Response Surface Techniques

Response surface methodology was developed by Box and coworkers in the fifties [25]. Thorough treatments have been given in the literature [26] and the discussion here is given as a brief introduction. 

At a true optimum, the response will have a maximum (or minimum) value. This means that there will be an extremum point on the response surface. To 

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In Section 2.2 it was shown that response surface methodology can be applied to enable a researcher to model the effect of multiple quantitative variables on a response with a low-degree polynomial. Frequently, response surface techniques have focused on the mean response as the only response of interest. However, by regarding the variation in the response as an additional response of interest, the researcher can investigate how to achieve a mean response that is on target with minimum variation. In particular, if a researcher replicates each design point in an experiment, then an estimate of the standard deviation at each point can be calculated and used to model the effect of the variables on the variability of the response. 

In the response surface strategy that was discussed in Section 2.3 standard response surface techniques are used to generate two response surface models, one for the mean response and one for the standard deviation of the response (or some function of the standard deviation). The standard deviation measures the stability of the response to the environmental variation. Standard analysis can reveal which factors affect the mean only, which only affect the variability, and which affect both the mean and the variability. The researcher can then apply optimization methods or construct contour plots of the mean and standard deviation response surfaces to determine settings of the design variables that will give a mean response that is close to the target with minimum variation. 

R.H. Myers and W.H. Carter, Jr., Response surface techniques for dual response systems. Technometrics, 15 (1973) 301-317. 

If we let columns in the design matrix define the constituents as follows 1, 2 define the substrate, columns 3, 4 define the amine co-substrate, and columns 5, 6 define the solvent, the first row in the design matrix in Table 11 would thus correspond to a selection of a substrate projected in the [( — ),( — )] quadrant, an amine from the [(—),( + )] quadrant, and a solvent from the [( + ),( + )] quadrant. The other rows define other combinations. The test items selected accordingly are shown in Table 13. To permit fair comparisons as to the performance of the reaction, it is necessary to adjust the experimental conditions for each system to yield an optimum result. The danger of using standardized conditions has been emphasized [1] and the arguments against such a technique are not repeated here. The conditions which afforded a maximum yield were determined by response surface techniques and these results are also shown in Table 13. 

An appealing aspect of the response surface technique is that the relations between the response and the experimental variables can be illustrated graphically. We have seen how three-dimensional projections can be drawn to show the shape of the response surface over the plane(s) spanned by the experimental variables taken... 

The pedagogical and psychological aspect of nice-looking figures should not be neglected, especially not when results obtained by response surface technique are to be presented to persons who have aversions to statistical methods. 

The synthesis of enamines by the modified titanium tetrachloride method was discussed in Chapter 12. The final yield and the rate of enamine formation depend on the molar ratios of TiCl4/substrate and amine/substrate. The optimum conditions with regard to these variables were determined by response surface technique and/or simplex technique for a series of carbonyl compounds. The results obtained for the morpholine enamines are summarized in Fig. 14.2. It is seen that the more crowded substrates require an excess of the reagents. The use of standardized conditions would have led to the wrong conclusions as to the utility of the method. For instance, when the optimum conditions for synthesis of the morpholine enamine from methyl isobutyl ketone were applied to diisopropyl ketone a yield of 12 % was obtained after 4 h. Under optimized conditions yields > 70 % could be obtained. 

To make fair comparisons, the experimental conditions were optimized to afford a maximum yield for each system. For this response surface technique was used. The optimum conditions and the yields obtained under these conditions are also given in Table 16.6. 

The experimental conditions for optimum yield were detennined by response surface technique for each of these substrates. The optimum conditions are summarized in Table 17.6, Entry 1-5. 

Factorial design and response surface techniques were used in combination with modeling and simulation to design and optimize an... 

Iwasaki and Todoroki [92] use a response surface technique to process the electrical resistance measurements. A large number of cross-ply and quasi-isotropic specimens were tested such that statistical data processing could be applied. Copper-foil electrodes mounted on one side of the CFRP specimens during prepreg layup were co-cured with the specimen (Figure 16.39). Impact-induced matrix cracking and delaminations were detected. Probability of location estimation and error bands were computed [92]. The extension of this method to woven CFRP composites is described in Hirano and Todoroki [93]. 

The essence of the SM technique is approximation of functions g by simple algebraic expressions s within a subset H of parameter space 0. The approximating functions for the responses are obtained using the methodology of the response surface technique [17,19,27], by means of a relatively small number of computer simulations, referred as computer experiments. They are performed at pre-selected combinations of the parameter values and the entire set of these combinations is called a design of computer experiments. The computer experiments are performed using the complete dynamic model (3) and the functions obtained in this manner are referred as surrogate models. 

The response-surface technique outlined above becomes less practical for a number of active variables exceeding --20. How can one deal with a significantly larger number of parameters Before... 

Lattice Search Techniques versus Response Surface Techniques... 

Both the pattern search and the response surface techniques allow the use of discrete decision variables. Topological changes can always be represented by changes in discrete decision variables (such as those defining the flowsheet topology), so these techniques can accommodate both parametric and topological optimization. 

The response surface techniques are preferred during the early phases of design. They serve the multiple uses of prioritizing the decision variables, scoping the optimization problem to determine an estimate of... 

Applications of Response Surface Techniques to Uncertainty Analysis in Gas Kinetic Models... 

From this fact, a corresponding approximation quality of the failure probability can be deduced, if variance-reduced sampling techniques which rely on the MPP together with the abovementioned response surface techniques are employed. This can be clearly seen from Fig. 7, where the probability of failure obtained with various approximation techniques has been plotted over the threshold value Fq. Importance sampling at the predicted MPP with 30 batches of 10,000 samples has been employed for each failure probability estimate. For global approximations, again polynomials of third degree are necessary in order to obtain accurate predictions. On the other hand, local approximations with linear polynomials lead already to quite accurate results, if only the principle direction is partitioned. 

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