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A Column Generation-Based Heuristic for the Split Delivery Vehicle Routing Problem with Time Windows

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Abstract

The vehicle routing problem with time windows (VRPTW) is one of the most studied variants of routing problems. We consider the split delivery VRPTW (SDVRPTW), an extension in which customers can be visited multiple times, if advantageous. While this additional flexibility can result in significant cost reductions, it also results in additional modeling and computational challenges. Indeed, the branch-and-price algorithms used to successfully solve VRPTW instances require substantial modifications before they can be used to solve SDVRPTW instances, and then often exhibit inferior performance when compared with branch-and-cut algorithms for solving SDVRPTW instances (whereas branch-and-price algorithms tend to perform better than branch-and-cut algorithms for the VRPTW). We propose a new route-based formulation for the SDVRPTW that differs fundamentally from others presented in the literature. It is the first formulation in which the number of decision variables related to delivery quantities as well as the number of constraints is polynomial in the number of customers. We use this formulation as the basis for a column generation-based heuristic that produces high-quality solutions for a wide range of benchmark instances with 50 and 100 customers and vehicle capacity equal to 50 and 100. It finds many new best known solutions, and, for the first time, we report upper bounds for all 100-customer instances.

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Acknowledgments

The authors are thankful to the anonymous reviewers for their insightful comments and suggestions.

Funding

This work was supported by the Sao Paulo Research Foundation (FAPESP) (grant numbers 18/23555-1, 16/01860-1, and 13/07375-0) and the National Council for Scientific and Technological Development (CNPq) (grant number 304601/2017-9).

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Authors and Affiliations

  1. Federal University of Sao Carlos, Sao Carlos, SP, Brazil

    Pedro Munari

  2. Georgia Institute of Technology, Atlanta, GA, USA

    Martin Savelsbergh

Authors
  1. Pedro Munari
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  2. Martin Savelsbergh
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Correspondence to Pedro Munari.

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The authors declare that there is no conflict of interest.

Additional information

This article is part of the Topical Collection on Decomposition at 70

Appendix

Appendix

Detailed results of the computational experiments reported in Section 4. The instances and meaning of the column headers are the same as in the body of the paper. We use “tl” in column Ttotal to indicate that the time limit was reached by the corresponding solution approach. Additionally, we highlight in bold when the proposed CG-based heuristic achieved the best result.

Table 3 Results of the state-of-the-art BC methods and the CG-based heuristic on the SDVRPTW instances with n = 50 and Q = 50
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Table 4 Results of the state-of-the-art BC methods and the CG-based heuristic on the SDVRPTW instances with n = 50 and Q = 100
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Table 5 Results of the state-of-the-art BC methods and the CG-based heuristic on the SDVRPTW instances with n = 100 and Q = 50
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Table 6 Results of the state-of-the-art BC methods and the CG-based heuristic on the SDVRPTW instances with n = 100 and Q = 100
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Munari, P., Savelsbergh, M. A Column Generation-Based Heuristic for the Split Delivery Vehicle Routing Problem with Time Windows. SN Oper. Res. Forum 1, 26 (2020). https://doi.org/10.1007/s43069-020-00026-z

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  • DOI: https://doi.org/10.1007/s43069-020-00026-z

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