Skip to main content
Log in

Estimation-based ant colony optimization and local search for the probabilistic traveling salesman problem

  • Published:
Swarm Intelligence Aims and scope Submit manuscript

Abstract

The use of ant colony optimization for solving stochastic optimization problems has received a significant amount of attention in recent years. In this paper, we present a study of enhanced ant colony optimization algorithms for tackling a stochastic optimization problem, the probabilistic traveling salesman problem. In particular, we propose an empirical estimation approach to evaluate the cost of the solutions constructed by the ants. Moreover, we use a recent estimation-based iterative improvement algorithm as a local search. Experimental results on a large number of problem instances show that the proposed ant colony optimization algorithms outperform the current best algorithm tailored to solve the given problem, which also happened to be an ant colony optimization algorithm. As a consequence, we have obtained a new state-of-the-art ant colony optimization algorithm for the probabilistic traveling salesman problem.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Applegate, D., Bixby, R. E., Chvatal, V., & Cook, W. J. (2001). Concorde—a code for solving traveling salesman problems. URL http://www.math.princeton.edu/tsp/concorde.html.

  • Balaprakash, P., Birattari, M., & Stützle, T. (2007). Improvement strategies for the F-Race algorithm: Sampling design and iterative refinement. In T. Bartz-Beielstein, M. Blesa, C. Blum, B. Naujoks, A. Roli, G. Rudolph, & M. Sampels (Eds.), LNCS : Vol. 4771. Hybrid metaheuristics, HM 2007 (pp. 113–127). Berlin: Springer.

    Chapter  Google Scholar 

  • Balaprakash, P., Birattari, M., Stützle, T., Yuan, Z., & Dorigo, M. (2008). Estimation-based ant colony optimization and local search for the probabilistic traveling salesman problem. IRIDIA Supplementary page. URL http://iridia.ulb.ac.be/supp/IridiaSupp2008-018/.

  • Balaprakash, P., Birattari, 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.

    Article  Google Scholar 

  • Bentley, J. L. (1992). Fast algorithms for geometric traveling salesman problems. ORSA Journal on Computing, 4(4), 387–411.

    MATH  MathSciNet  Google Scholar 

  • Bertsimas, D., & Howell, L. (1993). Further results on the probabilistic traveling salesman problem. European Journal of Operational Research, 65(1), 68–95.

    Article  MATH  Google Scholar 

  • Bianchi, L. (2006). Ant colony optimization and local search for the probabilistic traveling salesman problem: a case study in stochastic combinatorial optimization. Ph.D. thesis, Université Libre de Bruxelles, Brussels, Belgium.

  • Bianchi, L., & Campbell, A. (2007). Extension of the 2-p-opt and 1-shift algorithms to the heterogeneous probabilistic traveling salesman problem. European Journal of Operational Research, 176(1), 131–144.

    Article  MATH  MathSciNet  Google Scholar 

  • Bianchi, L., & Gambardella, L. M. Ant colony optimization and local search based on exact and estimated objective values for the probabilistic traveling salesman problem (Technical Report IDSIA-06-07). IDSIA, USI-SUPSI, Manno, Switzerland, June 2007.

  • Bianchi, L., Gambardella, L., & Dorigo, M. (2002a). Solving the homogeneous probabilistic travelling salesman problem by the ACO metaheuristic. In M. Dorigo, G. Di Caro, & M. Sampels (Eds.), LNCS : Vol. 2463. Ant algorithms, third international workshop, ANTS 2002 (pp. 176–187). Berlin: Springer.

    Google Scholar 

  • Bianchi, L., Gambardella, L. M., & Dorigo, M. (2002b). An ant colony optimization approach to the probabilistic traveling salesman problem. In J. J. Guervós, P. Adamidis, H. Beyer, J. L. Martín, & H. P. Schwefel (Eds.), LNCS : Vol. 2439. 7th international conference on parallel problem solving from nature, PPSN VII (pp. 883–892). Berlin: Springer.

    Chapter  Google Scholar 

  • Bianchi, L., Knowles, J., & Bowler, N. (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.

    Article  MATH  MathSciNet  Google Scholar 

  • Birattari, M. (2004). The problem of tuning metaheuristics as seen from a machine learning perspective. Ph.D. thesis, Université Libre de Bruxelles, Brussels, Belgium.

  • Birattari, M. (2009). Tuning metaheuristics: a machine learning perspective. Studies in computational intelligence (Vol. 197). Berlin: Springer.

    MATH  Google Scholar 

  • 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.), Operations research/computer science interfaces series : Vol. 44. Metaheuristics—progress in complex systems optimization (pp. 189–203). Berlin: Springer.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Bonabeau, E., Dorigo, M., & Theraulaz, G. (1999). Swarm intelligence: from natural to artificial systems. London: Oxford University Press.

    MATH  Google Scholar 

  • Branke, J., & Guntsch, M. (2004). Solving the probabilistic TSP with ant colony optimization. Journal of Mathematical Modelling and Algorithms, 3(4), 403–425.

    Article  MATH  MathSciNet  Google Scholar 

  • Bullnheimer, B., Hartl, R. F., & Strauss, C. (1999). A new rank based version of the ant system: A computational study. Central European Journal for Operations Research and Economics, 7(1), 25–38.

    MATH  MathSciNet  Google Scholar 

  • Cordón, O., de Viana, I. F., & Herrera, F. (2002). Analysis of the best-worst ant system and its variants on the TSP. Mathware and Soft Computing, 9(2–3), 177–192.

    MATH  MathSciNet  Google Scholar 

  • Di Caro, G., & Dorigo, M. (1998). AntNet: Distributed stigmergetic control for communications networks. Journal of Artificial Intelligence Research, 9, 317–365.

    MATH  Google Scholar 

  • Dorigo, M., & Birattari, M. (2007). Swarm intelligence. Scholarpedia, 2(9), 1462.

    Google Scholar 

  • Dorigo, M., & Gambardella, L. M. (1997). Ant colony system: A cooperative learning approach to the traveling salesman problem. IEEE Transactions on Evolutionary Computation, 1(1), 53–66.

    Article  Google Scholar 

  • Dorigo, M., & Stützle, T. (2004). Ant colony optimization. Cambridge: MIT Press.

    MATH  Google Scholar 

  • Fisher, R. A. (1925). Statistical methods for research workers. Edinburgh: Oliver and Boyd.

    Google Scholar 

  • Gendreau, M., Laporte, G., & Séguin, R. (1996). Stochastic vehicle routing. European Journal of Operational Research, 88, 3–12.

    Article  MATH  Google Scholar 

  • Gutjahr, W. J. (2003). A converging ACO algorithm for stochastic combinatorial optimization. In A. Albrecht & K. Steinhofl (Eds.), LNCS : Vol. 2827. Stochastic algorithms: foundations and applications (pp. 10–25). Berlin: Springer.

    Google Scholar 

  • Gutjahr, W. J. (2004). S-ACO: An ant based approach to combinatorial optimization under uncertainty. In M. Dorigo, M. Birattari, C. Blum, L. M. Gambardella, F. Mondada, & T. Stützle (Eds.), LNCS : Vol. 3172. Ant colony optimization and swarm intelligence, 5th international workshop, ANTS 2004 (pp. 238–249). Berlin: Springer.

    Google Scholar 

  • Hoos, H., & Stützle, T. (2005). Stochastic local search: foundations and applications. San Mateo: Morgan Kaufmann.

    MATH  Google Scholar 

  • Jaillet, P. (1985). Probabilistic traveling salesman problems. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.

  • Johnson, D. S., & McGeoch, L. A. (1997). The travelling salesman problem: a case study in local optimization. In E. H. L. Aarts & J. K. Lenstra (Eds.), Local search in combinatorial optimization (pp. 215–310). Wiley: New York.

    Google Scholar 

  • Johnson, D. S., McGeoch, L.A., Rego, C, & Glover, F. (2001). 8th DIMACS implementation challenge. URL http://www.research.att.com/~dsj/chtsp/.

  • Laporte, G., Louveaux, F., & Mercure, H. (1994). A priori optimization of the probabilistic traveling salesman problem. Operations Research, 42, 543–549.

    Article  MATH  MathSciNet  Google Scholar 

  • Martin, O., Otto, S. W., & Felten, E. W. (1991). Large-step Markov chains for the traveling salesman problem. Complex Systems, 5(3), 299–326.

    MATH  MathSciNet  Google Scholar 

  • Stützle, T. (2002). ACOTSP: A software package of various ant colony optimization algorithms applied to the symmetric traveling salesman problem. URL http://www.aco-metaheuristic.org/aco-code/.

  • Stützle, T., & Hoos, H. (2000). \(\mathcal{MAX}\)\(\mathcal{MIN}\) ant system. Future Generation Computer Systems, 16(8), 889–914.

    Article  Google Scholar 

  • Tukey, J. W. (1949). Comparing individual means in the analysis of variance. Biometrics, 5(2), 99–114.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Prasanna Balaprakash.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Balaprakash, P., Birattari, M., Stützle, T. et al. Estimation-based ant colony optimization and local search for the probabilistic traveling salesman problem. Swarm Intell 3, 223–242 (2009). https://doi.org/10.1007/s11721-009-0031-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11721-009-0031-y

Keywords

Navigation