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Dynamic Programming for the Minimum Tour Duration Problem

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  • Christian Tilk

    (Gutenberg School of Management and Economics, Johannes Gutenberg University Mainz, D-55128 Mainz, Germany)

  • Stefan Irnich

    (Gutenberg School of Management and Economics, Johannes Gutenberg University Mainz, D-55128 Mainz, Germany)

Abstract

The minimum tour duration problem (MTDP) is a variant of the traveling salesman problem with time windows, which consists of finding a time window-feasible Hamiltonian path minimizing the tour duration. We present a new effective dynamic programming (DP)-based approach for the MTDP. When solving the traveling salesman problem with time windows with DP, two independent resources are propagated along partial paths, one for costs and one for earliest arrival times. For dealing with tour duration minimization, we provide a new DP formulation with three resources for which effective dominance and bounding procedures are applicable. This is a nontrivial task because in the MTDP at least two resources depend on each other in a nonadditive and nonlinear way. In particular, we define consistent resource extension functions (REFs) so that dominance is straightforward using componentwise comparison for the respective resource vectors. Moreover, one of the main advantages of the new REF definition is that the DP can be reversed consistently such that the forward DP or any of its relaxations provide bounds for the backward DP, and vice versa. Computational tests confirm the effectiveness of the proposed approach.

Suggested Citation

  • Christian Tilk & Stefan Irnich, 2017. "Dynamic Programming for the Minimum Tour Duration Problem," Transportation Science, INFORMS, vol. 51(2), pages 549-565, May.
    • Handle: RePEc:inm:ortrsc:v:51:y:2017:i:2:p:549-565
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    Cited by:

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    4. Ann-Kathrin Rothenbächer, 2017. "Branch-and-Price-and-Cut for the Periodic Vehicle Routing Problem with Flexible Schedule Structures," Working Papers 1714, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.
    5. Ann-Kathrin Rothenbächer, 2019. "Branch-and-Price-and-Cut for the Periodic Vehicle Routing Problem with Flexible Schedule Structures," Transportation Science, INFORMS, vol. 53(3), pages 850-866, May.
    6. Jeanette Schmidt & Christian Tilk & Stefan Irnich, 2023. "Exact Solution of the Vehicle Routing Problem With Drones," Working Papers 2311, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.
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    8. Fuying Liu & Chen Liu & Qi Zhao & Chenhao He, 2021. "A Hybrid Teaching-Learning-Based Optimization Algorithm for the Travel Route Optimization Problem alongside the Urban Railway Line," Sustainability, MDPI, vol. 13(3), pages 1-17, January.
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    10. Selvaprabu Nadarajah & Andre A. Cire, 2020. "Network-Based Approximate Linear Programming for Discrete Optimization," Operations Research, INFORMS, vol. 68(6), pages 1767-1786, November.
    11. Vu, Duc Minh & Hewitt, Mike & Vu, Duc D., 2022. "Solving the time dependent minimum tour duration and delivery man problems with dynamic discretization discovery," European Journal of Operational Research, Elsevier, vol. 302(3), pages 831-846.

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