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OR Spectrum

, Volume 35, Issue 1, pp 75–105 | Cite as

Dynamic capacitated lot sizing with random demand and dynamic safety stocks

  • Stefan Helber
  • Florian Sahling
  • Katja Schimmelpfeng
Regular Article

Abstract

We present a stochastic version of the single-level, multi-product dynamic lot-sizing problem subject to a capacity constraint. A production schedule has to be determined for random demand so that expected costs are minimized and a constraint based on a new backlog-oriented δ-service-level measure is met. This leads to a non-linear model that is approximated by two different linear models. In the first approximation, a scenario approach based on the random samples is used. In the second approximation model, the expected values of physical inventory and backlog as functions of the cumulated production are approximated by piecewise linear functions. Both models can be solved to determine efficient, robust and stable production schedules in the presence of uncertain and dynamic demand. They lead to dynamic safety stocks that are endogenously coordinated with the production quantities. A numerical analysis based on a set of (artificial) problem instances is used to evaluate the relative performance of the two different approximation approaches. We furthermore show under which conditions precise demand forecasts are particularly useful from a production–scheduling perspective.

Keywords

Stochastic demand Lot sizing Service level Dynamic safety stocks 

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References

  1. Absi N, Kedad-Sidhoum S (2009) The multi-item capacitated lot-sizing problem with safety stocks and demand shortage costs. Comput Oper Res 36(11): 2926–2936CrossRefGoogle Scholar
  2. Bihlmaier R, Koberstein A, Obst R (2009) Modeling and optimizing of strategic and tactical production planning in the automotive industry under uncertainty. OR Spectr 31(2): 311–336CrossRefGoogle Scholar
  3. Bookbinder JH, Tan JY (1988) Strategies for the probabilistic lot-sizing problem with service-level constraints. Manage Sci 34(9): 1096–1108CrossRefGoogle Scholar
  4. Brandimarte P (2006) Multi-item capacitated lot-sizing with demand uncertainty. Int J Prod Res 44(15): 2997–3022CrossRefGoogle Scholar
  5. Buschkühl L, Sahling F, Helber S, Tempelmeier H (2010) Dynamic capacitated lot-sizing problems: a classification and review of solution approaches. OR Spectr 32(2): 231–261CrossRefGoogle Scholar
  6. Di Summa M, Wolsey LA (2008) Lot-sizing on a tree. Oper Res Lett 36: 7–13CrossRefGoogle Scholar
  7. Fleischmann B (2003) Bestandsmanagement zwischen zero stock und inventory control. OR News (19):22–27Google Scholar
  8. Florian M, Lenstra JK, Kan AHGR (1980) Deterministic production planning: algorithms and complexity. Manage Sci 26: 669–679CrossRefGoogle Scholar
  9. Freimer MB, Linderoth JT, Thomas DJ (2010) The impact of sampling methods on bias and variance in stochastic linear programs. Comput Optim Appl. doi: 10.1007/s10589-010-9322-x
  10. Guan Y, Ahmed S, Nemhauser GL, Miller AJ (2006) A branch-and-cut algorithm for the stochastic uncapacitated lot-sizing problem. Math Program Ser A 105: 55–84CrossRefGoogle Scholar
  11. Guan Y, Miller AJ (2008) Polynomial-time algorithms for stochastic uncapacitated lot-sizing problems. Oper Res 56(5): 1172–1183CrossRefGoogle Scholar
  12. Helber S, Henken K (2010) Profit-oriented shift scheduling of inbound contact centers with skills-based routing, impatient customers, and retrials. OR Spectr 32(1): 109–134CrossRefGoogle Scholar
  13. Helber S, Sahling F (2010) A fix-and-optimize approach for the multi-level capacitated lot sizing problem. Int J Prod Econ 123(2): 247–256CrossRefGoogle Scholar
  14. Jans R, Degraeve Z (2008) Modeling industrial lot sizing problems: a review. Int J Prod Res 46(6): 1619–1643CrossRefGoogle Scholar
  15. Kanet JJ, Gorman MF, Stoesslein M (2010) Dynamic planned safety stocks in supply networks. Int J Prod Res 48(22): 6859–6880CrossRefGoogle Scholar
  16. Karimi B, Fatemi Ghomi SMT, Wilson JM (2003) The capacitated lot sizing problem: a review of models and algorithms. Omega 31(5): 365–378CrossRefGoogle Scholar
  17. Maes J, van Wassenhove LN (1986) A simple heuristic for the multi item single level capacitated lotsizing problem. Oper Res Lett 4(6): 265–273CrossRefGoogle Scholar
  18. Martel A, Diaby M, Boctor F (1995) Multiple items procurement under stochastic nonstationary demands. Eur J Oper Res 87(1): 74–92CrossRefGoogle Scholar
  19. Mißler-Behr M (1993) Methoden der Szenario-Analyse. Deutscher Universitätsverlag, WiesbadenGoogle Scholar
  20. Pochet Y, Wolsey LA (2006) Production planning by mixed integer programming. Springer, New YorkGoogle Scholar
  21. Robinson P, Narayanan A, Sahin F (2009) Coordinated deterministic dynamic demand lot-sizing problem: a review of models and algorithms. Omega 37(1): 3–15CrossRefGoogle Scholar
  22. Sahling F (2010) Mehrstufige Losgrößenplanung bei Kapazitätsrestriktionen. Gabler Research: Produktion und Logistik. Gabler, WiesbadenCrossRefGoogle Scholar
  23. Sahling F, Buschkühl L, Tempelmeier H, Helber S (2009) Solving a multi-level capacitated lot sizing problem with multi-period setup carry-over via a fix-and-optimize heuristic. Comput Oper Res 36(9): 2546–2553CrossRefGoogle Scholar
  24. Saliby E (1990) Descriptive sampling: a better approach to monte carlo simulation. J Oper Res Soc 41: 1133–1142Google Scholar
  25. Sox CR, Jackson PL, Bowman A, Muckstadt JA (1999) A review of the stochastic lot scheduling problem. Int J Prod Econ 62(3): 181–200CrossRefGoogle Scholar
  26. Sox CR, Muckstadt JA (1997) Optimization-based planning for the stochastic lot-scheduling problem. IIE Trans 29(5): 349–357Google Scholar
  27. Tempelmeier H (2006) Inventory management in supply networks: problems, models, solutions. Books on Demand, NorderstedtGoogle Scholar
  28. Tempelmeier H (2011) A column generation heuristic for dynamic capacitated lot sizing with random demand under a fill rate constraint. Omega 39(6): 627–633CrossRefGoogle Scholar
  29. Tempelmeier H, Herpers S (2010) ABC β—a heuristic for dynamic capacitated lot sizing with random demand under a fillrate constraint. Int J Prod Res 48(17): 5181–5193CrossRefGoogle Scholar
  30. Tunc H, Kilic OA, Tarim SA, Eksioglu B (2011) The cost of using stationary inventory policies when demand is non-stationary. Omega 39(4): 410–415CrossRefGoogle Scholar
  31. Winands EMM, Adan IJBF, van Houtum GJ (2011) The stochastic economic lot scheduling problem: a survey. Eur J Oper Res 210: 1–9CrossRefGoogle Scholar
  32. Zhao X, Xie J, Jiang Q (2001) Lot-sizing rule and freezing the master production schedule under capacity constraint and determinstic demand. Prod Oper Manage 10(1): 45–67CrossRefGoogle Scholar
  33. Zipkin PH (2000) Foundations of inventory management. McGraw-Hill, BostonGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Stefan Helber
    • 1
  • Florian Sahling
    • 1
  • Katja Schimmelpfeng
    • 2
  1. 1.Department of Production ManagementLeibniz UniversityHannoverGermany
  2. 2.Chair of Accounting and Control, Brandenburg University of TechnologyCottbusGermany

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