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The Cost-Balanced Path Problem: A Mathematical Formulation and Complexity Analysis

Author

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  • Daniela Ambrosino

    (Department of Economics and Business Studies, University of Genoa, 16126 Genoa, Italy
    These authors contributed equally to this work.)

  • Carmine Cerrone

    (Department of Economics and Business Studies, University of Genoa, 16126 Genoa, Italy
    These authors contributed equally to this work.)

Abstract

This paper introduces a new variant of the Shortest Path Problem ( S P P ) called the Cost-Balanced Path Problem ( C B P P ). Various real problems can either be modeled as B C P P or include B C P P as a sub-problem. We prove several properties related to the complexity of the C B P P problem. In particular, we demonstrate that the problem is NP-hard in its general version, but it becomes solvable in polynomial time in a specific family of instances. Moreover, a mathematical formulation of the C B P P , as a mixed-integer programming model, is proposed, and some additional constraints for modeling real requirements are given. This paper validates the proposed model and its extensions with experimental tests based on random instances. The analysis of the results of the computational experiments shows that the proposed model and its extension can be used to model many real applications. Obviously, due to the problem complexity, the main limitation of the proposed approach is related to the size of the instances. A heuristic solution approach should be required for larger-sized and more complex instances.

Suggested Citation

  • Daniela Ambrosino & Carmine Cerrone, 2022. "The Cost-Balanced Path Problem: A Mathematical Formulation and Complexity Analysis," Mathematics, MDPI, vol. 10(5), pages 1-13, March.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:5:p:804-:d:763262
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    References listed on IDEAS

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    1. Zhang, Dongqing & Wallace, Stein W. & Guo, Zhaoxia & Dong, Yucheng & Kaut, Michal, 2021. "On scenario construction for stochastic shortest path problems in real road networks," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 152(C).
    2. Prakash, A. Arun, 2018. "Pruning algorithm for the least expected travel time path on stochastic and time-dependent networks," Transportation Research Part B: Methodological, Elsevier, vol. 108(C), pages 127-147.
    3. Dell'Amico, Mauro & Hadjicostantinou, Eleni & Iori, Manuel & Novellani, Stefano, 2014. "The bike sharing rebalancing problem: Mathematical formulations and benchmark instances," Omega, Elsevier, vol. 45(C), pages 7-19.
    4. He, Fang & Yin, Yafeng & Lawphongpanich, Siriphong, 2014. "Network equilibrium models with battery electric vehicles," Transportation Research Part B: Methodological, Elsevier, vol. 67(C), pages 306-319.
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    Cited by:

    1. Antonin Ponsich & Bruno Domenech & Mariona Vilà, 2023. "Preface to the Special Issue “Mathematical Optimization and Evolutionary Algorithms with Applications”," Mathematics, MDPI, vol. 11(10), pages 1-6, May.

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