Simplex design approach

Finally, the simplex design has also been adopted for crystallization purposes (Prater et ah, 1999). This is an iterative approach starting with one more combination than factors under investigation. In an example with three factors at three equally spaced levels, 0, p and q, the first set consists of... 

It is always preferable to optimise by varying all the factors simultaneously. The latter approach leads to experimental design and to the simplex method, which is described in a number of specialised monographs. 

In the case of constraints on proportions of components the approach is known, simplex-centroid designs are constructed with coded or pseudocomponents [23]. Coded factors in this case are linear functions of real component proportions, and data analysis is not much more complicated in that case. If upper and lower constraints (bounds) are placed on some of the X resulting in a factor space whose shape is different from the simplex, then the formulas for estimating the model coefficients are not easily expressible. In the simplex-centroid x 23 full factorial design or simplex-lattice x 2n design [5], the number of points increases rapidly with increasing numbers of mixture components and/or process factors. In such situations, instead of full factorial we use fractional factorial experiments. The number of experimental trials required for studying the combined effects of the mixture com-... 

The following table lists the parameters for FAAS, EAAS, and FAES, which are both dependent and independent. A yes in any column indicates that the listed parameter is appropriate for that technique. If an optimization is necessary when independent parameters are involved, it is important to use a systematic approach that permits one to vary all parameter values to develop the optimum for each. If the variables are simply varied one at a time, false optimum values and poor results will be obtained. Experimental design techniques are required for good results one of the best approaches is the SIMPLEX technique, which has been fully discussed in the literature.15... 

One of die most popular approaches is called simplex optimisation. A simplex is file simplest possible object in A-dimensional space, e.g. a line in one dimension and a triangle in two dimensions, as introduced previously (Figure 2.32). Simplex optimisation implies that a series of experiments are performed on the comers of such a figure. Most simple descriptions are of two factor designs, where the simplex is a triangle, but, of course, there is no restriction on the number of factors. 

There is some controversy as to whedier simplex methods should genuinely be considered as experimental designs, rather than algorithms for optimisation. Some statisticians often totally ignore this approach and, indeed, many books and courses of... 

The large number and variety of factors on which SFE performance relies makes optimizing it rather a difficult task. Multivariate optimization approaches have been used from the beginning of this technique to both minimize the processing time and increase the extraction efficiency [18,19]. Simplex models [20] were the first to be used to examine the influence of interdependent variables. Two- and three-level orthogonal factor designs were developed to optimize up to nine extraction variables (viz. CO, flow-rate, fluid... 

Some of the approaches included in DOE include factorial design, simplex optimization (self-directed optimization), and response surfaces [24]. In factorial design, two (or more) values are assigned to key variables, and experiments are selected at random and run to determine the optimal results. Thus for a system comprising five variables at two settings each, the total number of runs would be 25, or 32. Usually 2n 1 runs will allow effective optimization from only half the number of runs, 16 in this case [25]. If a variable can be eliminated because it probably has little impact, the number of experiments can be reduced further to 2n 2, or only eight runs for initial optimization. Computer programs are available to assist in DOE optimization [21],... 

The experimental regions we have treated, when in the form of a simplex, have had the same shape as the total factor space. Thus they have been equilateral triangles, regular tetrahedra, etc. However this is not a necessary limitation. Any point in an irregular simplex, like the simplex of figure 9.17a, may be defined in terms of a mixture of the pseudocomponents at eaeh of the vertices. A response may then be analysed within that simplex by means of the Scheffe models and designs. This approach was used by the authors of references (7) and (8). 

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