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An efficient branch-and-cut algorithm for the parallel drone scheduling traveling salesman problem

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Abstract

This paper proposes an efficient branch-and-cut algorithm to exactly solve the parallel drone scheduling traveling salesman problem. The problem is first formulated as a mixed integer linear program with truck-flow variables defined on undirected edges, not on directed arcs as in existing models. The formulation is then strengthened by valid inequalities and the branch-and-cut algorithm is developed. The experimental results show that our algorithm can find optimal solutions for all existing instances, but two in a reasonable running time. To make the problem more challenging for future solution methods, we introduce two new sets of 120 larger instances with the number of customers varying from 318 to 783 and test our algorithm and investigate the performance of state-of-the-art metaheuristics on these instances. We show that the proposed algorithm can steadily solve the instances with up to 400 customers to optimality. Optimal solutions of several cases with 600 and 783 customers are also found by our algorithm. This is the first time problems of such a large size are optimally solved.

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Acknowledgements

This research was supported by the Science & Technology 4.0 Program grant funded by the Vietnam Ministry of Science & Technology (MOST) (No. ĐTCT-KC-4.003/19-25). The manuscript of the paper was finished during the research stay of the corresponding author (Minh Hoàng Hà) at the Vietnamese Institute for Advanced Studies in Mathematics (VIASM). He wishes to thank this institution for their kind hospitality and support.

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Authors and Affiliations

  1. ORLab, Faculty of Computer Science, Phenikaa University, Hanoi, Vietnam

    Minh Anh Nguyen, Hai Long Luong & Minh Hoàng Hà

  2. School of Information and Communication Technology, Hanoi University of Science and Technology, Hanoi, Vietnam

    Ha-Bang Ban

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  1. Minh Anh Nguyen
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  2. Hai Long Luong
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  3. Minh Hoàng Hà
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  4. Ha-Bang Ban
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Correspondence to Minh Hoàng Hà.

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Appendix

Appendix

The following tables report the detailed results for the experiment. The values in bold present the best known solutions found by all the three methods while the symbols ‘\(*\)’ imply the optimal solutions proved by the B &C. The column headings are as follows:

  • Instance: the characteristics of instances;

  • el: the percentage of drone-eligible customers;

  • sp: the drone speed;

  • \(\#\): the number of drones;

  • dp: the position of the depot;

  • HACO, SISSR, B &C: the objective values of the best solutions found by HACO, SISSR, and B &C, respectively (Tables 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15). The numbers marked with “*” imply the optimal solutions proved by the B &C;

  • gap: the gap values returned by CPLEX when performing the B &C;

  • time: the running time in seconds of the B &C;

  • node size: the number of nodes in the branch-and-bound search tree generated by CPLEX;

  • Number of added constraints: the number of constraints for each type added to the model during the search of the B &C.

Table 4 Results on the TSPLIB instance att48 from Mbiadou Saleu et al. (2021)
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Table 5 Results on the TSPLIB instance berlin52 from Mbiadou Saleu et al. (2021)
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Table 6 Results on the TSPLIB instance eil101 from Mbiadou Saleu et al. (2021)
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Table 7 Results on the TSPLIB instance gr120 from Mbiadou Saleu et al. (2021)
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Table 8 Results on the TSPLIB instance pr152 from Mbiadou Saleu et al. (2021)
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Table 9 Results on the TSPLIB instance gr229 from Mbiadou Saleu et al. (2021)
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Table 10 Results on the new TSPLIB instance lin318
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Table 11 Results on the new TSPLIB instance pr439
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Table 12 Results on the new TSPLIB instance rat575
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Table 13 Results on the new TSPLIB instance rat783
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Table 14 Results on the new randomly generated instances with 400 customers
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Table 15 Results on the new randomly generated instances with 600 customers
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Nguyen, M.A., Luong, H.L., Hà, M.H. et al. An efficient branch-and-cut algorithm for the parallel drone scheduling traveling salesman problem. 4OR-Q J Oper Res 21, 609–637 (2023). https://doi.org/10.1007/s10288-022-00527-z

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  • DOI: https://doi.org/10.1007/s10288-022-00527-z

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