Optimization when there is more than one response variable

Optimization when there is more than one response variable 

In the sections above it was discussed how response surface models can be used to optimize a single response. It is, however, common that there is more than one response of interest. Examples of this are  

Sometimes it is possible to find experimental conditions which satisfy all the specified criteria sometimes there are conflicts between certain criteria and a compromise solution must be found. It is evident that such problems can be difficult to solve. A special branch of mathematics, Optimization theory is devoted to this type of problem. In this area it is assumed that the object function (the theoretical response function) is perfectly known, and that the final solution can be reached by using mathematical and numerical methods. [17] From the discussions in Chapter 3, it is evident that mathematical optimization theory is difficult to apply in the area of organic synthesis, especially when new ideas are explored. Conclusions must be drawn from observations in suitably designed experiments. The response surface models thus obtained are local and approximate, and definitefy not perfectly known. Nevertheless, we shall see that we can use experimentally determined models to find solutions to the problems sketched above. 

When there are relatively few response variables to consider, it is possible to map each response separate by a response surface model. These models can then be evaluated simultaneously to determine suitable settings of the experimental variables. Aspects of this are discussed below. 

When there are many response variables to consider, simultaneous evaluation of response surface models from each response becomes cumbersome. In such cases, a considerable simplification can often be achieved by multivariate analysis of the response matrix. For such purposes, principal components analysis and/or multivariate correlation by PLS are useful. These methods are discussed in Chapters 15 and 17. 

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