  In this paper we study general transportation problems in $\mathbb{R}^n$, in
which $m$ different goods are moved simultaneously. The initial and final
positions of the goods are prescribed by measures $\mu^-$, $\mu^+$ on
$\mathbb{R}^n$ with values in $\mathbb{R}^m$. When the measures are finite
atomic, a discrete transportation network is a measure $T$ on $\mathbb{R}^n$
with values in $\mathbb{R}^{n\times m}$ represented by an oriented graph
$\mathcal{G}$ in $\mathbb{R}^n$ whose edges carry multiplicities in
$\mathbb{R}^m$. The constraint is encoded in the relation ${\rm
div}(T)=\mu^--\mu^+$. The cost of the discrete transportation $T$ is obtained
integrating on $\mathcal{G}$ a general function
$\mathcal{C}:\mathbb{R}^m\to\mathbb{R}$ of the multiplicity. When the initial
data $\left(\mu^-,\mu^+\right)$ are arbitrary (possibly diffuse) measures, the
cost of a transportation network between them is computed by relaxation of the
functional on graphs mentioned above. Our main result establishes the existence
of cost-minimizing transportation networks for arbitrary data
$\left(\mu^-,\mu^+\right)$. Furthermore, under additional assumptions on the
cost integrand $\mathcal{C}$, we prove the existence of transportation networks
with finite cost and the stability of the minimizers with respect to variations
of the given data. Finally, we provide an explicit integral representation
formula for the cost of rectifiable transportation networks, and we
characterize the costs such that every transportation network with finite cost
is rectifiable.
