I need to do a weighted matching between two sets (say students and professors). The set of students is much larger than set of professors. So multiple students can be matched to professors. However, each professor also has a minimum number of students he needs. So even though all students maybe matched best with one professor, some need to distributed amongst the rest, as long as they can be matched of course -- since not all professors and students will have edges between them.
Standard Bipartite matching won't work since they match one to one. Max Flow won't work since they don't consider weights.