Chaughule, Thorat - Statistical Analysis/Design of ExperimentsII. Once in the vicinity of the optimum response the design needs to fit a more ela-
borate model between the response and the factors. Special experimental designs, re-
ferred to as RSM designs are employed. The fitted model is used to arrive at the best op-
erating conditions that result in either a maximum or minimum response.
III. It is possible that a number of responses may have to be optimized at the same
time. For example, in a drying experiment the goal may to optimize drying temperature,
product quality parameters like rehydration ratio, % retention of active nutrient etc. The
optimum settings for each of the responses in such cases may lead to conflicting settings
of the factors. Hence, a balanced selection has to be found that will give the most appro-
priate values for all the responses. Desirability functions are the ones used in these cases.
Overview of the Method- Suppose y = y (a, b, c ...), where, y is the outcome or result or response which is
to be optimized, and there are ‘n’ parameters, a, b, c ... which can be varied. - In these notes, it is assumed that the optimum ‘y’ is the maximum ‘y’. Similar
analysis can be done for minimizing ‘y’. - The goal of RSM is to efficiently hunt for the optimum values of a, b, c ... such that
‘y’ is maximized or minimized.
The response can be represented graphically, either in the three-dimensional space
or as contour plots that help visualize the shape of a response surface. Contours are
curves of constant response drawn in any two variables' plane (xi, xj), keeping all other
variables fixed. Each contour corresponds to a particular height of the response surface.
Generally, the structure of the relationship between the response and the independent
variables is unknown. The first step in RSM is to find a suitable approximation to the
true relationship. The most common forms are low-order polynomials (first or second-
order).
RSM is an important tool for optimization of a response from 3-4 important factors.
Criteria for optimal design of experiments by response surface methodology are asso-
ciated with the mathematical model of the process. Generally, these mathematical mod-
els are polynomials with an unknown structure, so the corresponding experiments are
designed only for a particular problem. The choice of design of experiments can have a
large influence on the accuracy of the approximation and the cost of constructing the
response surface.
8.6.3.1. Central Composite Design
Central composite design (CCD) is the most widely used response surface design. Al-
though rotatability is a desirable property of a central composite design where there is a
difficulty in extending the star points beyond the experimental region defined by the
upper and lower limits of each factor, a face-centered design can be used (Tsapatsaris &
Kotzekidou, 2004). A second-order model can be constructed efficiently with central
composite designs (CCD) (Montgomery and Douglas, 1997). CCD are first-order (2N)
designs augmented by additional centre and axial points to allow estimation of the tun-
ing parameters of a second-order model. Figure 8. 3 shows a CCD for 3 design variables.
In Figure 8. 2 , the design involves 2N factorial points, 2N axial points and 1 central point.
CCD presents an alternative to 3N designs in the construction of second-order models