  Motivated by an error-correcting generalization of Bachet's weights problem,
we define and classify relaxed complete partitions. We show that these
partitions enjoy a succinct description in terms of lattice points in
polyhedra, with adjustments in the error being commensurate with translations
in the defining hyperplanes. Our main result is that the enumeration of the
minimal such partitions (those with fewest possible parts) is achieved via
Brion's formula. This generalizes work of Park on classifying complete
partitions and that of R{\o}dseth on enumerating minimal complete partitions.
