We study the polyhedral structure of simple mixed integer sets that generalize the two variable set {(s, z) ∈ ℝ1+ × ℤ1 : s ≥ z − b}. These sets form basic building blocks that can be used to derive tight formulations for more complicated mixed integer programs. For four such sets we give a complete description by valid inequalities and/or an integral extended formulation, and we also indicate what constraints can be added without destroying integrality. We apply these results to provide tight formulations for certain piecewise–linear convex objective integer programs, and in a companion paper we exploit them to provide polyhedral descriptions and computationally effective mixed integer programming formulations for discrete lot-sizing problems.
