Simplex optimization criteria

Simplex Optimization Criteria. For chromatographic optimization, it is necessary to assign each chromatogram a numerical value, based on its quality, which can be used as a response for the simplex algorithm. Chromatographic response functions (CRFs), used for this purpose, have been the topics of many books and articles, and there are a wide variety of such CRFs available (33,34). The criteria employed by CRFs are typically functions of peak-valley ratio, fractional peak overlap, separation factor, or resolution. After an extensive (but not exhaustive) survey, we... 

The situation is different if a sequential method of optimization (e.g. Simplex optimization, section 5.3) is used. In this case a criterion value is assigned to every chromatogram and the result is compared with previously obtained values. Hence, if the number of observed peaks increases, this may lead to incorrect comparisons. For example, if in one chromatogram three fully separated peaks were observed, the value of TIP for that chromatogram would equal one. However, if in the next chromatogram four peaks were observed which were not completely resolved (e.g. P values between each pair of successive peaks of 0.9), then the resulting value for TIP would only be 0.73. However, the second chromatogram is clearly to be preferred to the first one. 

To deal with this problem, it appears to be more correct to update the previously found criteria values than it is to increase the value of the new one. To do so, it is not only required to keep track of the criterion values of previous chromatograms, but also of the number of observed peaks. In the case of Simplex optimization this is especially easy, since only the criterion values of three chromatograms need to be remembered (see section 5.3). Hence, for the above example the previous result might be IIP= 1 with n = 3. If the new result is IIP= 0.73 with n = 4, then the previous result needs to be updated to I1P = 0 with n = 4. 

As with programmed temperature GC, the application of the Simplex optimization procedure to programmed solvent LC is relatively straightforward. The same procedure can be used both for isocratic and for gradient optimization, as long as an appropriate criterion is selected for each case. ... 

An obvious and easy to calculate optimization criterion is the recognition rate. Because only integer numbers of patterns are classified, the response surface is discontinuous and consists of a series of plateaus. It is possible for the simplex to become stranded on such a plateau. Another problem arises if a weight vector with 100 % recognition is found which does not necessarily occupy the optimum position between the clusters. Therefore, it is necessary to use a second optimization criterion to smooth the surface. 

For the optimization of, for instance, a tablet formulation, two strategies are available a sequential or a simultaneous approach. The sequential approach consists of a series of measurements where each new measurement is performed after the response of the previous one is knovm. The new experiment is planned according to a direction in the search space that looks promising with respect to the quality criterion which has to be optimized. Such a strategy is also called a hill-climbing method. The Simplex method is a well known example of such a strategy. Textbooks are available that describe the Simplex methods [20]. 

The model [C] can be evaluated by comparing with [D] using a -criterion. The parameters of the model can be optimized by some optimization procedure (Simplex). 

When the criterion optimized is a linear function of the operating variables, the feasibility problem is said to be one in linear programming. Being the simplest possible feasibility problem, it was the first one studied, and the publication in 1951 of G. Dantzig s simplex method for solving linear-programming problems (D2) marked the beginning of contemporary research in optimization theory. Section IV,C is devoted to this important technique. 

Fig. 1.22. Opiimisation of two operation parameters by simplex melhtxl. The dolled contour lines correspond to equal values of the optimisation criterion. ABC = original simplex. ACD = new simplex obtained by rejection of the point B with the worst value of ihe opiimisation criterion and reflecting the original simplex in the opposite field O = the combination of the iwo optimised operation parameters for the highest (optimal) value of the optimisation criterion.

The main disadvantage of the simplex method consists in the laige number of experiments required to find optimal working conditions. Further, the optimisation criterion characterises the separation of the sample mixture by a single number, so that the detailed information on the separation of the individual sample components is lost and because of the high probability that the search method will slide into a region with a local maximum of the optimisation criterion, the simplex optimisation method can be expected to be fully successful only with the separations of relatively simple samples. 

Examples of optimizations in HPLC using the simplex approach can be found in [28,84]. In [28] the mobile phase composition for the chiral separation of (6/ )- and (65)-leucovorin on a BSA (bovine serum albumin) stationary phase is optimized by means of a variable-size simplex. Three factors were examined, the pH of the mobile phase buffer, the ionic strength of the buffer and the percentage of 1-propanol in the mobile phase. Table 6.19 shows the experimental origin, the initial step size and the acceptable limits for the factors. The criterion optimized is the valley-to-peak ratio (Section 6.2). The points selected and the results are pre.sented in Table 6.20 and... 

The Hausdorff distance was applied in chemistry in various chirality studies. The Hausdorff distance was used to measure the deviation of a chiral nuclear arrangement from some arbitrary reference arrangement, as proposed by Rassat. Mislow and co-workers used the Hausdorff distance between the object and its optimally overlapping mirror image to provide a chirality measure of the second type. - Using this Hausdorff distance criterion, Buda and Mislow determined the most chiral constrained and unconstrained simplexes in two and three dimensions, that is, the most chiral triangles and tetrahedra." ... 

The method of steepest ascent and the simplex search can handle only one criterion, while the resportse surface methods allow simultaneous mapping of several responses. Response surface modelling can therefore be used to optimize several responses simultaneously. The problem of multiple responses is elegantly handled by PLS modelling. This technique is discussed in Chapter 17. 

Automatic optimization requires an efficient computer system to establish whether one separation is better than another. A so-called resolution function , for which various dihnitions have been proposed, is used as an objective criterion, covering resolution between neighbouring peaks and the analysis time involved. Identihcation by the computer of individual peaks, whether by the injection of standards or by spectroscopic means, is a distinct advantage. The simplex process may be used to optimize automatically the separation of complex acid-base mixtures by simultaneous variation of pH, ion pair reagent concentration, composition of ternary mobile phase, flow rate and temperature. Some automatic optimization systems are available commercially and, although they do leave some room for improvement, there is no doubt that they provide satisfactory answers to a great many problems. 

Efficient experimentation is based on the methods of experimental design and its quantitative evaluation. The latter can be performed by means of mathematical models or graphical representations. Alternatively, sequential methods are apphed, such as the simplex method, instead of these simultaneous methods of experimental optimization. There, the optimum conditions are found by systematic search for the objective criterion, for example, the maximum yield of a chemical reaction, in the space of all experimental variables. 

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