Initial sequential simplex optimization

Comparing the experiments with the advanced simplex method and the general sequential simplex method, the former requires only ten while the latter needs about fifty-four (RP-HPTLC) or forty (NP-HPTLC) experiments. In addition, the advanced simplex method can select repeatedly initial simplex experiments from preliminary experiments without any additional experiments. Therefore the advanced simplex method has distinct advantages over the general sequental simplex method for two-factor optimization in HPTLC. 

Bindschaedler and Gurny [12] published an adaptation of the simplex technique to a TI-59 calculator and applied it successfully to a direct compression tablet of acetaminophen (paracetamol). Janeczek [13] applied the approach to a liquid system (a pharmaceutical solution) and was able to optimize physical stability. In a later article, again related to analytical techniques, Deming points out that when complete knowledge of the response is not initially available, the simplex method is probably the most appropriate type [14]. Although not presented here, there are sets of rules for the selection of the sequential vertices in the procedure, and the reader planning to carry out this type of procedure should consult appropriate references. 

Section 5.3 describes sequential methods of optimization, in particular the Simplex method. In sequential methods the optimization procedure starts with some initial experiments, inspects the data and defines the location of a new data point which is expected to yield an improved chromatogram. The idea is to approach the optimum step by step in this way. 

In contrast to the simultaneous optimization procedures described in the previous section, the Simplex method is a sequential one. A minimum number of initial experiments is performed, and based on the outcome of these a decision is made on the location of a subsequent data point. This simplest form of a sequential optimization scheme can be characterized by the path 1012 in figure 5.4. 

In Figure 2.15, an example is given of the modified simplex procedure for the determination of fluticasone propionate with flow injection analysis (104). The initial simplex is formed by points 1, 2, and 3. Points 4-14 represent the sequentially selected vertices. Point 6 seems to be situated close to the optimum because it is maintained in many simplexes. It is observed that again, as in the classic procedure, the simplexes circle around the optimum, but here also their size decreases as the procedure continues. To optimize three or more factors, the simplex procedures can be generalized, as described in Reference 8. 

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