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Lot-Sizing with Start-Up Times

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  • François Vanderbeck

    (Cambridge University, Engineering Department and Judge Institute of Management Studies, Trumpington St., Cambridge CB2 1AG, United Kingdom)

Abstract

Many practical applications of lot-sizing and scheduling problems involve start-up times. Operations research literature contains but few studies of lot-sizing models that take start-up times explicitly into account. Here, we review some of these studies, discuss the models and their complexity, and we propose further models. We consider in particular a single-stage single-mode multi-item lot-sizing model with continuous set-ups and sequence independent start-up times, which we solve using an integer programming column generation algorithm and we develop a dynamic programming procedure for the single-item subproblem that treats the initial stock as a decision variable. We also use cutting planes developed by Constantino for the multi-item polyhedra. By combining column and cut generation, the lower bounds that we obtain before branching are on average less than 2% from an optimal solution. Our algorithm solves instances with 3 to 5 items and 24 periods in an average of 50 seconds on a modern workstation, and problems with 36 periods in an average of 750 seconds. Solutions guaranteed to be within 2% of optimality are obtained in less than 75% of these times.

Suggested Citation

  • François Vanderbeck, 1998. "Lot-Sizing with Start-Up Times," Management Science, INFORMS, vol. 44(10), pages 1409-1425, October.
  • Handle: RePEc:inm:ormnsc:v:44:y:1998:i:10:p:1409-1425
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    File URL: http://dx.doi.org/10.1287/mnsc.44.10.1409
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    References listed on IDEAS

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    1. William W. Trigeiro & L. Joseph Thomas & John O. McClain, 1989. "Capacitated Lot Sizing with Setup Times," Management Science, INFORMS, vol. 35(3), pages 353-366, March.
    2. du Merle, O & Goffin, J-L & Trouiller, C & Vial, J-Ph, 1997. "A Lagrangian Relaxation of the Capacitated Multi-Item Lot Sizing Problem Solved with an Interior Point Cutting Plan Algorithm," Papers 97.11, Ecole des Hautes Etudes Commerciales, Universite de Geneve-.
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    4. Michael Florian & Morton Klein, 1971. "Deterministic Production Planning with Concave Costs and Capacity Constraints," Management Science, INFORMS, vol. 18(1), pages 12-20, September.
    5. van Hoesel, Stan & Kolen, Antoon, 1994. "A linear description of the discrete lot-sizing and scheduling problem," European Journal of Operational Research, Elsevier, vol. 75(2), pages 342-353, June.
    6. Fleischmann, Bernhard, 1990. "The discrete lot-sizing and scheduling problem," European Journal of Operational Research, Elsevier, vol. 44(3), pages 337-348, February.
    7. 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.
    8. Wolsey, Laurence A., 1997. "MIP modelling of changeovers in production planning and scheduling problems," European Journal of Operational Research, Elsevier, vol. 99(1), pages 154-165, May.
    9. 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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    Citations

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    Cited by:

    1. BELVAUX, Gaetan & WOLSEY, Laurence A., 2000. "Modelling practical lot-sizing problems as mixed integer programs," CORE Discussion Papers 2000009, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Jans, R.F. & Degraeve, Z., 2005. "Modeling Industrial Lot Sizing Problems: A Review," ERIM Report Series Research in Management ERS-2005-049-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.
    3. Raf Jans, 2009. "Solving Lot-Sizing Problems on Parallel Identical Machines Using Symmetry-Breaking Constraints," INFORMS Journal on Computing, INFORMS, vol. 21(1), pages 123-136, February.
    4. Silvio Alexandre de Araujo & Bert De Reyck & Zeger Degraeve & Ioannis Fragkos & Raf Jans, 2015. "Period Decompositions for the Capacitated Lot Sizing Problem with Setup Times," INFORMS Journal on Computing, INFORMS, vol. 27(3), pages 431-448, August.
    5. Karina Copil & Martin Wörbelauer & Herbert Meyr & Horst Tempelmeier, 2017. "Simultaneous lotsizing and scheduling problems: a classification and review of models," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 39(1), pages 1-64, January.
    6. Gaetan Belvaux & Laurence A. Wolsey, 2001. "Modelling Practical Lot-Sizing Problems as Mixed-Integer Programs," Management Science, INFORMS, vol. 47(7), pages 993-1007, July.
    7. Degraeve, Z. & Jans, R.F., 2003. "A New Dantzig-Wolfe Reformulation And Branch-And-Price Algorithm For The Capacitated Lot Sizing Problem With Set Up Times," ERIM Report Series Research in Management ERS-2003-010-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.
    8. Laurence A. Wolsey, 2002. "Solving Multi-Item Lot-Sizing Problems with an MIP Solver Using Classification and Reformulation," Management Science, INFORMS, vol. 48(12), pages 1587-1602, December.
    9. Ioannis Fragkos & Zeger Degraeve & Bert De Reyck, 2016. "A Horizon Decomposition Approach for the Capacitated Lot-Sizing Problem with Setup Times," INFORMS Journal on Computing, INFORMS, vol. 28(3), pages 465-482, August.
    10. Zeger Degraeve & Raf Jans, 2007. "A New Dantzig-Wolfe Reformulation and Branch-and-Price Algorithm for the Capacitated Lot-Sizing Problem with Setup Times," Operations Research, INFORMS, vol. 55(5), pages 909-920, October.
    11. Jans, Raf, 2010. "Classification of Dantzig-Wolfe reformulations for binary mixed integer programming problems," European Journal of Operational Research, Elsevier, vol. 204(2), pages 251-254, July.
    12. Mütze, Torsten, 2014. "Scheduling with few changes," European Journal of Operational Research, Elsevier, vol. 236(1), pages 37-50.
    13. 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.
    14. 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.

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