Response Surface Methodology

On many occasions, a system (an industrial process, a method of analysis, a synthesis, etc.) is in operation and the researcher wonders about the possibility of improving the results by only changing the values of the parameters that control the operation of the system. This demands exploration of an admissible region of the space where the parameters vary. This involves two stages: (i) to experiment in some points of this region and to obtain the results, and (ii) to make a prediction of the results that would be obtained in the rest of the points by evaluating the reliability of this prediction. After that, the researcher will know if the results from his/her system can be improved, and to what extent and the values of the parameters that will have to be used for it. The response surface methodology (RSM) is, above all, a collection of criteria to decide in what points of the feasible region the experimentation should be made, in such a way that the prediction is most precise (whenever possible). Once the researcher has defined the problem, the domain of experimentation and the response, the RSM provides, on the one hand, alternative experimental strategies and, on the other hand, the criteria to evaluate them. The great advantage is that this task of adapting the experimentation to the problem under study is accomplished before carrying out the experiments. Following this point of view, the focus of this chapter is on the criteria and the methodology rather than the detailed list of designs already described in the literature, which in most of the cases are specially adapted to a specific problem. This chapter is thus an introduction to exploration and optimization of response surfaces by fitting an empirical model to some experimental data. In Section 1.12.2, the elements of the RSM are defined. In Section 1.12.3, the most important optimality criteria based on the variance are exposed. The statistical validation of the empirical model is discussed in Section 1.12.4, and in Section 1.12.5 the most often used designs for the fitting of the first- and second-order models are described. The methods for the analysis of the fitted response surface model are discussed in Section 1.12.6. The chapter concludes with some consideration about the practical use of the RSM and bibliographic references are given for more advanced questions, impossible to be tackled in a single chapter. Nowadays, there are several efficient programs that allow the researcher to adapt the experimental design to his/her needs, by using the optimality criteria that she/she considers relevant for the problem under study. Although these programs are an essential tool, their use has to be guided by a clear understanding of the methodology, the criteria and their meaning in practice.