[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

A study of the lot-sizing polytope

  • Published:
Mathematical Programming Submit manuscript

Abstract.

The lot-sizing polytope is a fundamental structure contained in many practical production planning problems. Here we study this polytope and identify facet–defining inequalities that cut off all fractional extreme points of its linear programming relaxation, as well as liftings from those facets. We give a polynomial–time combinatorial separation algorithm for the inequalities when capacities are constant. We also report computational experiments on solving the lot–sizing problem with varying cost and capacity characteristics.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs NJ, 1993

  2. Barany, I., Van Roy, T.J., Wolsey, L.A.: Uncapacitated lot sizing: The convex hull of solutions. Math. Program. Study 22, 32–43 (1984)

    MathSciNet  MATH  Google Scholar 

  3. Belvaux, G., Wolsey., L.A.: bc–prod: a specialized branch–and–cut system for lot–sizing problems. Manage. Sci. 46, 724–738 (2000)

    Article  Google Scholar 

  4. Belvaux, G., Wolsey., L.A.: Modelling practical lot-sizing problems as mixed integer programs. Manage. Sci. 47, 993–1007 (2001)

    Article  Google Scholar 

  5. Bitran, G.R., Yanasse., H.H.: Computational complexity of the capacitated lot size problem. Manage. Sci. 28, 1174–1186 (1982)

    MathSciNet  MATH  Google Scholar 

  6. Constantino., M.: A cutting plane approach to capacitated lot–sizing with start–up costs. Math. Program. 75, 353–376 (1996)

    Article  Google Scholar 

  7. Federgruen, A., Tzur., M.: A simple forward algorithm to solve general dynamic lot sizing models with n periods in O(n logn) or O(n) time. Manage. Sci. 37, 909–925 (1991)

    MATH  Google Scholar 

  8. Florian, M., Klein., M.: Deterministic production planning with concave costs and capacity constraints. Manage. Sci. 18, 12–20 (1971)

    MATH  Google Scholar 

  9. Florian, M., Lenstra, J.K., Rinnooy Kan, H.G.: Deterministic production planning: Algorithms and complexity. Manage. Sci. 26, 669–679 (1980)

    MathSciNet  MATH  Google Scholar 

  10. Gu, Z., Nemhauser, G.L., Savelsbergh., M.W.P.: Sequence independent lifting in mixed integer programming. J. Combinatorial Optim. 4, 109–129 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Leung, J.M.Y., Magnanti, T.L., Vachani., R.: Facets and algorithms for capacitated lot sizing. Math. Program. 45, 331–359 (1989)

    MathSciNet  MATH  Google Scholar 

  12. Loparic, M., Marchand, H., Wolsey., L.A.: Dynamic knapsack sets and capacitated lot–sizing. Math. Program. 95, 53–69 (2003)

    Article  Google Scholar 

  13. Miller, A.,: Polyhedral Approaches to Capacitated Lot-Sizing Problems. PhD thesis, School of Industrial and Systems Engineering, Georgia Institute of Technology, December 1999

  14. Miller, A., Nemhauser, G.L., Savelsbergh., M.W.P.: On the capacitated lot–sizing and continuous 0–1 knapsack polyhedra. Eur. J. Oper. Res. 125, 298–315 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Padberg, M.W., Van Roy, T.J., Wolsey., L.A.: Valid linear inequalities for fixed charge problems. Oper. Res. 32, 842–861 (1984)

    Google Scholar 

  16. Pochet., Y.: Valid inequalities and separation for capacitated economic lot sizing. Oper. Res. Lett. 7, 109–115 (1988)

    Article  Google Scholar 

  17. Pochet, Y., Wolsey., L.A.: Solving multi–item lot–sizing problems using strong cutting planes. Manage. Sci. 37, 53–67 (1991)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. Pochet, Y., Wolsey., L.A.: Polyhedra for lot-sizing with Wagner-Whitin costs. Math. Program. 67, 297–323 (1994)

    MathSciNet  MATH  Google Scholar 

  20. Van Hoesel, C.P.M., Wagelmans., A.P.M.: An O(T 3) algorithm for the economic lot-sizing problem with constant capacities. Manage. Sci. 42, 142–150 (1996)

    MATH  Google Scholar 

  21. Wagelmans, A., Van Hoesel, S., Kolen., A.: Economic lot sizing: An O(n log n) algorithm that runs in linear time in the Wagner-Whitin case. Oper. Res. 40, S145–S156 (1992)

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

    MATH  Google Scholar 

  23. Wolsey., L.A.: Submodularity and valid inequalities in capacitated fixed charge networks. Oper. Res. Lett. 8, 119–124 (1989)

    Article  Google Scholar 

  24. Wolsey., L.A.: Solving multi-item lot-sizing problems with an mip solver using classification and reformulation. Manage. Sci. 48, 1587–1602 (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Industrial Engineering and Operations Research, University of California, 4135 Etcheverry Hall, Berkeley CA, 94720–1777, USA

    Alper Atamtürk

  2. Department of Civil and Environmental Engineering, University of California, 416G McLaughlin Hall, Berkeley, CA, 94720–1720, USA Currently in Pontificia Universidad Católica de Chile.

    Juan Carlos Muñoz

Authors
  1. Alper Atamtürk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juan Carlos Muñoz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alper Atamtürk.

Additional information

Supported, in part, by NSF Grants 0070127 and 0218265, and a grant from ILOG, Inc.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-003-0465-8

Keywords

Search

Navigation