Total flow using minimum number of edges on a bipartite network
Your problem is NP-hard.
Consider a partition problem instance with input $a_1,\ldots,a_n$.
We create a complete bipartite graph with $2$ source vertices each with capacity $\frac{1}{2} \sum_{i=1}^n a_i$, and $n$ sink vertices, where the $i$th vertex has capacity $a_i$. Each edge has infinite capacity. 

It's easy to verify there is a partition with equal sum if and only if the solution to your problem on $G$ has $n$ edges.