I'm interested in a variant of the transportation problem and cannot find a reference for the problem I'm thinking of.
In the original Monge-Kantorovich problem about continuous transport, the total mass of $m$ objects are to be moved to the same mass of destinations. Assuming a quadratic euclidean distance cost, the solution is well known and has some nice properties.
What happens if the total masses of each side are different? e.g., if mass $1$ objects should be moved to total mass of $x$ destinations, $x\neq 1$? So if $x<1$ some of the objects cannot be moved because of this capacity constraint. On the other hand, if $x>1$, what would be the optimal transport? (assuming quadratic costs)
Anyone familiar with this variant? I can't find a reference that deals with this problem..
This variant of the optimal transport problem is called "optimal partial transport".
See here and here.