Two classical transportation problems revisited: Pure constant fixed charges and the paradox

We analyze degeneracy characterizations for two classical problems: the transportation paradox in linear transportation problems and the pure constant fixed charge transportation problem. Solving the pure constant fixed charge problem is equivalent to finding a basic tree solution with maximum degree of degeneracy. Problems possess degenerate solutions if the equal subsum property is satisfied for the supplies and demands. Determining the existence of degeneracy is an NP-complete problem. But this NP-hardness remains even if all equal subsums are known in advance. For the second problem, the transportation paradox, there exists a vast literature that typically describes methods, derived within the framework of the classical transportation algorithm, for determining solutions where the more-for-less phenomenon occurs. We show how to solve this problem as a simple standard network flow problem. The paradox is linked to overshipment solutions, which belong to supply and demand configurations that tend to have a high degree of degeneracy.