The results on optimal diallel cross designs are based on standard
linear model assumptions where the general combining ability effects
are taken as fixed. In many practical situations, this assumption may
not be tenable since often one studies only a sample of inbred lines
from a possibly large hypothetical population. A random effects model
is proposed in this paper that allows us to obtain an interval
estimate of a ratio of the variance components. We address the issue
of optimal designs by considering the $D_l$-optimality criteria.
Designs that are $D_l$-optimal for the estimation of heredity are
obtained in the sense that the designs minimize the maximum expected
length of the $h$ confidence intervals. The approach leads to certain
connections with the optimization problem under the fixed effects
model.