Abstract
In open vehicle routing problems, the vehicles are not required to return to the depot after completing service. In this paper, we present the first exact optimization algorithm for the open version of the well-known capacitated vehicle routing problem (CVRP). The algorithm is based on branch-and-cut. We show that, even though the open CVRP initially looks like a minor variation of the standard CVRP, the integer programming formulation and cutting planes need to be modified in subtle ways. Computational results are given for several standard test instances, which enables us for the first time to assess the quality of existing heuristic methods, and to compare the relative difficulty of open and closed versions of the same problem.
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Appendix A
Appendix A
To show validity of the mixed comb inequalities, it is helpful to prove the following lemma.
Lemma 1
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For any set S such that 0∈S, the following three inequalities are valid.
Proof
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Owing to the degree equations, the inequality (A.1) is equivalent to the capacity inequality on V\S and the inequalities (A.2) and (A.3) are equivalent to the balancing inequalities on V\S and S\{0}, respectively. □
Proof of Theorem 1
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We follow the standard Chvátal–Gomory integer rounding argument. If we sum together the following inequalities:
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the degree equations for all i∈H,
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the inequality (6) on H∩T i for 1⩽i⩽t,
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the inequality (6) on T i and T i \H for i∈𝒩,
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the inequalities (A.1) for T i and T i \H, for i∈𝒟,
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the inequalities (A.2) for T i and T i \H, for i∈𝒮, and
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the inequalities (A.3) for T i and T i \H, for i∈ℛ,
we obtain (after some re-arranging):
Dividing this inequality by two and rounding down yields the result. □
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Letchford, A., Lysgaard, J. & Eglese, R. A branch-and-cut algorithm for the capacitated open vehicle routing problem. J Oper Res Soc 58, 1642–1651 (2007). https://doi.org/10.1057/palgrave.jors.2602345
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DOI: https://doi.org/10.1057/palgrave.jors.2602345