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

Given two sets A={a_1,a_2,…,a_s} and {b_1,b_2,…,b_t}, a many-to-many matching with demands and capacities (MMDC) between A and B matches each element a_i in A to at least α_i and at most α’_i elements in B, and each element b_j in B to at least β_j 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.