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New cutting-planes for the time- and/or precedence-constrained ATSP and directed VRP

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Abstract

In this paper, we introduce five classes of new valid cutting planes for the precedence-constrained (PC) and/or time-window-constrained (TW) Asymmetric Travelling Salesman Problems (ATSPs) and directed Vehicle Routing Problems (VRPs). We show that all five classes of new inequalities are facet-defining for the directed VRP-TW, under reasonable conditions and the assumption that vehicles are identical. Similar proofs can be developed for the VRP-PC. As ATSP-TW and PC-ATSP can be formulated as directed identical-vehicle VRP-TW and PC-VRP, respectively, this provides a link to study the polyhedral combinatorics for the ATSP-TW and PC-ATSP. The first four classes of these new cutting planes are cycle-breaking inequalities that are lifted from the well-known \({D^-_k}\) and \({D^+_k}\) inequalities (see Grötschel and Padberg in Polyhedral theory. The traveling salesman problem: a guided tour of combinatorial optimization, Wiley, New York, 1985). The last class of new cutting planes, the TW 2 inequalities, are infeasible-path elimination inequalities. Separation of these constraints will also be discussed. We also present prelimanry numerical results to demonstrate the strengh of these new cutting planes.

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Authors and Affiliations

  1. School of Engineering and Information Technology, Deakin University, Burwood, VIC, 3125, Australia

    Vicky Mak

  2. Mathematical and Information Science, CSIRO, Private Bag 33, Clayton, VIC, 3169, Australia

    Andreas T. Ernst

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  1. Vicky Mak
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  2. Andreas T. Ernst
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Correspondence to Vicky Mak.

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Mak, V., Ernst, A.T. New cutting-planes for the time- and/or precedence-constrained ATSP and directed VRP. Math Meth Oper Res 66, 69–98 (2007). https://doi.org/10.1007/s00186-006-0141-x

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  • DOI: https://doi.org/10.1007/s00186-006-0141-x

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