Mathematical Programming

, Volume 154, Issue 1–2, pp 113–144 | Cite as

\(n\)-Step cycle inequalities: facets for continuous multi-mixing set and strong cuts for multi-module capacitated lot-sizing problem

  • Manish Bansal
  • Kiavash Kianfar
Full Length Paper Series B


In this paper, we introduce a generalization of the continuous mixing set, which we refer to as the continuous multi-mixing set. This set is closely related to the feasible set of the multi-module capacitated lot-sizing (MML) problem with(out) backlogging. We present a family of valid inequalities for the continuous multi-mixing set and identify conditions under which they are facet-defining. The cycle inequalities, \(n\)-step MIR inequalities, and mixed \(n\)-step MIR inequalities are special cases of these inequalities. We also present an exact separation algorithm for our inequalities. We then use these inequalities to generate valid inequalities for MML with(out) backlogging. Our computational results show that our cuts are very effective in solving the MML instances, resulting in substantial reduction in the integrality gap, number of nodes, and total solution time.


\(n\)-Step cycle inequalities \(n\)-Step MIR Continuous multi-mixing Multi-module capacitated lot-sizing with/without backlogging Mixed integer programming Cutting planes 

Mathematics Subject Classification

90C10 90C11 90C57 90C90 



This work was supported by the National Science Foundation Grants CMMI-1100343 and CMMI-1435526, which are gratefully acknowledged.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015

Authors and Affiliations

  1. 1.Department of Industrial and Systems EngineeringTexas A&M UniversityCollege StationUSA

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