Repairing constrained experimental regions and designs for mixture or nonmixture variables when some design points are unacceptable
- Journal of Quality Technology
- January 2020
- Volume 52 Issue 1
- pp. 1-13
- Piepel, Greg F., Cooley, Scott K.
A constrained experimental region (CER) involving mixture variables or nonmixture variables is generally specified by lower and/or upper bounds on the variables and possibly by lower and/or upper bounds on linear combinations of the variables. Additionally, the proportions of mixture variables typically are constrained to sum to unity. In some cases, an experimental design generated to explore the CER may contain unacceptable points (e.g., infeasible points or points with undesirable response values). This may not be discovered until after the experiment has been completed. In such cases, it can be desirable to repair the experimental design and the CER so that they contain only acceptable points. Two approaches for performing such repairs are discussed. The first approach repairs the CER to exclude unacceptable subregions, then uses existing optimal design methodology to select design points to repair the design. The new second approach (i) replaces each unacceptable design point with an acceptable design point and (ii) adds one new constraint associated with each replacement design point to repair the CER. Both approaches are illustrated with a three-component mixture experiment example that permits visualization. A 15-component mixture experiment example is also used to illustrate the new second approach, which is reasonable for experiments with many variables. Finally, an example involving three nonmixture variables is provided in supplementary material. Applying the approaches for repairing experimental designs and CERs to problems with both mixture and nonmixture variables is briefly discussed.