OR Spectrum

, Volume 37, Issue 4, pp 983–1006 | Cite as

Models and Lagrangian heuristics for a two-level lot-sizing problem with bounded inventory

  • Nadjib Brahimi
  • Nabil Absi
  • Stéphane Dauzère-Pérès
  • Safia Kedad-Sidhoum
Regular Article


We consider a two-level dynamic lot-sizing problem where the first level consists of N finished products competing for a single type of purchased raw material in the second level. While the procurement and production capacities are unlimited, the storage capacity of the raw material is limited and must be carefully managed. The goal is to simultaneously determine a replenishment plan for the raw material and optimal production plans for the finished products on a horizon of T periods while minimizing production, purchasing, setup and inventory holding costs. The problem is modeled using mixed-integer linear programs and solved using both a Lagrangian relaxation-based heuristic and a commercial mixed-integer linear programming solver. Learning capabilities are integrated in the Lagrangian relaxation to update step size in the subgradient algorithm. The computational results show that the Lagrangian heuristic outperforms the solver on different formulations, in particular for large problems with long time horizons.


Production planning Lot sizing Lagrangian relaxation Multi-level Bounded inventory 



The authors would like to thank anonymous referees for their careful reading, thorough reviews and valuable comments that helped us improving the quality of the paper. The first author would like to express his gratitude to the University of Sharjah (U.A.E.) for its financial support while the author was a faculty member at the department of Industrial Engineering and Management.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Nadjib Brahimi
    • 1
  • Nabil Absi
    • 2
  • Stéphane Dauzère-Pérès
    • 2
  • Safia Kedad-Sidhoum
    • 3
  1. 1.Département d’Automatique, Productique et InformatiqueEcole des Mines de Nantes, IRCCyN (CNRS UMR 6597)Nantes Cedex 3France
  2. 2.Department of Manufacturing Sciences and LogisticsEcole des Mines de Saint-Etienne (CNRS UMR 6158 LIMOS)GardanneFrance
  3. 3.LIP6 - UPMCSorbonne University, Univ. Paris 06 (CNRS UMR 7606)ParisFrance

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