Computing discrete optimal transport

I am trying to find a combinatorial approach to solve the following optimization problem.

begin{align}
&max_{x_{ij}} C_{ij} x_{ij}, \
&text{such that},\
&sum_{j} x_{ij} leq r_i~forall i in [N],\
&sum_{i} x_{ij} leq c_j~forall j in [M],\
&x_{ij} geq 0~forall i,j,
end{align}
where the constants satisfy: $C_{ij} geq 0$, $sum_i r_i = 1$.

If $sum_j c_j = 1$ then, I think, this problem is similar to the discrete (finite) optimal transport problem.

I am not interested in the most efficient solution approach. I am interested in an approach that reveals any interesting structure of a solution if such a structure exists.

In particular, is there a greedy algorithm (after sorting the weights $C_{ij}$) that solves this problem?