1 -1
1 1 0 -1 1 1
-1 1 1 0 1 -1
1 -1 -1 0 -1 1
1 -1 1 -1 0 -1
10 -1 1 -1 1 0 1
11 1 1 1 1 -1 0
12 -1 -1 -1 -1 1 0
13 0 0 0 0 0 0
the engineer to detect curvature in
the relation between response and
the factors. At that moment, star
points (at levels -a -and +a) can
be added to the design afterward
to allow the engineer to properly
quantify quadratic effects. This flexibility
makes a CCD very popular in
industrial process development [6].
From the CCD representation shown
in Figure 2, it is clear that the design
points are uniformly distributed in the
experimental space.
Definitive screening design
In contrast to the most familiar
screening designs where input variables
are set at only two levels (-1
and +1, or low and high level), the
definitive screening design (DSD)
introduced in 2011 by Jones and
Nachtsheim, employs three levels for
the variables: -1, 0 and +1; or low,
center and high level [7]. For N variables
the DSD requires only 2N+1
experimental runs. In Table 1, the
designs for the case of 4 to 6 factors
are shown, as presented in the
paper from the DSD-inventors Jones
and Nachtsheim. The DSDs are
comprised of N fold-over pairs plus
one overall center run consisting of
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the center values of all variables (This
is indicated in red color on the last
rows of Table 1).
The following design pattern further
characterizes a DSD [7]:
1. Regarding the location of the zeros
(highlighted in grey color in Table
1) - The first two runs have zeros
in the column of the first variable
X1; the next two runs have zeros in
the column of the second variable
X2 and so on
2. Regarding the pair of runs -
These are mirrored (folded over),
which means that the second
run of a pair is found by multiplying
the first run of this pair by
-1. Hence, the first 2N runs have
exactly one variable at its center
value (0), while all other variables
are at their extremes (-1 or +1) and
are referred to as " edge runs, " because
the 3-D projections involving
these variables, they are on the
edges of the cube [8]. In case the
number of factors N is uneven, it is
recommended to choose the DSD
design for N+1 factors and then
to drop the extraneous column,
which results then again in an Nfactor
design with 2N+3 runs [8].
The two extra runs are, inherently,
55
X2
Type
(-1, 1, 1)
Center point
Cube point
Star point
(0, -1, 1)
(1, 0, 1)
Type
Center point
Edge point
Vertex point
(1, 1, 0)
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Chemical Engineering January 2019

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