Mixture experiments simplex lattice

The most frequently used mixture-"composition-property designs of experiments belong to simplex-lattice designs suggested by Scheffe [5], The basis of this kind of designing experiments is a uniform scatter of experimental points on the so-called simplex lattice. Points, or design points form a [q,n] lattice in a (q-1) simplex, where q is the number of components in a composition and n is the degree of a polynomial. For each component there exist (n+1) similar levels Xp0,l/n,2/n.1 and all... 

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 first designs for mixture experiments were described by Scheffe [3] in the form of a grid or lattice of points uniformly distributed on the simplex. They are called q, i j simplex-lattice designs. The notation q, v implies a simplex lattice for q components used to construct a mixture polynomial of degree v. The term mixture polynomial is introduced to distinguish it from the polynomials applicable for mutually independent or process variables, which are described later in our discussion of factorial designs (section 8.4). In this way, we distinguish mixture polynomials from classical polynomials. 

This type of experimental design still has one of the drawbacks of the initial simplex lattice design, since most of the responses used to determine the coefficients are obtained from mixtures formed of q components (q < q) and these experiments are situated at the limits of the domain (which is the major drawback),... 

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. 

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