Soft Computing

, Volume 13, Issue 5, pp 419–436 | Cite as

Adaptation and application of multi-objective evolutionary algorithms for rule reduction and parameter tuning of fuzzy rule-based systems

  • María José GactoEmail author
  • Rafael Alcalá
  • Francisco Herrera


Recently, multi-objective evolutionary algorithms have been applied to improve the difficult tradeoff between interpretability and accuracy of fuzzy rule-based systems. It is known that both requirements are usually contradictory, however, these kinds of algorithms can obtain a set of solutions with different trade-offs. This contribution analyzes different application alternatives in order to attain the desired accuracy/interpr-etability balance by maintaining the improved accuracy that a tuning of membership functions could give but trying to obtain more compact models. In this way, we propose the use of multi-objective evolutionary algorithms as a tool to get almost one improved solution with respect to a classic single objective approach (a solution that could dominate the one obtained by such algorithm in terms of the system error and number of rules). To do that, this work presents and analyzes the application of six different multi-objective evolutionary algorithms to obtain simpler and still accurate linguistic fuzzy models by performing rule selection and a tuning of the membership functions. The results on two different scenarios show that the use of expert knowledge in the algorithm design process significantly improves the search ability of these algorithms and that they are able to improve both objectives together, obtaining more accurate and at the same time simpler models with respect to the single objective based approach.


Pareto Front Marginal Utility External Population Binary Tournament Selection Linear Utility Function 
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.



Supported in part by the Spanish Ministry of Education and Science under grant no. TIN2005-08386-C05-01, and the Andalusian government under grant no. P05-TIC-00531.


  1. Alcalá R, Alcalá-Fdez J, Casillas J, Cordón O, Herrera F (2006) Hybrid learning models to get the interpretability-accuracy trade-off in fuzzy modeling. Soft Comput 10(9):717–734CrossRefGoogle Scholar
  2. Alcalá R, Alcalá-Fdez J, Herrera F, Otero J (2007a) Genetic learning of accurate and compact fuzzy rule based systems based on the 2-tuples linguistic representation. Int J Approx Reason 44(1):45–64zbMATHCrossRefGoogle Scholar
  3. Alcalá R, Alcalá-Fdez J, Gacto MJ, Herrera F (2007b) Rule base reduction and genetic tuning of fuzzy systems based on the linguistic 3-tuples representation. Soft Computing 11(5):401–419CrossRefGoogle Scholar
  4. Alcalá R, Alcalá-Fdez J, Herrera F (2007c) A proposal for the genetic lateral tuning of linguistic fuzzy systems and its interaction with rule selection. IEEE Trans Fuzzy Syst 15(4):616–635CrossRefGoogle Scholar
  5. Alcalá R, Gacto MJ, Herrera F, Alcalá-Fdez J (2007d) A multi-objective genetic algorithm for tuning and rule selection to obtain accurate and compact linguistic fuzzy rule-based systems. Int J Uncertain Fuzziness Knowl Based Syst 15(5):539–557zbMATHCrossRefGoogle Scholar
  6. Branke J, Deb K, Dierolf H, Osswald M (2004) Finding knees in multi-objective optimization. In: Proc Parallel Problem Solving from Nature Conf.—PPSN VIII. LNCS, vol 3242. Birmingham, UK, pp 722–731Google Scholar
  7. Casillas J, Cordón O, Herrera F, Magdalena L (eds) (2003a) Interpretability issues in fuzzy modeling. Studies in Fuzziness and Soft Computing. Springer, Heidelberg, pp 128Google Scholar
  8. Casillas J, Cordón O, Herrera F, Magdalena L (eds) (2003b) Accuracy improvements in linguistic fuzzy modelling. Studies in Fuzziness and Soft Computing. Springer, Heidelberg, pp 129Google Scholar
  9. Casillas J, Cordón O, del Jesus MJ, Herrera F (2005) Genetic tuning of fuzzy rule deep structures preserving interpretability and its interaction with fuzzy rule set reduction. IEEE Trans Fuzzy Syst 13(1):13–29CrossRefGoogle Scholar
  10. Cococcioni M, Ducange P, Lazzerini B, Marcelloni F (2007) A Pareto-based multi-objective evolutionary approach to the identification of mamdani fuzzy systems. Soft Comput 11:1013–1031CrossRefGoogle Scholar
  11. Coello CA, Toscano G (2001) A Micro-Genetic Algorithm for multiobjective optimization. In: First international conference on evolutionary multi-criterion optimization. LNCS 1993, London, UK, pp 126–140Google Scholar
  12. Coello CA, Van Veldhuizen DA, Lamont GB (2002) Evolutionary algorithms for solving multi-objective problems. Kluwer, DordrechtGoogle Scholar
  13. Cordón O, Herrera F, Sánchez L (1999) Solving electrical distribution problems u-sing hybrid evolutionary data analysis techniques. Appl Intell 10:5–24CrossRefGoogle Scholar
  14. Cordon O, Herrera F, del Jesus MJ, Villar P (2001) A multiobjective genetic algorithm for feature selection and granularity learning in fuzzy-rule based cla-ssi-fication systems. In: Proceedings of IX IFSA World Congress and XX NAFIPS Int. Conf., Vancouver, Canada, pp 1253–1258Google Scholar
  15. Corne D, Knowles J, Oates M (2000) The Pareto Envelope-based Selection Algorithm for multiobjective optimization. In: Proc. Parallel Problem Solving from Nature Conf.—PPSN VI, LNCS 1917. Paris, France, pp 839–848Google Scholar
  16. Corne D, Jerram N, Knowles J, Oates M (2001) PESA-II: Region based selection in evolutionary multiobjective optimization. In: Proceedings of genetic and evolutionary computation conf., San Francisco, pp 283–290Google Scholar
  17. Deb K (2001) Multi-objective optimization using evolutionary algorithms. Wiley, New YorkzbMATHGoogle Scholar
  18. Deb K, Agrawal S, Pratab A, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evolut Comput 6(2):182–197CrossRefGoogle Scholar
  19. Erickson M, Mayer A, Horn J (2001) The Niched Pareto Genetic Algorithm 2 a-pplied to the design of groundwater remediation systems. In: Proceedings of first international conference on evolutionary multi-criterion optimization. LNCS, vol 1993. London, UK, pp 681–695Google Scholar
  20. Eshelman LJ (1991) The CHC adaptive search algorithm: How to have safe search when engaging in nontraditional genetic recombination. Found Genetic Algorithms 1:265–283Google Scholar
  21. Eshelman LJ, Schaffer JD (1993) Real-coded genetic algorithms and interval-schemata. Found Genetic Algorithms 2:187–202Google Scholar
  22. Fonseca CM, Fleming PJ (1993) Genetic algorithms for multiobjective optimization: Formulation, discussion and generalization. In: Proceedings of 5th international conference on genetic algorithms, San Mateo, pp 416–423Google Scholar
  23. Goldberg D (2000) A meditation on the computational intelligence and its future. Illigal Report #2000019, Department of General Engineering, University of Illinois at Urbana-Champaign, Foreword Proc. of the 2000 Int. Symposium on Computational IntelligenceGoogle Scholar
  24. Horn J, Nafpliotis N, Goldberg DE (1994) A niched Pareto genetic algorithm for multiobjective optimization. In: Proceedings of first IEEE conference on evolutionary computation, IEEE World Congress on Computational Intelligence, vol 1, Piscataway, pp 82–87Google Scholar
  25. Ishibuchi H, Murata T (1996) Multi-objective genetic local search algorithm. In: Proceedings of third IEEE international conference on evolutionary computation, Japan, pp 119–124Google Scholar
  26. Ishibuchi H, Nojima Y (2007) Analysis of interpretability-accuracy tradeoff of fuzzy systems by multiobjective fuzzy genetics-based machine learning. Int J Approx Reason 44(1):4–31zbMATHCrossRefMathSciNetGoogle Scholar
  27. Ishibuchi H, Yamamoto T (2003) Interpretability issues in fuzzy genetics-based machine learning for linguistic modelling. In: Lawry J, Shanahan JG, Ralescu AL (ed) Modelling with words: learning, fusion, and reasoning within a formal liguistic representation framework. LNCS, vol 2873. Springer, Berlin, pp 209–228Google Scholar
  28. Ishibuchi H, Yamamoto T (2004) Fuzzy rule selection by multi-ob-jective genetic local search algorithms and rule evaluation measures in data mining. Fuzzy Sets Syst 141(1):59–88zbMATHCrossRefMathSciNetGoogle Scholar
  29. Ishibuchi H, Murata T, Turksen IB (1997) Single-objective and two-objective genetic algorithms for selecting linguistic rules for pattern classification problems. Fuzzy Sets Syst 89(2):135–150CrossRefGoogle Scholar
  30. Ishibuchi H, Nakashima T, Murata T (2001) Three-objective genetics-based machine learning for linguistic rule extraction. Inf Sci 136:109–133zbMATHCrossRefGoogle Scholar
  31. Jimenez F, Gomez-Skarmeta AF, Roubos H, Babuska R (2001) Accurate, transparent, and compact fuzzy models for function approximation and dynamic modeling through multi-objective evolutionary optimization. In: Proceedings of first international conference on evolutionary multi-criterion optimization. LNCS, vol 1993, Zurich, Switzerland, pp 653–667Google Scholar
  32. Knowles JD, Corne DW (2000) Approximating the non dominated front using the Pareto archived evolution strategy. Evolut Comput 8(2):149–172CrossRefGoogle Scholar
  33. Narukawa K, Nojima Y, Ishibuchi H (2005) Modification of evolutionary multiobjective optimization algorithms for multiobjective design of fuzzy rule-based classification systems. In: Proceedings of 2005 IEEE international conference on fuzzy systems, Reno, USA, pp 809–814Google Scholar
  34. Mamdani EH, Assilian S (1975) An experiment in linguistic synthesis with a fuzzy logic controller. Int J Man Mach Stud 7:1–13zbMATHCrossRefGoogle Scholar
  35. Rosenberg RS (1967) Simulation of genetic populations with biochemical properties. MA Thesis, Univ. Michigan, Ann Harbor, MichiganGoogle Scholar
  36. Schaffer JD (1985) Multiple objective optimization with vector evaluated genetic algorithms. In: Proceedings of first international conference on genetic algorithms, Pittsburgh, pp 93–100Google Scholar
  37. Srinivas N, Deb K (1994) Multiobjective optimization using nondominated sorting in genetic algorithms. Evolut Comput 2:221–248CrossRefGoogle Scholar
  38. Takagi T, Sugeno M (1985) Fuzzy identification of systems and its application to modeling and control. IEEE Trans Syst Man Cybern 15(1):116–132zbMATHGoogle Scholar
  39. Wang LX, Mendel JM (1992) Generating fuzzy rules by learning from examples. IEEE Trans Syst Man Cybern 22(6):1414–1427CrossRefMathSciNetGoogle Scholar
  40. Wang HL, Kwong S, Jin YC, Wei W, Man KF (2005a) Multi-objective hierarchical genetic algorithm for interpretable fuzzy rule-based knowledge extraction. Fuzzy Sets Syst 149(1):149–186zbMATHCrossRefMathSciNetGoogle Scholar
  41. Wang HL, Kwong S, Jin YC, Wei W, Man KF (2005b) Agentbased evolutionary approach for interpretable rule-based knowledge extraction. IEEE Trans Syst Man Cybern C Appl Rev 35(2):143–155CrossRefGoogle Scholar
  42. Waugh S (1995) Extending and Benchmarking Cascade-Correlation. PhD Thesis, Computer Science Department, University of TasmaniaGoogle Scholar
  43. Zitzler E, Thiele L (1999) Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans Evolut Comput 3(4):257–271CrossRefGoogle Scholar
  44. Zitzler E, Laumanns M, Thiele L (2001) SPEA2: Improving the strength Pareto evolutionary algorithm for multiobjective optimization. In: Proc evolutionary methods for design, optimization and control with app to industrial problems, Barcelona, Spain, pp 95–100Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • María José Gacto
    • 1
    Email author
  • Rafael Alcalá
    • 1
  • Francisco Herrera
    • 1
  1. 1.Department of Computer Science and A.IUniversity of GranadaGranadaSpain

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