Consider this problem:
$$ \begin{align} \min_{y,z,l \geq 0} \quad & g(y,z,l) := \sum_{(i,j)\in E} \sum_p (-w_{ijp}) y_{ijp} & \\ \textrm{s.t.} \quad & \left( \sum_{(i,j)\in E} y_{ijp} + l_{ip} \right)- \left( \sum_{(j,i)\in E} y_{jip} + z_{ip} \right)= \begin{cases} +1 \quad i=1 \\ -1 \quad i=n \\ 0 \qquad i\neq 1,n \end{cases} & \forall i \in I, p\in P \\ & \sum_p (z_{ip} - l_{ip}) = 0 & \forall i\in I \end{align} $$
Sets $I=\{1,\ldots,n\}$ and $P=\{1,\ldots,m\}$ correspond to $n$ nodes and $m$ commodities.
Set $E \subseteq I\times I$ corresponds to the arcs of a directed acyclic graph where node $1$ is the source (no predecessors) and node $n$ is the sink (no successors).
Sending $y_{ijp}$ units of commodity $p$ from node $i$ to $j$ costs $-w_{ijp}y_{ijp}$, with all $w_{ijp}$ positive.
Note that all commodities have $1$ as the source and $n$ as the sink.
If you ignore the $z_{ip}$ and $l_{ip}$ variables the problem becomes a set of $m$ independent shortest-path problems. Specifically, let $C = \{(y,z,l)\}$ denote the class of all solutions with $z_{ip} = l_{ip}$ for all $i, p$. Then, the best $(y,z,l) \in C$ can be found by solving $m$ independent shortest-path problems, i.e. by assigning $y_{ijp} \in \{0,1\}$.
It is this remark that makes me hope that we are dealing with a special-case of multicommodity flow that can be solved more efficiently. But I haven't found anything in the literature.
Does anyone recognize this problem or have an idea for a clever heuristic?