Skip to main content
SpringerLink
Log in

On the recoverable robust traveling salesman problem

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

We consider an uncertain traveling salesman problem, where distances between nodes are not known exactly, but may stem from an uncertainty set of possible scenarios. This uncertainty set is given as intervals with an additional bound on the number of distances that may deviate from their expected, nominal values. A recoverable robust model is proposed, that allows a tour to change a bounded number of edges once a scenario becomes known. As the model contains an exponential number of constraints and variables, an iterative algorithm is proposed, in which tours and scenarios are computed alternately. While this approach is able to find a provably optimal solution to the robust model, it also needs to solve increasingly complex subproblems. Therefore, we also consider heuristic solution procedures based on local search moves using a heuristic estimate of the actual objective function. In computational experiments, these approaches are compared.

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

Fig. 1

Similar content being viewed by others

Notes

  1. Naturally, many other meta-heuristic approaches are possible here. A simulated annealing algorithm has also been tested; however, results are not discussed, as tabu search showed a better performance.

References

  1. Aissi, H., Bazgan, C., Vanderpooten, D.: Mimax and minmax regret versions of combinatorial optimization problems: a survey. Eur. J. Oper. Res. 197(2), 427–438 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bouman, P.C., Akker, J.M., van den Hoogeveen, J.A.: Recoverable robustness by column generation. In: European symposium on algorithms. Lecture Notes in Computer Science, vol. 6942, pp. 215–226. Springer (2011)

  3. Bertsimas, D., Brown, D., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53(3), 464–501 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ben-Tal, A., Goryashko, A., Guslitzer, E., Nemirovski, A.: Adjustable robust solutions of uncertain linear programs. Math. Programm. A 99, 351–376 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust optimization. Princeton University Press, Princeton, Oxford (2009)

    Book  MATH  Google Scholar 

  7. Büsing, C.: The exact subgraph recoverable robust shortest path problem. In: Ahuja, R.K., Möhring, R.H., Zaroliagis, C.D. (eds.) Robust and online large-scale optimization. Lecture notes in computer science, vol. 5868, pp. 231–248. Springer, Berlin, Heidelberg (2009)

  8. Büsing, C.: Recoverable robust shortest path problems. Networks 59(1), 181–189 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fischetti, M., Monaci, M.: Cutting plane versus compact formulations for uncertain (integer) linear programs. Math. Programm. Comput. 4(3), 239–273 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Goerigk, M. Heße, S., Müller-Hannemann, M., Schmidt, M., Schöbel, A.: Recoverable robust timetable information. In: Frigioni, D., Stiller, S. (eds.) Proc. of ATMOS 13. OASIcs, vol. 33, pp. 1–14 (2013)

  11. Goerigk, M.: A note on upper bounds to the robust knapsack problem with discrete scenarios. Ann. Oper. Res. 223(1), 461–469 (2014)

  12. Goerigk, M., Schöbel, A.: Recovery-to-optimality: a new two-stage approach to robustness with an application to aperiodic timetabling. Comput. Oper. Res. 52(A(0)), 1 – 15 (2014)

  13. Goerigk, M., Schöbel, A.: Algorithm engineering in robust optimization. LNCS State-of-the-Art Surveys Springer (2015, To appear)

  14. Helsgaun, K.: An effective implementation of the Lin-Kernighan traveling salesman heuristic. Eur. J. Oper. Res. 126(1), 106–130 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kouvelis, P., Yu, G.: Robust discrete optimization and its applications. Kluwer Academic Publishers (1997)

  16. Lawler, E.L., Lenstra, J.K., Kan, A.H.G.R., Shmoys, D.B.: The traveling salesman problem: a guided tour of combinatorial optimization, vol. 3. Wiley, New York (1985)

  17. Liebchen, C., Lübbecke, M., Möhring, R.H., Stiller, S.: The concept of recoverable robustness, linear programming recovery, and railway applications. In: Ahuja, R.K., Möhring, R.H., Zaroliagis, C.D., (eds.) Robust and online large-scale optimization. Lecture Note on Computer Science, vol. 5868, pp. 1–27. Springer (2009)

  18. Montemanni, R., Barta, J., Mastrolilli, M., Gambardella, L.M.: The robust traveling salesman problem with interval data. Transp. Sci. 41(3), 366–381 (2007)

    Article  Google Scholar 

  19. Montemanni, R.: A benders decomposition approach for the robust spanning tree problem with interval data. Eur. J. Oper. Res. 174(3), 1479–1490 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zeng, B., Zhao, L.: Solving two-stage robust optimization problems using a column-and-constraint generation method. Oper. Res. Lett. 41(5), 457–461 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Effort sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under Grant Number FA8655-13-1-3066. The U.S Government is authorized to reproduce and distribute reprints for Governmental purpose notwithstanding any copyright notation thereon.

Author information

Authors and Affiliations

  1. Technische Universität Kaiserslautern, Paul-Ehrlich-Str. 14, 67653, Kaiserslautern, Germany

    André Chassein & Marc Goerigk

Authors
  1. André Chassein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marc Goerigk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marc Goerigk.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chassein, A., Goerigk, M. On the recoverable robust traveling salesman problem. Optim Lett 10, 1479–1492 (2016). https://doi.org/10.1007/s11590-015-0949-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-015-0949-5

Keywords

Search

Navigation