  Let $G$ and $H$ be graphs, and $G\boxtimes H$ the strong product of $G$ and
$H$. We prove that for any connected graphs $G$ and $H$ there is a strongly
connected orientation $D$ of $G\boxtimes H$ such that ${\rm diam}(D)\leq
2r+15$, where $r$ is the radius of $G\boxtimes H$.This improves the general
bound ${\rm diam}(D)\leq 2r^2+2r$ for arbitrary graphs, proved by Chv${\rm
\acute{a}}$tal and Thomassen.
