  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.
