Equiradial designs

If prior knowledge of the influence of the variables is available, the size of the simplex can be chosen to give a good progression along the path of the steepest ascent. Close to an optimum, the simplex will encircle the optimum conditions. As these experimental points are evenly distributed, they can be incorporated in a equiradial design (see Chapter 12) which can be used to fit a response surface model in the optimum domain. 

The equiradial designs are useful when two experimental variables are studied. The coded experimental settings are evenly distributed on the periphery of the unit circle, and with at least one experiment at the center point. Without the center point the X X matrix will be singular. The designs are defined by the following relations ... 

A quadratic model in two variables contains six parameters, and the smallest design which can be used must therefore contain six experiments. This wiU correspond to an equiradial design in which the experiments on the periphery (m = 5) define a regular pentagon, see Fig. 12.17a. 

The most commonly used equiradial design is with m = 6, which defines a regular hexagon, see Fig. 12.17b. Such a distribution of experimental points is also obtained when a regular simplex with two variables has reached an optimum domain and have encircled the optimum point. It is therefore possible to establish a response surface model from the simplex experiments and use the model to locate the optimum. 

Fig. 12.17 Equiradial designs fa) Pentagon design (b) Hexagon design.

An equiradial design with m = 8 defines an octagon. This is also the distribution which is obtained by a central composite rotatable design. 

The equiradial designs are rotatable. It is possible to adjust the number of center point experiments to achieve uniform precision as well as near-orthogonal properties. Uniform precision for the pentagon and the hexagon is obtained with three center point experiments. Orthogonal properties are obtained with five center point experiments for the pentagon, and with six center point experiments for the hexagon. 

Example Response surface by an equiradial design for determination of the optimum conditions for the synthesis of 4-(7V -dimethylamino)acetophenone[13]... 

Equiradial designs for more than 2 factors - the simplex design... 

An equiradial design for 2 factors - the regular pentagon Box-Behnken designs... 

Figure 5.5 Equiradial designs for 2 factors, (a) Equilateral triangle, (b) square, (c) regular pentagon, and (d) regular hexagon.

B. Equiradial Designs for More than 2 Factors - The Simplex Design... 

III). The second is an equiradial design, a regular pentagon with one centre point (table 5.8, column 3, and figure 5.5), and it is this one that was chosen. 

The equiradial designs describe a circular experimental domain with the points distributed along the circle periphery and at least one point in the center of the circle. Equiradial designs are rotatable and, depending on the number of replicates of the center point, uniform precision and nearorthogonality can be achieved. 

---