  A mixed multigraph is a multigraph which may contain both undirected and
directed edges. An orientation of a mixed multigraph $G$ is an assignment of
exactly one direction to each undirected edge of $G$. A mixed multigraph $G$
can be oriented to a strongly connected digraph if and only if $G$ is
bridgeless and strongly connected [Boesch and Tindell, Am. Math. Mon., 1980].
For each $r \in \mathbb{N}$, let $f(r)$ denote the smallest number such that
any strongly connected bridgeless mixed multigraph with radius $r$ can be
oriented to a digraph of radius at most $f(r)$. We improve the current best
upper bound of $4r^2+4r$ on $f(r)$ [Chung, Garey and Tarjan, Networks, 1985] to
$1.5 r^2 + r + 1$. Our upper bound is tight upto a multiplicative factor of
$1.5$ since, $\forall r \in \mathbb{N}$, there exists an undirected bridgeless
graph of radius $r$ such that every orientation of it has radius at least $r^2
+ r$ [Chv\'atal and Thomassen, J. Comb. Theory. Ser. B., 1978]. We prove a
marginally better lower bound, $f(r) \geq r^2 + 3r + 1$, for mixed multigraphs.
While this marginal improvement does not help with asymptotic estimates, it
clears a natural suspicion that, like undirected graphs, $f(r)$ may be equal to
$r^2 + r$ even for mixed multigraphs. En route, we show that if each edge of
$G$ lies in a cycle of length at most $\eta$, then the oriented radius of $G$
is at most $1.5 r \eta$. All our proofs are constructive and lend themselves to
polynomial time algorithms.
