Skip to main content

Metaheuristic algorithms and probabilistic behaviour: a comprehensive analysis of Ant Colony Optimization and its variants

Abstract

The application of metaheuristic algorithms to combinatorial optimization problems is on the rise and is growing rapidly now than ever before. In this paper the historical context and the conducive environment that accelerated this particular trend of inspiring analogies or metaphors from various natural phenomena are analysed. We have implemented the Ant System Model and the other variants of ACO including the 3-Opt, Max–Min, Elitist and the Rank Based Systems as mentioned in their original works and we converse the missing pieces of Dorigo’s Ant System Model. Extensive analysis of the variants on Travelling Salesman Problem and Job Shop Scheduling Problem shows how much they really contribute towards obtaining better solutions. The stochastic nature of these algorithms has been preserved to the maximum extent to keep the implementations as generic as possible. We observe that stochastic implementations show greater resistance to changes in parameter values, still obtaining near optimal solutions. We report how Polynomial Turing Reduction helps us to solve Job Shop Scheduling Problem without making considerable changes in the implementation of Travelling Salesman Problem, which could be extended to solve other NP-Hard problems. We elaborate on the various parallelization options based on the constraints enforced by strong scaling (fixed size problem) and weak scaling (fixed time problem). Also we elaborate on how probabilistic behaviour helps us to strike a balance between intensification and diversification of the search space.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

References

  • Abidin ZZ, Ngah UK, Arshad MR, Ping OB (2010) A novel y optimization algorithm for swarming application. In: IEEE conference on robotics, automation and mechatronics (RAM), pp 425–428

  • Amdahl Gene M (1967) Validity of the single processor approach to achieving Large-Scale Computing Capabilities (PDF). AFIPS Conference Proceedings 30: 483–485. doi:10.1145/1465482.1465560

  • Apostolopoulos T, Vlachos A (2011) Application of the firefly algorithm for solving the economic emissions load dispatch problem. Int J Comb 2011: Article ID 523806

  • Barricelli NA (1954) Esempi numerici di processi di evoluzione. Methodos 6:45–68

    MathSciNet  Google Scholar 

  • Barricelli NA (1957) Symbiognetic evolution processes realized by artificial methods. Methodos 9:143–182

    Google Scholar 

  • Beni G, Wang J (1989) Swarm intelligence in cellular robotic systems. In: Proceedings of NATO advanced workshop on robots and biological systems. Tuscany, Italy

  • Bottou Lon (1998) Online algorithms and stochastic approximations. Online Learning and Neural Networks. Cambridge University Press. ISBN 978-0-521-65263-6

  • Bullnheimer B, Richard F, Hartl, Strau C (1997) A new rank based version of the ant system: a computational study. Working Paper No. 1

  • Coelho L, Bernert DL, Mariani VC (2011) A chaotic firefly algorithm applied to reliability-redundancy optimization. In: 2011 IEEE congress on evolutionary computation (CEC’11), pp 517–521

  • Colorni A, Dorigo M, Maniezzo V (1991) Distributed optimization by ant colonies. In: ECAL91—European conference on artificial life, pp 134–142

  • Denebourg JL, Goss S (1989) Collective patterns and decision-making. Ethol Ecol Evol 1(4):295–311

    Article  Google Scholar 

  • Deneubourg J-L, Pasteels JM, Verhaeghe JC (1983) Probabilistic behaviour in ants: a strategy of errors. J Theoret Biol 105(2):259–271

    Article  Google Scholar 

  • Dorigo M, Gambardella LM (1997b) Ant colonies for the traveling salesman problem. BioSystems 43(2):73–81

    Article  Google Scholar 

  • Dorigo M, Gambardella LM (1997a) Ant colony system: a cooperative learning approach to the traveling salesman problem. IEEE Trans Evol Comput 1(1):53–66

    Article  Google Scholar 

  • Dorigo M, Maniezzo V, Colorni A (1991) The ant system: an autocatalytic optimizing process. In: No. 91-016. Technical report

  • Dorigo M, Maniezzo V, Colorni A (1996) Ant system: optimization by a colony of cooperating agents. Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on 26(1):29–41

    Article  Google Scholar 

  • Finilla AB, Gomez MA, Sebenik C, Doll DJ (1994) Quantum annealing: a new method for minimizing multidimensional functions. Chem Phys Lett 219:343

    Article  Google Scholar 

  • Fraser AS (1960) Simulation of genetic systems by automatic digital computers vi. epistasis. Aust J Biol Sci 13(2):150–162

    Google Scholar 

  • Glover F (1977) Heuristics for integer programming, using surrogate constraints. Decis Sci 8(1):156–166

    Article  Google Scholar 

  • Glover F (1986) Future paths for integer programming and links to artificial intelligence. Comput Operat Res 13(5):533549. doi:10.1016/0305-0548(86)90048-1

    MathSciNet  Google Scholar 

  • Glover F (1989) Tabu search-part I. ORSA J Comput 1(3):190

    MATH  Article  Google Scholar 

  • Goldberg DE (1991) The theory of virtual alphabets-parallel problem solving from nature. Springer, Berlin

    Google Scholar 

  • Goldberg DE, Holland JH (1988) Genetic algorithms and machine learning. Mach Learn 32:95–99

    Article  Google Scholar 

  • Green DG, Liu J, Abbass H (2014) Dual phase evolution: from theory to practice. Springer, Berlin ISBN 978-1441984227

    Book  Google Scholar 

  • Gupta DK, Arora Y, Singh UK, Gupta JP (2012) Recursive Ant Colony Optimization for estimation of parameters of a function. In: Recent advances in Information Technology (RAIT), International conference. doi:10.1109/RAIT.2012.6194620, pp 448–454

  • Gustafson JL (1988) Reevaluating Amdahl’s Law. Commun ACM 31(5):532–533

    Article  Google Scholar 

  • Gutjahr WJ (2000) A graph-based Ant System and its convergence. Future Gener Comput Syst 16:873–888

    Article  Google Scholar 

  • Haddad OB, Afshar A, Marino AB (2006) Honey-bees mating optimization (HBMO) algorithm: a new heuristic approach for water resources optimization. Water Resour Manag 20(5):661–680

    Article  Google Scholar 

  • Hedayatzadeh R, Salmassi F, Keshtgari M, Akbari R, Ziarati K (2010) Termite colony optimization: a novel approach for optimizing continuous problems. In: 18th Iranian conference on electrical engineering (ICEE), pp 553–558

  • Heppner F, Grenander U, Krasner S (1990) A stochastic nonlinear model for coordinated bird flocks. In: The Ubiquity of Chaos. AAAS Publications, Washington, DC

  • Holland JH (1975) Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence. University of Michigan Press, Michigan

    Google Scholar 

  • http://www.metaheuristics.net/, 2000. Visited in (January 2003)

  • James K, Eberhart R (1995) Particle swarm optimization. In: Proceedings of IEEE international conference on neural networks, vol 4(2)

  • Karaboga D (2005) An idea based on honey bee swarm for numerical optimization. In: Technical Report TR06, Erciyes University Press, Erciyes

  • Karaboga D (2010) Artificial bee colony algorithm. Scholarpedia 5(3):6915. doi:10.4249/scholarpedia.6915

    Article  Google Scholar 

  • Kennedy J, Eberhart R (1995) Particle Swarm Optimization. Proceedings of IEEE International Conference on Neural Networks IV. doi:10.1109/ICNN.1995.488968

  • Kirkpatrick S, Gelattr SD, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671

    MATH  MathSciNet  Article  Google Scholar 

  • Krishnanand KN, Ghose D (2005) Detection of multiple source locations using a glowworm metaphor with applications to collective robotics. In: IEEE Swarm intelligence symposium, pp 84–91

  • Laarhoven PJM, Aarts EHL (1987) Simulated annealing: theory and applications. Springer, Berlin

    MATH  Book  Google Scholar 

  • Lukasik S, Zak S (2009) Firefly algorithm for continuous constrained optimization tasks. Computational collective intelligence. In: Semantic Web, Social networks and multi-agent systems, pp 97–106

  • Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21(6):1087

    Article  Google Scholar 

  • Mirjalili S, Mirjalili SM, Lewis A (2014) Grey Wolf Optimizer. Adv Eng Softw 69:46–61

    Article  Google Scholar 

  • Mirjalili S (2015) The Ant Lion optimizer. Adv Eng Softw 83:8098

    Article  Google Scholar 

  • Nakrani S, Tovey C (2004) On honey bees and dynamic server allocation in internet hosting centers. Adapt Behav 12(3–4):223240

    Google Scholar 

  • Niu B (2012) Bacterial colony optimization. Dis Dyn Nat Soc. Article ID 698057

  • Osman IH, Laporte G (1996) Metaheuristics: a bibliography. Ann Operat Res 63(513):623

    MathSciNet  Google Scholar 

  • Pan WT (2011) A new fruitfly optimization algorithm: taking the financial distress model as an example. Knowl Based Syst 26:69–74

    Article  Google Scholar 

  • Rampriya B, Mahadevan K, Kannan S (2010) Unit commitment in deregulated power system using Lagrangian firefly algorithm. In: Proceedings of IEEE international conference on communication control and computing technologies (ICCCCT), pp 389–393

  • Reynolds CW (1987) Flocks, herds and schools: a distributed behavioral model. Comput Graph 21(4):25–34

    Article  Google Scholar 

  • Rosengren R (1971) Route fidelity, visual memory and recruitment behaviour in foraging wood ants of the genus Formica (Hymenoptera, Formicidae). Acta Zool Fenn 133:1–106

    Google Scholar 

  • Sayadi MK, Ramezanian R, Ghaffari-Nasab N (2010) A discrete firefly meta-heuristic with local search for makespan minimization in permutation flow shop scheduling problems. Int J Ind Eng Comput 1:110

    Google Scholar 

  • Schmidt G (2000) Scheduling with limited machine availability. Eur J Oper Res 121(1):1–15

    MATH  Article  Google Scholar 

  • Shi Y, Eberhart R (1998) A modified particle swarm optimizer. Evolutionary computation proceedings. In: IEEE World congress on computational intelligence, the 1998 IEEE international conference on IEEE

  • Snyder L (1986) Type architectures, shared memory, and the corollary of modest potential. Ann Rev Comput Sci 1:289–317

    Article  Google Scholar 

  • Sorensen K (2012) Metaheuristics the metaphor exposed. In: International transactions of operations research. Pub Online: Feb 08, 2013. doi:10.1111/itor.12001 (p)

  • Sorin CN, Oprean C, Kifor CV, Carabulea I (2008) Elitist ant system for route allocation problem. In: World scientific and engineering academy and society (WSEAS) Stevens Point, Wisconsin, USA

  • Sttzle T, Hoos HH (2000) Max Min Ant System. Future Gener Comput Syst 16:889–914

    Article  Google Scholar 

  • Talreja S (2013) A heuristic proposal in the dimension of Ant colony Optimization. Appl Math Sci 7(41):2017–2026

    MathSciNet  Google Scholar 

  • Whitley D (1994) A genetic algorithm tutorial. Stat Comput 4(2):6585. doi:10.1007/BF00175354

  • Xiao-Min H, Zhang J, Li Y (2008) Orthogonal methods based Ant Colony search for solving continuous optimization problems. J Comput Sci Technol 23(1):2–18

    Article  Google Scholar 

  • Yang XS (2008) Nature-inspired metaheuristic algorithms. Frome. In: Luniver Press. ISBN 1-905986-10-6

  • Yang XS (2010) A new metaheuristic bat-inspired algorithm. In: Gonzalez JR et al (eds) Nature inspired cooperative strategies for optimization (NISCO 2010), Studies in computational intelligence, vol 284. Springer, Berlin, pp 65–74

    Chapter  Google Scholar 

Download references

Acknowledgments

The authors would like to acknowledge the infrastructure support provided by the Massively Parallel Programming Laboratory (CUDA Teaching Centre), Department of Computer Applications, National Institute of Technology, Trichy. The authors would like to thank Dr. Hemalatha Thiagarajan, Professor, Department of Mathematics, National Institute of Technology, Trichy for providing valuable insights on the simulation of various probability distribution functions. The authors would also like to thank their fellow researchers Mr.S.Thiruselvan, Mr.P.Arish and Ms.R.Eswari for their valuable suggestions and support.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Anandkumar Prakasam.

Additional information

The code is licensed and made available under the open source Creative Commons License to enable constructive criticism, reproducible research and to ensure maximum dissemination of the research work. The code, results and graphs are provided as supplementary materials which will also be made available as a project in Google code and Git-Hub.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (rar 53260 KB)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Prakasam, A., Savarimuthu, N. Metaheuristic algorithms and probabilistic behaviour: a comprehensive analysis of Ant Colony Optimization and its variants. Artif Intell Rev 45, 97–130 (2016). https://doi.org/10.1007/s10462-015-9441-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10462-015-9441-y

Keywords

  • Ant Colony Optimization
  • Metaheuristics
  • Ant System
  • ACO variants
  • Travelling Salesman Problem
  • Job Shop Scheduling Problem