N-step Mingling Inequalities: New Facets for the Mixed-integer Knapsack Set

The n-step mixed integer rounding (MIR) inequalities of Kianfar and Fathi (Math Program 120(2):313–346, 2009) are valid inequalities for the mixedinteger knapsack set that are derived by using periodic n-step MIR functions and define facets for group problems. The mingling and 2-step mingling inequalities of Atamtürk and Günlük (Math Program 123(2):315–338, 2010) are also derived based on MIR and they incorporate upper bounds on the integer variables. It has been shown that these inequalities are facet-defining for the mixed integer knapsack set under certain conditions and generalize several well-known valid inequalities. In this paper, we introduce new classes of valid inequalities for the mixed-integer knapsack set with bounded integer variables, which we call n-step mingling inequalities (for positive integer n). These inequalities incorporate upper bounds on integer variables into nstep MIR and, therefore, unify the concepts of n-step MIR and mingling in a single class of inequalities. Furthermore, we show that for each n, the n-step mingling inequality defines a facet for the mixed integer knapsack set under certain conditions. For n = 2, we extend the results of Atamtürk and Günlük on facet-defining properties of 2-step mingling inequalities. For n ≥ 3, we present new facets for the mixed integer knapsack set. As a special case we also derive conditions under which the n-step MIR inequalities define facets for the mixed integer knapsack set. In addition, we show that A. Atamtürk Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA 94720-1777, USA e-mail: [email protected] K. Kianfar (B) Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX 77843-3131, USA e-mail: [email protected]