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

  • Manish Bansal
  • Kiavash Kianfar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8494)

Abstract

In this paper, we introduce a generalization of the well-known continuous mixing set (which we refer to as the continuous n-mixing set) \(Q^{m,n}:=\{ (y, v,s) \in ( {\mathbb{Z}} \times{\mathbb{Z}}_{+}^{n-1})^{m} \times{\mathbb R}^{m+1}_{+}: \sum_{t=1}^n{\alpha_t y_t^i} + v_i + s\geq{\beta}_i, i=1,\ldots,m\}\). 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 the n′-step cycle inequalities, and show that they are facet-defining for conv(Q m,n ) in many cases. We also present a compact extended formulation for Q m,n and an exact separation algorithm for the n′-step cycle inequalities. We then use these inequalities to generate valid inequalities for the MML problem with(out) backlogging. Our computational results show that our cuts are very effective in solving the MML instances with backlogging, resulting in substantial reduction in the integrality gap, number of nodes, and total solution time.

Keywords

n-step cycle inequalities n-step MIR continuous n-mixing multi-module capacitated lot-sizing with backlogging 

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References

  1. 1.
    Atamtürk, A., Günlük, O.: Mingling: mixed-integer rounding with bounds. Mathematical Programming 123(2), 315–338 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Atamtürk, A., Kianfar, K.: n-step mingling inequalities: new facets for the mixed-integer knapsack set. Mathematical Programming 132(1), 79–98 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Cherkassky, B.V., Goldberg, A.V.: Negative-cycle detection algorithms. Mathematical Programming 85(2), 277–311 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Cormen, T.H., Stein, C., Rivest, R.L., Leiserson, C.E.: Introduction to Algorithms, 3rd edn. McGraw-Hill Higher Education (2009)Google Scholar
  5. 5.
    Dash, S., Günlük, O.: Valid inequalities based on simple mixed-integer sets. Mathematical Programming 105(1), 29–53 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Günlük, O., Pochet, Y.: Mixing mixed-integer inequalities. Mathematical Programming 90(3), 429–457 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Kianfar, K., Fathi, Y.: Generalized mixed integer rounding inequalities: facets for infinite group polyhedra. Mathematical Programming 120(2), 313–346 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Nemhauser, G.L., Wolsey, L.A.: A recursive procedure to generate all cuts for 0–1 mixed integer programs. Mathematical Programming 46(1-3), 379–390 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Pochet, Y., Wolsey, L.A.: Lot-Sizing with constant batches: Formulation and valid inequalities. Mathematics of Operations Research 18, 767–785 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Ralph, E.: Gomory: An algorithm for the mixed integer problem. Tech. Rep. RM-2597, RAND Corporation (1960)Google Scholar
  11. 11.
    Sanjeevi, S., Kianfar, K.: Mixed n-step MIR inequalities: Facets for the n-mixing set. Discrete Optimization 9(4), 216–235 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Shigeno, M., Iwata, S., McCormick, S.T.: Relaxed most negative cycle and most positive cut canceling algorithms for minimum cost flow. Mathematics of Operations Research 25(1), 76–104 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Tarjan, R.E.: Data structures and network algorithms. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (1983)Google Scholar
  14. 14.
    Vyve, M.V.: The continuous mixing polyhedron. Mathematics of Operations Research 30(2), 441–452 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Wolsey, L.A.: Integer Programming. Wiley, New York (1998)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Manish Bansal
    • 1
  • Kiavash Kianfar
    • 1
  1. 1.Department of Industrial and Systems EngineeringTexas A&M UniversityCollege StationUSA

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