OR Spectrum

, Volume 30, Issue 4, pp 773–785 | Cite as

Equivalence of the LP relaxations of two strong formulations for the capacitated lot-sizing problem with setup times

  • Meltem Denizel
  • F. Tevhide Altekin
  • Haldun Süral
  • Hartmut Stadtler
Regular Article


The multi-item Capacitated Lot-Sizing Problem (CLSP) has been widely studied in the literature due to its relevance to practice, such as its application in constructing a master production schedule. The problem becomes more realistic with the incorporation of setup times since they may use up significant amounts of the available resource capacity. In this paper, we present a proof to show the linear equivalence of the Shortest Path (SP) formulation and the Transportation Problem (TP) formulation for CLSP with setup costs and times. Our proof is based on a linear transformation from TP to SP and vice versa. In our proof, we explicitly consider the case when there is no demand for an item in a period, a case that is frequently observed in the real world and in test problems in the literature. The equivalence result in this paper has an impact on the choice of model formulation and the development of solution procedures.


Production Capacitated lot-sizing Strong formulation Linear relaxation 


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Meltem Denizel
    • 1
  • F. Tevhide Altekin
    • 1
  • Haldun Süral
    • 2
  • Hartmut Stadtler
    • 3
  1. 1.Faculty of ManagementSabancı UniversityTuzla, İstanbulTurkey
  2. 2.Industrial Engineering DepartmentMiddle East Technical UniversityAnkaraTurkey
  3. 3.Institute for Logistics and TransportUniversity of HamburgHamburgGermany

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