Solving Probabilistic Traveling Salesman Problem

  • Chapter
  • First Online:
The Traveling Salesman Problem
  • 155 Accesses

Abstract

Problem solving under uncertainty has a high impact on the real-world applications, since most real-world optimization problems are inherently dynamic and stochastic. Today, uncertainty and dynamism have become more relevant in many practical applications. This chapter describes how to use the ABSS to solve the probabilistic TSP, based on simulation.

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

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
EUR 29.95
Price includes VAT (Germany)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
EUR 37.44
Price includes VAT (Germany)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
EUR 48.14
Price includes VAT (Germany)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Applegate, D. L., Bixby, R. E., Chvátal, V., & Cook, W. J. (2006). The traveling salesman problem: A computational study. Princeton University Press.

    MATH  Google Scholar 

  2. Balaprakash, P., Piratari, M., Stützle, T., & Dorigo, M. (2009). Adaptive sample size and importance sampling in estimation-based local search for the probabilistic traveling salesman problem. European Journal of Operational Research, 199(1), 98–110. https://doi.org/10.1016/j.ejor.2008.11.027

    Article  MathSciNet  MATH  Google Scholar 

  3. Balaprakash, P., Piratari, M., Stützle, T., & Dorigo, M. (2010). Estimation-based metaheuristics for the probabilistic traveling salesman problem. Computer & Operations Research, 37(1), 1939–1951. https://doi.org/10.1016/j.cor.2009.12.005

    Article  MathSciNet  MATH  Google Scholar 

  4. Bertsimas, D. (1988). Probabilistic combinatorial optimization problems. PhD Dissertation. Boston: Department of Mathematics, Massachusetts Institute of Technology

    Google Scholar 

  5. Bertsimas, D., Jaillet, P., & Odoni, A. R. (1990). A priori optimization. Operations Research, 38(6), 1019–1033. https://doi.org/10.1287/opre.38.6.1019

    Article  MathSciNet  MATH  Google Scholar 

  6. Bertsimas, D., & Howell, L. H. (1993). Further results on the probabilistic traveling salesman problem. European Journal of Operational Research, 65(1), 68–95. https://doi.org/10.1016/0377-2217(93)90145-D

    Article  MATH  Google Scholar 

  7. Besu, M. M., & Raghavarao, D. (1990). Sample size methodology. Academic Press.

    Google Scholar 

  8. Binachi, L. (2006). Ant colony optimization and local search for the probabilistic traveling salesman problem: A case study in stochastic combinatorial optimization. Ph.D Dissertation. Brussels, Belgium: Universite Libre de Bruxelles

    Google Scholar 

  9. Binachi, L., & Campbell, A. M. (2007). Extension of the 2-p-opt and 1-shift algorithm to the heterogeneous probabilistic traveling salesman problem. European Journal of Operational Research, 176(1), 131–144. https://doi.org/10.1016/j.ejor.2005.05.027

    Article  MathSciNet  Google Scholar 

  10. Binachi, L., Dorigo, M., Gambardella, L. M., & Gutjahr, W. J. (2009). A survey on metaheuristics for stochastic combinatorial optimization. Natural Computing, 8, 239–287. https://doi.org/10.1007/s11047-008-9098-4

    Article  MathSciNet  MATH  Google Scholar 

  11. Binachi, L., Gambardella, L. M., & Dorigo, M. (2002). An ant colony optimization approach to the probabilistic traveling salesman problem. In: Proceedings of the 7th International Conference on Parallel Problem Solving from Nature, LNCS (vol. 2439, pp. 883–892). Berlin: Springer. https://doi.org/10.1007/3-540-45712-7_85

  12. Binachi, L., Gambardella, L. M., & Dorigo, M. (2002). Solving the homogeneous probabilistic traveling salesman problem by the ACO metaheuristic. In: M. Marco Dorigo, G. Di Caro, & M. Sampels (Eds.), Proceedings of the 3rd International Workshop on Ant Algorithms, LNCS (vol. 2463, pp. 176–187). London: Springer

    Google Scholar 

  13. Binachi, L., Knowles, J., & Bowler, N. E. (2005). Local search for the probabilistic traveling salesman problem: Correction to the 2-p-opt and 1-shift algorithms. European Journal of Operational Research, 162(1), 206–219. https://doi.org/10.1016/j.ejor.2003.10.016

    Article  MathSciNet  MATH  Google Scholar 

  14. Birattari, M., Balaprakash, P., & Dorigo, M. (2005). ACO/F-Race: ant colony optimization and racing techniques for combinatorial optimization under uncertainly. In: K. F. Doerner, M. Gendreau, P. Greistorfer, W. J. Gutjahr, R. F. Hartl, & M. Reimann (Eds.), Proceedings of the 6th Metaheuristics International Conference (pp. 107–112)

    Google Scholar 

  15. Birattari, M., Balaprakash, P., & Dorigo, M. (2006). The ACO-F-race algorithm for combinatorial optimization under uncertainty. In: K. F. Doerner, M. Gendreau, P. Greistorfer, W. J. Gutjahr, R. F. Hartl, & M. Reimann (Eds.), Metaheuristics–progress in complex systems optimization. operations research/computer science interfaces series (pp. 189–203). Berlin: Springer

    Google Scholar 

  16. Birattari, M., Balaprakash, P., Stützle, T., & Dorigo, M. (2008). Estimation-based local search for stochastic combinatorial optimization using delta evaluations: A case study on the probabilistic traveling salesman problem. INFORMS Journal on Computing, 20(4), 644–658. https://doi.org/10.1287/ijoc.1080.0276

    Article  MathSciNet  MATH  Google Scholar 

  17. Bowler, N. E., Fink, T. M., & Ball, R. C. (2003). Characterization of the probabilistic traveling salesman problem. Physical Review E, 68(3), 1–7. https://doi.org/10.1103/PhysRevE.68.036703

    Article  Google Scholar 

  18. Branke, J., & Guntsch, M. (2005). Solving the probabilistic TSP with ant colony optimization. Journal Mathematical Modeling and Algorithms, 3, 403–425. https://doi.org/10.1007/s10852-005-2585-z

    Article  MathSciNet  MATH  Google Scholar 

  19. Campbell, A. M. (2006). Aggregation for the probabilistic traveling salesman problem. Computers & Operations Research, 33(9), 2703–2724. https://doi.org/10.1016/j.cor.2005.02.024

    Article  MathSciNet  MATH  Google Scholar 

  20. Choi, J., Lee, J. H., & Realff, M. J. (2004). An algorithmic framework for improving heuristic solutions: Part II, a new version of the stochastic traveling salesman problem. Computers & Chemical Engineering, 28(8), 1297–1307. https://doi.org/10.1016/j.compchemeng.2003.09.002

    Article  Google Scholar 

  21. Gutjahr, W. J. (2003). A converging ACO algorithm for stochastic combinatorial optimization. In: A. Albrecht, K. Steinhöfel (Eds.), Proceedings of the 2nd Symposium on Stochastic Algorithms, Foundations and Applications, LNCS (vol. 2827, pp. 10–25). Berlin: Springer

    Google Scholar 

  22. Gutjahr, W. J. (2004). S-ACO: an ant-based approach to combinatorial optimization under uncertainty. In: M. Dorigo, M. Biraattari, C. Blum, L. M. Gambardella, F. Mondada, & T. Stützle (Eds.), Proceedings of the 4th International Workshop on Ant Colony Optimization and Swarm Intelligence, LNCS (vol. 3172, pp. 238–249). Berlin: Springer

    Google Scholar 

  23. Homem-de-Mello, T. (2003). Variable-sample methods for stochastic optimization. ACM Transactions on Modeling and Computer Simulation 13(2), 108–133.https://doi.org/10.1145/858481.858483

  24. Jaillet, P. (1985). Probabilistic traveling salesman problems. PhD Thesis. Massachusetts Institute of Technology

    Google Scholar 

  25. Jaillet, P. (1988). A priori solution of a traveling salesman problem in which a random subset of the customers are visited. Operations Research 36(6), 929–936.https://doi.org/10.1287/opre.36.6.929

  26. Jaillet, P. (1993). Analysis of probabilistic combinatorial optimization problems in Euclidean spaces. Mathematics of Operations Research, 18(1), 51–70

    Google Scholar 

  27. Jaillet, P., & Odoni, A. R. (1988). The probabilistic vehicle routing problem. Vehicle Routing: Methods and Studies (pp. 293–318). North-Holland.

    MATH  Google Scholar 

  28. Jézéquel, A. (1985). Probabilistic Vehicle Routing Problems. Master thesis. Boston: Massachusetts Institute of Technology

    Google Scholar 

  29. Kleywegy, A. J., Shapiro, A., & Homem-de-Mello, T. (2001). The sample average approximation method for stochastic discrete optimization. SIAM Journal on Optimization, 12(2), 479–502. https://doi.org/10.1137/S1052623499363220

    Article  MathSciNet  MATH  Google Scholar 

  30. Laporte, G., Louveaux, F. V., & Mercure, H. (1994). A priori optimization of the probabilistic traveling salesman problem. Operations Research, 42(3), 543–549. https://doi.org/10.1287/opre.42.3.543

    Article  MathSciNet  MATH  Google Scholar 

  31. Lindley, D. V. (1997). The choice of sample size. The Statistician, 46(2), 129–138. https://doi.org/10.1111/1467-9884.00068

    Article  Google Scholar 

  32. Liu, Y.-H. (2007). A hybrid scatter search for the probabilistic traveling salesman problem. Computers & Operations Research, 34(10), 2949–2963. https://doi.org/10.1016/j.cor.2005.11.008

    Article  MATH  Google Scholar 

  33. Liu, Y.-H. (2008). Diversified local search strategy under scatter search framework for the probabilistic traveling salesman problem. European Journal of Operational Research, 192(2), 332–346. https://doi.org/10.1016/j.ejor.2007.08.023

    Article  MATH  Google Scholar 

  34. Liu, Y.-H. (2008). Solving the probabilistic traveling salesman problem based on genetic algorithm with queen selection scheme. In F. Greco (Ed.), Traveling Salesman Problem (pp. 157–172). Intech.

    Google Scholar 

  35. Liu, Y. -H. (2008). A memetic algorithm for the probabilistic traveling salesman problem. In: IEEE Congress on Evolutionary Computation (CEC2008), pp. 146–152, IEEE Press

    Google Scholar 

  36. Liu, Y.-H., Jou, R.-C., Wang, C.-C., & Chiu, C.-S. (2007). An evolutionary algorithm with diversified crossover operator for the heterogeneous probabilistic TSP. In J. G. Carbonell & J. Siekmann (Eds.), Modeling decisions for artificial intelligence MDAI 2007, LNCS (Vol. 4617, pp. 351–360). Springer.

    Google Scholar 

  37. Marinakis, Y., & Marinaki, M. Y. (2010). A hybrid multi-swarm particle swarm optimization algorithm for the probabilistic traveling salesman problem. Computer & Operations Research, 37(3), 432–442. https://doi.org/10.1016/j.cor.2009.03.004

    Article  MathSciNet  MATH  Google Scholar 

  38. Marinakis, Y., Migdalas, A., & Pardalos, P. M. (2008). Expanding neighborhood search GRASP for the probabilistic traveling salesman problem. Optimization Letters, 2, 351–361. https://doi.org/10.1007/s11590-007-0064-3

    Article  MathSciNet  MATH  Google Scholar 

  39. Rossi, F., & Gavioli, F. (1987). Aspects of heuristic methods in the probabilistic traveling salesman problem. Advanced school on stochastic in combinatorial optimization (pp. 214–227). World Scientific.

    Google Scholar 

  40. Shapiro, A., & Homem-de-Mello, T. (1998). A simulation-based approach to two-stage stochastic programming with recourse. Mathematical Programming 81, 301–325.https://doi.org/10.1007/BF01580086

  41. Sudman, S. (1976). Applied sampling. Academic Press.

    Google Scholar 

  42. Verweij, B., Ahmed, S., Kleywegt, A. J., Nemhauser, G., & Shapiro, A. (2003). The sample average approximation method applied to stochastic routing problems: A computational study. Computational Optimization and Applications, 24, 289–333. https://doi.org/10.1023/A:1021814225969

    Article  MathSciNet  MATH  Google Scholar 

  43. Walson. (2001). How to determine a sample size. Tipsheet #60. Penn State Cooperative extension, University Park

    Google Scholar 

  44. Weiler, C., Biesinger, B., Hu, B., & Raidl, G. R. (2015). Heuristic approaches for the probabilistic traveling salesman problem. In: Computer Aided Systems Theory–EUROCAST 2015. LNCS (vol. 9520, pp. 342–349). Berlin: Springer. https://doi.org/10.1007/978-3-319-27340-2_43

  45. Weyland, D., Bianchi, L., & Gambardella, L. M. (2009). New approximation-based local search algorithms for the probabilistic traveling salesman problem. In: Computer Aided Systems Theory–EUROCAST 2009, LNCS (vol. 5717, pp. 681–688). Heidelberg: Springer. https://doi.org/10.1007/978-3-642-04772-5_88

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Weiqi Li .

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Li, W. (2024). Solving Probabilistic Traveling Salesman Problem. In: The Traveling Salesman Problem. Synthesis Lectures on Operations Research and Applications. Springer, Cham. https://doi.org/10.1007/978-3-031-35719-0_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-35719-0_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-35718-3

  • Online ISBN: 978-3-031-35719-0

  • eBook Packages: Synthesis Collection of Technology (R0)

Publish with us

Policies and ethics

Navigation