Soft Computing

, Volume 18, Issue 4, pp 619–629 | Cite as

Behaviour of pseudo-random and chaotic sources of stochasticity in nature-inspired optimization methods



Stochasticity, noisiness, and ergodicity are the key concepts behind many natural processes and its modeling is an important part of their implementation. There is a handful of soft-computing methods that are directly inspired by nature or stochastic natural processes. The implementation of such a nature-inspired optimization and search methods usually depends on streams of integer and floating point numbers generated in course of their execution. The pseudo-random numbers are utilized for in-silico emulation of probability-driven natural processes such as arbitrary modification of genetic information (mutation, crossover), partner selection, and survival of the fittest (selection, migration) and environmental effects (small random changes in motion direction and velocity). Deterministic chaos is a well known mathematical concept that can be used to generate sequences of seemingly random real numbers within selected interval in a predictable and well controllable way. In the past, it has been used as a basis for various pseudo-random number generators (PRNGs) with interesting properties. This work provides an empirical comparison of the behavior of selected nature-inspired optimization algorithms using different PRNGs and chaotic systems as sources of stochasticity.


Pseudo-random number generators Deterministic chaos  Simulation Genetic algorithms Differential evolution  Particle swarm optimization 



This work was supported by the European Regional Development Fund in the IT4-Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070), by the Bio-Inspired Methods: research, development and knowledge transfer project, Reg. No. CZ.1.07/2.3.00/20.0073, and by the Development of human resources in research and development of latest soft computing methods and their application in practice project, Reg. No. CZ.1.07/2.3.00/20.0072 funded by Operational Programme Education for Competitiveness, co-financed by ESF and state budget of the Czech Republic. The following grant is also acknowledged for the financial support provided for this research: Grant Agency of the Czech Republic-GACR P103/13/08195S. The work was also partially supported by Grants of SGS No. SP2013/70 and No. SP2013/114, VŠB-Technical University of Ostrava.


  1. Affenzeller M, Winkler S, Wagner S, Beham A (2009) Genetic algorithms and genetic programming: modern concepts and practical applications. Chapman and Hall/CRC, Boca RatonCrossRefGoogle Scholar
  2. Andrecut M (1998) Logistic map as a random number generator. Int J Modern Phys B 12(09):921–930. doi: 10.1142/S021797929800051X. Google Scholar
  3. Bastos-Filho CJA, Andrade J, Pita M, Ramos A (2009) Impact of the quality of random numbers generators on the performance of particle swarm optimization. In: IEEE international conference on systems, man and cybernetics. SMC 2009, pp 4988–4993 (2009). doi: 10.1109/ICSMC.2009.5346366
  4. Bastos-Filho CJA, Oliveira M, Nascimento DNO, Ramos AD (2010) Impact of the random number generator quality on particle swarm optimization algorithm running on graphic processor units. In: 2010 10th international conference on hybrid intelligent systems (HIS), pp 85–90. doi: 10.1109/HIS.2010.5601073
  5. Bland IM, Megson G (1996) Systolic random number generation for genetic algorithms. Electr Lett 32(12):1069–1070. doi: 10.1049/el:19960709 CrossRefGoogle Scholar
  6. Blum C, Merkle D (2008) Swarm intelligence: introduction and applications. Springer Publishing Company (incorporated)Google Scholar
  7. Cantú-Paz E (2002) On random numbers and the performance of genetic algorithms. In: Proceedings of the genetic and evolutionary computation conference, GECCO ’02, Morgan Kaufmann Publishers Inc., San Francisco, pp 311–318.
  8. Cárdenas-Montes M, Vega-Rodríguez MA, Gómez-Iglesias A (2011) Sensitiveness of evolutionary algorithms to the random number generator. In: Proceedings of the 10th international conference on adaptive and natural computing algorithms-volume part I, ICANNGA’11, Springer, Berlin, pp 371–380.
  9. Chen SL, Hwang T, Lin WW (2010) Randomness enhancement using digitalized modified logistic map. IEEE transactions on circuits and systems II: express briefs 57(12):996–1000. doi: 10.1109/TCSII.2010.2083170 Google Scholar
  10. Cheng MY, Huang KY, Chen HM (2012) Dynamic guiding particle swarm optimization with embedded chaotic search for solving multidimensional problems. Optim Lett 6(4):719–729 (2012). doi: 10.1007/s11590-011-0297-z. Google Scholar
  11. Clerc M (2010) Particle swarm optimization. ISTE, Wiley.
  12. Czarn A, MacNish C, Vijayan K, Turlach BA (2004) Statistical exploratory analysis of genetic algorithms: the influence of gray codes upon the difficulty of a problem. In: Webb GI, Yu X (eds) Australian conference on artificial intelligence. Lecture notes in computer science, vol 3339. Springer, pp 1246–1252Google Scholar
  13. Determan J, Foster J (1999) Using chaos in genetic algorithms. In: Proceedings of the 1999 congress on evolutionary computation, 1999. CEC 99. vol 3, p 2101. doi: 10.1109/CEC.1999.785533
  14. Engelbrecht A (2007) Computational intelligence: an introduction, 2nd edn. Wiley, New YorkCrossRefGoogle Scholar
  15. Gordon DM (2010) Ant encounters: interaction networks and colony behavior. Primers in complex systems. Princeton University Press, Princeton.
  16. Hu W, Liang H, Peng C, Du B, Hu Q (2013) A hybrid chaos-particle swarm optimization algorithm for the vehicle routing problem with time window. Entropy 15(4):1247–1270. doi: 10.3390/e15041247.
  17. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of IEEE international conference on neural networks, 1995, vol 4, pp 1942–1948. doi: 10.1109/ICNN.1995.488968
  18. Lee CY, Yao X (2004) Evolutionary programming using mutations based on the levy probability distribution. IEEE Trans Evol Comput 8(1):1–13. doi: 10.1109/TEVC.2003.816583 Google Scholar
  19. Li-Jiang Y, Tian-Lun C (2002) Application of chaos in genetic algorithms. Commun Theory Phys 38(2):168–172Google Scholar
  20. Luscher M (1994) A portable high-quality random number generator for lattice field theory simulations. Comput Phys Commun 79(1): 100–110. doi: 10.1016/0010-4655(94)90232-1. Google Scholar
  21. Ma Z, Vandenbosch G (2012) Impact of random number generators on the performance of particle swarm optimization in antenna design. In: 2012 6th European conference on antennas and propagation (EUCAP), pp 925–929. doi: 10.1109/EuCAP.2012.6205998
  22. Maheshkumar Y, Ravi V, Abraham A (2013) A particle swarm optimization-threshold accepting hybrid algorithm for unconstrained optimization. Neural Netw World 23(1):17–30Google Scholar
  23. Masuda K, Kurihara K (2009) Particle swarm optimization with external chaotic noise. In: ICCAS-SICE, pp 5002–5007Google Scholar
  24. Matsumoto M, Nishimura T (1998) Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Trans Model Comput Simul 8(1):3–30. doi: 10.1145/272991.272995.
  25. Maucher M, Schning U, Kestler H (2011) Search heuristics and the influence of non-perfect randomness: examining genetic algorithms and simulated annealing. Comput Stat 26(2):303–319. doi: 10.1007/s00180-011-0237-5.
  26. Meysenburg MM, Foster JA (1997) The quality of pseudo-random number enerations and simple genetic algorithm performance. In: Bäck T (ed) ICGA. Morgan Kaufmann, Burlington, pp 276– 282Google Scholar
  27. Meysenburg MM, Foster JA (1999) Randomness and GA performance, revisited. In: Banzhaf W, Daida J, Eiben AE, Garzon MH, Honavar V, Jakiela M, Smith RE (eds) Proceedings of the genetic and evolutionary computation conference, vol 1. Morgan Kaufmann, Orlando, pp. 425–432.
  28. Meysenburg MM, Hoelting D, McElvain D, Foster JA (2002) How random generator quality impacts ga performance. In: Langdon WB, Cantú-Paz E, Mathias KE, Roy R, Davis D, Poli R, Balakrishnan K, Honavar V, Rudolph G, Wegener J, Bull L, Potter MA, Schultz AC, Miller JF, Burke EK, Jonoska N (eds) GECCO. Morgan Kaufmann, Burlington, pp 480–487Google Scholar
  29. Mitchell M (1996) An introduction to genetic algorithms. MIT Press, CambridgeGoogle Scholar
  30. Persohn K, Povinelli R (2012) Analyzing logistic map pseudorandom number generators for periodicity induced by finite precision floating-point representation. Chaos Solitons Fractals 45(3):238–245. doi: 10.1016/j.chaos.2011.12.006.
  31. Poláková R, Tvrdík J (2013) A combined approach to adaptive differential evolution. Neural Netw World 23(1):3–15Google Scholar
  32. Price KV, Storn RM, Lampinen JA (2005) Differential evolution a practical approach to global optimization. Natural Computing Series. Springer, Berlin.
  33. Sabeti M, Boostani R, Zoughi T (2012) Using genetic programming to select the informative eeg-based features to distinguish schizophrenic patients. Neural Netw World 22(1):3–20Google Scholar
  34. Schuster H, Just W (2006) Deterministic chaos. Wiley.
  35. Shastry MC, Nagaraj N, Vaidya PG (2006) The b-exponential map: a generalization of the logistic map, and its applications in generating pseudo-random numbers. CoRR abs/cs/0607069Google Scholar
  36. Storn R (1996) Differential evolution design of an IIR-filter. In: Proceeding of the IEEE conference on evolutionary computation ICEC. IEEE Press, Piscataway. pp 268–273Google Scholar
  37. Storn R, Price K (1995) Differential evolution: a simple and efficient adaptive scheme for global optimization over continuous spaces. Tech. rep.
  38. Suganthan PN, Hansen N, Liang JJ, Deb K, Chen YP, Auger A, Tiwari S (2005) Problem definitions and evaluation criteria for the CEC 2005 special session on real parameter optimization. Tech rep, Nanyang Technological UniversityGoogle Scholar
  39. Szczepański J, Kotulski Z (2001) Pseudorandom number generators based on chaotic dynamical systems. Open Syst Inf Dyn 8(2):137–146. doi: 10.1023/A:1011950531970.
  40. Tirronen V, Ayramo S, Weber M (2011) Study on the effects of pseudorandom generation quality on the performance of differential evolution. In: Dobnikar A, Lotri U, Ster B (eds) Adaptive and natural computing algorithms. Lecture notes in computer science, vol 6593. Springer, Berlin, pp 361–370. doi: 10.1007/978-3-642-20282-_37.
  41. Wagner NR (1993) The logistic lattice in random number generation. In: Proceedings of the thirtieth annual allerton conference on communications, control, and computing. Coordinated Science Lab and Department of Electrical and Computer Engineering, University of Illinois at Urbabn-Champaign, pp 922–931Google Scholar
  42. Wu AS, Lindsay RK, Riolo R (1997) Empirical observations on the roles of crossover and mutation. In: Bäck T (ed) Proceedings of the seventh international conference on genetic algorithms. Morgan Kaufmann, San Francisco, pp 362–369.
  43. Yang M, Guan J, Cai Z, Wang L (2010) Self-adapting differential evolution algorithm with chaos random for global numerical optimization. In: Proceedings of the 5th international conference on advances in computation and intelligence, ISICA’10. Springer, Berlin, pp 112–122.
  44. Yao JB, Yao BZ, Li L, Jiang YL (2012) Hybrid model for displacement prediction of tunnel surrounding rock. Neural Netw World 22(3):263–275Google Scholar
  45. Zhang S, Hu Q, Wang X, Zhu Z (2009) Application of chaos genetic algorithm to transformer optimal design. In: International workshop on chaos-fractals theories and applications, 2009. IWCFTA ’09, pp 108–111. doi: 10.1109/IWCFTA.2009.30
  46. Zhao S, Xu G, Tao T, Liang L (2009) Real-coded chaotic quantum-inspired genetic algorithm for training of fuzzy neural networks. Computers and mathematics with applications 57(11–12):2009–2015. doi: 10.1016/j.camwa.2008.10.048. In: Proceedings of the international conference on bio-inspired computing-theories and applications BIC-TA 2007 Zhengzhou

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.IT4Innovations and Department of Computer ScienceVŠB-Technical University of OstravaOstrava PorubaCzech Republic

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