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Heuristic Approaches for the Probabilistic Traveling Salesman Problem

  • Christoph Weiler
  • Benjamin Biesinger
  • Bin Hu
  • Günther R. Raidl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9520)

Abstract

The Probabilistic Traveling Salesman Problem (PTSP) is a variant of the classical Traveling Salesman Problem (TSP) where each city has a given probability requiring a visit. We aim for an a-priori tour including every city that minimizes the expected length over all realizations. In this paper we consider different heuristic approaches for the PTSP. First we analyze various popular construction heuristics for the classical TSP applied on the PTSP: nearest neighbor, farthest insertion, nearest insertion, radial sorting and space filling curve. Then we investigate their extensions to the PTSP: almost nearest neighbor, probabilistic farthest insertion, probabilistic nearest insertion. To improve the constructed solutions we use existing 2-opt and 1-shift neighborhood structures for which exact delta evaluation formulations exist. These are embedded within a Variable Neighborhood Descent framework into a Variable Neighborhood Search. Computational results indicate that this approach is competitive to already existing heuristic algorithms and able to find good solutions in low runtime.

Keywords

Probabilistic traveling salesman problem Variable neighborhood search Construction heuristics 

Notes

Acknowledgments

The authors thank Dennis Weyland for providing the source code of his EACS for better comparison.

References

  1. 1.
    Balaprakash, P., Birattari, M., Stützle, T., Dorigo, M.: Adaptive sample size and importance sampling in estimation-based local search for the probabilistic traveling salesman problem. Eur. J. Oper. Res. 199(1), 98–110 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Balaprakash, P., Birattari, M., Stützle, T., Yuan, Z., Dorigo, M.: Estimation-based ant colony optimization and local search for the probabilistic traveling salesman problem. Swarm Intel. 3(3), 223–242 (2009)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bartholdi III, J.J., Platzman, L.K.: An \({O}(n log n)\) planar travelling salesman heuristic based on spacefilling curves. Oper. Res. Lett. 1(4), 121–125 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bertsimas, D., Howell, L.H.: Further results on the probabilistic traveling salesman problem. Eur. J. Oper. Res. 65(1), 68–95 (1993)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bertsimas, D.J., Chervi, P., Peterson, M.: Computational approaches to stochastic vehicle routing problems. Transp. Sci. 29(4), 342–352 (1995)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bertsimas, D.J., Jaillet, P., Odoni, A.R.: A priori optimization. Oper. Res. 38(6), 1019–1033 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bianchi, L., Gambardella, L.M., Dorigo, M.: An ant colony optimization approach to the probabilistic traveling salesman problem. In: Guervós, J.J.M., Adamidis, P.A., Beyer, H.-G., Fernández-Villacañas, J.-L., Schwefel, H.-P. (eds.) PPSN 2002. LNCS, vol. 2439, pp. 883–892. Springer, Heidelberg (2002) Google Scholar
  8. 8.
    Bianchi, L., Knowles, J., Bowler, N.: Local search for the probabilistic traveling salesman problem: correction to the 2-p-opt and 1-shift algorithms. Eur. J. Oper. Res. 162(1), 206–219 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chervi, P.: A computational approach to probabilistic vehicle routing problems. Master’s thesis, Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science (1988)Google Scholar
  10. 10.
    Jaillet, P.: Probabilistic traveling salesman problems. Ph.D. thesis, Massachusetts Institute of Technology (1985)Google Scholar
  11. 11.
    Marinakis, Y., Marinaki, M.: A hybrid multi-swarm particle swarm optimization algorithm for the probabilistic traveling salesman problem. Comput. Oper. Res. 37(3), 432–442 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mladenović, N., Hansen, P.: Variable neighborhood search. Comput. Oper. Res. 24(11), 1097–1100 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
  14. 14.
    Weyland, D., Bianchi, L., Gambardella, L.M.: New approximation-based local search algorithms for the probabilistic traveling salesman problem. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds.) EUROCAST 2009. LNCS, vol. 5717, pp. 681–688. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  15. 15.
    Weyland, D., Montemanni, R., Gambardella, L.M.: An enhanced ant colony system for the probabilistic traveling salesman problem. In: Di Caro, G.A., Theraulaz, G. (eds.) BIONETICS 2012. LNICST, vol. 134, pp. 237–249. Springer, Heidelberg (2014) Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Christoph Weiler
    • 1
  • Benjamin Biesinger
    • 1
  • Bin Hu
    • 2
  • Günther R. Raidl
    • 1
  1. 1.Institute of Computer Graphics and AlgorithmsTU WienViennaAustria
  2. 2.Mobility Department - Dynamic Transportation SystemsAIT Austrian Institute of TechnologyViennaAustria

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