Skip to main content
Log in

Hybrid Heuristic for Solving the Euclidean Travelling Salesman Problem

  • Original Research
  • Published:
SN Computer Science Aims and scope Submit manuscript

Abstract

This study introduces a hybrid methodology that integrates the ant colony optimization (ACO) with genetic algorithm (GA) techniques. ACO is employed first to create an initial population and to derive a sub-optimal solution for the TSP using a newly designed inver-over (IO) operator. The Proposed IO operator is utilized to improve the solution derived from the ACO. This refined solution is then employed in the GA, where a genetic operator is applied alongside other randomly selected members from the initial population during the second phase. GA is used with the proposed crossover operator and the 2-opt heuristic in this phase to achieve optimal solution refinement towards a global optimum. Our evaluation of the algorithm’s efficacy uses benchmark datasets from TSPLIB. The proposed approach gives superior solution quality, both the average and the best solution metrics, demonstrating enhanced performance with a lower percentage of best error and percentage of average error. Experimental results indicate that the hybrid approach outperforms the efficiency of other state-of-the-art techniques.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Algorithm 2
Fig. 5

Similar content being viewed by others

Data Availability

http://elib.zib.de/pub/mp-testdata/tsp/tsplib/tsp/index.html.

Notes

  1. http://elib.zib.de/pub/mp-testdata/tsp/tsplib/tsp/index.html.

References

  1. Akhand M, Ayon SI, Shahriyar S, Siddique N, Adeli H. Discrete spider monkey optimization for travelling salesman problem. Appl Soft Comput. 2020;86: 105887.

    Article  Google Scholar 

  2. Akhand M, Ayon SI, Shahriyar S, Siddique N, Adeli H. Discrete spider monkey optimization for travelling salesman problem. Appl Soft Comput. 2020;86: 105887.

    Article  Google Scholar 

  3. Al-Gaphari GH, Al-Amry R, Al-Nuzaili AS. Discrete crow-inspired algorithms for traveling salesman problem. Eng Appl Artif Intell. 2021;97: 104006.

    Article  Google Scholar 

  4. Albayrak M, Allahverdi N. Development a new mutation operator to solve the traveling salesman problem by aid of genetic algorithms. Expert Syst Appl. 2011;38:1313–20.

    Article  Google Scholar 

  5. Ali IM, Essam D, Kasmarik K. A novel design of differential evolution for solving discrete traveling salesman problems. Swarm Evol Comput. 2020;52: 100607.

    Article  Google Scholar 

  6. Castelli M, Cattaneo G, Manzoni L, Vanneschi L. A distance between populations for n-points crossover in genetic algorithms. Swarm Evol Comput. 2019;44:636–45.

    Article  Google Scholar 

  7. Chen S-M, Chien C-Y. Solving the traveling salesman problem based on the genetic simulated annealing ant colony system with particle swarm optimization techniques. Expert Syst Appl. 2011;38:14439–50.

    Article  Google Scholar 

  8. Chitty DM. Applying aco to large-scale tsp instances. In: UK workshop on computational intelligence. Springer. 2017. pp. 104–18.

  9. Choong SS, Wong L-P, Lim CP. An artificial bee colony algorithm with a modified choice function for the traveling salesman problem. Swarm Evol Comput. 2019;44:622–35.

    Article  Google Scholar 

  10. Croes GA. A method for solving traveling-salesman problems. Oper Res. 1958;6:791–812.

    Article  MathSciNet  Google Scholar 

  11. Deng W, Chen R, He B, Liu Y, Yin L, Guo J. A novel two-stage hybrid swarm intelligence optimization algorithm and application. Soft Comput. 2012;16:1707–22.

    Article  Google Scholar 

  12. Dong X, Cai Y. A novel genetic algorithm for large scale colored balanced traveling salesman problem. Future Gener Comput Syst. 2019;95:727–42.

    Article  Google Scholar 

  13. Ebadinezhad S. Deaco: adopting dynamic evaporation strategy to enhance aco algorithm for the traveling salesman problem. Eng Appl Artif Intell. 2020;92: 103649.

    Article  Google Scholar 

  14. Eskandari L, Jafarian A, Rahimloo P, Baleanu D. A modified and enhanced ant colony optimization algorithm for traveling salesman problem. In: Mathematical methods in engineering. Springer; 2019. pp. 257–65.

  15. Ezugwu AE-S, Adewumi AO. R. Expert Syst Appl. 2017;87:70–8.

    Article  Google Scholar 

  16. Ezugwu AE-S, Adewumi AO, Frîncu ME. Simulated annealing based symbiotic organisms search optimization algorithm for traveling salesman problem. Expert Syst Appl. 2017;77:189–210.

    Article  Google Scholar 

  17. FReisleben B, Merz P. A genetic local search algorithm for solving symmetric and asymmetric traveling salesman problems. In: Proceedings of IEEE international conference on evolutionary computation. IEEE. 1996. pp. 616–21.

  18. Geng X, Chen Z, Yang W, Shi D, Zhao K. Solving the traveling salesman problem based on an adaptive simulated annealing algorithm with greedy search. Appl Soft Comput. 2011;11:3680–9.

    Article  Google Scholar 

  19. Gil-Gala FJ, Durasević M, Sierra MR, Varela R. Evolving ensembles of heuristics for the travelling salesman problem. Nat Comput. 2023;22:671–84.

    Article  MathSciNet  Google Scholar 

  20. Gong X, Rong Z, Wang J, Zhang K, Yang S. A hybrid algorithm based on state-adaptive slime mold model and fractional-order ant system for the travelling salesman problem. Complex Intell Syst. 2023;9:3951–70.

    Article  Google Scholar 

  21. Hatamlou A. Solving travelling salesman problem using black hole algorithm. Soft Comput. 2018;22:8167–75.

    Article  Google Scholar 

  22. Helsgaun K. An effective implementation of the lin-kernighan traveling salesman heuristic. Eur J Oper Res. 2000;126:106–30.

    Article  MathSciNet  Google Scholar 

  23. Huang Y, Shen X-N, You X. A discrete shuffled frog-leaping algorithm based on heuristic information for traveling salesman problem. Appl Soft Comput. 2021;102: 107085.

    Article  Google Scholar 

  24. Kanna SR, Sivakumar K, Lingaraj N. Development of deer hunting linked earthworm optimization algorithm for solving large scale traveling salesman problem. Knowl Based Syst. 2021;107199.

  25. Khan I, Maiti MK. A swap sequence based artificial bee colony algorithm for traveling salesman problem. Swarm Evol Comput. 2019;44:428–38.

    Article  Google Scholar 

  26. Khan I, Maiti MK. A swap sequence based artificial bee colony algorithm for traveling salesman problem. Swarm Evol Comput. 2019;44:428–38.

    Article  Google Scholar 

  27. Kirimtat A, KRejcar, O., TasgetiRen, M. F., & HerRera-Viedma, E. Multi-performance based computational model for the cuboid open traveling salesman problem in a smart floating city. Build Environ. 2021;196: 107721.

    Article  Google Scholar 

  28. Lu Y, Hao J-K, Wu Q. Hybrid evolutionary search for the traveling repairman problem with profits. Inf Sci. 2019;502:91–108.

    Article  MathSciNet  Google Scholar 

  29. Lu Y, Hao J-K, Wu Q. Hybrid evolutionary search for the traveling repairman problem with profits. Inf Sci. 2019;502:91–108.

    Article  MathSciNet  Google Scholar 

  30. Mahi M, Baykan ÖK, Kodaz H. A new hybrid method based on particle swarm optimization, ant colony optimization and 3-opt algorithms for traveling salesman problem. Appl Soft Comput. 2015;30:484–90.

    Article  Google Scholar 

  31. Marinakis Y, Marinaki M. A hybrid multi-swarm particle swarm optimization algorithm for the probabilistic traveling salesman problem. Comput Oper Res. 2010;37:432–42.

    Article  MathSciNet  Google Scholar 

  32. Masutti TA, de Castro LN. A self-organizing neural network using ideas from the immune system to solve the traveling salesman problem. Inf Sci. 2009;179:1454–68.

    Article  MathSciNet  Google Scholar 

  33. Mavrovouniotis M, Yang S. Ant colony optimization with immigrants schemes for the dynamic travelling salesman problem with traffic factors. Appl Soft Comput. 2013;13:4023–37.

    Article  Google Scholar 

  34. Osaba E, Yang X-S, Diaz F, Lopez-Garcia P, Carballedo R. An improved discrete bat algorithm for symmetric and asymmetric traveling salesman problems. Eng Appl Artif Intell. 2016;48:59–71.

    Article  Google Scholar 

  35. Pandiri V, Singh A. Swarm intelligence approaches for multidepot salesmen problems with load balancing. Appl Intell. 2016;44:849–61.

    Article  Google Scholar 

  36. Pandiri V, Singh A. A swarm intelligence approach for the colored traveling salesman problem. Appl Intell. 2018;48:4412–28.

    Article  Google Scholar 

  37. Panwar K, Deep K. Discrete grey wolf optimizer for symmetric travelling salesman problem. Appl Soft Comput. 2021;105: 107298.

    Article  Google Scholar 

  38. Pop PC, Cosma O, Sabo C, Sitar CP. A comprehensive survey on the generalized traveling salesman problem. Eur J Oper Res. 2023.

  39. Rokbani N, Abraham A, Twir I, Haqiq A. Solving the travelling salesman problem using fuzzy and simplified variants of ant supervised by pso with local search policy, fas-pso-ls, sas-pso-ls. Int J Hybrid Intell Syst. 2019;15:17–26.

    Google Scholar 

  40. Rokbani N, Kumar R, Abraham A, Alimi AM, Long HV, Priyadarshini I, Son LH. Bi-heuristic ant colony optimization-based approaches for traveling salesman problem. Soft Comput. 2021;25:3775–94.

    Article  Google Scholar 

  41. Roy A, Manna A, Maity S. A novel memetic genetic algorithm for solving traveling salesman problem based on multi-parent crossover technique. Decis Mak Appl Manag Eng. 2019;2:100–11.

    Article  Google Scholar 

  42. Saenphon T, PhimoltaRefs S, Lursinsap C. Combining new fast opposite gradient search with ant colony optimization for solving travelling salesman problem. Eng Appl Artif Intell. 2014;35:324–34.

    Article  Google Scholar 

  43. Sahana SK. Hybrid optimizer for the travelling salesman problem. Evol Intell. 2019;12:179–88.

    Article  Google Scholar 

  44. Sahin M. Solving tsp by using combinatorial bees algorithm with nearest neighbor method. Neural Comput Appl. 2022;1–17.

  45. Saji Y, Barkatou M. A discrete bat algorithm based on levy flights for Euclidean traveling salesman problem. Expert Syst Appl. 2021;172: 114639.

    Article  Google Scholar 

  46. Singh DR, Singh MK, Singh T. A hybrid algorithm with modified inver-over operator and ant colony optimization for traveling salesman problem. Adv Comput Control Commun Technol. 2016;1:29.

    Google Scholar 

  47. Singh DR, Singh MK, Singh T. A hybrid algorithm with modified inver-over operator and genetic algorithm search for traveling salesman problem. In: Advanced computing and communication technologies. Springer; 2016b. pp. 141–50.

  48. Stodola P, Michenka K, Nohel J, Rybansky M. Hybrid algorithm based on ant colony optimization and simulated annealing applied to the dynamic traveling salesman problem. Entropy. 2020;22:884.

    Article  MathSciNet  Google Scholar 

  49. Taillard ÉD, Helsgaun K. Popmusic for the travelling salesman problem. Eur J Oper Res. 2019;272:420–9.

    Article  MathSciNet  Google Scholar 

  50. Tao G, Michalewicz Z. Inver-over operator for the tsp. In: International conference on parallel problem solving from nature. Springer. 1998. pp. 803–12.

  51. Tawanda T, Nyamugure P, Kumar S, Munapo E. A labelling method for the travelling salesman problem. Appl Sci. 2023;13:6417.

    Article  Google Scholar 

  52. Tawhid MA, Savsani P. Discrete sine-cosine algorithm (dsca) with local search for solving traveling salesman problem. Arab J Sci Eng. 2019;44:3669–79.

    Article  Google Scholar 

  53. Tuani AF, Keedwell E, Collett M. Heterogenous adaptive ant colony optimization with 3-opt local search for the travelling salesman problem. Appl Soft Comput. 2020;97: 106720.

    Article  Google Scholar 

  54. Wang Y, Li C, Yin M. A two-phase removing algorithm for minimum independent dominating set problem. Appl Soft Comput. 2020;88: 105949.

    Article  Google Scholar 

  55. Wang Y, Sun J, Li J, Gao K. A modified inver-over operator for the traveling salesman problem. In: International conference on intelligent computing. Springer. 2011a. pp. 17–23.

  56. Wang Y, Wu Y, Xu N. Discrete symbiotic organism search with excellence coefficients and self-escape for traveling salesman problem. Comput Ind Eng. 2019;131:269–81.

    Article  Google Scholar 

  57. Wang Y-T, Li J-Q, Gao K-Z, Pan Q-K. Memetic algorithm based on improved inver-over operator and lin-kernighan local search for the euclidean traveling salesman problem. Comput Math Appl. 2011;62:2743–54.

    Article  MathSciNet  Google Scholar 

  58. Wei F-F, Chen W-N, Hu X-M, Zhang J. An empirical study on evolutionary algorithms for traveling salesman problem. In: 2019 9th international conference on information science and technology (ICIST). IEEE. 2019, pp. 273–80.

  59. Yang K, You X, Liu S, Pan H. A novel ant colony optimization based on game for traveling salesman problem. Appl Intell. 2020;50:4529–42.

    Article  Google Scholar 

  60. Yun H-Y, Jeong S-J, Kim K-S. Advanced harmony search with ant colony optimization for solving the traveling salesman problem. J Appl Math. 2013;2013.

  61. Zhong Y, Lin J, Wang L, Zhang H. Hybrid discrete artificial bee colony algorithm with threshold acceptance criterion for traveling salesman problem. Inf Sci. 2017;421:70–84.

    Article  MathSciNet  Google Scholar 

  62. Zhou Y, Luo Q, Chen H, He A, Wu J. A discrete invasive weed optimization algorithm for solving traveling salesman problem. Neurocomputing. 2015;151:1227–36.

    Article  Google Scholar 

  63. Zhou Y, Ouyang X, Xie J. A discrete cuckoo search algorithm for travelling salesman problem. Int J Collab Intell. 2014;1:68–84.

    Google Scholar 

Download references

Funding

The third author extends their appreciation to the Institute of Excellence, Banaras Hindu University (IoE BHU), for their support and collaborative efforts in carrying out this research.

Author information

Authors and Affiliations

Authors

Contributions

Dharm Raj Singh: Implementation, Validation, and Writing-original draft. Manoj Kumar Singh: Conceptualization, Supervision, Writing-review and editing. Sachchida Nand Chaurasia: Writing- draft review and editing, and statistical analysis of results. Pradeepika Verma: Investigation, Visualization, Writing - Review and Editing.

Corresponding author

Correspondence to Sachchida Nand Chaurasia.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Singh, D.R., Singh, M.K., Chaurasia, S.N. et al. Hybrid Heuristic for Solving the Euclidean Travelling Salesman Problem. SN COMPUT. SCI. 5, 1050 (2024). https://doi.org/10.1007/s42979-024-03417-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s42979-024-03417-9

Keywords

Navigation