factors can be written as:
(8)Once a second order model is fit to the response, the next step is to
locate the point of maximum or minimum response. The second order model
for factors can be written as:(8)
The point for which the response, 
, is optimized is the point at which
the partial derivatives, 
, 
,
are all equal to zero. This point
is called the stationary point. The stationary point may be a
point of maximum response, minimum response or a saddle point. These three
conditions are shown in Figures 9.19 (a), (b) and
(c) respectively. Notice that these conditions are easy to identify, in
the case of two factor experiments, by the inspection of the contour plots.
However, when more than two factors exist in an experiment, then the general
mathematical solution for the location of the stationary point has to
be used. Eqn. (8) can be written in matrix notation
as:
(9)where:
Then the stationary point can be determined as follows:
(10)
Thus, the stationary point is:
(11)
The optimum response is the response corresponding to 
. The optimum response can be obtained
by substituting 
from Eqn. (11)
into Eqn. (9) to get:
(12)
|
Figure 9.19: Types of second order response surfaces and their contour plots - (a) shows the surface with a maximum point, (b) shows the surface with a minimum point and (c) shows the surface with a saddle point. |
Once the stationary point is known, it is necessary to determine if
it is a maximum or minimum or saddle point. To do this, the second order
model has to be transformed to the canonical form. This is done
by transforming the model to a new coordinate system such that the origin
lies at the stationary point and the axes are parallel to the principal
axes of the fitted response surface (see Figure 9.20).
The resulting model equation then takes the following form: 
(13)
where the 
s are the transformed independent variables,
and 
s are constants that are also the eigenvalues
of the matrix 
. The nature of the stationary point
is known by looking at the signs of the 
s. If the 
s are all negative, then 
is a point of maximum response. If
the 
s are all positive then 
is a point of minimum response. If
the 
s have different signs, then 
is a saddle point.
|
Figure 9.20: The second order model in canonical form. |
Example 9.3
Continuing with Example 9.2
in Chapter 9, RSM Designs, the second order
model fitted to the response, in terms of the coded variables, was obtained
as:
Then the 
and 
matrices for this model are:
From Eqn. (11) the stationary point is:
Then, in terms of the actual values, the stationary point can be found
as:
To find the nature of the stationary point the eigenvalues of the 
matrix can be obtained as follows
using the determinant of the matrix 
:
This gives us:
Solving the quadratic equation in 
returns the eigenvalues 
and 
. Since both the eigenvalues are negative,
it can be concluded that the stationary point is a point of maximum response.
The predicted value of the maximum response can be obtained using Eqn.
(12) as:
In DOE++, the maximum response can be obtained using the Optimization icon in the Control Panel by entering the required values as shown in Figure 9.21. In the figure, the Goal is set to Maximize and the limits of the search range for maximizing the response are entered as 90 and 100. The value of the maximum response and the corresponding values of the factors obtained are shown in Figure 9.22. These values match the values calculated in this example.
|
Figure 9.21: Settings to obtain the maximum value of the response in Example 9.3. |
|
Figure 9.22: Plot of the maximum response in Example 9.3 against the factors, temperature and time. |
