Design of Experiments
Optimization Strategies
Many ways to design experiments
Experimental optimization can be carried out in several
ways. Most popular is the one-variable-at-a-time approach. This approach is however
extremely inefficient in locating the true optimum when interaction effects are present.
Multivariable design of experiments are since many years
used to overcome the problems with interaction effects. There are two general groups of
designs to choose from: Sequential or simultaneous experiment designs. The choice depends
of the purpose of the study.
Sequential design of experiments
Sequential design of experiments are very useful for
optimization studies. Experiments are successively performed in a direction of improvement
until the optimum is reached. The far most useful method is the simplex approach,
described in many hundreds of publications during recent years. The simplex method can
handle many variables with only a few trials, and does not require any assumptions with
regard to the underlying model. The simplex optimization also starts with a design,
consisting of as many trials as there are variables plus one (a simplex is a k+1 geometric
figure in a k-dimensional space). Subsequent trials are calculated by reflection towards
improved conditions. The modified simplex method also adjusts the size of the design to
further increase the rate of improvement.
Simultaneous (statistical) design of experiments
Empirical models are best built with traditional
simultaneous (statistical) experiment designs, e.g. response surface designs. The optimal
variable settings should then be known beforehand, with understanding as the primary goal.
These designs are sometimes also used for the purpose of optimization, but this involves
several steps with many trials. As a consequence these studies are usually limited to only
a few variables, with a substantial risk of missing the true optimum. If the model does
not include all relevant variables and/or does not cover the optimum its value is
diminished. Empirical model building is therefore generally inefficient as primary
optimization technique, but useful to gain scientific insight.
A joint approach
Are these both approaches, sequential and simultaneous
design of experiments, competing alternatives or can they be joined into a comprehensive
and effective optimization and model-building strategy? Our application partner in the US,
Statistical Designs, may have come up with the answer. Their approach is first to optimize
and then to study variable effects, significance, etc. (i.e. model-building):
"In the past, optimization usually required answers
to three ordered questions:
- What variables are the most significant?
- In what way do they affect the quality of the product or
process?
- What is the optimal combination of settings for these
significant variables?
This historical approach to optimization is slow and
expensive. An alternative approach to optimization answers the same three questions in
reverse order:
- What is the optimal combination of settings of the
variables?
- In what way do the variables affect the quality of this
product or process in the region of the optimum?
- What variables are most significant in the region of the
optimum?
Clearly, this approach requires efficient optimization
strategies. For many optimization projects in research, development, and manufacturing,
the sequential simplex (an EVOP technique) is the method of choice."
Conclusions
Some general conclusions that can be drawn from this:
- Use the sequential simplex method for experimental
optimization.
- Use traditional statistical design of experiments for
model-building.
- Always optimize before applying the traditional statistical
design of experiments.
Literature
Statistical Methods and the Chemist by R. M.
Driver, Chem. Brit.
6:4 pp. 154-158 (1970).
Sequential Simplex Optimization. A Technique for Improving Quality and Productivity
in Research, Development, and Manufacturing by Walters, Parker, Morgan and Deming,
CRC Press 1991.
Chemometrics - Application of Mathematics and Statistics to Laboratory Systems
by R. G. Brereton, Ellis Horwood Ltd. 1990.
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