Mixture Design and Mixture
Optimization
Sequential simplex
In a mixture the sum of the components always add up to
100%. This is known as the mixture constraint. Here we will explain how mixtures can be
optimized with the sequential simplex method, not to be confused with the simplex-lattice
or simplex-centroid mixture designs.
One component left out
The sequential simplex method can easily handle the
mixture constraint and the dependence between the control variables. The work procedure is
straightforward. The first simplex is defined in a mixture component space with one of the
components left out. In MultiSimplex the classical first simplex may be easier to fit
within the mixture boundaries, than the standard modified design.
The last component is then calculated as the difference
between 100% and the other components. Violation of the mixture constraint is treated as
easily as other boundary violations, by just assigning an infinitely bad response. In
MultiSimplex this is done automtically, without user intervention.
Other non-mixture variables can also be added without any
complications for the simplex optimization procedure.
A constrained mixture component space
Another way to carry out the mixture experiments is to
keep all components as control variables, but forcing the simplex to move within a
constrained mixture component space. In MultiSimplex you can use the "User
Defined" option to construct the first simplex. In all trials the sum of the
components must add up to 100%. One way of doing this is to specify the extreme vertices
and the center point as the first simplex trials. This will result in a degenerated
k-dimensional first simplex in a k+1 dimensional component space. See "Sequential
Simplex Optimization" by Walters et al (1991) for more details.
Conclusions
The sequential simplex method is an easy and
straightforward approach to the optimization of mixtures. There is no need for assumptions
about underlying models or elaborated procedures to deal with the mixture constraint. The
lack of independence between the components makes statistical modeling and design of
mixtures rather difficult and cumbersome. "Experiments with Mixtures" by Cornell
(1990) is recommended as reading for those interested in that subject.
Literature:
Palasota, J. A.; Leonidou, I.; Palasota, J. M.; Chang,
H.-L.;, Deming, S. N. Sequential simplex optimization in a constrained simplex
mixture space in liquid chromatography. Analytica
Chimica Acta 270:101-106 (1992).
Walters, F. H., Parker, L. R., Morgan, S. L., Deming, S.
N. Sequential Simplex Optimization. CRC
Press, Boca Raton, Florida, 1991.
Cornell, J. A. Experiments with Mixtures. John Wiley & Sons, New York, 1990.
- Simplex Introduction
- The Basic Simplex Method
- The Modified Simplex Method
- Evolutionary Operation
- Mixture Optimization
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