Mixing polyhedra with two non divisible coefficients

Abstract

We consider the mixed-integer set ${X=\{(s,x,y) \in \mathbb{R}\times \mathbb{Z}^n \times \mathbb{Z}^m: s + a_1 x_j \geq b_j, \forall j \in N_1, s + a_2 y_j \geq d_j, j \in N_2\}}$ where ${N_1 = \{1,\ldots,n\},\; N_2 = \{1,\ldots,m\}}$ and ${a_1, a_2 \in\mathbb{Z}_+\setminus\{0\}}$. This set may arise in a relaxation of mixed-integer problems such as lot-sizing problems. We decompose X into a small number of subsets whose convex hull description is trivial. The convex hull of X is equal to the closure of the convex hull of the union of those polyhedra. Using a projection theorem from Balas (Discret Appl Math 89:3–44, 1998) we obtain a compact characterization of the facets of the convex hull of X. Then by studying the projection cone we characterize all the facet-defining inequalities of the convex hull of X in the space of the original variables. Each of those inequalities is either a mixed MIR inequality (Günlük and Pochet in Math Programm 90:429–457, 2001), or it is based on a directed cycle on a special bipartite graph. When a ₁ and a ₂ are relative prime, the convex hull of X is described by the mixed MIR inequalities.