Network Flow - Simulating a network of water pipes

I'm trying to devise an algorithm that will simulate a network of pipes with multiple sources and multiple sinks of specific capacity.

So far I have tried using the classic Ford-Fulkerson algorithm but the issue i run into is this, given the following graph:

    S
    |
    a
   / \
  B   C

Given S with a source capacity of 1, and both B and C with a sink capacity of 1 - a flow will result S - a - B, saturating B to 1 and leaving C with flow 0.

I'm trying to distribute flow uniformly across the network, so that both B and C receive 0.5. Any ideas?

Thanks!

Say you have n sources s₁, ..., s_n with source capacities c_i and m sinks t₁, ..., t_m. Let f = sum_i c_i. We want to find a feasible flow in the network where every source i has a net flow of -c_i and every sink has a net flow of f / m.

We can solve this by introducing a super source S and a super sink T and connecting each of the sources i to S via an edge (s_i, S) of capacity c_i. We connect each t_i to T via an edge of capacity f / m. Then we just run max-flow with source S and sink T.

If it is not possible to push exactly f / m units of flow to each sink, it is not clear what you want to optimize, but you might find the following two approaches useful:

• Choose e and connect the sinks to T via edges with capacities f / m + e. Use binary search to find the minimum e such that the total flow will be f. This minimizes max_i inflow(t_i)
• Choose e and add the sinks to T via edges with lower bound e. Use binary search to find the maximum e that still allows for a feasible flow. This maximizes min_i inflow(t_i). The feasible flow problem with lower bounds can be reduced to max-flow: http://www.cs.uiuc.edu/~jeffe/teaching/algorithms/2009/notes/18-maxflowext.pdf In this case the construction is really simple: Just add an additional super-source S' and connect S' to S via an edge of capacity m * e. Connect all sinks to T via edges of capacity e. Check if the maximum flow between S' and T is m * e