Mixed n-step MIR inequalities: Facets for the n-mixing set

Günlük and Pochet [O. Günlük , Y. Pochet: Mixing mixed integer inequalities. Mathematical Programming 90(2001) 429-457] proposed a procedure to mix mixed integer rounding (MIR) inequalities. The mixed MIR inequalities define the convex hull of the mixing set {(y, . . . , y, v) ∈ Z × R+ : α1y + v ≥ βi, i = 1, . . . ,m} and can also be used to generate valid inequalities for general as well as several special mixed integer programs (MIPs). In another direction, Kianfar and Fathi [K. Kianfar, Y. Fathi: Generalized mixed integer rounding inequalities: facets for infinite group polyhedra. Mathematical Programming 120(2009) 313-346] introduced the n-step MIR inequalities for the mixed integer knapsack set through a generalization of MIR. In this paper, we generalize the mixing procedure to the n-step MIR inequalities and introduce the mixed n-step MIR inequalities. We prove that these inequalities define facets and high-dimensional faces for a generalization of the mixing set with n integer variables in each row (which we refer to as the n-mixing set), i.e. {(y, . . . , y, v) ∈ (Z × Zn−1 + ) × R+ : ∑n j=1 αjy i j + v ≥ βi, i = 1, . . . ,m}. The mixed MIR inequalities are simply the special case of n = 1. We then show that mixed n-step MIR can generate multi-row valid inequalities for general MIP and can be used to generalize well-known inequalities for capacitated lot-sizing and facility location problems to the multi-capacity case.