  We introduce and investigate properties of a variant of the semi-discrete
optimal transport problem. In this problem, one is given an absolutely
continuous source measure and cost function, along with a finite set which will
be the support of the target measure, and a "storage fee" function. The goal is
then to find a map for which the total transport cost plus the storage fee
evaluated on the masses of the pushforward of the source measure is minimized.
We prove existence and uniqueness for the problem, derive a dual problem for
which strong duality holds, and give a characterization of dual maximizers and
primal minimizers. Additionally, we find some stability results for minimizers.
