Skip to main content
SpringerLink
Log in

Fitness Landscape Ruggedness Impact on PSO in Dealing with Three Variants of the Travelling Salesman Problem

  • Conference paper
  • First Online:
Learning and Intelligent Optimization (LION 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13621))

Included in the following conference series:

  • 532 Accesses

Abstract

Fitness landscape analysis has gained quite some attention in understanding the behaviour of metaheuristics. Swarm intelligence is a type of metaheuristics that has grown considerably on the algorithmic side over the past decade. Nevertheless, only little attention has been paid to understanding the behaviour of algorithms on different fitness landscapes, especially in combinatorial optimization. Our aim in this paper is to re-motivate the importance of this issue. Moreover, by considering particle swarm optimization (PSO), we present a first investigation on its adaptation to three variants of the travelling salesman problem and how its performance is correlated with the ruggedness of the problem instances. The results show that PSO performance deteriorates with the increase in the number of cities and the ruggedness of the instances.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aiex, R.M., Resende, M.G., Ribeiro, C.C.: TTT plots: a Perl program to create time-to-target plots. Optim. Lett. 1(4), 355–366 (2007). https://doi.org/10.1007/s00291-020-00604-x

  2. de Armas, J., Lalla-Ruiz, E., Tilahun, S.L., Voß, S.: Similarity in metaheuristics: a gentle step towards a comparison methodology. Natural Comput. 21, 265–287 (2021). https://doi.org/10.1007/s11047-020-09837-9

  3. Bernardino, R., Paias, A.: Heuristic approaches for the family traveling salesman problem. Int. Trans. Oper. Res. 28(1), 262–295 (2021). https://doi.org/10.1111/itor.12771

    Article  MathSciNet  Google Scholar 

  4. Boese, K.D., Kahng, A.B., Muddu, S.: A new adaptive multi-start technique for combinatorial global optimizations. Oper. Res. Lett. 16(2), 101–113 (1994). https://doi.org/10.1016/0167-6377(94)90065-5

    Article  MathSciNet  MATH  Google Scholar 

  5. Camacho Villalón, C.L., Stützle, T., Dorigo, M.: Grey wolf, firefly and bat algorithms: three widespread algorithms that do not contain any novelty. In: Dorigo, M., et al. (eds.) ANTS 2020. LNCS, vol. 12421, pp. 121–133. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-60376-2_10

    Chapter  Google Scholar 

  6. Cárdenas-Montes, M.: Creating hard-to-solve instances of travelling salesman problem. Appl. Soft Comput. 71, 268–276 (2018). https://doi.org/10.1016/j.asoc.2018.07.010

    Article  Google Scholar 

  7. Caserta, M., Voß, S.: Metaheuristics: intelligent problem solving. In: Maniezzo, V., Stützle, T., Voß, S. (eds.) Matheuristics. Annals of Information Systems, vol. 10, pp. 1–38. Springer, Boston (2009). https://doi.org/10.1007/978-1-4419-1306-7_1

  8. Chen, W.N., Zhang, J., Chung, H., Zhong, W.L., Wu, W.G., Shi, Y.: A novel set-based particle swarm optimization method for discrete optimization problems. IEEE Trans. Evol. Comput. 14(2), 278–300 (2010). https://doi.org/10.1109/tevc.2009.2030331

    Article  Google Scholar 

  9. Daniel, W.W.: Applied Nonparametric Statistics. PWS-KENT Pub, Boston (1990)

    Google Scholar 

  10. Dokeroglu, T., Sevinc, E., Kucukyilmaz, T., Cosar, A.: A survey on new generation metaheuristic algorithms. Comput. Industr. Eng. 137, 106040 (2019). https://doi.org/10.1016/j.cie.2019.106040

    Article  Google Scholar 

  11. Engelbrecht, A.P., Bosman, P., Malan, K.M.: The influence of fitness landscape characteristics on particle swarm optimisers. Nat. Comput. (2021). https://doi.org/10.1007/s11047-020-09835-x

  12. Glover, F.: A template for scatter search and path relinking. In: Hao, J.-K., Lutton, E., Ronald, E., Schoenauer, M., Snyers, D. (eds.) AE 1997. LNCS, vol. 1363, pp. 1–51. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0026589

    Chapter  Google Scholar 

  13. Glover, F., Sörensen, K.: Metaheuristics. Scholarpedia 10(4), 6532 (2015). https://doi.org/10.4249/scholarpedia.6532

    Article  Google Scholar 

  14. Goldbarg, E.F.G., Goldbarg, M.C., de Souza, G.R.: Particle swarm optimization algorithm for the traveling salesman problem. In: Greco, F. (ed.) Traveling Salesman Problem, pp. 75–96. InTech (2008). https://doi.org/10.5772/5580

  15. Goldbarg, E.F.G., de Souza, G.R., Goldbarg, M.C.: Particle swarm for the traveling salesman problem. In: Gottlieb, J., Raidl, G.R. (eds.) EvoCOP 2006. LNCS, vol. 3906, pp. 99–110. Springer, Heidelberg (2006). https://doi.org/10.1007/11730095_9

    Chapter  Google Scholar 

  16. Greistorfer, P., Voß, S.: Controlled pool maintenance for metaheuristics. In: Rego, C., Alidaee, B. (eds.) Metaheuristic Optimization via Memory and Evolution, pp. 387–424. Kluwer Academic Publishers (2005). https://doi.org/10.1007%2F0-387-23667-8_18

  17. Hains, D.R., Whitley, L.D., Howe, A.E.: Revisiting the big valley search space structure in the TSP. J. Oper. Res. Soc. 62(2), 305–312 (2011). https://doi.org/10.1057/jors.2010.116

    Article  Google Scholar 

  18. Huang, Y., Li, W., Tian, F., Meng, X.: A fitness landscape ruggedness multiobjective differential evolution algorithm with a reinforcement learning strategy. Appl. Soft Comput. 96, 106693 (2020). https://doi.org/10.1016/j.asoc.2020.106693

    Article  Google Scholar 

  19. Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of ICNN 1995 - International Conference on Neural Networks. IEEE (1995). https://doi.org/10.1109/icnn.1995.488968

  20. Kennedy, J., Eberhart, R.: A discrete binary version of the particle swarm algorithm. In: IEEE International Conference on Systems, Man, and Cybernetics. Computational Cybernetics and Simulation. IEEE (1997). https://doi.org/10.1109/icsmc.1997.637339

  21. Liefooghe, A., Daolio, F., Verel, S., Derbel, B., Aguirre, H., Tanaka, K.: Landscape-aware performance prediction for evolutionary multi-objective optimization. IEEE Trans. Evol. Comput. 24(6), 1063–1077 (2019)

    Article  Google Scholar 

  22. Lloyd, S.: Least squares quantization in PCM. IEEE Trans. Inf. Theory 28(2), 129–137 (1982). https://doi.org/10.1109%2Ftit.1982.1056489

  23. Lu, Y., Hao, J.K., Wu, Q.: Solving the clustered traveling salesman problem via TSP methods. arXiv preprint arXiv:2007.05254 (2020). https://doi.org/10.48550/arXiv.2007.05254

  24. Malan, K.M., Engelbrecht, A.P.: Characterising the searchability of continuous optimisation problems for PSO. Swarm Intell. 8(4), 275–302 (2014). https://doi.org/10.1007/s11721-014-0099-x

    Article  Google Scholar 

  25. Malan, K.M., Engelbrecht, A.P.: Quantifying ruggedness of continuous landscapes using entropy. In: 2009 IEEE Congress on Evolutionary Computation. IEEE (2009). https://doi.org/10.1109/cec.2009.4983112

  26. Malan, K.M., Engelbrecht, A.P.: Ruggedness, funnels and gradients in fitness landscapes and the effect on PSO performance. In: 2013 IEEE Congress on Evolutionary Computation. IEEE (2013). https://doi.org/10.1109/cec.2013.6557671

  27. Malan, K.M., Engelbrecht, A.P.: A survey of techniques for characterising fitness landscapes and some possible ways forward. Inf. Sci. 241, 148–163 (2013). https://doi.org/10.1016/j.ins.2013.04.015

    Article  Google Scholar 

  28. Malan, K.M., Engelbrecht, A.P.: Fitness landscape analysis for metaheuristic performance prediction. In: Richter, H., Engelbrecht, A. (eds.) Recent Advances in the Theory and Application of Fitness Landscapes. ECC, vol. 6, pp. 103–132. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-41888-4_4

    Chapter  Google Scholar 

  29. Merz, P., Freisleben, B.: Fitness landscape analysis and memetic algorithms for the quadratic assignment problem. IEEE Trans. Evol. Comput. 4(4), 337–352 (2000). https://doi.org/10.1109/4235.887234

    Article  Google Scholar 

  30. Merz, P., Freisleben, B.: Memetic algorithms for the traveling salesman problem. Complex Syst. 13(4), 297–346 (2001)

    MathSciNet  MATH  Google Scholar 

  31. Ochoa, G., Veerapen, N.: Deconstructing the big valley search space hypothesis. In: Chicano, F., Hu, B., García-Sánchez, P. (eds.) EvoCOP 2016. LNCS, vol. 9595, pp. 58–73. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-30698-8_5

    Chapter  Google Scholar 

  32. Ochoa, G., Veerapen, N.: Mapping the global structure of TSP fitness landscapes. J. Heurist. 24(3), 265–294 (2017). https://doi.org/10.1007/s10732-017-9334-0

    Article  Google Scholar 

  33. Pitzer, E., Affenzeller, M.: A comprehensive survey on fitness landscape analysis. In: Fodor, J., Klempous, R., Suárez Araujo, C.P. (eds.) Recent Advances in Intelligent Engineering Systems. SCI, vol. 378, pp. 161–191. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-23229-9_8

  34. Poursoltan, S., Neumann, F.: Ruggedness quantifying for constrained continuous fitness landscapes. In: Datta, R., Deb, K. (eds.) Evolutionary Constrained Optimization. ISFS, pp. 29–50. Springer, New Delhi (2015). https://doi.org/10.1007/978-81-322-2184-5_2

    Chapter  Google Scholar 

  35. PyPI: mlrose, 13 March 2022. https://pypi.org/project/mlrose/

  36. Reeves, C.R.: Landscapes, operators and heuristic search. Ann. Oper. Res. 86, 473–490 (1999). https://doi.org/10.1007/bf01165154

    Article  MathSciNet  MATH  Google Scholar 

  37. Reinelt, G.: TSPLIB—a traveling salesman problem library. ORSA J. Comput. 3(4), 376–384 (1991). https://doi.org/10.1287/ijoc.3.4.376

    Article  MATH  Google Scholar 

  38. Richter, H.: Fitness landscapes: from evolutionary biology to evolutionary computation. In: Richter, H., Engelbrecht, A. (eds.) Recent Advances in the Theory and Application of Fitness Landscapes. Emergence, Complexity and Computation, vol. 6, pp. 3–31. Springer, Heidelberg (2014). https://doi.org/10.1007%2F978-3-642-41888-4_1

  39. Shi, Y., Eberhart, R.: A modified particle swarm optimizer. In: 1998 IEEE International Conference on Evolutionary Computation Proceedings. IEEE World Congress on Computational Intelligence (Cat. No.98TH8360). IEEE (1998). https://doi.org/10.1109/icec.1998.699146

  40. Sun, Y., Ernst, A., Li, X., Weiner, J.: Generalization of machine learning for problem reduction: a case study on travelling salesman problems. OR Spectrum 43(3), 607–633 (2020). https://doi.org/10.1007/s00291-020-00604-x

    Article  MathSciNet  MATH  Google Scholar 

  41. Sutton, A.M., Whitley, D., Lunacek, M., Howe, A.: PSO and multi-funnel landscapes: how cooperation might limit exploration. In: Keijzer, M. (ed.) Genetic and Evolutionary Computation Conference, pp. 75–82. Association for Computing Machinery, New York (2006). https://doi.org/10.1145/1143997.1144008

  42. Sörensen, K.: Metaheuristics-the metaphor exposed. Int. Trans. Oper. Res. 22(1), 3–18 (2015). https://doi.org/10.1111/itor.12001

    Article  MathSciNet  MATH  Google Scholar 

  43. Varadarajan, S., Whitley, D., Ochoa, G.: Why many travelling salesman problem instances are easier than you think. In: Proceedings of the 2020 Genetic and Evolutionary Computation Conference, pp. 254–262. ACM (2020). https://doi.org/10.1145/3377930.3390145

  44. Vassilev, V.K., Fogarty, T.C., Miller, J.F.: Smoothness, ruggedness and neutrality of fitness landscapes: from theory to application. In: Ghosh, A., Tsutsui, S. (eds.) Advances in Evolutionary Computing: Theory and Applications, pp. 3–44. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-642-18965-4_1

  45. Vaz, A.I., Vicente, L.N.: PSwarm: a hybrid solver for linearly constrained global derivative-free optimization. Optim. Methods Softw. 24(4–5), 669–685 (2009). https://doi.org/10.1080/10556780902909948

  46. Voß, S.: Tabu search: applications and prospects. In: Du, D., Pardalos, P.M. (eds.) Network Optimization Problems, vol. 2, pp. 333–353. World Scientific, Singapore (1993). https://doi.org/10.1142/9789812798190_0017

  47. Voss, S.: Book review: Marco Dorigo and Thomas Stützle: Ant colony optimization. Math. Methods Oper. Res. 63(1), 191–192 (2006). https://doi.org/10.1007/s00186-005-0050-4

    Article  Google Scholar 

  48. Wang, K.P., Huang, L., Zhou, C.G., Pang, W.: Particle swarm optimization for traveling salesman problem. In: Proceedings of the 2003 International Conference on Machine Learning and Cybernetics (IEEE Cat. No.03EX693), pp. 1583–1585. IEEE (2003). https://doi.org/10.1109/icmlc.2003.1259748

  49. Watson, J.P.: Empirical modeling and analysis of local search algorithms for the job-shop scheduling problem. Ph.D. thesis, Colorado State University (2003)

    Google Scholar 

  50. Watson, J.P.: An introduction to fitness landscape analysis and cost models for local search. In: Handbook of Metaheuristics, pp. 599–623. Springer, US (2010). https://doi.org/10.1007/978-1-4419-1665-5_20

  51. Weinberger, E.: Correlated and uncorrelated fitness landscapes and how to tell the difference. Biol. Cybern. 63(5), 325–336 (1990). https://doi.org/10.1007/bf00202749

    Article  MATH  Google Scholar 

  52. Wright, S.: The roles of mutation, inbreeding, crossbreeding and selection in evolution. In: Proceedings of the Sixth International Congress of Genetics, vol. 1, pp. 356–366 (1932)

    Google Scholar 

  53. Xin, B., Chen, J., Pan, F.: Problem difficulty analysis for particle swarm optimization. In: Proceedings of the First ACM/SIGEVO Summit on Genetic and Evolutionary Computation - GEC 2009, pp. 623–630. ACM Press (2009). https://doi.org/10.1145/1543834.1543919

  54. Yang, X.-S. (ed.): Recent Advances in Swarm Intelligence and Evolutionary Computation. SCI, vol. 585. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-13826-8

    Book  Google Scholar 

Download references

Acknowledgement

Malek Sarhani was supported by the Alexander von Humboldt Foundation.

Author information

Authors and Affiliations

  1. Institute of Information Systems, University of Hamburg, Hamburg, Germany

    Abtin Nourmohammadzadeh, Malek Sarhani & Stefan Voß

  2. School of Business Administration, Al Akhawayn University, Ifrane, Morocco

    Malek Sarhani

Authors
  1. Abtin Nourmohammadzadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Malek Sarhani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefan Voß
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abtin Nourmohammadzadeh .

Editor information

Editors and Affiliations

  1. SBA Research, Vienna, Austria

    Dimitris E. Simos

  2. Moscow Aviation Institute (National Research University), Moscow, Russia

    Varvara A. Rasskazova

  3. Università degli Studi di Milano-Bicocca, Milan, Italy

    Francesco Archetti

  4. Wilfrid Laurier University, Waterloo, ON, Canada

    Ilias S. Kotsireas

  5. University of Florida, Gainesville, FL, USA

    Panos M. Pardalos

Rights and permissions

Reprints and permissions

Copyright information

© 2022 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Nourmohammadzadeh, A., Sarhani, M., Voß, S. (2022). Fitness Landscape Ruggedness Impact on PSO in Dealing with Three Variants of the Travelling Salesman Problem. In: Simos, D.E., Rasskazova, V.A., Archetti, F., Kotsireas, I.S., Pardalos, P.M. (eds) Learning and Intelligent Optimization. LION 2022. Lecture Notes in Computer Science, vol 13621. Springer, Cham. https://doi.org/10.1007/978-3-031-24866-5_31

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-24866-5_31

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-24865-8

  • Online ISBN: 978-3-031-24866-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation