Skip to main content

Advertisement

SpringerLink
Log in

New valid inequalities and formulations for the static joint Chance-constrained Lot-sizing problem

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

We study the static joint chance-constrained lot-sizing problem, in which production decisions over a planning horizon are made before knowing random future demands, and the inventory variables are then determined by the demand realizations. The joint chance constraint imposes a service level requirement that the probability that all demands are met on time be above a threshold. We model uncertain outcomes with a finite set of scenarios and begin by applying existing results about chance-constrained programming to obtain an initial extended mixed-integer programming formulation. We further strengthen this formulation with a new class of valid inequalities that generalizes the classical (\(\ell ,S\)) inequalities for the deterministic uncapacitated lot-sizing problem. In addition, we prove an optimality condition of the solutions under a modified Wagner-Whitin condition, and based on this derive a new extended mixed-integer programming formulation. This formulation is further extended to the case with constant capacities. We conduct a thorough computational study demonstrating the effectiveness of the new valid inequalities and extended formulation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Abdi, A., Fukasawa, R.: On the mixing set with a knapsack constraint. Math. Program. 157(1), 191–217 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahmed, S., Sahinidis, N.V.: An approximation scheme for stochastic integer programs arising in capacity expansion. Oper. Res. 51(3), 461–471 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ahmed, S., King, A.J., Parija, G.: A multi-stage stochastic integer programming approach for capacity expansion under uncertainty. J. Global Optim. 26(1), 3–24 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Atamtürk, A., Muñoz, J.C.: A study of the lot-sizing polytope. Math. Program. 99(3), 443–465 (2004). https://doi.org/10.1007/s10107-003-0465-8

    Article  MathSciNet  MATH  Google Scholar 

  5. Atamtürk, A., Nemhauser, G.L., Savelsbergh, M.W.: The mixed vertex packing problem. Math. Program. 89(1), 35–53 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barany, I., Van Roy, T., Wolsey, L.A.: Uncapacitated lot-sizing: The convex hull of solutions, pp. 32–43. Springer, Berlin Heidelberg, Berlin, Heidelberg (1984)

    MATH  Google Scholar 

  7. Beraldi, P., Ruszczyński, A.: A branch and bound method for stochastic integer problems under probabilistic constraints. Optimization Methods and Software 17(3), 359–382 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bookbinder, J.H., Tan, J.Y.: Strategies for the probabilistic lot-sizing problem with service-level constraints. Manage. Sci. 34(9), 1096–1108 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  9. Charnes, A., Cooper, W.W.: Deterministic equivalents for optimizing and satisficing under chance constraints. Oper. Res. 11(1), 18–39 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  10. Charnes, A., Cooper, W.W., Symonds, G.H.: Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil. Manage. Sci. 4(3), 235–263 (1958)

    Article  Google Scholar 

  11. Eppen, G., Martin, R.: Solving multi-item lot-sizing problems using variable definition. Oper. Res. 35, 832–848 (1987)

    Article  MATH  Google Scholar 

  12. Gicquel, C., Cheng, J.: A joint chance-constrained programming approach for the single-item capacitated lot-sizing problem with stochastic demand. Ann. Oper. Res. 264, 1–33 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Guan, Y., Miller, A.J.: Polynomial-time algorithms for stochastic uncapacitated lot-sizing problems. Oper. Res. 56(5), 1172–1183 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Guan, Y., Ahmed, S., Nemhauser, G.L., Miller, A.J.: A branch-and-cut algorithm for the stochastic uncapacitated lot-sizing problem. Math. Program. 105(1), 55–84 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Günlük, O., Pochet, Y.: Mixing mixed-integer inequalities. Math. Program. 90(3), 429–457 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Huang, K., Küçükyavuz, S.: On stochastic lot-sizing problems with random lead times. Oper. Res. Lett. 36(3), 303–308 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jiang, R., Guan, Y.: An o\((n^2)\)-time algorithm for the stochastic uncapacitated lot-sizing problem with random lead times. Oper. Res. Lett. 39(1), 74–77 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jiang, Y., Xu, J., Shen, S., Shi, C.: Production planning problems with joint service-level guarantee: a computational study. Int. J. Prod. Res. 55(1), 38–58 (2017)

    Article  Google Scholar 

  19. Jiang, Y., Shi, C., Shen, S.: Service level constrained inventory systems. Prod. Oper. Manag. 28, 2365–2389 (2019)

    Article  Google Scholar 

  20. Küçükyavuz, S.: On mixing sets arising in chance-constrained programming. Math. Program. 132(1), 31–56 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Küçükyavuz, S., Pochet, Y.: Uncapacitated lot sizing with backlogging: the convex hull. Math. Program. 118(1), 151–175 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Liu, X., Küçükyavuz, S.: A polyhedral study of the static probabilistic lot-sizing problem. Ann. Oper. Res. 261(1), 233–254 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Luedtke, J., Ahmed, S., Nemhauser, G.L.: An integer programming approach for linear programs with probabilistic constraints. Math. Program. 122(2), 247–272 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pochet, Y., Wolsey, L.A.: Lot-size models with backlogging: Strong reformulations and cutting planes. Math. Program. 40(1), 317–335 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pochet, Y., Wolsey, L.A.: Lot-sizing with constant batches: Formulation and valid inequalities. Math. Oper. Res. 18, 767–785 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pochet, Y., Wolsey, L.A.: Polyhedra for lot-sizing with wagner–whitin costs. Math. Program. 67(1), 297–323 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pochet, Y., Wolsey, L.A.: Production Planning by Mixed Integer Programming. Springer, New York (2006)

    MATH  Google Scholar 

  28. Tarim, S.A., Kingsman, B.G.: The stochastic dynamic production/inventory lot-sizing problem with service-level constraints. Int. J. Prod. Econ. 88, 105–119 (2004)

    Article  Google Scholar 

  29. Tarim, S.A., Dogru, M.K., Özen, U., Rossi, R.: An efficient computational method for a stochastic dynamic lot-sizing problem under service-level constraints. Eur. J. Oper. Res. 215, 563–571 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Van Vyve, M., Wolsey, L.A.: Approximate extended formulations. Math. Program. 105, 501–522 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wagner, H.M., Whitin, T.M.: Dynamic version of the economic lot size model. Manage. Sci. 5(1), 89–96 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhang, M., Küçükyavuz, S., Goel, S.: A branch-and-cut method for dynamic decision making under joint chance constraints. Manage. Sci. 60(5), 1317–1333 (2014)

    Article  Google Scholar 

  33. Zhao, M., Huang, K., Zeng, B.: A polyhedral study on chance constrained program with random right-hand side. Math. Program. 166(1), 19–64 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank two anonymous referees and the associate editor for helpful comments that led to significant improvements of the paper. Z. Zhang and C. Gao thank the financial support from the National Nature Science Foundation of China under Grant No. 12071428 and 62111530247, and the Zhejiang Provincial Natural Science Foundation of China under Grant No. LZ20A010002.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Zhejiang University, Hangzhou, People’s Republic of China

    Zeyang Zhang & Chuanhou Gao

  2. Department of Industrial & Systems Engineering, University of Wisconsin-Madison, Madison, US

    James Luedtke

Authors
  1. Zeyang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chuanhou Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. James Luedtke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to James Luedtke.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Z., Gao, C. & Luedtke, J. New valid inequalities and formulations for the static joint Chance-constrained Lot-sizing problem. Math. Program. 199, 639–669 (2023). https://doi.org/10.1007/s10107-022-01847-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-022-01847-y

Mathematics Subject Classification

Search

Navigation