  Alphabetic optimality criteria, such as the $D$, $A$, and $I$ criteria,
require specifying a model to select optimal designs. They are not model free
and the optimal designs selected by them are not robust to model uncertainty.
Recently, many extensions of the $D$ and $A$ criteria have been proposed for
selecting robust designs with high estimation efficiency. However, approaches
for finding robust designs with high prediction efficiency are rarely studied
in the literature. In this paper, we propose the $P_\alpha$ criterion and
develop its approximation version for two-level designs, called the ${\tilde
P_\alpha}$ criterion. They are useful for selecting robust designs with high
estimation, high prediction, or balanced estimation and prediction efficiency
for projective submodels. Computational studies show that the ${\tilde
P}_\alpha$ criterion is a good approximation of the $P_\alpha$ criterion and
can reduce great computation time when we search designs over a wide range of
models. The connection between the ${\tilde P_\alpha}$ criterion and the
generalized minimum aberration (GMA) criterion is studied. Result shows that
${\tilde P_\alpha}$ plays a great role to link the alphabetic optimality
criteria and the aberration-based criteria.
