Packing Problems in Semigroup Programming

We define an optimization problem in the discrete semimodule over the natural numbers given by an ordered commutative semi-group and show that a canonical order induced in any semi-group by the right-hand-side element gives an ordered semi-group for which the optimization problem is equivalent to the (Equality) Semigroup Program, therefore the extension is consistent. Packing Programs correspond to positively ordered semigroups satisfying a self-positive condition, in these cases the semimodule is ordered. Packing Programs give a generalization of Integer Packing Programs. We show that the facets of the convex hull of solutions to a Packing Program are super-additives and we characterize the polars and neopolars of Master Packing Programs.