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.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
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.
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.
Bentley, J. L. (1992). Fast algorithms for geometric traveling salesman problems. ORSA Journal on Computing, 4(4), 387–411.
Bertsimas, D., & Howell, L. (1993). Further results on the probabilistic traveling salesman problem. European Journal of Operational Research, 65(1), 68–95.
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.
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.
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.
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.
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.
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.
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.
Bonabeau, E., Dorigo, M., & Theraulaz, G. (1999). Swarm intelligence: from natural to artificial systems. London: Oxford University Press.
Branke, J., & Guntsch, M. (2004). Solving the probabilistic TSP with ant colony optimization. Journal of Mathematical Modelling and Algorithms, 3(4), 403–425.
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.
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.
Di Caro, G., & Dorigo, M. (1998). AntNet: Distributed stigmergetic control for communications networks. Journal of Artificial Intelligence Research, 9, 317–365.
Dorigo, M., & Birattari, M. (2007). Swarm intelligence. Scholarpedia, 2(9), 1462.
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.
Dorigo, M., & Stützle, T. (2004). Ant colony optimization. Cambridge: MIT Press.
Fisher, R. A. (1925). Statistical methods for research workers. Edinburgh: Oliver and Boyd.
Gendreau, M., Laporte, G., & Séguin, R. (1996). Stochastic vehicle routing. European Journal of Operational Research, 88, 3–12.
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.
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.
Hoos, H., & Stützle, T. (2005). Stochastic local search: foundations and applications. San Mateo: Morgan Kaufmann.
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.
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.
Martin, O., Otto, S. W., & Felten, E. W. (1991). Large-step Markov chains for the traveling salesman problem. Complex Systems, 5(3), 299–326.
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.
Tukey, J. W. (1949). Comparing individual means in the analysis of variance. Biometrics, 5(2), 99–114.
Author information
Authors and Affiliations
Corresponding author
Rights 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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11721-009-0031-y