  Let $\Gamma$ be a finite group acting linearly on $\C^n$, freely outside the
origin. In previous work a generalisation of Kronheimer's construction of
moduli of Hermitian-Yang-Mills bundles with certain invariance properties was
given. This produced varieties $X_\zeta$ (parameterised by $\zeta\in\Q^N$)
which are partial resolutions of $\C^n/\Gamma$. In this article, it is shown
the same $X_\zeta$ can be described as moduli spaces of representations of the
McKay quiver associated to the action of $\Gamma$. It it shown that, for
abelian groups, $X_\zeta$ are toric varieties defined by convex polyhedra which
are the solution sets for a generalisation of the transportation problem on the
McKay quiver. The generalised transportation problem is solved for an arbitrary
quiver to give a description of the extreme points, faces, and tangent cones to
the solution polyhedra in terms of certain distinguished trees in the quiver.
Applied the McKay quiver, this gives an explicit procedure for calculating
$X_\zeta $, its Euler number, and its singularities for any $\zeta$. The
$\zeta$-parameter-space is thus partitioned into a finite disjoint union of
cones inside which the biregular type of $X_\zeta$ remains constant. Finally,
the example $\C^3/\Z_5$ (weights $1,2,3$) is worked out in detail, and figures
of smooth and singular $X_\zeta$ and their corresponding flows are drawn. A
further example of smooth crepant resolution $X_\zeta$ is drawn for the
singularity ${1/11}(1,4,6)$.
