The first thing you need to do is find a feasible flow in the network
with upper and lower bounds on the edges. To handle the lower bounds on
the edges, you introduce additional sources and sinks. If an edge has lower
bound K, its neighboring vertices get connected to sources and sinks with
edges of capacity K. (I'm not going to spell out all the details).
Its also convenient to get rid of the orginal source and sink by adding
an infinite capacity edge from t back to s.
The second part is to convert the feasible flow to a minimum flow. Use the
residual graph. Find a maximum flow from the sink to the source.