Independently adjustable factors

The experiments that were considered in earlier sections all involved factors that could be independently controlled — any combination of factor values within the experimental region of interest was possible. For example, in an experiment to assess the effects of temperature and addition of an enzyme to the strength of a plastic, these two factors can be independently adjusted to give a run of the experiment at any desired combination of factor levels.

Mixtures

In contrast, some experiments are conducted to compare different mixtures of ingredients in a product. If there are k ingredients whose proportions in the mixture are denoted by x, z, ..., then the proportions are constrained to sum to exactly 1.0 (i.e. 100% of the product) so they cannot be independently adjusted.

Ingredient   Proportion  
A x
B z
... ...
Total 1.0

As an extreme example, if there is 100% of ingredient A in a run of the experiment, then the only allowable proportion for the other ingredients is zero.

Mixtures in which one ingredient dominates
If one ingredient contributes nearly 100% of the mixture (e.g. when a product consists of one base ingredient with small amounts of (k - 1) additives), then it may be possible to independently adjust the proportions of the (k - 1) additives within the range of interest and a standard factorial (or fractional factorial) experimental design is possible for these (k - 1) factors.
Other mixtures
If several ingredients can be moderately large, a standard design for (k - 1) of the proportions as factors is not possible. Increasing the proportion of any ingredient reduces the range of allowable proportions for the others. The remainder of this section develops designs for this type of experiment.

Experimental designs for mixtures

When designing experiments for mixtures, we usually treat all k proportions symmetrically. For k = 3 ingredients whose proportions are x, z and u, the experimental region consists of values such that

0 ≤ x ≤ 1

0 ≤ z ≤ 1

0 ≤ u ≤ 1

x + z + u = 1

These constraints define an equilateral triangle in 3 dimensions, and the experimental design should choose design points within this triangle.

Mixtures of three fruit juices

Consider an experiment to compare the perceived sweetness of a drink that consists of a mixture of apple, orange and pineapple juices. The factors that can be controlled in the experiment are the proportions of these three juices.

The triangle in the diagram contains all possible proportions of the juices that sum to 1.0.

The 3 vertices of the triangle are mixtures containing only a single ingredient. Points on the edges are mixtures of 2 of the 3 ingredients.