Fixed values for some proportions

In some experiments, we are not interested in the effects of some ingredients in the mixture, so they are held at fixed values. The sum of the proportions of the ingredients of interest are therefore constrained to some value less than 1.0.

The 'trick' is to remove these fixed proportions from consideration in the mixture and deal only with the proportions within the remainder of the product. The proportions within this varying part of the product sum to one, so standard designs can be used

As an example, it may have been decided that exactly 80% of a bread mix will be white flour. An experiment is to be conducted to assess how bread texture depends on the proportions of wholemeal flour, rye flour and gluten in the rest of the mix. The proportions of wholemeal, rye and gluten within the 20% of the mix that is not white flour will sum to 100% and can each be anywhere between 0% and 100%.

Fat content of biscuits

A large commercial bakery produces a wide range of biscuits, including an almond biscuit whose recipe contains 8% fats by weight. The biscuits are traditionally made with butter, but the baker is interested in replacing some of the butter with cheaper margarine and canola oil. The baker therefore wants to conduct an experiment to compare the quality of the biscuits made with different proportions of the three fats in the recipe, but the remaining ingredients (92% by weight) staying the same.

Other ingredients 92%
Butter/Margarine/Canola mixture    8%

The triangle on the left shows the design space for the 8% of the product that can be varied.

The triangle on the right translates this design space into proportions of the fats in the biscuits. These proportions add to one.

Click anywhere in either triangle to see how the proportions of the whole product and of the fats are related. (For example, verify that 8% Butter in the whole mixture corresponds to 100% Butter within the fat content of the recipe.)

A design

The diagram below shows one possible design for this experiment.

The triangle on the right shows a standard simplex-centroid design. The triangle on the left translates each point into percentages of the mixture for the whole product.

Minimum values for some proportions

In many experiments, there is a restriction that one or more of the ingredients in the mixture must be at least some minimum value. For example, in a blended fruit juice, it may be decided that at least 60% of the product must be apple juice.

A similar 'trick' can be used to design the mixture experiment. The minimum proportions can be considered 'fixed' and we can create a design for the remainder of the product.

For example, if at least 60% of a blended fruit juice must be apple juice, the proportions of juices within the remaining 40% do not have any such constraints — within this 40% of the product, the proportion of apple juice could be anywhere between 0% and 100%.

Fat content of biscuits

For marketing reasons, it was decided that the percentage of Butter in the product should be at least 3.4%.

Other ingredients 92.0%
Basic Butter 3.4%
Rest of Butter/Margarine/Canola mixture    4.6%

With this constraint, we only need to consider an experimental design for the 4.6% of the recipe that can vary.

The diagram below shows simplex-centroid design for this experiment. The triangle on the right shows the proportions of the 3 ingredients in the 4.6% of the product that can be varied.

Click on the design points to see how the design on the right (for the 4.6% of the mixture that varies) translates into a design for proportions in the whole mixture.