Response Surfaces, Sampling, and Optimization

A response surface is a representation of the output or response of a chemical system or instrument expressed as a function of the relevant independent variables. When possible, the response surface is depicted as a two-dimensional or three-dimensional plot. It is a representation of the instrumental or systemic response as the independent variables are varied, and this representation is widely used in optimization. Much of experimentation can be considered as an exercise in the exploration and exploitation of the relevant response surfaces. 

The sequential simplex method of optimization was proposed by Nelder and Mead. With a number of improvements and enhancements the simplex method has found great utility in real situations in analytical laboratory experiments and process control situations. The simplex method is a hill-climbing method that seeks to climb the response surface depending on the features of the response surface in its immediate neighborhood. Only one new experiment is done for each step in the optimization sequence, and the location of this new experiment on the response surface is completely determined by the previous few experiments. The method of Nelder and Mead using a variable-size simplex is the most commonly used. A complete description of how the simplex method works is beyond the scope of this review, since the information is contained elsewhere.Many applications of simplex optimization have appeared, and a few examples follow. 

Chubb and co-workers applied simplex optimization to increasing the yield of the Bucherer-Berg reaction. In this reaction, a ketone reacts with NH3, COS, and HCN to give a complex, heterocyclic product. Eight variables are related to the yield obtained the initial concentrations of the four reactants, pH, temperature, time of the reaction, and the solvent used. A mixed solvent of ethanol and water was used, with the ratio varied as part of the optimization. Results were reported for several sets of experiments using cyclohexanone and adamantanone as the starting material. The yields were improved rapidly using a variable-size simplex procedure. In one experiment, the yield was improved from 49 to 88%. 

Routh et al. applied simplex optimization to a flame spearoscopy experiment. Leary and co-workers used simplex optimization to improve the instrumental operating conditions for an inductively coupled plasma spectrometer. Harper and co-workers reported using a simplex search to locate an optimum ultrasonic extraction of trace elements from atmospheric particulates collected on glass-fiber high-volume sample filters. The method was quantitative for 13 elements, and was designated a U.S. Environmental Protection Agency reference method. 

Shaw and co-workers have recently reported using the simplex opti- 

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