  A general method for obtaining highly efficient factorial designs of
relatively small sizes is developed for cDNA microarray experiments. The method
allows the main effects and interactions of successive orders to be of possibly
unequal importance. First, the approximate theory is em-ployed to get an
optimal design measure which is then discretized. It is, however, observed that
a na\"ive discretization may fail to yield an exact design of the stipulated
size and, even when it yields such an exact design, there is often scope for
improvement in efficiency. To address these issues, we propose a step-up/down
procedure which is seen to work very well. The resulting highly efficient
designs are found to remain almost free from possible dye-color effects under a
suitable dye-color assignment. They are also seen to be quite robust to
heteroscedasticity as may be caused by biological variability. We focus on the
baseline and all-to-next parametrizations but our method works equally well
also for hybrids of the two and other parametrizations.
