n-Step Cycle Inequalities: Facets for Continuous n-Mixing Set and Strong Cuts for Multi-Module Capacitated Lot-Sizing Problem

In this paper, we introduce a generalization of the continuous mixing set (which we refer to as the continuous n-mixing set) Qm,n := {(y, v, s) ∈ ( Z× Zn−1 + )m×Rm+1 + : ∑nt=1 αty t + vi + s ≥ βi, i = 1, . . . ,m}. This set is closely related to the feasible set of the multi-module capacitated lot-sizing (MML) problem with(out) backlogging. For each n′ ∈ {1, . . . , n}, we develop a class of valid inequalities for this set, referred to as n′-step cycle inequalities, and show that they are facetdefining for conv(Qm,n) in many cases. The cycle inequalities of Van Vyve (Math of OR 30:441-452, 2005), the n-step MIR inequalities of Kianfar and Fathi (Math Progam 120:313-346, 2009), and the mixed n-step MIR inequalities of Sanjeevi and Kianfar (Discrete Optimization 9:216-235, 2012) form special cases of the n-step cycle inequalities for Qm,n. We also present a compact extended formulation for Qm,n and an exact separation algorithm over the set of all n′-step cycle inequalities for any given n′ ∈ {1, . . . , n}. We then use these inequalities to generate valid inequalities for the MML problem with(out) backlogging, referred to as the n′-step (k, l, S, C) cycle inequalities for n′ ∈ {1, . . . , n}. Our computational results show that our cuts are very effective in solving the MML instances with(out) backlogging, resulting in substantial reduction in the integrality gap, number of nodes, and total solution time.