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Relaxations for two-level multi-item lot-sizing problem

Author

Listed:
  • VAN VYVE, Mathieu

    (Université catholique de Louvain, CORE, B-1348 Louvain-la-Neuve, Belgium.)

  • WOLSEY, Laurence

    (Université catholique de Louvain, CORE, B-1348 Louvain-la-Neuve, Belgium.)

  • YAMAN, Hande

    (Université catholique de Louvain, CORE, B-1348 Louvain-la-Neuve, Belgium and Bilkent University)

Abstract

We consider several variants of the two-level lot-sizing problem with one item at the upper level facing dependent demand, and multiple items or clients at the lower level, facing independent demands. We first show that under a natural cost assumption, it is sufficient to optimize over a stock-dominant relaxation. We further study the polyhedral structure of a strong relaxation of this problem involving only initial inventory variables and setup variables. We consider several variants: uncapacitated at both levels with or without start-up costs, uncapacitated at the upper level and constant capacity at the lower level, constant capacity at both levels. We finally demonstrate how the strong formulations described improve our ability to solve instances with up to several dozens of periods and a few hundred products.

Suggested Citation

  • VAN VYVE, Mathieu & WOLSEY, Laurence & YAMAN, Hande, 2012. "Relaxations for two-level multi-item lot-sizing problem," LIDAM Discussion Papers CORE 2012031, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:2012031
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    References listed on IDEAS

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    1. POCHET, Yves & WOLSEY, Laurence A., 1988. "Lot-size models with backlogging: strong reformulations and cutting planes," LIDAM Reprints CORE 791, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. POCHET, Yves & WOLSEY, Laurence A., 2010. "Single item lot-sizing with non-decreasing capacities," LIDAM Reprints CORE 2145, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. VAN VYVE, Matthieu & WOLSEY, Laurence A., 2006. "Approximate extended formulations," LIDAM Reprints CORE 1807, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Yves Pochet & Laurence A. Wolsey, 1991. "Solving Multi-Item Lot-Sizing Problems Using Strong Cutting Planes," Management Science, INFORMS, vol. 37(1), pages 53-67, January.
    5. Harvey M. Wagner & Thomson M. Whitin, 1958. "Dynamic Version of the Economic Lot Size Model," Management Science, INFORMS, vol. 5(1), pages 89-96, October.
    6. Panayotis Afentakis & Bezalel Gavish, 1986. "Optimal Lot-Sizing Algorithms for Complex Product Structures," Operations Research, INFORMS, vol. 34(2), pages 237-249, April.
    7. 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.
    8. Willard I. Zangwill, 1969. "A Backlogging Model and a Multi-Echelon Model of a Dynamic Economic Lot Size Production System--A Network Approach," Management Science, INFORMS, vol. 15(9), pages 506-527, May.
    9. Yves Pochet & Laurence A. Wolsey, 1993. "Lot-Sizing with Constant Batches: Formulation and Valid Inequalities," Mathematics of Operations Research, INFORMS, vol. 18(4), pages 767-785, November.
    10. Stan van Hoesel & H. Edwin Romeijn & Dolores Romero Morales & Albert P. M. Wagelmans, 2005. "Integrated Lot Sizing in Serial Supply Chains with Production Capacities," Management Science, INFORMS, vol. 51(11), pages 1706-1719, November.
    11. POCHET, Yves & WOLSEY, Laurence A., 1993. "Lot-sizing with constant batches: formulation and valid inequalities," LIDAM Reprints CORE 1066, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    12. VERWEIJ, Bram & WOLSEY, Laurence A., 2004. "Uncapacitated lot-sizing with buying, sales and backlogging," LIDAM Reprints CORE 1704, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    13. Awi Federgruen & Michal Tzur, 1991. "A Simple Forward Algorithm to Solve General Dynamic Lot Sizing Models with n Periods in 0(n log n) or 0(n) Time," Management Science, INFORMS, vol. 37(8), pages 909-925, August.
    14. MELO, Rafael A. & WOLSEY, Laurence A., 2010. "Uncapacitated two-level lot-sizing," LIDAM Reprints CORE 2235, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    15. Alok Aggarwal & James K. Park, 1993. "Improved Algorithms for Economic Lot Size Problems," Operations Research, INFORMS, vol. 41(3), pages 549-571, June.
    16. van Hoesel, Stan & Wagelmans, Albert & Moerman, Bram, 1994. "Using geometric techniques to improve dynamic programming algorithms for the economic lot-sizing problem and extensions," European Journal of Operational Research, Elsevier, vol. 75(2), pages 312-331, June.
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    1. DUFAYS, Arnaud, 2012. "Infinite-state Markov-switching for dynamic volatility and correlation models," LIDAM Discussion Papers CORE 2012043, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).

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