# How to solve min cost perfect matching problems?

I'm trying to design an algorithm for the following generalized assignment problem. We converted the problem to a weighted bipartite graph constituted of two sets $$A$$ and $$B$$ where $$|A| \ne |B|$$. Any edge in $$A$$ has a size and an edge in $$B$$ has a capacity which means an edge in $$B$$ can accept more input vertices up to its maximum capacity. $$C$$ is the cost vector; $$C_{ij}$$ is the cost of a vertex from edge $$i$$ in $$A$$ to edge $$j$$ in $$B$$. The target is a minimum cost perfect matching. That is all edges in $$A$$ require matching with an edge in $$B$$ as long as the sum of vertices coming to an edge in $$B$$ does not exceed its capacity.

As the solution, I see min-cost flow networks can be solved by converting to a Ford-Fulkerson problem, but I see such problems are normally seeking for max flow targets, not min cost. Should I modify the Ford-Fulkerson to follow my problem and how? I am happy to hear about alternative solutions as well.