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When Probability Tables are examined it is found that Factors A, B and F show large effects which are very significant, whereas C shows a very low effect which is not significant and D shows no effect. A, B and F have been identified as the most important

Fig. 4.1. Optimal point of a response surface by one factor at a time.

over the response surface. In Fig. 4.1, the profile generated by fixing Xx and changing X2 and then using the best X2 value and changing Xx constitutes a cross which may not encroach upon the area in which the optimum resides.

The technique may be applied at different levels of sophistication. Hendrix applied the technique at its simplest level to predict the optimum combination of two variables. The values of the variables for the initial experiments are chosen randomly or with the guidance of previous experience of the process. There is little to be gained from using more than 15-20 experiments. The resulting contour map gives an indication of the area in which the optimum combination of variables resides. A new set of experiments may then be designed within the indicated zone. Hendrix proposed the following strategy to arrive at the optimum in an incremental fashion:

(i) McDaniel et al. (1976), Fig. 4.3. The variables under investigation were cerelose and soybean level, with the analysis indicating the optimum to be 6.2% cerelose and 3.2% soybean.

(ii) Saval et al. (1993). The medium for streptomycin production was optimized for four components resulting in a 52% increase in streptomycin yield, a 10% increase in mycelial dry weight and a 48% increase in specific growth rate (Table 4.18).

When further optimization experiments are necessary for medium development in large vessels, the number of experiments will normally be restricted because of the cost and the lack of spare large vessels (Spendley et al., 1962). The simplex search method attempts to optimize n variables by initially performing

1. Define the space on the plot to be explored.

### 2. Run five random experiments in this space.

3. Define a new space centred upon the best of the five experiments and make the new space smaller than the previous one, perhaps by cutting each dimension by one half.

4. Run five more random experiments' in this new space.

5. Continue doing this until no further improvement is observed, or until you cannot afford any more experiments!

The more sophisticated applications of the response surface technique use mathematical models to analyse the first round of experimental data and to predict the relationship between the response and the variables. These calculations then allow predictive contours to be drawn and facilitate a more rapid optimization with fewer experiments. If three or more variables are to be examined then several contour maps will have to be constructed. Hicks (1993) gives an excellent account of the development of equations to model the different interactions which may take place between the variables. Several computer software packages are now available which allow the operator to determine the equations underlying the responses and, thus, to determine the likely area on the surface in which the optimum resides. Some examples of the types of response surface profiles that may be generated are illustrated in Fig. 4.2.

The following examples illustrate the application of the technique:

Fig. 4.2. Typical response surfaces in two dimensions; (a) mound, (b) rising ridge, (c) saddle.

Fig. 4.2. Typical response surfaces in two dimensions; (a) mound, (b) rising ridge, (c) saddle.

Fig. 4.3. Contour plot of two independent variables, cerelose and soybean meal, for optimization of the candidin fermentation (Redrawn from McDaniel et al., 1976; Bull et al., 1990).

Fig. 4.3. Contour plot of two independent variables, cerelose and soybean meal, for optimization of the candidin fermentation (Redrawn from McDaniel et al., 1976; Bull et al., 1990).

tions of carbon and nitrogen sources in a medium for antibiotic production.

In our example a graph is constructed in which the x axis represents the concentration range of the carbon source (the first variable) and the y axis represents the concentration range of the nitrogen source (the second variable). The first vertex A (experimental point) of the simplex represents the current concentrations of the two variables which are producing the best yield of the antibiotic. The experiment for the second vertex B is planned using a new carbon-nitrogen mixture and the position of the third vertex C can now be plotted on the graph using lengths AC and BC equal to AB (the simplex equilateral triangle, Fig. 4.4a). The concentrations of the carbon and nitrogen sources to use in the third experiment can now be determined graphically and the experiment can be undertaken to determine the yield of antibiotic. The results of the three experiments are assessed and the worst response to antibiotic production indentified. In our example, experiment A was the worst and B the best. The simplex design is n + 1 experimental trials. The results of this initial set of trials are then used to predict the conditions of the next experiment and the situation is repeated until the optimum combination is attained. Thus, after the first set of trials the optimization proceeds as individual experiments. The prediction is achieved using a graphical representation of the trials where the experimental variables are the axes. Using this procedure, the experimental variables are plotted and not the results of the experiments. The initial experimental conditions are chosen such that the points on the graph are equidistant from one another and form the vertices of a polyhedron described as the simplex. Thus, with two variables the simplex will be an equilateral triangle. The results of the initial set of three experiments are then used to predict the next experiment enabling a new simplex to be constructed. The procedure will be explained using an example to optimize the concentra-

 Nutrient (g dm 3) Original Optimized