The similar problem of finding an orientation to minimize the *maximum* outdegree is discussed [here](https://mathoverflow.net/questions/56891/unidentified-combinatorial-problem). You can solve your range minimization problem via integer linear programming as follows. For each undirected edge $(i,j)$, let binary decision variable $x_{i,j}$ indicate whether direction $i\to j$ is chosen ($1$) or direction $j\to i$ is chosen ($0$). Let decision variables $M$ and $m$ represent the maximum and minimum outdegrees, respectively. The problem is to minimize $M-m$ subject to
\begin{align}
\delta^+(i)&=\sum_{\substack{j\in V:\\(i,j)\in E \lor (j,i)\in E}} x_{i,j} &&\text{for $i\in V$}\\
M &\ge \delta^+(i) &&\text{for $i\in V$}\\
m &\le \delta^+(i) &&\text{for $i\in V$}
\end{align}