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An Algorithm for Single-Item Capacitated Economic Lot Sizing with Piecewise Linear Production Costs and General Holding Costs


  • Dong X. Shaw

    (School of Industrial Engineering, Purdue University, West Lafayette, Indiana 47907)

  • Albert P. M. Wagelmans

    (Econometric Institute, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands)


We consider the Capacitated Economic Lot Size Problem with piecewise linear production costs and general holding costs, which is an NP-hard problem but solvable in pseudo-polynomial time. A straightforward dynamic programming approach to this problem results in an O(n 2 c\bar d\bar ) algorithm, where n is the number of periods, and d\bar and c\bar are the average demand and the average production capacity over the n periods, respectively. However, we present a dynamic programming procedure with complexity O(n 2 q\bar d\bar ), where q\bar is the average number of pieces required to represent the production cost functions. In particular, this means that problems in which the production functions consist of a fixed set-up cost plus a linear variable cost are solved in O(n 2 d\bar ) time. Hence, the running time of our algorithm is only linearly dependent on the magnitude of the data. This result also holds if extensions such as backlogging and startup costs are considered. Moreover, computational experiments indicate that the algorithm is capable of solving quite large problem instances within a reasonable amount of time. For example, the average time needed to solve test instances with 96 periods, 8 pieces in every production cost function, and average demand of 100 units is approximately 40 seconds on a SUN SPARC 5 workstation.

Suggested Citation

  • Dong X. Shaw & Albert P. M. Wagelmans, 1998. "An Algorithm for Single-Item Capacitated Economic Lot Sizing with Piecewise Linear Production Costs and General Holding Costs," Management Science, INFORMS, vol. 44(6), pages 831-838, June.
  • Handle: RePEc:inm:ormnsc:v:44:y:1998:i:6:p:831-838

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    References listed on IDEAS

    1. Gabriel R. Bitran & Horacio H. Yanasse, 1982. "Computational Complexity of the Capacitated Lot Size Problem," Management Science, INFORMS, vol. 28(10), pages 1174-1186, October.
    2. M. Florian & J. K. Lenstra & A. H. G. Rinnooy Kan, 1980. "Deterministic Production Planning: Algorithms and Complexity," Management Science, INFORMS, vol. 26(7), pages 669-679, July.
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    Cited by:

    1. Akbalik, A. & Pochet, Y., 2009. "Valid inequalities for the single-item capacitated lot sizing problem with step-wise costs," European Journal of Operational Research, Elsevier, vol. 198(2), pages 412-434, October.
    2. Akbalik, Ayse & Rapine, Christophe, 2013. "The single item uncapacitated lot-sizing problem with time-dependent batch sizes: NP-hard and polynomial cases," European Journal of Operational Research, Elsevier, vol. 229(2), pages 353-363.
    3. Ou, Jinwen, 2017. "Improved exact algorithms to economic lot-sizing with piecewise linear production costs," European Journal of Operational Research, Elsevier, vol. 256(3), pages 777-784.
    4. van den Heuvel, W. & Wagelmans, A.P.M., 2003. "A Geometric Algorithm to solve the NI/G/NI/ND Capacitated Lot-Sizing Problem in O(T^2) Time," ERIM Report Series Research in Management ERS-2003-066-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    5. Awi Federgruen & Joern Meissner & Michal Tzur, 2007. "Progressive Interval Heuristics for Multi-Item Capacitated Lot-Sizing Problems," Operations Research, INFORMS, vol. 55(3), pages 490-502, June.
    6. Akbalik, Ayse & Penz, Bernard, 2009. "Exact methods for single-item capacitated lot sizing problem with alternative machines and piece-wise linear production costs," International Journal of Production Economics, Elsevier, vol. 119(2), pages 367-379, June.
    7. Akbalik, Ayse & Hadj-Alouane, Atidel B. & Sauer, Nathalie & Ghribi, Houcem, 2017. "NP-hard and polynomial cases for the single-item lot sizing problem with batch ordering under capacity reservation contract," European Journal of Operational Research, Elsevier, vol. 257(2), pages 483-493.
    8. Koca, Esra & Yaman, Hande & Selim Akt├╝rk, M., 2015. "Stochastic lot sizing problem with controllable processing times," Omega, Elsevier, vol. 53(C), pages 1-10.
    9. Archetti, Claudia & Bertazzi, Luca & Grazia Speranza, M., 2014. "Polynomial cases of the economic lot sizing problem with cost discounts," European Journal of Operational Research, Elsevier, vol. 237(2), pages 519-527.
    10. Vernon Ning Hsu, 2000. "Dynamic Economic Lot Size Model with Perishable Inventory," Management Science, INFORMS, vol. 46(8), pages 1159-1169, August.
    11. Liu, X. & Tu, Yl., 2008. "Production planning with limited inventory capacity and allowed stockout," International Journal of Production Economics, Elsevier, vol. 111(1), pages 180-191, January.
    12. Milind Dawande & Srinagesh Gavirneni & Yinping Mu & Suresh Sethi & Chelliah Sriskandarajah, 2010. "On the Interaction Between Demand Substitution and Production Changeovers," Manufacturing & Service Operations Management, INFORMS, vol. 12(4), pages 682-691, September.
    13. Li, Hongyan & Meissner, Joern, 2011. "Competition under capacitated dynamic lot-sizing with capacity acquisition," International Journal of Production Economics, Elsevier, vol. 131(2), pages 535-544, June.
    14. Chung-Lun Li & Qingying Li, 2016. "Polynomial-Time Solvability of Dynamic Lot Size Problems," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 33(03), pages 1-20, June.
    15. Fleischhacker, Adam J. & Zhao, Yao, 2011. "Planning for demand failure: A dynamic lot size model for clinical trial supply chains," European Journal of Operational Research, Elsevier, vol. 211(3), pages 496-506, June.
    16. Lee, C.Y. & Cetinkaya, S. & Wagelmans, A.P.M., 1999. "A dynamic lot-sizing model with demand time windows," Econometric Institute Research Papers EI 9948-/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    17. Jans, Raf & Degraeve, Zeger, 2007. "Meta-heuristics for dynamic lot sizing: A review and comparison of solution approaches," European Journal of Operational Research, Elsevier, vol. 177(3), pages 1855-1875, March.
    18. Brahimi, Nadjib & Dauzere-Peres, Stephane & Najid, Najib M. & Nordli, Atle, 2006. "Single item lot sizing problems," European Journal of Operational Research, Elsevier, vol. 168(1), pages 1-16, January.
    19. Karimi, B. & Fatemi Ghomi, S. M. T. & Wilson, J. M., 2003. "The capacitated lot sizing problem: a review of models and algorithms," Omega, Elsevier, vol. 31(5), pages 365-378, October.


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