Computing a many-to-many matching with demands and capacities between two sets using the Hungarian algorithm

Given two sets A=a1,a2,...,as and b1,b2,...,bt, a many-to-many matching with demands and capacities (MMDC) between A and B matches each element ai in A to at least αi and at most α'i elements in B, and each element bj in B to at least βj and at most β'j elements in A for all 1=<i<=s and 1=<j<=t. In this paper, we present an algorithm for finding a minimum-cost MMDC between A and B using the well-known Hungarian algorithm.