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Stronger multi-commodity flow formulations of the Capacitated Vehicle Routing Problem

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  • Letchford, Adam N.
  • Salazar-González, Juan-José

Abstract

The Capacitated Vehicle Routing Problem is a much-studied (and strongly NP-hard) combinatorial optimization problem, for which many integer programming formulations have been proposed. We present two new multi-commodity flow (MCF) formulations, and show that they dominate all of the existing ones, in the sense that their continuous relaxations yield stronger lower bounds. Moreover, we show that the relaxations can be strengthened, in pseudo-polynomial time, in such a way that all of the so-called knapsack large multistar (KLM) inequalities are satisfied. The only other relaxation known to satisfy the KLM inequalities, based on set partitioning, is strongly NP-hard to solve. Computational results demonstrate that the new MCF relaxations are significantly stronger than the previously known ones.

Suggested Citation

  • Letchford, Adam N. & Salazar-González, Juan-José, 2015. "Stronger multi-commodity flow formulations of the Capacitated Vehicle Routing Problem," European Journal of Operational Research, Elsevier, vol. 244(3), pages 730-738.
    • Handle: RePEc:eee:ejores:v:244:y:2015:i:3:p:730-738
    DOI: 10.1016/j.ejor.2015.02.028
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    Cited by:

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    2. Michiel A. J. uit het Broek & Albert H. Schrotenboer & Bolor Jargalsaikhan & Kees Jan Roodbergen & Leandro C. Coelho, 2021. "Asymmetric Multidepot Vehicle Routing Problems: Valid Inequalities and a Branch-and-Cut Algorithm," Operations Research, INFORMS, vol. 69(2), pages 380-409, March.
    3. Abdulkader, M.M.S. & Gajpal, Yuvraj & ElMekkawy, Tarek Y., 2018. "Vehicle routing problem in omni-channel retailing distribution systems," International Journal of Production Economics, Elsevier, vol. 196(C), pages 43-55.
    4. Letchford, Adam N. & Salazar-González, Juan-José, 2016. "Stronger multi-commodity flow formulations of the (capacitated) sequential ordering problem," European Journal of Operational Research, Elsevier, vol. 251(1), pages 74-84.
    5. Dollevoet, T.A.B. & Munari, P. & Spliet, R., 2020. "A p-step formulation for the capacitated vehicle routing problem," Econometric Institute Research Papers EI2020-01, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    6. Alfandari, Laurent & Ljubić, Ivana & De Melo da Silva, Marcos, 2022. "A tailored Benders decomposition approach for last-mile delivery with autonomous robots," European Journal of Operational Research, Elsevier, vol. 299(2), pages 510-525.
    7. Bektaş, Tolga & Gouveia, Luis & Martínez-Sykora, Antonio & Salazar-González, Juan-José, 2019. "Balanced vehicle routing: Polyhedral analysis and branch-and-cut algorithm," European Journal of Operational Research, Elsevier, vol. 273(2), pages 452-463.
    8. Archetti, Claudia & Ljubić, Ivana, 2022. "Comparison of formulations for the Inventory Routing Problem," European Journal of Operational Research, Elsevier, vol. 303(3), pages 997-1008.
    9. Twan Dollevoet & Remy Spliet, 2023. "Preprocessing to Reduce Vehicle Capacity for Routing Problems," SN Operations Research Forum, Springer, vol. 4(2), pages 1-7, June.
    10. Rodríguez-Pereira, Jessica & Fernández, Elena & Laporte, Gilbert & Benavent, Enrique & Martínez-Sykora, Antonio, 2019. "The Steiner Traveling Salesman Problem and its extensions," European Journal of Operational Research, Elsevier, vol. 278(2), pages 615-628.
    11. Leggieri, Valeria & Haouari, Mohamed, 2017. "Lifted polynomial size formulations for the homogeneous and heterogeneous vehicle routing problems," European Journal of Operational Research, Elsevier, vol. 263(3), pages 755-767.
    12. Letchford, Adam N. & Salazar-González, Juan-José, 2019. "The Capacitated Vehicle Routing Problem: Stronger bounds in pseudo-polynomial time," European Journal of Operational Research, Elsevier, vol. 272(1), pages 24-31.
    13. Lee, Younsoo & Lee, Kyungsik, 2023. "Valid inequalities and extended formulations for lot-sizing and scheduling problem with sequence-dependent setups," European Journal of Operational Research, Elsevier, vol. 310(1), pages 201-216.
    14. Elena Fernández & Jessica Rodríguez-Pereira, 2017. "Multi-depot rural postman problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(2), pages 340-372, July.
    15. Khodakaram Salimifard & Sara Bigharaz, 2022. "The multicommodity network flow problem: state of the art classification, applications, and solution methods," Operational Research, Springer, vol. 22(1), pages 1-47, March.
    16. Bayliss, Christopher & Bektaş, Tolga & Tjon-Soei-Len, Vernon & Rohner, Remo, 2023. "Designing a multi-modal and variable-echelon delivery system for last-mile logistics," European Journal of Operational Research, Elsevier, vol. 307(2), pages 645-662.
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    18. Esteban Inga & Roberto Hincapié & Sandra Céspedes, 2019. "Capacitated Multicommodity Flow Problem for Heterogeneous Smart Electricity Metering Communications Using Column Generation," Energies, MDPI, vol. 13(1), pages 1-21, December.

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