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A Branch-First, Cut-Second Approach for Locomotive Assignment

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  • Koorush Ziarati

    (École Polytechnique (Montréal) Département de Mathématiques et Génie Industriel, C.P. 6079, Succ. Centre-Ville, Montréal, Canada H3C 3A7 and GERAD)

  • François Soumis

    (École Polytechnique (Montréal) Département de Mathématiques et Génie Industriel, C.P. 6079, Succ. Centre-Ville, Montréal, Canada H3C 3A7 and GERAD)

  • Jacques Desrosiers

    (École des Hautes Études Commerciales and GERAD, 3000 chemin de la Côte-Ste-Catherine, Montréal, Canada H3T 2A7)

  • Marius M. Solomon

    (Northeastern University, College of Business Administration, Department of Management Sciences, 314 Hayden Hall, 360 Huntingdon Avenue, Boston, Massachusetts 02115 and GERAD)

Abstract

The problem of assigning locomotives to trains consists of selecting the types and number of engines that minimize the fixed and operational locomotive costs resulting from providing sufficient power to pull trains on fixed schedules. The force required to pull a train is often expressed in terms of horsepower and tonnage requirements rather than in terms of number of engines. This complicates the solution process of the integer programming formulation and usually creates a large integrality gap. Furthermore, the solution of the linearly relaxed problem is strongly fractional. To obtain integer solutions, we propose a novel branch-and-cut approach. The core of the method consists of branching decisions that define on one branch the projection of the problem on a low-dimensional subspace. There, the facets of the polyhedron describing a restricted constraint set can be easily derived. We call this approach branch-first, cut-second. We first derive facets when at most two types of engines are used. We then extend the branching rule to cases involving additional locomotive types. We have conducted computational experiments using actual data from the Canadian National railway company. Simulated test-problems involving two or more locomotive types were solved over 1-, 2-, and 3-day rolling horizons. The cuts were successful in reducing the average integrality gap by 52% for the two-type case and by 34% when more than 25 types were used. Furthermore, the branch-first, cut-second approach was instrumental in improving the best known solution for an almost 2,000-leg weekly problem involving 26 locomotive types. It reduced the number of locomotives by 11, or 1.1%, at an equivalent savings of $3,000,000 per unit. Additional tests on different weekly data produced almost identical results.

Suggested Citation

  • Koorush Ziarati & François Soumis & Jacques Desrosiers & Marius M. Solomon, 1999. "A Branch-First, Cut-Second Approach for Locomotive Assignment," Management Science, INFORMS, vol. 45(8), pages 1156-1168, August.
  • Handle: RePEc:inm:ormnsc:v:45:y:1999:i:8:p:1156-1168
    DOI: 10.1287/mnsc.45.8.1156
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    References listed on IDEAS

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    1. Belgacem Bouzaiene-Ayari & Clark Cheng & Sourav Das & Ricardo Fiorillo & Warren B. Powell, 2016. "From Single Commodity to Multiattribute Models for Locomotive Optimization: A Comparison of Optimal Integer Programming and Approximate Dynamic Programming," Transportation Science, INFORMS, vol. 50(2), pages 366-389, May.
    2. Agostinho Agra & Marielle Christiansen & Alexandrino Delgado, 2013. "Mixed Integer Formulations for a Short Sea Fuel Oil Distribution Problem," Transportation Science, INFORMS, vol. 47(1), pages 108-124, February.
    3. Petr KOZLOV & Elena TIMUKHINA & Nikolay TUSHIN, 2018. "Coordination Of Locomotives Turnover And Servicing Modes," Transport Problems, Silesian University of Technology, Faculty of Transport, vol. 13(1), pages 19-26, March.
    4. Philimon Nyamugure & Siphosenkosi Dube Swene & Edward T. Chiyaka & Farikayi K. Mutasa, 2014. "Train Schedule Optimization: A Case Study of the National Railways of Zimbabwe," International Journal of Management Sciences, Research Academy of Social Sciences, vol. 3(1), pages 1-20.
    5. Rouillon, Stéphane & Desaulniers, Guy & Soumis, François, 2006. "An extended branch-and-bound method for locomotive assignment," Transportation Research Part B: Methodological, Elsevier, vol. 40(5), pages 404-423, June.
    6. Lin, Zhiyuan & Kwan, Raymond S.K., 2016. "A branch-and-price approach for solving the train unit scheduling problem," Transportation Research Part B: Methodological, Elsevier, vol. 94(C), pages 97-120.
    7. Camilo Ortiz-Astorquiza & Jean-François Cordeau & Emma Frejinger, 2021. "The Locomotive Assignment Problem with Distributed Power at the Canadian National Railway Company," Transportation Science, INFORMS, vol. 55(2), pages 510-531, March.
    8. Petr KOZLOV & Sergey VAKULENKO & Nikolay TUSHIN & Elena TIMUKHINA, 2017. "Model To Calculate The Optimal Mode Of Train Locomotives Turnover," Transport Problems, Silesian University of Technology, Faculty of Transport, vol. 12(3), pages 125-133, September.
    9. Piu, F. & Prem Kumar, V. & Bierlaire, M. & Speranza, M.G., 2015. "Introducing a preliminary consists selection in the locomotive assignment problem," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 82(C), pages 217-237.
    10. Balachandran Vaidyanathan & Ravindra K. Ahuja & James B. Orlin, 2008. "The Locomotive Routing Problem," Transportation Science, INFORMS, vol. 42(4), pages 492-507, November.
    11. Gao, Yuan & Schmidt, Marie & Yang, Lixing & Gao, Ziyou, 2020. "A branch-and-price approach for trip sequence planning of high-speed train units," Omega, Elsevier, vol. 92(C).
    12. Vaidyanathan, Balachandran & Ahuja, Ravindra K. & Liu, Jian & Shughart, Larry A., 2008. "Real-life locomotive planning: New formulations and computational results," Transportation Research Part B: Methodological, Elsevier, vol. 42(2), pages 147-168, February.
    13. Warren B. Powell & Belgacem Bouzaiene-Ayari & Coleman Lawrence & Clark Cheng & Sourav Das & Ricardo Fiorillo, 2014. "Locomotive Planning at Norfolk Southern: An Optimizing Simulator Using Approximate Dynamic Programming," Interfaces, INFORMS, vol. 44(6), pages 567-578, December.
    14. Armin Fügenschuh & Henning Homfeld & Andreas Huck & Alexander Martin & Zhi Yuan, 2008. "Scheduling Locomotives and Car Transfers in Freight Transport," Transportation Science, INFORMS, vol. 42(4), pages 478-491, November.
    15. Zhiyuan Lin & Raymond S. K. Kwan, 2016. "Local convex hulls for a special class of integer multicommodity flow problems," Computational Optimization and Applications, Springer, vol. 64(3), pages 881-919, July.
    16. Prashant Premkumar & P. N. Ram Kumar, 2022. "Locomotive assignment problem: integrating the strategic, tactical and operational level aspects," Annals of Operations Research, Springer, vol. 315(2), pages 867-898, August.
    17. Ravindra K. Ahuja & Jian Liu & James B. Orlin & Dushyant Sharma & Larry A. Shughart, 2005. "Solving Real-Life Locomotive-Scheduling Problems," Transportation Science, INFORMS, vol. 39(4), pages 503-517, November.
    18. Prashant Premkumar & P. N. Ram Kumar, 2019. "Literature Review of Locomotive Assignment Problem from Service Operations Perspective: The Case of Indian Railways," IIM Kozhikode Society & Management Review, , vol. 8(1), pages 74-86, January.
    19. Danial Davarnia & Jean-Philippe P. Richard & Ece Içyüz-Ay & Bijan Taslimi, 2019. "Network Models with Unsplittable Node Flows with Application to Unit Train Scheduling," Operations Research, INFORMS, vol. 67(4), pages 1053-1068, July.
    20. Xu, Xiaoming & Li, Chung-Lun & Xu, Zhou, 2018. "Integrated train timetabling and locomotive assignment," Transportation Research Part B: Methodological, Elsevier, vol. 117(PA), pages 573-593.
    21. Chung, Ji-Won & Oh, Seog-Moon & Choi, In-Chan, 2009. "A hybrid genetic algorithm for train sequencing in the Korean railway," Omega, Elsevier, vol. 37(3), pages 555-565, June.

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