Skip to main content

Advertisement

SpringerLink
Log in

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

  • Original Article
  • Published:
Mathematical Methods of Operations Research Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Achuthan NR, Caccetta L, Hill SP (1998) Capacitated vehicle routing problems: some new cutting planes. Asia Pac J Operat Res 15:109–123

    MATH  MathSciNet  Google Scholar 

  • Ascheuer N (1995) Hamiltonian path problems in the on-line optimization of flexible manufacturing systems. PhD thesis, Technische Universität Berlin, Germany

  • Ascheuer N, Fischetti M, Grötschel M (2000a) A polyhedral study of the asymmetric travelling salesman problem with time-windows. Networks 36:69–79

    Article  MATH  MathSciNet  Google Scholar 

  • Ascheuer N, Jünger N, Reinelt G (2000b) Branch and cut algorithm for the asymmetric travelling salesman problem with precedence constraints. Comput Optim Appl 17:61–84

    Article  MATH  MathSciNet  Google Scholar 

  • Ascheuer N, Fischetti M, Grötschel M (2001) Solving the asymmetric travelling salesman problem with time-windows by branch-and-cut. Math Program Ser A 90:475–506

    Article  MATH  Google Scholar 

  • Balas E (1999) New classes of efficiently solvable generalised travelling salesman problems. Ann Operat Res 86:529–558

    Article  MATH  MathSciNet  Google Scholar 

  • Balas E, Fischetti M (1993) A lifting procedure for the asymmetric travelling salesman problem and a large new class of facets. Math Program 58:325–352

    Article  MathSciNet  Google Scholar 

  • Balas E, Fischetti M (1997) On the monotonization of polyhedra. Math Program 78:59–84

    Article  MathSciNet  Google Scholar 

  • Balas E, Fischetti M (1999) Lifted cycle inequalities for the asymmetric travelling salesman problem. Math Operat Res 24(2):273–292

    MATH  MathSciNet  Google Scholar 

  • Balas E, Simonetti N (2001) Linear time dynamic-programming algorithms for new classes of restricted tsps: A computational study. INFORMS J Comput 13(1):56–75

    Article  MathSciNet  Google Scholar 

  • Balas E, Fischetti M, Pulleyblank WR (1989) The asymmetric assignment problem and some new facets of the travelling salesman polytope on a directed graph. SIAM J Disc Math 2(4):425–451

    Article  MATH  Google Scholar 

  • Balas E, Fischetti M, Pulleyblank WR (1995) The precedence-constrained asymmetric travelling salesman polytope. Math Program 68:241–265

    MathSciNet  Google Scholar 

  • Bard JF, Kontoravdis G, Gang Y (1999) A branch-and-cut procedure for the vehicle routing problem with time windows. Transp Sci 36:250–269

    Article  Google Scholar 

  • Belenguer JM, Martinez MC, Mota E (2000) A lower bound for the split delivery vehicle routing problem. Operat Res 48(5):801–810

    Article  MATH  MathSciNet  Google Scholar 

  • Berger J, Barkaoui M (2004) A parallel hybrid genetic algorithm for the vehicle routing problem with time windows. Comput Operat Res 31(12):2037–2053

    Article  MATH  MathSciNet  Google Scholar 

  • Bräysy O, Hasle G, Dullaert W (2003) A reactive variable neighborhood search for the vehicle routing problem with time windows. INFORMS J Comput 15(4):347–368

    Article  MathSciNet  Google Scholar 

  • Bräysy O, Hasle G, Dullaert W (2004) A multi-start local search algorithm for the vehicle routing problem with time windows. Eur J Operat Res 159(3):586–605

    Article  MATH  Google Scholar 

  • Chaovalitwongse W, Kim D, Pardalos P (2003) Grasp with a new local search scheme for vehicle routing problem with time windows. J Comb Optim 7(2):179–207

    Article  MATH  MathSciNet  Google Scholar 

  • Cornuejols G, Harche F (1993) Polyhedral study of the capacitated vehicle routing problem. Math Program 60:21–52

    Article  MathSciNet  Google Scholar 

  • Dantzig G, Fulkerson R, Johnson S (1954) Solution of a large scale travelling salesman problem. Operat Res 2:393–410

    MathSciNet  Google Scholar 

  • Desrochers M, Laporte G (1991) Improvements and extensions to the Miller-Tucker-Zemlin subtour elimination constraints. Operat Res Lett 10:27–36

    Article  MATH  MathSciNet  Google Scholar 

  • Desrochers M, Desrosiers J, Solomon MM (1992) A new optimization algorithm for the vehicle routing problem with time windows. Operat Res 40:342–354

    MATH  MathSciNet  Google Scholar 

  • Desrosiers J, Soumis F, Desrochers M, Sauvé M (1986) Vehicle routing and scheduling with time windows. Math Program Study 26:249–251

    MATH  Google Scholar 

  • Dumas Y, Desrosiers J, Gelinas E, Solomon MM (1995) An optimal algorithm for the traveling salesman problem with time windows. Operat Res 43(2):367–371

    MATH  MathSciNet  Google Scholar 

  • Ernst A, Mak V (2005) On the minimium makespan machine scheduling problem with sequence dependent setups and due dates. Tech. Report TR M05/01, School of Information Technology, Deakin University

  • Escudero LF, Guignard M, Malik K (1994) A lagrangian relax and cut approach for the sequential ordering with precedence constraints. Ann Operat Res 50:219–237

    Article  MATH  MathSciNet  Google Scholar 

  • Fischetti M (1991) Facets of the asymmetric travelling salesman polytope. Math Operat Res 16:42–56

    MATH  MathSciNet  Google Scholar 

  • Fischetti M (1995) Clique tree inequalities define facets of the asymmetric travelling salesman polytope. Disc Appl Math 56:9–18

    Article  MATH  MathSciNet  Google Scholar 

  • Fischetti M, Toth P (1997) A polyhedral approach to the asymmetric travelling salesman problem. Manage Sci 43(11):1520–1536

    MATH  Google Scholar 

  • Fisher ML, Jörnsten KO, Madsen OBG (1997) Vehicle routing problem with time windows: two optimization algorithms. Operat Res 45(3):488–492

    MATH  Google Scholar 

  • Focacci F, Lodi A, Milano M (2002) A hybrid exact algorithm for the atsptw. INFORMS J Comput 14(4):403–417

    Article  MathSciNet  Google Scholar 

  • Grötschel M, Padberg MW (1985) In: Lawler EL, Lenstra JK, Rinnooy Kan AHG, Shmoys DB (eds) Polyhedral theory The traveling salesman problem: a guided tour of combinatorial optimization. Wiley, New York

  • Kallehauge B, Boland N (2005) Path inequalities for the vehicle routing problem with time windows. Tech. report, Centre for Traffic and Transport, Technical University of Denmark

  • Kohl N, Madsen OBG (1997) An optimization algorithm for the vehicle routing problem with time windows based on lagrangean relaxation. Operat Res 45(3):395–406

    MATH  MathSciNet  Google Scholar 

  • Kohl N, Descrosiers J, Madsen OBG, Solomon MM, Soumis F (1999) 2-path cuts for the vehicle routing problem with time windows. Transp Sci 33:101–116

    MATH  Google Scholar 

  • Kolen AWJ, Rinnooy Kan AHG, Trienekens HWJM (1987) Vehicle routing problem with time windows. Operat Res 35(2):266–273

    MATH  MathSciNet  Google Scholar 

  • Laporte G, Norbert Y (1984) Comb inequalities for the vehicle routing problem. Methods Operat Res 51:271–276

    MATH  Google Scholar 

  • Larsen J (1999) Parallelization of the vehicle routing problem with time windows. PhD thesis, Department of Mathematical Modelling, Technical University of Danmark

  • Mak VH (2001) On the asymmetric travelling salesman problem with replenishment arcs. PhD thesis, The Department of Mathematics and Statistics, The University of Melbourne

  • Mingozzi A, Bianco L, Ricciardelli S (1997) Dynamic programming strategies for the traveling salesman problem with time window and precedence constraints. Operat Res 45(3):365–377

    MATH  MathSciNet  Google Scholar 

  • Nemhauser GL, Wolsey LA (1988) Integer and combinatorial optimization. Wiley, New York

    MATH  Google Scholar 

  • Schulze J, Fahle T (1999) A parallel algorithm for vehicle routing problem with time window constraints. Ann Operat Res 86:585–607

    Article  MATH  MathSciNet  Google Scholar 

  • Solomon MM (1987) Algorithms in solving vehicle routing and scheduling problems with time window constraints. Operat Res 35(2):254–265

    MATH  Google Scholar 

  • Solomon MM, Desrosiers J (1988) Time window constrained routing and scheduling problems. Transp Sci 22(1):1–13

    Article  MATH  MathSciNet  Google Scholar 

  • Tsitsklis JN (1992) Special cases of traveling salesman and repairman problems with time windows. Networks 22:263–282

    Article  MathSciNet  Google Scholar 

Download references

Author information

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

Authors
  1. Vicky Mak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andreas T. Ernst
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vicky Mak.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

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

Keywords

Search

Navigation