On the ramified optimal allocation problem

This paper proposes an optimal allocation problem with ramified transport technology in a spatial economy. Ramified transportation is used to model the transport economy of scale in group transportation observed widely in both nature and efficiently designed transport systems of branching structures. The ramified allocation problem aims at finding an optimal allocation plan as well as an associated optimal allocation path to minimize overall cost of transporting commodity from factories to households. This problem differentiates itself from existing ramified transportation literature in that the distribution of production among factories is not fixed but endogenously determined as observed in many allocation practices. It's shown that due to the transport economy of scale in ramified transportation, each optimal allocation plan corresponds equivalently to an optimal assignment map from households to factories. This optimal assignment map provides a natural partition of both households and allocation paths. We develop methods of marginal transportation analysis and projectional analysis to study properties of optimal assignment maps. These properties are then related to the search for an optimal assignment map in the context of state matrix.