Chaughule, Thorat - Statistical Analysis/Design of Experiments1994 ) work at two levels, and can be constructed on the basis of fractional replication of
a full factorial design. This design allows reliable short listing of a small number of in-
gredients for further optimization and allows one to obtain, unbiased estimates of linear
effects of all the factors with maximum accuracy for a given number of observations, the
accuracy being the same for all effects (Akhnazarova and Kafarov, 1982).
8.6.8. Doehlert design
Doehlert (1970) proposed an alternative and very useful experimental design for
second-order models is the uniform shell. Doehlert designs are easily applied to optim-
ize variables and offer advantages in relation to central composite and Box-Behnken de-
signs. They need fewer experiments, which are more efficient and can move through the
experimental domain. Despite these attractive features, it took some time until re-
searchers started to take notice of Doehlert designs (Ferreira et al, 2004).
For two variables, the Doehlert design consists of one central point and six points
forming a regular hexagon, and therefore situated on a circle. In three dimensions it can
be viewed in different ways, depending on the geometric structure selected (Garcıa
Campana et al, 1997). In Doehlert designs the number of levels is not the same for all
variables. In a two-variable Doehlert design, for example, one variable is studied at five
levels, while the other is studied at only three levels. This property allows a free choice
of the factors to be assigned to a large or small number of levels. Different criteria can be
used to assign the factors. As a general rule, it is preferable to choose the variable with
the stronger effect as the factor with five levels, in order to obtain maximum information
of the system.
The uniform shell design developed by Doehlert (1970) has the following characte-
ristics:
i. Uniform distributions of the experimental points are allocated on the surface of a
hypersphere.
ii. The number of experiments is given by N2 + N+1, where N is the number of va-
riables under study.
iii. Each factor is analyzed at different number of levels. This particular characteris-
tic is relevant when some factors have certain restrictions such as cost or instrumental
constraints, so their study with a small number of levels is necessary
iv. Extension of the experimental matrix to another experimental domain may be
done by using previous adjacent points.
Many of the papers reporting the use of a Doehlert matrix involve the optimization
of a process controlled by only two variables, for which seven experiments are required
(Ferreira et al, 2004). Doehlert design belongs to the category of simultaneous designs,
whose basic idea is to record one or more selected experimental responses for a set of
experiments carried out in a systematic way, in order to predict the optimum and the
interaction effects using regression analysis (Araujo and Brereton, 1996). This design
offers advantage in relation to other second order models such as central composite de-
sign (CCD) and Box Behnken design (BBD), as they need fewer experiments, which are
easier and more efficient.