Simplex experimental design study

Figure 2. Simplex experimental design study of the degree of cure of AC30LIC 1/ AIPE/UPE blends as measured by MEK double rubs.

An experimental design was applied to study which variables determined the measurement of total reduced sulfur species in natural waters and sediments. Then, a simplex optimisation of relevant variables was made... 

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-... 

There are a niunber of different experimental design techniques that can be used for medium optimization. Four simple methods that have been used successfully in titer improvement programs are discussed below. These should provide the basis for initial medium-improvement studies that can be carried out in the average laboratory. Other techniques requiring a deeper knowledge of statistics, including simplex optimization, multivariate analysis, and principle-component analysis, have been reviewed (5,6). 

The simplex method has been used widely over the past 30 years, its success as much owing to its simplicity as to its efficiency. Unlike the other methods described in this chapter and most of the others in this book, it assumes no mathematical model for the phenomenon or phenomena being studied. The often long and costly phase of determination of a model equation may therefore be avoided, and the method is thus economical in principle. It is sequential because the experiments are analysed one by one, as each is carried out. Because the method is not model-based we will not describe it in detail, but well indicate how it can "fit in" and complement statistical experimental design. 

Fig. 7.10. Experimental designs for studying four-component mixtures, (a) Simplex lattice design, (b) Simplex centroid design.

To obtain the estimates of the model coefficients allowing the best forecast quality in the experimental domain studied, Scheffe propo.ses an experimental design that he calls simple.x lattice design. In the case of q components and for a polynomial of degree m, the corresponding simplex lattice design is noted q, /n. The coordinates of each point are multiples of /m and such that ... 

As shown, mixture components are subject to the constraint that they must equal to the sum of one. In this case, standard mixture designs for fitting standard models such as simplex-lattice and simplex-centroid designs are employed. When mixtures are subject to additional constraints, constrained mixture designs (extreme-vertices) are then appropriate. Like the factorial experiments discussed above, mixture experimental errors are independent and identically distributed with zero mean and common variance. In addition, the true response surface is considered continuous over the region being studied. Overall, the measured response is assumed to depend only on the relative proportions of the components in the mixture and not on the amount. 

Direet generation of stibine from slurries and its determination by ETAAS was studied using simplex multivariate optimisation (eoneentrations of HCl and NaBH4, and Ar flow rate). Previous experimental designs were made to select those variables. 

Chemical mixture experiments have distinct characteristics that can preclude the use of traditional statistical analysis techniques. Mixture experimentation often poses unique exploratory questions that can be answered efficiently and economically with non-traditional statistical techniques. General statistical guidelines stress the importance of design, preliminary studies, action levels of variables, graphics, and appropriate statistical testing. Fractional ctorial and Simplex designs are just two of many statistical tools that are useful for analyses of mixture experiments. 

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