stable marriage/residency problem with multiple matches
Suppose you have a doctor that you want to assign to three hospitals.  Replace that doctor by three copies of him.  In other words, instead of a single vertex for Dr. Smith, you have three clones, Dr. Smith #1, Dr. Smith #2, and Dr. Smith #3.  They all have the same preference list.

Suppose you have a hospital that wants two doctors.  You can do the same thing: instead of a single hospital, you have one entity per open position at that hospital.  They all have the same preference rankings.

Now use the standard stable marriage algorithm on the resulting set of clones and open positions.  This will find a stable matching that obeys the multiplicity constraints you identified; that follows directly from the correctness property of the standard algorithm.