Skip to main content

Advertisement

SpringerLink
Log in

Polyhedral description of the integer single node flow set with constant bounds

  • Published:
Mathematical Programming Submit manuscript

Abstract

We consider the integer single node flow problem with constant lower and upper bounds on the variables. This problem arises as subproblem of more general mixed integer problems such as lotsizing problems. A complete description of the convex hull of the feasible solution set is given. This description is based on families of valid inequalities whose coefficients are obtained from the coefficients of the facet-defining inequalities for the convex hull of integer knapsack sets with two variables. All the coefficients can be computed in polynomial time in the size of the input data. We present polynomial separation algorithms and an 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

Similar content being viewed by others

References

  1. Agra, A., Constantino, M.: Polyhedral description of a mixed integer problem with constant bounds. Tech. Rep. 4/2000, University of Lisbon, Centro de Investigação Operacional, 2000

  2. Agra, A., Constantino, M.: Description of 2-integer continuous knapsack polyhedra. Tech. Rep. 3/2003, University of Lisbon, Centro de Investigação Operacional, 2003

  3. Agra, A.: Estudo poliédrico de modelos de programação inteira mista e extensões. Ph.D. thesis, Faculdade de Ciências da Universidade de Lisboa, 2004

  4. Balas, E., Oosten, M.: On the dimension of projected polyhedra. Discrete Applied Mathematics 87, 1–9 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Constantino, M.: Lower bounds in lot-sizing models: a polyhedral study. Mathematics of Operations Research 23 (1), 101–118 (1998)

    MathSciNet  Google Scholar 

  6. Goemans, M.: Valid inequalities and separation for mixed 0-1 constraints with variable upper bounds. Operations Research Letters 8, 315–322 (1985)

    MathSciNet  Google Scholar 

  7. Gu, Z., Nemhauser, G., Savelsbergh, M.: Sequence independent lifting in mixed integer programming. Journal of Combinatorial Optimization 4, 109–129 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hirschberg, D., Wong, C.: A polynomial-time algorithm for the knapsack problem with two variables. Journal of the Association for Computing Machinery 23 (1), 147–154 (1976)

    MathSciNet  Google Scholar 

  9. Louveaux, Q., Wolsey, L.: Lifting, superadditivity, mixed integer rounding and single node flow sets revisited. 4OR 1 (3), 173–207 (2003)

    Google Scholar 

  10. Martin, R.: Using separation algorithms to generate mixed integer model reformulations. Operations Research Letters 10, 119–128 (1991)

    MATH  MathSciNet  Google Scholar 

  11. Nemhauser, G., Wolsey, L.: Integer and Combinatorial Optimization. John Wiley & Sons, 1988

  12. Padberg, M., Van Roy, T., Wolsey, L.: Valid linear inequalities for fixed charge problems. Operations Research 33 (4), 842–861 (1985)

    MathSciNet  Google Scholar 

  13. Stallaert, J.: The complementary class of generalized flow cover inequalities. Discrete Applied Mathematics 77, 73–80 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  14. Van Roy, T., Wolsey, L.: Valid linear inequalities for mixed 0-1 programs. Discrete Applied Mathematics 14, 199–213 (1986)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and CEOC, University of Aveiro, Campus Universitário de Santiago, 3810-193, Aveiro, Portugal

    Agostinho Agra

  2. DEIO and CIO, University of Lisbon, Edifí cio C6, Campo Grande, 1749-016, Lisboa, Portugal

    Miguel F. Constantino

Authors
  1. Agostinho Agra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Miguel F. Constantino
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Agostinho Agra.

Additional information

Received: April, 2004

Rights and permissions

Reprints and permissions

About this article

Cite this article

Agra, A., Constantino, M. Polyhedral description of the integer single node flow set with constant bounds. Math. Program. 105, 345–364 (2006). https://doi.org/10.1007/s10107-005-0657-5

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-005-0657-5

Keywords

Search

Navigation