IDEAS home Printed from https://ideas.repec.org/a/inm/ormnsc/v61y2015i6p1275-1291.html
   My bibliography  Save this article

The Fixed Charge Transportation Problem: An Exact Algorithm Based on a New Integer Programming Formulation

Author

Listed:
  • Roberto Roberti

    (Department of Electrical, Electronic, and Information Engineering “Guglielmo Marconi” (DEI), University of Bologna, 40126 Bologna, Italy)

  • Enrico Bartolini

    (Department of Mathematics and Systems Analysis, Aalto University School of Science, 00076 Aalto, Finland)

  • Aristide Mingozzi

    (Department of Mathematics, University of Bologna, 40126 Bologna, Italy)

Abstract

The fixed charge transportation problem generalizes the well-known transportation problem where the cost of sending goods from a source to a sink is composed of a fixed cost and a continuous cost proportional to the amount of goods sent. In this paper, we describe a new integer programming formulation with exponentially many variables corresponding to all possible flow patterns to sinks. We show that the linear relaxation of the new formulation is tighter than that of the standard mixed integer programming formulation. We describe different classes of valid inequalities for the new formulation and a column generation method to compute a valid lower bound embedded into an exact branch-and-price algorithm. Computational results on test problems from the literature show that the new algorithm outperforms the state-of-the-art exact methods from the literature and can solve instances with up to 70 sources and 70 sinks.Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2014.1947 . This paper was accepted by Dimitris Bertsimas, optimization .

Suggested Citation

  • 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.
    • Handle: RePEc:inm:ormnsc:v:61:y:2015:i:6:p:1275-1291
    DOI: 10.1287/mnsc.2014.1947
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/mnsc.2014.1947
    Download Restriction: no
    File URL: https://libkey.io/10.1287/mnsc.2014.1947?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. J. A. Tomlin, 1971. "Technical Note—An Improved Branch-and-Bound Method for Integer Programming," Operations Research, INFORMS, vol. 19(4), pages 1070-1075, August.
    2. John William Stroup, 1967. "Letter to the Editor—Allocation of Launch Vehicles to Space Missions: A Fixed-Cost Transportation Problem," Operations Research, INFORMS, vol. 15(6), pages 1157-1163, December.
    3. Jeff Kennington & Ed Unger, 1976. "A New Branch-and-Bound Algorithm for the Fixed-Charge Transportation Problem," Management Science, INFORMS, vol. 22(10), pages 1116-1126, June.
    4. Katta G. Murty, 1968. "Solving the Fixed Charge Problem by Ranking the Extreme Points," Operations Research, INFORMS, vol. 16(2), pages 268-279, April.
    5. A. Victor Cabot & S. Selcuk Erenguc, 1986. "Improved Penalties for Fixed Cost Linear Programs Using Lagrangean Relaxation," Management Science, INFORMS, vol. 32(7), pages 856-869, July.
    6. Zonghao Gu & George L. Nemhauser & Martin W. P. Savelsbergh, 1999. "Lifted Cover Inequalities for 0-1 Integer Programs: Complexity," INFORMS Journal on Computing, INFORMS, vol. 11(1), pages 117-123, February.
    7. Yogesh Agarwal & Yash Aneja, 2012. "Fixed-Charge Transportation Problem: Facets of the Projection Polyhedron," Operations Research, INFORMS, vol. 60(3), pages 638-654, June.
    8. 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.
    9. Warren E. Walker, 1976. "A Heuristic Adjacent Extreme Point Algorithm for the Fixed Charge Problem," Management Science, INFORMS, vol. 22(5), pages 587-596, January.
    10. Zonghao Gu & George L. Nemhauser & Martin W. P. Savelsbergh, 1998. "Lifted Cover Inequalities for 0-1 Integer Programs: Computation," INFORMS Journal on Computing, INFORMS, vol. 10(4), pages 427-437, November.
    11. 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.
    12. 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).
    13. 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.
    14. 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.
    15. Paul Gray, 1971. "Technical Note—Exact Solution of the Fixed-Charge Transportation Problem," Operations Research, INFORMS, vol. 19(6), pages 1529-1538, October.
    16. Adlakha, Veena & Kowalski, Krzysztof, 2003. "A simple heuristic for solving small fixed-charge transportation problems," Omega, Elsevier, vol. 31(3), pages 205-211, June.
    17. 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.
    18. 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.
    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. Maria R. Ibanez & Jonathan R. Clark & Robert S. Huckman & Bradley R. Staats, 2018. "Discretionary Task Ordering: Queue Management in Radiological Services," Management Science, INFORMS, vol. 64(9), pages 4389-4407, September.
    2. Ortiz-Astorquiza, Camilo & Contreras, Ivan & Laporte, Gilbert, 2018. "Multi-level facility location problems," European Journal of Operational Research, Elsevier, vol. 267(3), pages 791-805.
    3. Alex Scott & Chris Parker & Christopher W. Craighead, 2017. "Service Refusals in Supply Chains: Drivers and Deterrents of Freight Rejection," Transportation Science, INFORMS, vol. 51(4), pages 1086-1101, November.
    4. 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.
    5. Enrico Bartolini & Mauro Dell’Amico & Manuel Iori, 2017. "Scheduling cleaning activities on trains by minimizing idle times," Journal of Scheduling, Springer, vol. 20(5), pages 493-506, October.
    6. Yixin Zhao & Torbjörn Larsson & Elina Rönnberg & Panos M. Pardalos, 2018. "The fixed charge transportation problem: a strong formulation based on Lagrangian decomposition and column generation," Journal of Global Optimization, Springer, vol. 72(3), pages 517-538, November.
    7. Juan Pablo Vielma, 2018. "Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139," Management Science, INFORMS, vol. 64(10), pages 4721-4734, October.

    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. 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.
    2. 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.
    3. 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.
    4. Jesús Sáez Aguado, 2009. "Fixed Charge Transportation Problems: a new heuristic approach based on Lagrangean relaxation and the solving of core problems," Annals of Operations Research, Springer, vol. 172(1), pages 45-69, November.
    5. Yixin Zhao & Torbjörn Larsson & Elina Rönnberg & Panos M. Pardalos, 2018. "The fixed charge transportation problem: a strong formulation based on Lagrangian decomposition and column generation," Journal of Global Optimization, Springer, vol. 72(3), pages 517-538, November.
    6. 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.
    7. 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.
    8. Yogesh Agarwal & Yash Aneja, 2012. "Fixed-Charge Transportation Problem: Facets of the Projection Polyhedron," Operations Research, INFORMS, vol. 60(3), pages 638-654, June.
    9. 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.
    10. Adlakha, Veena & Kowalski, Krzysztof & Wang, Simi & Lev, Benjamin & Shen, Wenjing, 2014. "On approximation of the fixed charge transportation problem," Omega, Elsevier, vol. 43(C), pages 64-70.
    11. Kowalski, Krzysztof & Lev, Benjamin & Shen, Wenjing & Tu, Yan, 2014. "A fast and simple branching algorithm for solving small scale fixed-charge transportation problem," Operations Research Perspectives, Elsevier, vol. 1(1), pages 1-5.
    12. Adlakha, Veena & Kowalski, Krzysztof & Lev, Benjamin, 2010. "A branching method for the fixed charge transportation problem," Omega, Elsevier, vol. 38(5), pages 393-397, October.
    13. V. Adlakha & K. Kowalski, 2015. "Fractional Polynomial Bounds for the Fixed Charge Problem," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 1026-1038, March.
    14. 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.
    15. Lev, Benjamin & Kowalski, Krzysztof, 2011. "Modeling fixed-charge problems with polynomials," Omega, Elsevier, vol. 39(6), pages 725-728, December.
    16. Gurwinder Singh & Amarinder Singh, 2021. "Solving fixed-charge transportation problem using a modified particle swarm optimization algorithm," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 12(6), pages 1073-1086, December.
    17. 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.
    18. Klose, Andreas & Drexl, Andreas, 2001. "Combinatorial optimisation problems of the assignment type and a partitioning approach," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 545, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    19. Alexey Sorokin & Vladimir Boginski & Artyom Nahapetyan & Panos M. Pardalos, 2013. "Computational risk management techniques for fixed charge network flow problems with uncertain arc failures," Journal of Combinatorial Optimization, Springer, vol. 25(1), pages 99-122, January.
    20. A. N. Balaji & J. Mukund Nilakantan & Izabela Nielsen & N. Jawahar & S. G. Ponnambalam, 2019. "Solving fixed charge transportation problem with truck load constraint using metaheuristics," Annals of Operations Research, Springer, vol. 273(1), pages 207-236, February.

    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:ormnsc:v:61:y:2015:i:6:p:1275-1291. 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.