Editing Maximum flow problem (section)

===Circulation–demand problem===
There are some factories that produce goods and some villages where the goods have to be delivered. They are connected by a networks of roads with each road having a capacity {{mvar|c}} for maximum goods that can flow through it. The problem is to find if there is a circulation that satisfies the demand.
This problem can be transformed into a maximum-flow problem.
# Add a source node {{mvar|s}} and add edges from it to every factory node {{mvar|f<sub>i</sub>}} with capacity {{mvar|p<sub>i</sub>}} where {{mvar|p<sub>i</sub>}} is the production rate of factory {{mvar|f<sub>i</sub>}}.
# Add a sink node {{mvar|t}} and add edges from all villages {{mvar|v<sub>i</sub>}} to {{mvar|t}} with capacity {{mvar|d<sub>i</sub>}} where {{mvar|d<sub>i</sub>}} is the demand rate of village {{mvar|v<sub>i</sub>}}.
Let ''G'' = (''V'', ''E'') be this new network. There exists a circulation that satisfies the demand if and only if :
: {{math|Maximum flow value(''G'')}} <math> = \sum_{i \in v} d_i </math>.
If there exists a circulation, looking at the max-flow solution would give the answer as to how much goods have to be sent on a particular road for satisfying the demands.

The problem can be extended by adding a lower bound on the flow on some edges.<ref>{{Cite web|url=https://www.cs.cmu.edu/~ckingsf/bioinfo-lectures/flowext.pdf|title=Max-flow extensions: circulations with demands|last=Carl Kingsford}}</ref>