IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v60y2012i3p638-654.html
   My bibliography  Save this article

Fixed-Charge Transportation Problem: Facets of the Projection Polyhedron

Author

Listed:
  • Yogesh Agarwal

    (Indian Institute of Management, Lucknow 226 013, Uttar Pradesh, India)

  • Yash Aneja

    (Odette School of Business, University of Windsor, Windsor, Ontario N9B 3P4, Canada)

Abstract

In this paper we consider the well-known fixed-charge transportation problem. To send any flow from source s i to destination t j , we incur a unit variable shipping cost of c ij and a fixed cost f ij . Here we study the structure of the projection polyhedron of this problem, in the space of 0-1 variables associated with fixed charges, and we develop several classes of valid inequalities and derive conditions under which they are facet defining. In some cases, if the conditions are not satisfied, we show how they can be lifted to define facets. Several heuristics for generating and adding these facets are presented. Using these results, we develop a computationally effective algorithm for solving the problem. The computational results clearly indicate the usefulness of this approach.

Suggested Citation

  • Yogesh Agarwal & Yash Aneja, 2012. "Fixed-Charge Transportation Problem: Facets of the Projection Polyhedron," Operations Research, INFORMS, vol. 60(3), pages 638-654, June.
    • Handle: RePEc:inm:oropre:v:60:y:2012:i:3:p:638-654
    DOI: 10.1287/opre.1120.1041
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.1120.1041
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.1120.1041?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Katta G. Murty, 1968. "Solving the Fixed Charge Problem by Ranking the Extreme Points," Operations Research, INFORMS, vol. 16(2), pages 268-279, April.
    2. Schaffer, Joanne R. & O'Leary, Daniel E., 1989. "Use of penalties in a branch and bound procedure for the fixed charge transportation problem," European Journal of Operational Research, Elsevier, vol. 43(3), pages 305-312, December.
    3. Zonghao Gu & George L. Nemhauser & Martin W.P. Savelsbergh, 2000. "Sequence Independent Lifting in Mixed Integer Programming," Journal of Combinatorial Optimization, Springer, vol. 4(1), pages 109-129, March.
    4. VAN ROY, Tony J. & WOLSEY, Laurence A., 1987. "Solving mixed integer programming problems using automatic reformulation," LIDAM Reprints CORE 782, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Tony J. Van Roy & Laurence A. Wolsey, 1987. "Solving Mixed Integer Programming Problems Using Automatic Reformulation," Operations Research, INFORMS, vol. 35(1), pages 45-57, February.
    6. VAN ROY, Tony J. & WOLSEY, Laurence A., 1985. "Valid inequalities and separation for uncapacitated fixed charge networks," LIDAM Reprints CORE 671, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. Richard S. Barr & Fred Glover & Darwin Klingman, 1981. "A New Optimization Method for Large Scale Fixed Charge Transportation Problems," Operations Research, INFORMS, vol. 29(3), pages 448-463, June.
    8. ORTEGA , Francisco & WOLSEY, Laurence A., 2003. "A branch-and-cut algorithm for the single-commodity, uncapacitated, fixed-charge network flow problem," LIDAM Reprints CORE 1611, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    9. Udatta S. Palekar & Mark H. Karwan & Stanley Zionts, 1990. "A Branch-and-Bound Method for the Fixed Charge Transportation Problem," Management Science, INFORMS, vol. 36(9), pages 1092-1105, September.
    10. Rardin, Ronald L. & Wolsey, Laurence A., 1993. "Valid inequalities and projecting the multicommodity extended formulation for uncapacitated fixed charge network flow problems," European Journal of Operational Research, Elsevier, vol. 71(1), pages 95-109, November.
    11. M. W. Padberg & T. J. Van Roy & L. A. Wolsey, 1985. "Valid Linear Inequalities for Fixed Charge Problems," Operations Research, INFORMS, vol. 33(4), pages 842-861, August.
    12. VAN ROY, Tony J. & WOLSEY, Laurence A., 1986. "Valid inequalities for mixed 0-1 programs," LIDAM Reprints CORE 697, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    13. Bruce W. Lamar & Chris A. Wallace, 1997. "Revised-Modified Penalties for Fixed Charge Transportation Problems," Management Science, INFORMS, vol. 43(10), pages 1431-1436, October.
    14. Padberg, M.W. & Van Roy, T.J. & Wolsey, L.A., 1985. "Valid linear inequalities for fixed charge problems," LIDAM Reprints CORE 656, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Bertazzi, Luca & Maggioni, Francesca, 2018. "A stochastic multi-stage fixed charge transportation problem: Worst-case analysis of the rolling horizon approach," European Journal of Operational Research, Elsevier, vol. 267(2), pages 555-569.
    2. Li, Xiangyong & Aneja, Y.P., 2017. "Regenerator location problem: Polyhedral study and effective branch-and-cut algorithms," European Journal of Operational Research, Elsevier, vol. 257(1), pages 25-40.
    3. Sagratella, Simone & Schmidt, Marcel & Sudermann-Merx, Nathan, 2020. "The noncooperative fixed charge transportation problem," European Journal of Operational Research, Elsevier, vol. 284(1), pages 373-382.
    4. Xu, Dongyang & Li, Kunpeng & Zou, Xuxia & Liu, Ling, 2017. "An unpaired pickup and delivery vehicle routing problem with multi-visit," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 103(C), pages 218-247.
    5. Agarwal, Y.K. & Aneja, Y.P., 2017. "Fixed charge multicommodity network design using p-partition facets," European Journal of Operational Research, Elsevier, vol. 258(1), pages 124-135.
    6. Erika Buson & Roberto Roberti & Paolo Toth, 2014. "A Reduced-Cost Iterated Local Search Heuristic for the Fixed-Charge Transportation Problem," Operations Research, INFORMS, vol. 62(5), pages 1095-1106, October.
    7. Roberto Roberti & Enrico Bartolini & Aristide Mingozzi, 2015. "The Fixed Charge Transportation Problem: An Exact Algorithm Based on a New Integer Programming Formulation," Management Science, INFORMS, vol. 61(6), pages 1275-1291, June.
    8. Aristide Mingozzi & Roberto Roberti, 2018. "An Exact Algorithm for the Fixed Charge Transportation Problem Based on Matching Source and Sink Patterns," Transportation Science, INFORMS, vol. 52(2), pages 229-238, March.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. ORTEGA, Francisco & WOLSEY, Laurence, 2000. "A branch-and-cut algorithm for the single commodity uncapacitated fixed charge network flow problem," LIDAM Discussion Papers CORE 2000049, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Erika Buson & Roberto Roberti & Paolo Toth, 2014. "A Reduced-Cost Iterated Local Search Heuristic for the Fixed-Charge Transportation Problem," Operations Research, INFORMS, vol. 62(5), pages 1095-1106, October.
    3. Roberto Roberti & Enrico Bartolini & Aristide Mingozzi, 2015. "The Fixed Charge Transportation Problem: An Exact Algorithm Based on a New Integer Programming Formulation," Management Science, INFORMS, vol. 61(6), pages 1275-1291, June.
    4. Dimitri J. Papageorgiou & Alejandro Toriello & George L. Nemhauser & Martin W. P. Savelsbergh, 2012. "Fixed-Charge Transportation with Product Blending," Transportation Science, INFORMS, vol. 46(2), pages 281-295, May.
    5. Mervat Chouman & Teodor Gabriel Crainic & Bernard Gendron, 2017. "Commodity Representations and Cut-Set-Based Inequalities for Multicommodity Capacitated Fixed-Charge Network Design," Transportation Science, INFORMS, vol. 51(2), pages 650-667, May.
    6. Gavin J. Bell & Bruce W. Lamar & Chris A. Wallace, 1999. "Capacity improvement, penalties, and the fixed charge transportation problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 46(4), pages 341-355, June.
    7. Richard Laundy & Michael Perregaard & Gabriel Tavares & Horia Tipi & Alkis Vazacopoulos, 2009. "Solving Hard Mixed-Integer Programming Problems with Xpress-MP: A MIPLIB 2003 Case Study," INFORMS Journal on Computing, INFORMS, vol. 21(2), pages 304-313, May.
    8. Jeffery L. Kennington & Charles D. Nicholson, 2010. "The Uncapacitated Time-Space Fixed-Charge Network Flow Problem: An Empirical Investigation of Procedures for Arc Capacity Assignment," INFORMS Journal on Computing, INFORMS, vol. 22(2), pages 326-337, May.
    9. Hultberg, Tim H. & Cardoso, Domingos M., 1997. "The teacher assignment problem: A special case of the fixed charge transportation problem," European Journal of Operational Research, Elsevier, vol. 101(3), pages 463-473, September.
    10. Mervat Chouman & Teodor Gabriel Crainic & Bernard Gendron, 2018. "The impact of filtering in a branch-and-cut algorithm for multicommodity capacitated fixed charge network design," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 6(2), pages 143-184, June.
    11. Quentin Louveaux & Laurence Wolsey, 2007. "Lifting, superadditivity, mixed integer rounding and single node flow sets revisited," Annals of Operations Research, Springer, vol. 153(1), pages 47-77, September.
    12. Alper Atamtürk & Martin Savelsbergh, 2005. "Integer-Programming Software Systems," Annals of Operations Research, Springer, vol. 140(1), pages 67-124, November.
    13. Sun, Minghe & Aronson, Jay E. & McKeown, Patrick G. & Drinka, Dennis, 1998. "A tabu search heuristic procedure for the fixed charge transportation problem," European Journal of Operational Research, Elsevier, vol. 106(2-3), pages 441-456, April.
    14. Jawahar, N. & Balaji, A.N., 2009. "A genetic algorithm for the two-stage supply chain distribution problem associated with a fixed charge," European Journal of Operational Research, Elsevier, vol. 194(2), pages 496-537, April.
    15. Aardal, K.I. & van Hoesel, S., 1995. "Polyhedral Techniques in Combinatorial Optimization," Other publications TiSEM ed028a07-eb6a-4c8d-8f21-d, Tilburg University, School of Economics and Management.
    16. Masoud Yaghini & Mohammad Karimi & Mohadeseh Rahbar & Mohammad Hassan Sharifitabar, 2015. "A Cutting-Plane Neighborhood Structure for Fixed-Charge Capacitated Multicommodity Network Design Problem," INFORMS Journal on Computing, INFORMS, vol. 27(1), pages 48-58, February.
    17. Alper Atamtürk & Simge Küçükyavuz, 2005. "Lot Sizing with Inventory Bounds and Fixed Costs: Polyhedral Study and Computation," Operations Research, INFORMS, vol. 53(4), pages 711-730, August.
    18. Binyuan Chen & Simge Küçükyavuz & Suvrajeet Sen, 2011. "Finite Disjunctive Programming Characterizations for General Mixed-Integer Linear Programs," Operations Research, INFORMS, vol. 59(1), pages 202-210, February.
    19. Wei-Kun Chen & Liang Chen & Mu-Ming Yang & Yu-Hong Dai, 2018. "Generalized coefficient strengthening cuts for mixed integer programming," Journal of Global Optimization, Springer, vol. 70(1), pages 289-306, January.
    20. Diego Klabjan & George L. Nemhauser, 2002. "A Polyhedral Study of Integer Variable Upper Bounds," Mathematics of Operations Research, INFORMS, vol. 27(4), pages 711-739, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:oropre:v:60:y:2012:i:3:p:638-654. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.