multi-commodity flow acyclic digraphs
The problem is at least as hard the maximum edge-disjoint paths problem (MEDP) since we can set $d(i) = 1$ for each pair. The approximability of MEDP has been investigated extensively and in particular it is known that MEDP in directed graphs is hard to approximate to within a factor of $m^{1/2-\epsilon}$ (technically it is $n^{1/2-\epsilon}$ since the graphs are sparse). This was shown in a paper of Guruswami etal. http://www.sciencedirect.com/science/article/pii/S0022000003000667. Roughly at the same time there was another paper by Ma and Wang established hardness of $2^{\log^{1-\epsilon} n}$ for acyclic graphs. More recent work by Chuzhoy etal showed hardness results even with congestion allowed; my understanding is that the hardness holds even for instances that are acyclic and these results are stronger than the one by Ma and Wang, however, one should look more carefully at the paper to make sure or ask the authors.