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Handbook of Stochastic Models and Analysis of Manufacturing System Operations pp 313–344Cite as

Stochastic Lot Sizing Problems

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Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 192))

Abstract

In this chapter dynamic lot sizing problems with random demands are discussed. Several approaches to handle uncertainty are presented. Single-item problems as well as multi-item lot sizing problems with limited capacities of a scarce resource are considered. Thereby the focus is on numerically tractable solution approaches.

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Notes

  1. 1.

    See [32].

  2. 2.

    This problem was first discussed by [34].

  3. 3.

    Note that several alternative model formulations are available. See [17].

  4. 4.

    Some model versions include also time-dependent variable production costs.

  5. 5.

    See [8].

  6. 6.

    See [2, 9, 31].

  7. 7.

    See [11].

  8. 8.

    See [5].

  9. 9.

    See [13, 20].

  10. 10.

    See [4], who also propose a heuristic solution procedure.

  11. 11.

    See [10].

  12. 12.

    See [5].

  13. 13.

    In case that the predetermined order-up-to level is smaller than the available inventory, the lot size is zero, in contrast to sending back the surplus stock to the supplier.

  14. 14.

    See [18].

  15. 15.

    See also [21].

  16. 16.

    See [33].

  17. 17.

    See [27].

  18. 18.

    See [11].

  19. 19.

    See [3, 5, 14, 23].

  20. 20.

    See [2, 9, 31].

  21. 21.

    See [25].

  22. 22.

    See [26, 29].

  23. 23.

    See [27, 29].

  24. 24.

    See [12].

  25. 25.

    See [24].

  26. 26.

    See [1].

  27. 27.

    See [24].

  28. 28.

    See [5, 23].

  29. 29.

    See [25].

  30. 30.

    See [27].

  31. 31.

    See [22, 35].

  32. 32.

    A literature overview over this type of models is given by [7].

  33. 33.

    See [11].

  34. 34.

    See [11].

  35. 35.

    See [28].

  36. 36.

    See [15].

  37. 37.

    See [16].

  38. 38.

    See [6].

  39. 39.

    See [26].

  40. 40.

    This means that the integrality requirements for the δ variables are omitted.

  41. 41.

    See [11].

  42. 42.

    See [19].

References

  1. Askin, R. (1981). A procedure for production lot sizing with probabilistic dynamic demand. AIIE Transactions, 13, 132–137.

    Article  Google Scholar 

  2. Banerjee, A., & Paul, A. (2005). Average fill rate and horizon length. Operations Research Letters, 33, 525–530.

    Article  Google Scholar 

  3. Bitran, G., & Yanasse, H. (1984). Deterministic approximations to stochastic production problems. Operations Research, 32, 999–1018.

    Article  Google Scholar 

  4. Bollapragada, S., & Morton, T. (1999). Simple heuristic for computing nonstationary (s, S) policies. Operations Research, 47, 576–584.

    Article  Google Scholar 

  5. Bookbinder, J., & Tan, J.-Y. (1988). Strategies for the probabilistic lot-sizing problem with service-level constraints. Management Science, 34, 1096–1108.

    Article  Google Scholar 

  6. Bradley, S., Hax, A., & Magnanti, T. (1977). Applied mathematical programming. Reading: Addison-Wesley.

    Google Scholar 

  7. Buschkühl, L., Sahling, F., Helber, S., & Tempelmeier, H. (2010). Dynamic capacitated lotsizing problems – a classification and review of solution approaches. OR Spectrum, 32(2), 231–261.

    Article  Google Scholar 

  8. Chen, F., & Krass, D. (2001). Inventory models with minimal service level constraints. European Journal of Operational Research, 134, 120–140.

    Article  Google Scholar 

  9. Chen, J., Lin, D., & Thomas, D. (2003). On the single item fill rate for a finite horizon. Operations Research Letters, 31, 119–123.

    Article  Google Scholar 

  10. Guan, Y., Ahmed, S., Miller, A., & Nemhauser, G. (2006). On formulations of the stochastic uncapacitated lot-sizing problem. Operations Research Letters, 34, 241–250.

    Article  Google Scholar 

  11. Helber, S., Sahling, F., & Schimmelpfeng, K. (2012). Dynamic capacitated lot sizing with random demand and dynamic safety stocks. OR Spectrum, 38, 75–105.

    Google Scholar 

  12. Herpers, S. (2009). Dynamische Losgrößenplanung bei stochastischer Nachfrage. Köln: Kölner Wissenschaftsverlag.

    Google Scholar 

  13. Iglehart, D. (1963). Dynamic programming and stationary analysis of inventory problems. In H. Scarf, D. Gilford, & M. Shelley (Eds.), Multistage inventory models and techniques. Stanford: Stanford University Press.

    Google Scholar 

  14. Lasserre, J., Bes, C., & Roubellat, F. (1985). The stochastic discrete dynamic lot size problem: An open-loop solution. Operations Research, 33, 684–689.

    Article  Google Scholar 

  15. Maes, J., & Van Wassenhove, L. (1986). A simple heuristic for the multi item single level capacitated lotsizing problem. OR Letters, 4, 265–273.

    Google Scholar 

  16. Manne, A. (1958). Programming of economic lot sizes. Management Science, 4, 115–135.

    Article  Google Scholar 

  17. Pochet, Y., & Wolsey, L. (2006). Production planning using mixed-integer programming. New York: Springer.

    Google Scholar 

  18. Rossi, R., Tarim, S., Hnich, B., & Prestwich, S. (2008). A global chance-constraint for stochastic inventory systems under service levels constraints. Constraints, 13, 490–517.

    Article  Google Scholar 

  19. Sahling, F., Buschkühl, L., Helber, S., & Tempelmeier, H. (2009). Solving a multi-level capacitated lot sizing problem with multi-period setup carry-over via a fix-and-optimize heuristic. Computers & Operations Research, 36, 2546–2553.

    Article  Google Scholar 

  20. Scarf, H. (1959). The optimality of (S, s) policies in the dynamic inventory problem. In K. Arrow, S. Karlin, & P. Suppes (Eds.), Mathematical methods in the social sciences (pp. 196–202). Stanford: Stanford University Press.

    Google Scholar 

  21. Sox, C. (1997). Dynamic lot sizing with random demand and non-stationary costs. Operations Research Letters, 20, 155–164.

    Article  Google Scholar 

  22. Sox, C., Jackson, P., Bowman, A., & Muckstadt, J. (1999). A review of the stochastic lot scheduling problem. International Journal of Production Economics, 62, 181–200.

    Article  Google Scholar 

  23. Tarim, S., & Kingsman, B. (2004). The stochastic dynamic production/inventory lot-sizing problem with service-level constraints. International Journal of Production Economics, 88, 105–119.

    Article  Google Scholar 

  24. Tarim, S., & Kingsman, B. (2006). Modelling and computing \(({R}^{n},{S}^{n})\) policies for inventory systems with non-stationary stochastic demands. European Journal of Operational Research, 174, 581–599.

    Article  Google Scholar 

  25. Tempelmeier, H. (2007). On the stochastic uncapacitated dynamic single-item lotsizing problem with service level constraints. European Journal of Operational Research, 181, 184–194.

    Article  Google Scholar 

  26. Tempelmeier, H. (2011). A column generation heuristic for dynamic capacitated lot sizing with random demand under a fill rate constraint. Omega, 39, 627–633.

    Article  Google Scholar 

  27. Tempelmeier, H. (2011). Inventory management in supply networks – problems, models, solutions (2nd ed.). Norderstedt: Books on Demand.

    Google Scholar 

  28. Tempelmeier, H., & Herpers, S. (2010). ABC β - a heuristic for dynamic capacitated lot sizing with random demand under a fill rate constraint. International Journal of Production Research, 48, 5181–5193.

    Article  Google Scholar 

  29. Tempelmeier, H., & Herpers, S. (2011). Dynamic uncapacitated lot sizing with random demand under a fillrate constraint. European Journal of Operational Research, 212(3), 497–507.

    Article  Google Scholar 

  30. Tempelmeier, H., & Hilger, T. (2013). Linear programming models for the stochastic dynamic capacitated lot sizing problem. Technical report, University of Cologne, Department of Supply Chain Management and Production, Cologne, Germany.

    Google Scholar 

  31. Thomas, D. (2005). Measuring item fill-rate performance in a finite horizon. Manufacturing & Service Operations Management, 7, 74–80.

    Article  Google Scholar 

  32. Tunc, H., Kilic, O., Tarim, S. A., & Eksioglu, B. (2011). The cost of using stationary inventory policies when demand is non-stationary. Omega, 39, 410–415.

    Article  Google Scholar 

  33. Vargas, V. (2009). An optimal solution for the stochastic version of the Wagner-Whitin dynamic lot-size model. European Journal of Operational Research, 198, 447–451.

    Article  Google Scholar 

  34. Wagner, H., & Whitin, T. (1958). Dynamic version of the economic lot size model. Management Science, 5, 89–96.

    Article  Google Scholar 

  35. Winands, E., Adan, I., & van Houtum, G. (2011). The stochastic economic lot scheduling problem: A survey. European Journal of Operational Research, 210, 1–9.

    Article  Google Scholar 

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Acknowledgements

The author is indebted to Timo Hilger, who cooperated in the preparation of the numerical experiments performed during the development and analysis of the different optimization models.

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  1. Department of Supply Chain Management and Production, University of Cologne, Cologne, Germany

    Horst Tempelmeier

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  1. Horst Tempelmeier
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Correspondence to Horst Tempelmeier .

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  1. , MIE Department, University of Massachusetts Amherst, Governors Drive 160, Amherst, 01003, Massachusetts, USA

    J. MacGregor Smith

  2. College of Admin. Sciences & Econ., Koç University, Rumelifeneri Yolu, Sarıyer, 34450, Istanbul, Turkey

    Barış Tan

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Tempelmeier, H. (2013). Stochastic Lot Sizing Problems. In: Smith, J., Tan, B. (eds) Handbook of Stochastic Models and Analysis of Manufacturing System Operations. International Series in Operations Research & Management Science, vol 192. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6777-9_10

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