Simplex disadvantages

Disadvantages of the simplex method are the number of experiments to reach an optimum is not known beforehand, this can lead to better but also to worse results compared to a simultaneous approach. If an optimum is reached nothing is known about that part of the response surface that has not been investigated, e.g. other, even higher optima can be present and, which is more important, the stability of the reached optimum against small variations of a criterion, is not known. 

The three methods described in the preceding paragraphs each offer distinct advantages and disadvantages. The first and most obvious difference between the methods is the distinction between the sequential methods (sequential simplex and prisma method) and the simultaneous method (mixture design). With the sequential method some experiments are performed, these are evaluated, and on the basis of this evaluation new design points are selected, these are evaluated etc. With the simultaneous... 

Although application of chemometrics in sample preparation is very uncommon, several optimisation techniques may be used to optimise sample preparation systematically. Those techniques can roughly be divided into simultaneous and sequential methods. The main restrictions of a sequential simplex optimisation [6,7] find their origin in the complexity of the optimisation function needed. This function is a predefined function, often composed of several criteria. Such a composite criterion may lead to ambiguous results [8]. Other important disadvantages of simplex optimisation methods are that not seldom local optima are selected instead of global optima and that the number of experiments needed is not known beforehand. 

Simplex has been used in analytical method development. Its advantages are that the response should improve with each round of experiments, allowing the experimenter to decide when to discontinue the experiments there are no arbitrary relations involved in the choice of the model equation and the methodology used to select the next point can easily be implemented in a spreadsheet. Disadvantages are that the Simplex method is an open-ended procedure, for which the number of experiments depends on... 

Details on the simplex algorithm are available elsewhere (33,34,47-50). The advantages and disadvantages of the simplex method pertinent to chromatographic applications are summarized in Table II. 

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Because the initial parameters are taken very close to the edges of the parameter space, the Simplex necessarily contracts immediately. It typically approaches the optimum area quickly, but spends much time locating its final optimum around a composition of 50% water, 20% methanol and (hence) 30% acetonitrile. The advantage of this is that most measurements are obtained in the location of the optimum. This could also turn into a disadvantage, because it implies that little information is obtained about the rest of the response surface, so that no idea can be formed about the merits of the located local optimum with respect to other optima. 

However, the Simplex procedure also has some considerable disadvantages, one of which we have already touched upon. A large number of experiments is usually required to locate the optimum. Typically, about 40 experiments appear to be required [508,509]. If the parameter space is reduced before the Simplex procedure is started, this number might be brought down to about 25 (see ref. [510] and section 5.4). 

The major disadvantage of simplex-centroid designs is that only one type of model can be applied, namely a model having the structure shown in Equation 8.20. 

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. 

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In the preceding chapter was discussed how a near-optimum experimental domain can be established through the method of steepest ascent. A disadvantage of this method is that it requires that the slopes of a linear approximation of the response surface model is established first. When it is reasonable to assume that the most influencing experimental variables are known, it is possible to locate a near-optimum domain through alternative procedures, viz. the simplex methods. In these methods, a minimum initial set of experiments is used to span a variation in the experimental domain. From these experiments, it is then possible to determine in which direction an improved response is to be expected. A new experiment is run in this direction. 

The alternative is to employ a multivariate optimization procedure such as Simplex. Simplex is an algorithm that seeks the vector of parameters that corresponds to the separation optimum within an n-dimensional experimental space. For example, a two-parameter CE separation optimized by Simplex would begin with three observations of the separation response at three different electrolyte conditions. These conditions are chosen by the analyst, often his or her best guess. From the evaluation of the response of each observation, the algorithm chooses the next experimental condition for investigation (4). As with the univariate method, the experiments continue until an optimal separation condition is determined. The disadvantage of such an approach is that it is unknown how many experiments are required to achieve an optimum, or if the optimum is local or global as the entire response surface is not known. 

The biggest disadvantage of the supermodified simplex is that the definition of each new simplex requires an extra experiment, made at point P. Some researchers have preferred to avoid this additional effort, using the average of the responses at all the vertexes of the simplex (except W) as an estimate of the response at P. This variant of the supermodified algorithm has been tested on several response surfaces (Brown, 1990). 

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. 

The modified simplex algorithm produces nearly optimum weight vectors even when linearly inseparable data sets are used for training but has the disadvantage of sometimes prohibitively large computer times C165I. 

Simplex optimization has become very popular and has been widely used in chromatography, but there are three major disadvantages (1) the relationship between the factor to be optimized and the parameters involved is seldomly revealed in detail, and the procedure therefore does not lead to a better understanding of the separation process, (2) a local optimum may be found and the optimization process stops there, and (3) a larger number of experiments are required. In order to overcome these drawbacks, recently a computer-assisted mixture design simplex method has been introduced by Wang and co-workers [18]. 

The window diagrams approach has a distinct disadvantage compared with this method, because in window diagrams solvent optimization methods can only be generated with binary and ternary solvents (plus the base solvent). The simplex method can be used for multicomponent solvent... 

Its main disadvantage is that it does not necessarily find the global optimum. However, this problem can be reduced if additional knowledge is integrated or if different starting points of the simplex procedure result in the same maximum. 

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