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Modelling of mixture experiments can be quite complex, largely because the value of each factor is related to other factors, so, for example, in the equation on page 45 x3 = 1 - x1 - x2, so an alternative expression that includes intercept and quadratic terms but not x3 could be conceived mathematically. However, it is simplest to stick to models of the type above, that include just single factor and interaction terms, which are called Sheffe models. Mixture models are discussed in more detail in Section 9.6.4.

### 2.12.2 Simplex Lattice Designs

A weakness of simplex centroid designs is the very large number of experiments required when the number of factors becomes large. For five factors, 31 experiments are necessary. However, just as for factorial designs (Section 2.6), it is possible to reduce the number of experiments. A full five factor simplex centroid design can be used to estimate all 10 possible three factor interactions, many of which are unlikely. An alternative class of designs called simplex lattice designs have been developed. These designs can be referenced by the notation {f, m} where there are f factors and m fold interactions. The smaller m, the less experiments.

### 2.12.3 Constrained Mixture Designs

In chemistry, it is often important to put constraints on the value of each factor. By analogy, we might be interested in studying the effect of changing the proportion of ingredients in a

100% Factor 1

100% Factor 1

Lower bound factor 1

100% Factor 2

100% Factor 3

Lower bound factor 1

100% Factor 2

100% Factor 3

100% Factor 1

100% Factor 1

Upper bound factor 3

Upper bound factor 1

100% Factor 2

100% Factor 3

Figure 2.21 Constrained mixture designs. (a) Lower bounds defined; (b) upper bounds defined; (c) upper and lower bounds defined: fourth factor as a filler; (d) upper and lower bounds defined

Upper bound factor 3

Upper bound factor 1

100% Factor 2

100% Factor 3

Figure 2.21 Constrained mixture designs. (a) Lower bounds defined; (b) upper bounds defined; (c) upper and lower bounds defined: fourth factor as a filler; (d) upper and lower bounds defined

cake. Sugar will be one ingredient, but there is no point baking a cake using 100 % sugar and 0 % of each other ingredient. A more sensible approach is to put a constraint on the amount of sugar, perhaps between 2 % and 5 % and look for solutions in this reduced mixture space. There are several situations, exemplified in Figure 2.21:

• A lower bound is placed on each factor. Providing the combined sum of the lower bounds is less than 100 %, a new mixture triangle is found, and standard mixture designs can then be employed.

• An upper bound is placed on each factor. The mixture space becomes rather more complex, and typically experiments are performed at the vertices, on the edges and in the centre of a new irregular hexagon.

• Each factor has an upper and lower bound. Another factor (the fourth in this example) is added so that the total is 100 %. An example might be where the fourth factor is water, the others being solvents, buffer solutions, etc. Standard designs such as the central composite design (Section 2.11) can be employed for the three factors in Figure 2.21, with the percentage of the final factor computed from the remainder.

Finally, it is possible to produce a mixture space where each factor has an upper and lower bound, although only certain combinations are possible. Often quite irregular polygons are formed by this mechanism. Note, interestingly, that in the case illustrated in Figure 2.21, the upper bound of factor 3 is never reached.

There is a great deal of literature on mixture designs, much of which is unfamiliar to chemists. A well known text is by Cornell [4]. Mixture designs are described further in Section 9.4.

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