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Uncapacitated lot sizing with backlogging: the convex hull

Abstract

An explicit description of the convex hull of solutions to the uncapacitated lot-sizing problem with backlogging, in its natural space of production, setup, inventory and backlogging variables, has been an open question for many years. In this paper, we identify valid inequalities that subsume all previously known valid inequalities for this problem. We show that these inequalities are enough to describe the convex hull of solutions. We give polynomial separation algorithms for some special cases. Finally, we report a summary of computational experiments with our inequalities that illustrates their effectiveness.

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References

  1. Agra A. and Constantino M. (1999). Lotsizing with backlogging and start-ups: The case of Wagner–Whitin costs. Oper. Res. Lett. 25: 81–88

    MATH  Article  MathSciNet  Google Scholar 

  2. Ahuja R.K., Magnanti T.L. and Orlin J.B. (1993). Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs

    Google Scholar 

  3. Constantino M. (2000). A polyhedral approach to a production planning problem. Ann. Oper. Res. 96: 75–95

    MATH  Article  MathSciNet  Google Scholar 

  4. Federgruen A. and Tzur M. (1993). The dynamic lot-sizing model with backlogging: A simple O(n log n) algorithm and minimal forecast horizon procedure. Naval Res. Logitics 40: 459–478

    MATH  Article  MathSciNet  Google Scholar 

  5. Guan Y., Ahmed S., Nemhauser G.L. and Miller A.J. (2006). A branch-and-cut algorithm for the stochastic uncapacitated lot-sizing problem. Math. Program. 105(1): 55–84

    MATH  Article  MathSciNet  Google Scholar 

  6. Ortega F. and Wolsey L.A. (2003). A branch-and-cut algorithm for the single-commodity, uncapacitated, fixed-charge network flow problem. Networks 41(3): 143–158

    MATH  Article  MathSciNet  Google Scholar 

  7. Pochet Y. and Wolsey L. (2006). Production Planning by Mixed Integer Programming. Springer, Heidelberg

    MATH  Google Scholar 

  8. Pochet Y. and Wolsey L.A. (1988). Lot-size models with backlogging: Strong reformulations and cutting planes. Math. Program. 40: 317–335

    MATH  Article  MathSciNet  Google Scholar 

  9. Pochet Y. and Wolsey L.A. (1991). Solving multi-item lot-sizing problems using strong cutting planes. Manage. Sci. 37: 53–67

    MATH  Article  Google Scholar 

  10. Pochet Y. and Wolsey L.A. (1994). Polyhedra for lot-sizing with Wagner–Whitin costs. Math. Program. 67: 297–323

    Article  MathSciNet  Google Scholar 

  11. Van Roy T.J. and Wolsey L.A. (1985). Valid inequalities and separation for uncapacitated fixed charge networks. Oper. Res. Lett. 4(3): 105–112

    MATH  Article  Google Scholar 

  12. Van Vyve M. (2006). Linear-programming extended formulations for the single-item lot-sizing problem with backlogging and constant capacity. Math. Program. 108(1): 53–77

    MATH  Article  MathSciNet  Google Scholar 

  13. Van Vyve M. and Wolsey L.A. (2006). Approximate extended formulations. Math. Program. 105(2–3): 501–522

    MATH  Article  MathSciNet  Google Scholar 

  14. Wolsey L.A. (2002). Solving multi-item lot-sizing problems with an MIP solver using classification and reformulation. Manage. Sci. 48(12): 1587–1602

    Article  Google Scholar 

  15. Zangwill W.I. (1966). A deterministic multi-period production scheduling model with backlogging. Manage. Sci. 13(1): 105–119

    Article  Google Scholar 

  16. Zangwill W.I. (1969). A backlogging model and a multi-echelon model of a dynamic economic lot size production system—A network approach. Manage. Sci. 15(9): 506–527

    MATH  Article  Google Scholar 

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Correspondence to Simge Küçükyavuz.

Additional information

The first author gratefully acknowledges partial financial support by a contract F49620-03-1-0477 from the AFOSR/MURI to the Department of Systems and Industrial Engineering and the Department of Management and Policy at the University of Arizona.

The work of Yves Pochet was partly carried out within the framework of ADONET, a European network in Algorithmic Discrete Optimization, contract no. MRTN-CT-2003-504438, and the text presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. The scientific responsibility is assumed by the authors.

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Küçükyavuz, S., Pochet, Y. Uncapacitated lot sizing with backlogging: the convex hull. Math. Program. 118, 151–175 (2009). https://doi.org/10.1007/s10107-007-0186-5

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  • DOI: https://doi.org/10.1007/s10107-007-0186-5

Keywords

  • Lot sizing
  • Backlogging
  • Convex hull
  • Separation algorithms
  • Computation

Mathematics Subject Classification (2000)

  • 90C11
  • 90C57