Modified Simplex optimization procedur

A variation of the latter optimization procedure known as modified simplex optimization has evolved to eliminate the limitations imposed by the simplex method. The... 

Figure 8. Resolution of racemic Doxazosin on a Chiral-APG column (see also Table 2) using an optimization strategy of modified simplex procedure. Reprinted with permission from ref 86.

The first results of optimization in chromatography were published in 1975 Since then a growing number of optimization experiments in HPLC using the Simplex procedure has been reported (table 9). The examples are mainly reversed-phase separations, in which the composition of the ternary or binary mobile phase composition is optimized. The factors optimized are usually a selection from flow rate, column temperature and length, the eluents constitution (e.g. organic modifier content, buffer concentration and pH), the gradient shape. Seven years after the first applications of Simplex optimization had appeared, the first fully automated optimization of HPLC separations was published by Berridge in 1982. This development coincid-... 

One way to include the less desirable variables of Table HI in the simplex optimization process would be to perform a simplex optimization for each of a limited number of user-selected combinations of the less suitable variables. The optimum results obtained for these selected combinations could then be compared, with the best overall result indicating which combination of fluid, modifier, stationary phase, etc. is best for that particular separation. A drawback of this approach, particularly if the individual optimization procedure is not automated, is its time-consuming, labor intensive nature. 

By far the most popular technique is based on simplex methods. Since its development around 1940 by DANTZIG [1951] the simplex method has been widely used and continually modified. BOX and WILSON [1951] introduced the method in experimental optimization. Currently the modified simplex method by NELDER and MEAD [1965], based on the simplex method of SPENDLEY et al. [1962], is recognized as a standard technique. In analytical chemistry other modifications are known, e.g. the super modified simplex [ROUTH et al., 1977], the controlled weighted centroid , the orthogonal jump weighted centroid [RYAN et al., 1980], and the modified super modified simplex [VAN DERWIEL et al., 1983]. CAVE [1986] dealt with boundary conditions which may, in practice, limit optimization procedures. 

In the following, the basic and modified simplex procedures are discussed for the optimization of two factors. The simplex is then a triangle (f + 1 = 3 vertices). For the basic procedure, it is an equilateral triangle, while for the modified procedure, it does not necessarily have to be. 

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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