Effect of Mutation to Distribution of Optimum Solution in Genetic Algorithm

  • Yu-an Zhang
  • QingLian Ma
  • Makoto Sakamoto
  • Hiroshi Furutani
Part of the Proceedings in Information and Communications Technology book series (PICT, volume 2)


Mutation plays an important role in the computing of Genetic Algorithms (GAs). In this paper, we use the success probability as a measure of the performance of GAs, and apply a method for calculating the success probability by means of Markov chain theory. We define the success probability as there is at least one optimum solution in a population. In this analysis, we assume that the population is in linkage equilibrium, and obtain the distribution of the first order schema. We calculate the number of copies of the optimum solution in the population by using the distribution of the first order schema. As an application of the method, we study the GA on the multiplicative landscape, and demonstrate the process to calculate the success probability for this example. Many researchers may consider that the success probability decreases exponentially as a function of the string length L. However, if mutation is included in the GA, it is shown that the success probability decreases almost linearly as L increases.


Genetic Algorithm Markov Chain Failure Probability Markov Chain Model Extinction State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Furutani, H., Zhang, Y., Sakamoto, M.: Study of the Distribution of Optimum Solution in Genetic Algorithm by Markov Chains. Transactions on Mathematical Modeling and its Applications (TOM) (to be published)Google Scholar
  2. 2.
    Ewens, J.W.J.: Mathematical Population Genetics. I. Theoretical Introduction, 2nd edn. Springer, New York (2004)MATHGoogle Scholar
  3. 3.
    Nix, A.E., Vose, M.D.: Modelling Genetic Algorithm with Markov Chains. Annals of Mathematical and Artificial Intelligence 5, 79–88 (1992)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Davis, T.E., Principe, J.C.: A Markov Chain Framework for the Simple Genetic Algorithm. Evolutionary Computation 1, 269–288 (1993)CrossRefGoogle Scholar
  5. 5.
    Imai, J., Shioya, H., Kurihara, M.: Modeling of Genetic Algorithms Based on the Viewpoint of Mixture Systems. Transactions on Mathematical Modeling and its Applications (TOM) 44(SIG 07), 51–60 (2003)Google Scholar
  6. 6.
    Asoh, H., Mühlenbein, H.: On the Mean Convergence Time of Evolutionary Algorithms without Selection and Mutation. In: Davidor, Y., Männer, R., Schwefel, H.-P. (eds.) PPSN 1994. LNCS, vol. 866, pp. 88–97. Springer, Heidelberg (1994)Google Scholar
  7. 7.
    Furutani, H.: Schema Analysis of Average Fitness in Multiplicative Landscape. In: Cantú-Paz, E., Foster, J.A., Deb, K., Davis, L., Roy, R., O’Reilly, U.-M., Beyer, H.-G., Kendall, G., Wilson, S.W., Harman, M., Wegener, J., Dasgupta, D., Potter, M.A., Schultz, A., Dowsland, K.A., Jonoska, N., Miller, J., Standish, R.K. (eds.) GECCO 2003. LNCS, vol. 2723, pp. 934–947. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Crow, J.F., Kimura, M.: An Introduction to Population Genetics Theory. Harper and Row, New York (1970)MATHGoogle Scholar
  9. 9.
    Zhang, Y., Sakamoto, M., Furutani, H.: Effects of Population Size on Performance of Genetic Algorithms and Roles of Crossover and Mutation. In: The Thirteenth International Symposium on Artificial Life and Robotics (AROB 13th 2008), pp. 861–864 (2008)Google Scholar
  10. 10.
    Zhang, Y., Sakamoto, M., Furutani, H.: Probability of Obtaining Optimum Solutions in Genetic Algorithm and Roles of Mutation. IPSJ SIG Technical Reports, 2008-MPS-68, vol. 2008(17), pp. 161–164 (2008)Google Scholar

Copyright information

© Springer Tokyo 2010

Authors and Affiliations

  • Yu-an Zhang
    • 1
  • QingLian Ma
    • 2
  • Makoto Sakamoto
    • 2
  • Hiroshi Furutani
    • 2
  1. 1.Interdisciplinary Graduate School of Agriculture and EngineeringUniversity of MiyazakiMiyazaki CityJapan
  2. 2.Faculty of EngineeringUniversity of MiyazakiMiyazaki CityJapan

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