Simplex sequential approach

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]. 

A limited number of variables (<. 3) is evaluated, but the experimental region within which the optimal result is situated is not known a priori. A sequential approach, called simplex can be used (Section 6.7). The same methodology can be applied when one is not interested in modelling the response but only in finding the optimal conditions. 

They allow a sequential approach, at least up to mode of degree 3, since the approach is strictly sequential only up to the reduced cubic model included. When the models of a higher degree are studied, many already obtained points do not fit the new simplex lattice design. However, this disadvantage can be overcome by taking these points as test points. 

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. 

The second approach to optimisation is a model independent one. One of these model independent methods is the sequential simplex [24,25] used by Shek et al. [26], The method is claimed to be ideally suited for the optimisation of formulations [27] because of the relatively low number of experiments to be performed. 

Given the mere handful of reports in the published literature (6,38,39,52), there are many avenues open in the development of systematic approaches to optimization in SFC. In addition to the opportunities mentioned in the sections on the simplex method and window diagram approach, others include the exploration of other sequential or simultaneous optimization strategies such as optiplex, simulated annealing, method of steepest ascent, etc. that are potentially useful in SFC. 

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. 

Simplex optimization of the primary (program) parameters in programmed temperature GC analysis has been demonstrated [612]. A systematic sequential search [613] may be used as an alternative. The Simplex method may be used to optimize a limited number of program parameters, whereas the latter approach was developed for the optimization of multisegment gradients. The use of interpretive methods has so far only been suggested [614,615]. 

On the other hand, in situations where the experimental region containing the optimum is not a priori known, a sequential optimization method, for example, a simplex approach, can be applied. Then, the following steps are considered ... 

Different sequential optimization methods can be distinguished, of which the simplex approaches are most commonly applied. They can be further... 

A sequential injection system could, in principle, be modelled by considering the semi-volume (S0.5) concept [97], However, even in situations where the effects of reaction equilibria and kinetics are ignored, the profiles obtained using the dye approach are rather different from the experimental profiles [98]. Therefore, system dimensioning is better accomplished by varying the main parameters and applying optimisation algorithms such as simplex. 

The implicit approach of chapter 5 was to optimize the process or formulation by examining the response surface directly. But other methods are both useful and necessary when there are many factors (desirability) or when the optimum is outside the experimental region (steepest ascent and optimum path). Also there are direct optimization methods available (sequential simplex) which do not involve mapping the response surface at all. 

Other sequential simplex methods and approaches The basic (fixed step) method... 

Chemometrics has been defined in some texts [155] as the entire process whereby data are transformed into information used for decision-making. It is this definition that is the most applicable to separation sciences, more specifically in method development and optimisation in liquid chromatography. In this example, chemometrics has been used to predict optimum separation conditions based on empirical data and other separation information. Chemometric approaches to method development can be based on either sequential simplex models [156] or simultaneous fixed factorial designs [157] or interactive mixture designs [158] which combine the advantages of simultaneous and simplex models. 

There are yet further sophistications such as the supermodifled simplex, which allows mathematical modeling of the shape of the response surface to provide guidelines as to the choice of the next simplex. Simplex optimization is only one of several computational approaches to optimization, including evolutionary optimization, and steepest ascent methods, however, it is the most commonly used sequential method in analytical chemistry, with diverse applications ranging from autoshimming of instruments to chromatographic optimizations, and can easily be automated. 

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