Response Surface Procedure

In seeking the optimal point, it is common to use the method of steepest ascent or the gradient method. In this approach, based on the resulting relationship, the new process operating point wDl be determined based on the direction with the steepest gradient. This approach ensures that the optimal point can be reached the fastest. 

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General Design and Analysis of Experiments. Most of these references contain 

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The relationship between output variables, called the response, and the input variables is called the response function and is associated with a response surface. When the precise mathematical model of the response surface is not known, it is still possible to use sequential procedures to optimize the system. One of the most popular algorithms for this purpose is the simplex method and its many variations (63,64). 

For this last stage, the one-at-a-time procedure may be a very poor choice. At Union Carbide, use of the one-at-a-time method increased the yield in one plant from 80 to 83% in 3 years. When one of the techniques, to be discussed later, was used in just 15 runs the yield was increased to 94%. To see why this might happen, consider a plug flow reactor where the only variables that can be manipulated are temperature and pressure. A possible response surface for this reactor is given in Figure 14-1. The response is the yield, which is also the objective function. It is plotted as a function of the two independent variables, temperature and pressure. The designer does not know the response surface. Often all he knows is the yield at point A. He wants to determine the optimum yield. The only way he usually has to obtain more information is to pick some combinations of temperature and pressure and then have a laboratory or pilot plant experimentally determine the yields at those conditions. 

Note, however, there are two critical limitations to these "predicting" procedures. First, the mathematical models must adequately fit the data. Correlation coefficients (R ), adjusted for degrees of freedom, of 0.8 or better are considered necessary for reliable prediction when using factorial designs. Second, no predictions outside the design space can be made confidently, because no data are available to warn of unexpectedly abrupt changes in direction of the response surface. The areas covered by Figures 8 and 9 officially violate this latter limitation, but because more detailed... 

Finally, a review of robustness testing of CE methods was made and the tests were critically discussed (Section IX). Some researchers use the OVAT procedure, which seems less appropriate for a number of reasons. Some use response surface designs, which also seems less preferable in this context. Another remarkable observation from the case smdies is that only in a minority the quantitative aspect of the method is considered in the responses smdied, even though that was the initial idea of proposing the robustness tests. 

Coenegracht et al. [3] have introduced a four solvent system to compose mobile phases for the separation of the parent alkaloids in different medicinal dry plant materials, like Cinchona bark and Opium. Through the use of mixture designs and response surface modeling an optimal mobile phase was found for each type of plant material. These new mobile phases resulted in equally good or better separations than obtained by the procedures of the Pharmacopeias. Although separations were as predicted, the accuracy of the quantitative predictions needed to be improved. 

The most economical procedure for a liquid-liquid extraction would be a single step extraction, since extraction procedures including several steps with the same or with different solvents are laborious and economically disadvantageous. Optimisation of extraction of more than one solute, which give different selective interactions (different response surfaces in the same mixture space), may require several extraction steps with different optimal extraction solvents or separate analysis of each analyte. However, procedures can be used, which select a composition of the extraction liquid that provides satisfactory partition coefficients or extraction yields for all solutes to be extracted. 

In order to develop testing procedures that will yield a more meaningful characterization of the sensitivity of primary expls, a testing method that involves the use of the so-response surface is reproduced in Fig 12, Ref 35 and briefly described on p 25. No description is given here since the authors of Ref 35 admit, that, f ohviously one needs a better procedure ... 

Five factors for hydride generation were studied to develop a method to determine As in gasolines. A central composite design was used to develop the response surface and, thus, optimise the extraction procedure... 

FAAS, ETAAS The variables implied on an ultrasound-assisted acid leaching procedure were evaluated by experimental designs. The relevant variables were subsequently optimised by a central composite design and response surface... 

The most striking differences between the two response surfaces are the size of the feasible region and the location of the optimal solution. The optimal solution under the confined assumption is not feasible under unconfined conditions. The infeasible space of the two problems is very similar and without modifications to the optimization solution algorithm, the search trajectory for the unconfined case is similar to that for the confined case. This results because there is no feedback in the infeasible space of the unconfined problem to prevent the search from identifying well W-2 as the more promising of the two wells. Without additional search procedures, the optimization algorithm will fail when extraction at W-2 leads to dewatering. 

CONTENTS 1. Chemometrics and the Analytical Process. 2. Precision and Accuracy. 3. Evaluation of Precision and Accuracy. Comparison of Two Procedures. 4. Evaluation of Sources of Variation in Data. Analysis of Variance. 5. Calibration. 6. Reliability and Drift. 7. Sensitivity and Limit of Detection. 8. Selectivity and Specificity. 9. Information. 10. Costs. 11. The Time Constant. 12. Signals and Data. 13. Regression Methods. 14. Correlation Methods. 15. Signal Processing. 16. Response Surfaces and Models. 17. Exploration of Response Surfaces. 18. Optimization of Analytical Chemical Methods. 19. Optimization of Chromatographic Methods. 20. The Multivariate Approach. 21. Principal Components and Factor Analysis. 22. Clustering Techniques. 23. Supervised Pattern Recognition. 24. Decisions in the Analytical Laboratory. 

The selected criterion will vary as a function of the selected parameters. This function is called the response surface. Depending on the selected criterion, the optimization procedure will be aimed at locating either the maximum or the minimum value of the response surface. The optimum is defined by those values of the parameters that correspond to this maximum or minimum. 

For the important case of the optimization of the mobile phase composition in reversed phase LC (RPLC), a typical two-dimensional response surface tends to be much less rugged, especially if the number of sample components is relatively small (n<10). A typical example is shown in figure 5.5. The selection of the normalized resolution product (r, eqn.4.19) as the criterion has also contributed to the smoother appearance of figure 5.5 relative to figure 5.1. Note that the criterion r has been recommended in chapter 4 for optimization processes in which the dimensions of the column are to be optimized after completion of the procedure (table 4.11). Therefore, the grid search approach is more appropriate for this kind of optimization than for optimization processes on the final analytical column. 

If the response surface is simple, the true optimum can be approached without the need for a compromise between the required accuracy of the resulting optimum and the number of experiments required, as was the case for the use of a constant step size in figure 5.7. On the other hand, a stop criterion for the Simplex needs to be defined carefully, because it will be clear from figure 5.8 that many experiments can easily be wasted in the close vicinity of the optimum if the requirements are too tight. For example, to locate an optimum with an accuracy of 0.1% in composition will require much more time (many more experiments) than if the procedure is stopped when the changes in the composition in successive steps start to fall well below 1%. 

Once the retention surfaces are known, any criterion may in principle be used to calculate the response surface and to locate the optimum composition. One of the criteria used by Glajch et al. is the threshold minimum resolution criterion (section 4.3.3). This is done by means of a graphical procedure, referred to as overlapping resolution mapping or ORM. This procedure involves the location of areas in the triangle where the resolution Rs exceeds a certain threshold value. This is repeated for all pairs of solutes and the results are combined to form a single figure. 

The meaning of this complex definition is illustrated in figure 5.28. The procedure starts with a (small) set of initial experiments. The next step is the application of a model to the data. This model can be a graphical or a mathematical one, but may also be a simple linear interpolation between the individual data points. Typically, the model is applied to the retention surfaces of the individual solutes, and not to the response surface. Alternatively [537], it may describe relative retentions with respect to a reference component in the... 

The third solution to the problem may be found in the use of more efficient computers, algorithms and computational methods. For instance, if segmentation of the parameter space (linear interpolation) is used, large parts of the retention surfaces and hence of the response surface may remain unaltered when a new data point is added to the existing set. The use of simple model equations instead of linear segmentation may also be more efficient from a computational point of view. However, such simple equations may only be used for the description of the retention behaviour in a limited number of cases and if the model equations become more complex the advantage quickly disappears. For example, d Agostino et al. used up to sixth order polynomial equations [537] and their procedure also led to excessive calculation times. 

The Simplex method (and related sequential search techniques) suffers mainly from the fact that a local optimum will be found. This will especially be the case if complex samples are considered. Simplex methods require a large number of experiments (say 25). If the global optimum needs to be found, then the procedure needs to be repeated a number of times, and the total number of experiments increases proportionally. A local optimum resulting from a Simplex optimization procedure may be entirely unacceptable, because only a poor impression of the response surface is obtained. 

Because this optimization only concerned program parameters and not selectivity parameters, the response surface will have been relatively simple. Therefore, the probability that the Simplex procedure would arrive at the global optimum rather than at a local one was greater than it was in section 5.3, where we described the use of the Simplex method for selectivity optimization. 

The response surface for the optimization of the primary (program) parameters in programmed temperature GC is less convoluted than a typical response surface obtained in selectivity optimization procedures (see section 5.1). This will increase the possibility of a Simplex procedure locating the global optimum. 

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