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

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  • Vicky Mak
  • Andreas Ernst

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. Copyright Springer-Verlag 2007

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  • Vicky Mak & Andreas Ernst, 2007. "New cutting-planes for the time- and/or precedence-constrained ATSP and directed VRP," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(1), pages 69-98, August.
    • Handle: RePEc:spr:mathme:v:66:y:2007:i:1:p:69-98
    DOI: 10.1007/s00186-006-0141-x
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