Abstract
Background: Fuzzy systems are employed in several fields like data processing, regression, pattern recognition, classification and management as a result of their characteristic of handling uncertainty and explaining the feature of the advanced system while not involving a particular mathematical model. Fuzzy rule-based systems (FRBS) or fuzzy rule-based classifiers (mainly designed for classification purpose) are primarily the fuzzy systems that consist of a group of fuzzy logical rules and these FRBS are unit annexes of ancient rule-based systems, containing the "If-then" rules. During the design of any fuzzy systems, there are two main objectives, interpretability and accuracy, which are conflicting with each another, i.e., improvement in any of those two options causes the decrement in another. This condition is termed as Interpretability –Accuracy Trade-off. To handle this condition, Multi-Objective Evolutionary Algorithms (MOEA) are often applied within the design of fuzzy systems. This paper reviews the approaches to the problem of developing fuzzy systems victimization evolutionary process Multi-Objective Optimization (EMO) algorithms considering ‘Interpretability-Accuracy Trade-off, current research trends and improvement in the design of fuzzy classifier using MOEA in the future scope of authors.
Methods: The state-of-the-art review has been conducted for various fuzzy classifier designs, and their optimization is reviewed in terms of multi-objective.
Result: This article reviews the different Multi-Objective Optimization (EMO) algorithms in the context of Interpretability -Accuracy tradeoff during fuzzy classification.
Conclusion: The evolutionary multi-objective algorithms are being deployed in the development of fuzzy systems. Improvement in the design using these algorithms include issues like higher spatiality, exponentially inhabited solution, I-A tradeoff, interpretability quantification, and describing the ability of the system of the fuzzy domain, etc. The focus of the authors in future is to find out the best evolutionary algorithm of multi-objective nature with efficiency and robustness, which will be applicable for developing the optimized fuzzy system with more accuracy and higher interpretability. More concentration will be on the creation of new metrics or parameters for the measurement of interpretability of fuzzy systems and new processes or methods of EMO for handling I-A tradeoff.
Keywords: FRBS, I-A tradeoff, evolutionary algorithm, fuzzy rules, multi-objective, MOEA, EMO.
Graphical Abstract
[1]
J. Casillas, F. Herrera, R. P’erez, M.J. del Jesus, and P. Villar, "Preface: Special issue on genetic fuzzy systems and the interpretability-accuracy trade-off", Int. J. Approx. Reason., vol. 44, no. 1, pp. 1-3, 2007.
[2]
J. Casillas, O. Cord’on, F. Herrera, and L. Magdalena, Accuracy improvements in linguistic fuzzy modeling., Springer-Verlag, 2003.
[3]
J. Casillas, O. Cordón, and F.H. Triguero, L. Magdalena and EditorsInterpretability issues in fuzzy modeling, Springer-Verlag, 2003.
[4]
R. Mikut, J. J¨akel, and L. Gr¨oll, "Interpretability issues in data-based learning of fuzzy systems", Fuzzy Sets Syst., vol. 150, pp. 179-197, 2005.
[5]
O. Cord’on, F. Herrera, F. Hoffmann, and L. Magdalena, "Genetic fuzzy systems evolutionary tuning and learning of fuzzy knowledge bases, ser. advances in fuzzy systems applications and theory", In: World Sci, vol. Vol. 19. 2001.
[6]
H. Roubos, and M. Setnes, "Compact and transparent fuzzy models and classifiers through iterative complexity reduction", IEEE Trans. Fuzzy Syst., vol. 9, no. 4, pp. 516-524, 2001.
[7]
X. Chang, and J.H. Lilly, "Evolutionary design of a fuzzy classifier from data", IEEE Trans. Syst. Man Cybern. B Cybern., vol. 34, no. 4, pp. 1894-1906, 2004.
[8]
J. Cassilas, O. Cordon, F. Herrera, and L. Magdalena, Interpretability improvements to find the balance interpretability-accuracy in fuzzy modeling: An overview in Interpretability issues in fuzzy modeling, studies in fuzziness and soft computing., vol. Vol. 128. Springer: Berlin, Heidelberg, 2003.
[9]
P.K. Shukla, and S.P. Tripathi, "A review on the interpretability-accuracy trade-off in Evolutionary Multi-Objective Fuzzy Systems (EMOFS)", Information, vol. 3, no. 3, pp. 256-277, 2012.
[10]
P.K. Shukla, and S.P. Tripathi, A survey on interpretability accuracy trade-off in evolutionary fuzzy systemsIn IEEE fifth International Conference on Genetic and Evolutionary Computing, 2011.
[11]
P.K. Shukla, and S.P. Tripathi, Interpretability issues in evolutionary multi-objective fuzzy knowledge base systemsIn Seventh International Conference on Bio-Inspired Computing: Theories and Applications, Advances in Intelligent Systems and Computing, Springer: India, 2013.
[12]
O. Cordon, F. Herrera, M.J. Del Jesus, and P. Villar, A multi-objective genetic algorithm for feature selection and granularity learning in fuzzy rule based classification systemsIn Proceedings of 9th IFSA World Congress, 2001, pp. 1253-1258.
[13]
H. Ishibuchi, and Y. Nojima, Accuracy-complexity trade-off algorithms by multi objective rule selectionIn Proceedings of the Workshop on Computational Intelligence in Data Mining, 2005, pp. 39-48.
[14]
R. Alcala, M.J. Gacto, and F. Herrera, "A multi-objective genetic algorithm for tuning and rule selection to obtain accurate and compact linguistic fuzzy rule-based systems", Int. J. Uncertain. Fussiness Knowledge Syst., vol. 15, no. 5, pp. 539-557, 2005.
[15]
R. Alcala, J. A-Fdez, M.J. Gacto and F. Herrera, “A multi-objective evolutionary algorithm for rule-selection and tuning on fuzzy rule based systemsIn Proceedings of Fuzzy Systems Conference, 2005, pp. 1367-1372.
[16]
M.J. Gacto, R. Alcala, and F. Herrera, An improved multi-objective genetic algorithm for tuning linguistic fuzzy systemsIn Proceedings of the International Conference on Information Processing Management, Uncertainty and Knowledge-Based System, 2008, pp. 1121-1128.Malaga, Spain
[17]
R. Alcala, J. Alcala-Fdez, M.J. Gacto, and F. Herrera, On the usefulness of MOEAs for getting compact FRBSs under parameter tuning and rule selection.In Multi-Objective Evolutionary Algorithms for Knowledge Discovery from Databases., vol. 98. Springer: Berlin, Heidelberg, 2008, pp. 91-107.
[18]
R. Alcala, P. Ducange, F. Herrera, B. Lazzerini, and F. Marcelloni, "A multi-objective evolutionary approach to concurrently learn rule and databases of linguistic fuzzy rule based systems", IEEE Trans. Fuzzy Syst., vol. 17, no. 5, pp. 1106-1121, 2009.
[19]
M. Antonelli, P. Ducange, B. Lazzerini, and F. Marcelloni, "Learning concurrently partition granularities and rule bases of mamdani fuzzy systems in a multi-objective evolutionary framework", Int. J. Approximate. Reason., vol. 50, pp. 1066-1080, 2009.
[20]
M.J. Gacto, R. Alcala, and F. Herrera, Multi-objective genetic fuzzy systems: On the necessity of including expert knowledge in the MOEA design processIn Proceedings of IPMU, 2008, pp. 1446-1453.
[21]
M.J. Gacto, R. Alcala, and F. Herrera, "Adaptation and application of multi-objective evolutionary algorithms for rule reduction and parameter tuning of fuzzy rule based systems", Soft Comput., vol. 13, pp. 419-436, 2009.
[22]
A.G. Di Nuovo, and V. Catania, Linguistic modifiers to improve the accuracy interpretability trade-off in multi-objective genetic design of fuzzy rule based classifier systems In 9th International Conference on Intelligent Systems Design and Applications, 2009, pp. 128-133.
[23]
R. Alcala, Y. Nojima, F. Herrera, and H. Ishibuchi, "Multi-objective genetic fuzzy rule selection of single granularity based fuzzy classification rules and its interaction with lateral tuning of membership functions", Soft Comput., vol. 15, no. 12, pp. 2303-2318, 2011.
[24]
M. Antonelli, P. Ducange, B. Lazzerini, and F. Marcelloni, Multi-objective evolutionary generation of mamdani fuzzy rule based systems based on rule and condition selectionIn Proceedings of the 5th IEEE International Workshop on Genetic and Evolutionary Fuzzy Systems, 2011, pp. 47-53.
[25]
H. Ishibuchi, Y. Nakashima, and Y. Nojima, Effects of fine fuzzy partitions on the generalization ability of evolutionary multi-objective fuzzy rule based classifiersIn International Conference on Fuzzy Systems, 2010, pp. 1-8.
[26]
M. Cococcioni, P. Ducange, B. Lazzerini, and F. Marcelloni, "A pareto based multi objective evolutionary approach to the identification of mamdani fuzzy systems", Soft Comput., vol. 11, pp. 1013-1031, 2007.
[28]
J.R. Koza, Genetic programming on the programming of computers by means of natural selection., The MIT Press: Cambridge, Massachusetts, 1992.
[29]
D.E. Goldberg, Genetic algorithms in search optimization and machine learning., Addison Wesley Publishing Company Reading Massachusetts, 1989.
[31]
N. Srinivas, and K. Deb, "Multiobjective optimization using non-dominated sorting in genetic algorithms", Evol. Comput., vol. 2, no. 3, pp. 221-248, 1994.
[32]
J. Horn, N. Nafpliotis, and D.E. Goldberg, "A niched pareto genetic algorithm for multiobjective optimization", In: Proceedings of the Ist IEEE Conference on Evolutionary Computation, vol. 1. pp. 82-87.1994,
[33]
C.M. Fonseca, and P.J. Fleming, "Genetic algorithms for multiobjective optimization: Formulation, discussion and generalization", In: Proceedings of the 5th International Conference on Genetic Algorithms, 1993, pp. 416-423.
[34]
E. Zitzler, and L. Thiele, "Multi objective evolutionary algorithms: A comparative case study and the strength pareto approach", IEEE Trans. Evol. Comput., vol. 3, no. 4, pp. 257-271, 1999.
[35]
E. Zitzler, M. Laumanns, and L. Thiele, " SPEA2: Improving the strength pareto evolutionary algorithms. Technical Report 103. Computer Engineering & Networks Laboratory (TIK). Swiss Federal Institute of Technology (ETH). Zurich, Switzerland", 2001
[36]
J.D. Knowles, and D.W. Corne, "Approximating the non-dominated front using the pareto achieved evolution strategy", Evol. Comput., vol. 8, no. 2, pp. 149-172, 2000.
[37]
K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, "A fast and elitist multiobjective genetic algorithm: NSGA II", IEEE Trans. Evol. Comput., vol. 6, no. 2, pp. 182-197, 2002.
[38]
M. Erickson, A. Mayer, and J. Horn, "The niched pareto genetic algorithms applied to the design of ground water remediation system", In: Ist International Conference on Evolutionary Multi Criteria Optimization, Springer-Verlag, 2001, pp. 681-695.
[39]
D.W. Corne, J.D. Knowles, and M.J. Oates, "Thepareto envelop based selection algorithm for multi-objective optimization", In: Proceedings of VI Conference of Parallel Problem Solving from Nature, Paris, France, Springer LNCS 1917, 2000, pp. 839-848.
[40]
C.A.C. Coello, and G.T. Pulido, "A micro genetic algorithm for Multi objective optimization", In: Proceedings of the First International Conference on Evolutionary Multi-Criteria Optimization, 2001, pp. 126-140.
[41]
C.A.C. Coello, and G.T. Pulido, "Multi-objective optimization using a micro-genetic algorithm", In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO’ 2001), Morgan Kaufmann Publishers, pp. 274-282. 2001
[42]
H. Ishibuchi, T. Murata, and I.B. Turksen, "Single objective and two objective genetic algorithms for selecting linguistic rules for pattern classification problems", Fuzzy Sets Syst., pp. 135-150, 1997.
[43]
O. Cordon, F. Herrera, M.J. Del Jesus, and P. Villar, "A multi-objective genetic algorithm for feature selection and granularity learning in fuzzy rule based classification systems", In: Proceedings of 9th International Fuzzy Systems Associations (IFSA) World Congress, Vancouver, Canada, 2001, pp. 1253-1258.
[44]
H. Ishibuchi, and Y. Nojima, "Accuracy-complexity trade-off algorithms by multi-objective rule selection", In: Proceedings of 2005 Workshop on Computational Intelligence in Data Mining, Houston, TX, USA, 2005, pp. 39-48.
[45]
R. Alcala, M.J. Gacto, and F. Herrera, "A multi-objective genetic algorithm for tuning and rule selection to obtain accurate and compact linguistic fuzzy rule-based systems", Int. J. Uncertain. Fussiness Knowl. Based Syst, pp. 539-557, 2007.
[46]
M. Cococcioni, P. Ducange, B. Lazzerini, and F. Marcelloni, "A pareto based multi-objective evolutionary approach to the identification of Mamdani fuzzy systems", In: Soft Comput., 2007, pp. 1013-1031.
[47]
R. Alcala, "J. A-Fdez, M.J. Gacto and F. Herrera, “A multi-objective evolutionary algorithm for rule-selection and tuning on fuzzy rule based systems", In: Proceedings of IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2007), London, UK, 2007, pp. 1367-1372.
[48]
M.J. Gacto, R. Alcala, and F. Herrera, "An improved multiobjective genetic algorithm for tuning linguistic fuzzy systems", In: Proceedings of Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU’08)M, Málaga, Spain, 2008, pp. 1121-1128.
[49]
M. Anotonelli, P. Ducange, B. Lazzerini, and F. Marcelloni, "A multi-objective genetic approach to concurrently learn partition granularity and rule bases of Mamdani fuzzy systems", In: Proceedings of 8th International Conference on Hybrid Intelligent Systems, Barcelona, Spain, 2008, pp. 278-283.
[50]
R. Alcala, J. Alcala-Fdez, M.J. Gacto, and F. Herrera, "On the usefulness of MOEAs for getting compact FRBSs under parameter tuning and rule selection", In: Stud. Comput. Intell, pp. 91-107. 2008
[51]
R. Alcala, P. Ducange, F. Herrera, B. Lazzerini, and F. Marcelloni, "A multi-objective evolutionary approach to concurrently learn rule and databases of linguistic fuzzy rule based systems", IEEE Trans. Fuzzy Syst., vol. 17, pp. 1106-1121, 2009.
[52]
M.J. Gacto, R. Alcala, and F. Herrera, "A multi-objective evolutionary algorithm for tuning fuzzy rule based systems with measures for preserving interpretability", In: Proceedings of International Fuzzy Systems Association World Congress and European Society of Fuzzy Logic and Technology Conference (IFSA-EUSFLAT), Lisbon, Portugal, pp. 1146-1151. 2009
[53]
M. Antonelli, P. Ducange, B. Lazzerini, and F. Marcelloni, "Learning concurrently partition granularities and rule bases of Mamdani fuzzy systems in a multi-objective evolutionary framework", Int. J. Approx. Reason., vol. 50, pp. 1066-1080, 2009.
[54]
M.J. Gacto, R. Alcala, and F. Herrera, "Multi-objective genetic fuzzy systems: On the necessity of including expert knowledge in the MOEA design process", In: Proceedings of Information Processing and Management of Uncertainty in Knowledge-Based Systems., (IPMU): Malaga, Spain, 2008, pp. 1446-1453.
[55]
M.J. Gacto, R. Alcala, and F. Herrera, "Adaptation and application of multi-objective evolutionary algorithms for rule reduction and parameter tuning of fuzzy rule based systems", Soft Comput., vol. 13, pp. 419-436, 2009.
[56]
R. Alcala, M.J. Gacto, and F. Herrera, 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., 2007, pp. 539-557.
[57]
A.G. Di Nuovo, and V. Catania, "Linguistic modifiers to improve the accuracy-interpretability trade-off in multi-objective genetic design of fuzzy rule based classifier systems", In: Proceedings of 9th International Conference on Intelligent Systems Design and Applications, Pisa, Italy, 2009, pp. 128-133.
[58]
R. Alcala, Y. Nojima, F. Herrera, and H. Ishibuchi, "Multi-objective genetic fuzzy rule selection of single granularity-based fuzzy classification rules and its interaction with lateral tuning of membership functions", Soft Comput., pp. 2303-2318, 2011.
[59]
M. Antonelli, P. Ducange, B. Lazzerini, and F. Marcelloni, "Multi-objective evolutionary generation of mamdani fuzzy rule based systems based on rule and condition selection", In: Proceedings of 5th IEEE International Workshop on Genetic and Evolutionary Fuzzy Systems, Paris, France, 2011, pp. 47-53.
[60]
M. Galende-Hernández, G.I. Sainz-Palmero, and M.J. Fuente-Aparicio, "Complexity reduction and interpretability improvement for fuzzy rule systems based on simple interpretability measures and indices by bi-objective evolutionary rule selection", Soft Comput., vol. 16, no. 3, pp. 451-470, 2012.
[61]
J. Gonzalez, I. Rojas, H. Pomares, F. Rojas, and J.M. Palomares, Multi-objective evolution of fuzzy systems.Soft Comput., vol. 10. 2006, pp. 735-748.
[62]
J. Gonzalez, I. Rojas, H. Pomares, L.J. Herrera, A. Guillen, J.M. Palomares, and F. Rojas, "Improving the accuracy while preserving the interpretability of fuzzy function approximators by means of Multi-objective evolutionary algorithms", Int. J. Approx. Reason Vol. 44, pp. 58-63, 2007.
[63]
H. Ishibuchi, "Evolutionary multi-objective optimization for fuzzy knowledge extraction", In: Proceedings of International Symposium on Advanced Intelligent Systems, Sokcho, Korea, 2007, pp. 58-63.
[64]
M. Antonelli, P. Ducange, and F. Marcelloni, "Exploiting a coevolutionary approach to concurrently select training instances and learn rule bases of mamdani fuzzy systems", In: Proceedings of IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Barcelona, Spain, 2010, pp. 1-7.
[65]
M. Antonelli, P. Ducange, and F. Marcelloni, Genetic training instance selection in multi-objective evolutionary fuzzy systems: A co-evolutionary approach.IEEE Trans. Fuzzy Syst., vol. 20. 2012, pp. 276-290.
[66]
A.A. Marquez, F.A. Marquez, and A. Peregrin, "A multi-objective evolutionary algorithm with an interpretability improvement mechanism for linguistic fuzzy systems with adaptive defuzzification", In: Proceedings of IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Barcelona, Spain, 2010, pp. 1-7.
[67]
H. Ishibuchi, T. Nakashima, and T. Murata, Three-objectives genetics-based machine learning for linguistic rule extraction.Inf. Sci., vol. 136. 2001, pp. 109-133.
[68]
H. Ishibuchi, and T. Yamamoto, Fuzzy rule selection by multi-objective genetic local search algorithms and rule evaluation measures in data mining.Fuzzy Sets Syst. Vol. 141, 2004, pp. 59-88.
[69]
M.R. Delgado, "F.V, Zuben and F. Gomide, “Multi-Objective decision making: Towards improvement of accuracy, interpretability and design autonomy in hierarchical genetic fuzzy systems", In: Proceedings of IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Honolulu, HI, USA, 2002, pp. 1222-1227.
[70]
M. Antonelli, P. Ducange, B. Lazzrini, and F. Mareclloni, "Multi-objective evolutionary learning of granularity, membership function parameters and rules of Mamdani fuzzy systems", Evol. Intell. Vol. 2, pp. 21-37, 2009.
[71]
Z.Y. Xing, Y. Zhang, Y.L. Hou, and G.Q. Cai, "Multi-objective fuzzy modeling using NSGA-II", In: Proceedings of IEEE Conference on Cybernetics and Intelligent Systems, Chengdu, China 21-24 September, 2008, pp. 119-124.
[72]
A.A. Marquez, F.A. Marquez, and A. Peregrin, A mechanism to improve the interpretability of linguistic fuzzy systems with adaptive defuzzification based on the use of a multi-objective evolutionary algorithms.Int. J. Comput. Intell. Syst. Vol. 5, 2012, pp. 297-321.
[73]
M. Antonelli, P. Ducange, B. Lazzerini, and F. Marcelloni, Learning knowledge bases of multi-objective evolutionary fuzzy systems by simultaneously optimizing accuracy, complexity and partition integrity.Soft Comput. Vol. 15, 2011, pp. 2335-2354.
[74]
M. Antonelli, P. Ducange, and B. Lazzerini, A Three-Objective Evolutionary Approach to Gene Rate Mamdani Fuzzy Rule Based Systems., Springer-Verlag: Berlin/Heidelberg, Germany, 2009, pp. 613-620.
[75]
P. Pulkkinen, and H. Koivisto, A dynamically constrained multiobjective genetic fuzzy systems for regression problems.IEEE Trans. Fuzzy Syst. Vol. 18,, 2010, pp. 161-167.
[76]
J.M. Alonso, L. Magdalena, and O. Cordon, "Embedding HILK in a three objective evolutionary algorithm with the aim of modeling highly interpretable fuzzy rule-based classifiers", In: Proceedings of 4th International Workshop on Genetic and Evolutionary Fuzzy Systems, Asturias, Spain, 2010, pp. 15-20.
[77]
M.J. Gacto, R. Alcala, and F. Herrera, "Analysis of the performance of a semantic interpretability-based tuning and rule selection of fuzzy rule-based systems by means of a multi-objective evolutionary algorithm", In: Proceedings of the 23rd International Conference on Industrial Engineering and Other Applications of Applied Intelligent Systems (IEA/AIE), Cordoba, Spain, 2010, pp. 228-238.
[78]
M.J. Gacto, R. Alcala, and F. Herrera, Integration of Index to preserve the semantic interpretability in the multi-objective evolutionary rule selection and tuning of linguistic fuzzy systems.IEEE Trans. Fuzzy Syst. Vol. 18, 2010, pp. 515-531.
[79]
M. Gonzalez, J. Cassilas, and C. Morell, "Dealing with three uncorrelated criteria by multi-objective genetic fuzzy systems", In: Proceedings of 5th International Workshop on Genetic and Evolutionary Fuzzy Systems, Paris, France, 2011, pp. 39-46.
[80]
Y. Zhang, X.B. Wu, Z.Y. Xing, and W.L. Hu, On generating interpretable and precise fuzzy systems based on pareto multi-objective cooperating co-evolutionary algorithm.Appl. Soft Comput., 2011, pp. 1289-1294.
[81]
H. Ishibuchi, and Y. Nojima, "Analysis of interpretability-accuracy trade-off of fuzzy systems by multi-objective fuzzy genetics-based machine learning", Int. J. Approx. Reason., vol. 44, pp. 4-31, 2007.
[82]
K. Nakukawa, Y. Nojima, and H. Ishibuchi, "Modification of evolutionary multi objective optimization algorithms for multi-objective design of fuzzy rule-based classification systems", In: Proceedings of IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Reno, NV, USA, 2005, pp. 809-814.
[83]
H. Ishibuchi, Y. Nakashima, and Y. Nojima, "Simple changes in problem formulations make a difference in multi-objective genetic fuzzy systems", In: Proceedings of 4th International Workshop on Genetic and Evolutionary Fuzzy Systems, Mieres, Spain, 2010, pp. 3-8.
[84]
H. Ishibuchi, Y. Nakashima, and Y. Nojima, "Double cross validation for performance evaluation of multi-objective genetic fuzzy systems", In: Proceedings of 5th International Workshop on Genetic and Evolutionary Fuzzy Systems, Paris, France, 2011, pp. 31-38.
[85]
Y. Nojima, H. Ishibuchi, and I. Kuwajima, "Comparison of search ability between genetic fuzzy rule selection and fuzzy genetics based machine learning", In: Proceedings of International Symposium on Evolving Fuzzy Systems, Ambelside, UK, 2006, pp. 125-130.
[86]
H. Ishibuchi, Y. Nakashima, and Y. Nojima, "Search ability of evolutionary multi-objective optimization algorithms for multi-objective fuzzy genetics based machine learning", In: Proceedings of IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Jeju Island, Korea, 2009, pp. 1724-1729.
[87]
H. Ishibuchi, Y. Nakashima, and Y. Nojima, Performance evaluation of evolutionary multi-objective optimization algorithms for multi-objective fuzzy genetics based machine learning.Soft Comput., vol. 15. 2011, pp. 2415-2434.
[88]
Q. Zhang, and H. Li, "MOEA/D: 2007 A multi-objective evolutionary algorithm based on decomposition", IEEE Trans. Evol. Comput., vol. 11, pp. 712-731, 2011.
[89]
H. Ishibuchi, and Y. Nojima, "Evolutionary multi-objective optimization for the design of fuzzy rule based ensemble classifiers", Int. J. Hybrid Intell. Syst., vol. 3, pp. 129-145, 2006.
[90]
H. Ishibuchi, and Y. Nojima, Fuzzy ensemble design thorough multi-objective fuzzy rule selection.Stud. Comput. Intell. Vol 16, 2006, pp. 507-530.
[91]
H. Ishibuchi, and Y. Nojima, "Optimization of scalarizing functions through evolutionary multi-objective optimization", In: Proceedings of the 4th International Conference on Evolutionary Multi-criterion Optimization (EMO), Matsishima, Japan, 2007, pp. 51-65.
[92]
H. Ishibuchi, Y. Nakashima, and Y. Nojima, "Effects of fine fuzzy partitions on the generalization ability of evolutionary multi-objective fuzzy rule based classifiers", In: Proceedings of IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Barcelona, Spain, 2010, pp. 1-8.
[93]
J. Branke, and K. Deb, "Integrating user preferences into evolutionary multi-objective optimization. Knowledge incorporation in evolutionary computation", In: , vol. Vol. 167. Jin, Y., Ed.; Springer: Berlin, Germany, 2004.
[94]
Y. Nojima, and H. Ishibuchi, "Interactive fuzzy modeling by evolutionary multi-objective optimization with user preferences", In: Proceedings of International Fuzzy Systems Association World Congress and European Society of Fuzzy Logic and Technology Conference (IFSA-EUSFLAT), Lisbon, Portugal, 2009, pp. 1839-1844.
[95]
Y. Nojima, and H. Ishibuchi, "Incorporation of user preference into multi-objective genetic fuzzy rule selection for pattern classification problems", In: Proceedings of 14th International Symposium on Artificial Life and Robotics, Ooita, Japan, 2009, pp. 186-189.
[96]
Y. Nojima, and H. Ishibuchi, "Interactive genetic fuzzy rule selection through evolutionary multi-objective optimization with user preference", In: Proceedings of IEEE Symposium on Computational Intelligence in Multicriteria Decision-Making, 2009, pp. 141-148.
[97]
H. Ishibuchi, and S. Namba, Evolutionary multi-objective knowledge extraction for high dimensional pattern classification problems., vol. 3242. Springer: Heidelberg, Germany, 2004, pp. 1123-1132.
[98]
M. Antonelli, P. Ducange, and F. Marcelloni, "A new approach to handle high dimensional and large data sets in multi-objective evolutionary fuzzy systems", In: Proceedings of IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Taipei, Taiwan, 2011, pp. 1286-1293.
[99]
R. Alcala, M.J. Gacto, and F. Herrera, "A fa, t and scalable multi-objective genetic fuzzy system for linguistic fuzzy modeling in high dimensional regression problems", IEEE Trans. Fuzzy Syst., vol. 19, pp. 666-681, 2011.
[100]
A. Botta, B. Lazzerini, F. Marcelloni, and D.C. Stefanescu, "Context adaptation of fuzzy systems through a multi-objective evolutionary approach based on a novel interpretability index", Soft Comput., vol. 13, pp. 437-449, 2009.
[101]
R. Cannone, J.M. Alonso, and L. Magdalena, "Multi-objective design of highly interpretable fuzzy rule based classifiers with semantic co-intention", In: Proceedings of 5th International Workshop on Genetic and Evolutionary Fuzzy Systems, Paris, France, 2011, pp. 1-8.
[102]
H. Ishibuchi, I. Kuwajima, and Y. Nojima, "Relation between pareto-optimal fuzzy rules and pareto optimal fuzzy rule sets", In: Proceedings of IEEE Symposium on Computational Intelligence in Multicriteria Decision Making (MCDM), Honolulu, HI, USA, 2007, pp. 42-49.
[103]
H. Ishibuchi, I. Kuwajima, and Y. Nojima, "Evolutionary multi-objective rule selection for classification rule mining", Stud. Comput. Intell., vol. 98, pp. 47-70, 2008.
[104]
H. Ishibuchi, I. Kuwajima, and Y. Nojima, Multi-objective classification rule mining. Multi-Objective Problem Solving from Nature., World Scientific: Ackensack, NJ, USA, 2008, pp. 219-240.
[105]
C.J. Carmona, P. Gonzalez, M.J. Del Jesus, and F. Herrera, "NMEEF-SD: Non-dominated multi-objective evolutionary algorithm for extracting fuzzy rules in subgroup discovery", IEEE Trans. Fuzzy Syst., vol. 18, pp. 958-970, 2010.
[106]
C.J. Carmona, R. Gonzalaez, R. Gacto, and M.J. Del Jesus, "Genetic lateral tuning for subgroup discovery with fuzzy rules using the algorithm NMEEF-SD", Int. J. Comput. Intell. Syst., vol. 5, no. 2, pp. 355-367, 2012.
[107]
A. Ghandar, and Z. Michalewicz, "An experimental study of multi-objective evolutionary algorithms for balancing interpretability and accuracy in fuzzy rule base classifiers for financial prediction", In: Proceedings of IEEE Symposium on Computational Intelligence for Financial Engineering and Economics, Paris, France, 2011, pp. 1-6.
[108]
M.J. Gacto, R. Alcala, and F. Herrera, "A multi-objective evolutionary algorithm for an effective tuning of fuzzy logic controllers in heating, ventilating and air conditioning systems", Appl. Intell., vol. 36, pp. 330-347, 2012.
[109]
S. Fukuda, J. Nakajima, B. de Beats, W. Wargeman, T. Mukai, A.M. Mouton, and N. Onikura, "A discussion on the accuracy-complexity relationship in modeling fish habitat preference using genetic takagi-sugeno fuzzy systems", In: Proceedings of 5th International Workshop on Genetic and Evolutionary Fuzzy Systems, Paris, France, 2011, pp. 81-86.
[110]
M. Nasir, S. Sengupta, S. Das, and P.N. Suganthan, "An improved multi-objective optimization algorithm based on fuzzy dominance for risk minimization in biometric sensor network", In: IEEE Congress on Evolutionary Computation, 2012, pp. 1-8.
[111]
M.B. Gorzałczany, and F. Rudziński, "An improved multi-objective evolutionary optimization of data-mining-based fuzzy decision support systems", In: IEEE International Conference on Fuzzy Systems (FUZZ), 2016, pp. 2227-2234.
[112]
A. Ferranti, F. Marcelloni, and A. Segatori, "A multi-objective evolutionary fuzzy system for big data", In: IEEE International Conference on Fuzzy Systems (FUZZ), 2016, pp. 1562-1569.
[113]
C. Mencar, C. Castiello, R. Cannone, and A.M. Fanelli, "Interpretability assessment of fuzzy knowledge bases: A cointension based approach", Int. J. Approx. Reason., vol. 52, pp. 501-518, 2011.
[114]
C. Mencar, and A.M. Fanelli, "Interpretability constraints for fuzzy information granulation", Inf. Sci., vol. 178, pp. 4585-4618, 2008.
[115]
R. Mikut, J. Jakel, and L. Groll, "Interpretability issues in data based learning of fuzzy systems", Fuzzy Sets Syst., vol. 150, pp. 179-197, 2005.
[116]
J.M. Alonso, and L. Magdalena, "Special issues on interpretable fuzzy systems", Inf. Sci., vol. 181, pp. 4331-4339, 2011.
[117]
R. Alcala, J. Alcala-Fdez, J. Cassilas, O. Cordon, and F. Herrera, "Hybrid learning models to get the interpretability accuracy trade-off in fuzzy modeling", Soft Comput., vol. 10, pp. 717-734, 2006.
[118]
Y. Jin, "Fuzzy modeling of high dimensional systems: Complexity reduction and interpretability improvement", IEEE Trans. Fuzzy Syst., vol. 8, pp. 212-221, 2000.
[119]
K. Deb, and J. Sundar, "Reference point based multi-objective optimization using evolutionary algorithms", In: Proceedings of Genetic and Evolutionary Computation Conference (GECCO), Seattle, DC, USA, 2006, pp. 635-642.
Related Journals
International Journal of Sensors, Wireless Communications and Control
Related Books
Challenges and Opportunities for Deep Learning Applications in Industry 4.0
Recent Advances in IoT and Blockchain Technology
Augmented Intelligence: Deep Learning, Machine Learning, Cognitive Computing, Educational Data Mining
Human-Computer Interaction and Beyond: Advances Towards Smart and Interconnected Environments Part II
Machine Learning and Its Application: A Quick Guide for Beginners
Computational Intelligence For Data Analysis
Applied Machine Learning and Multi-Criteria Decision-Making in Healthcare
Advanced Computing Techniques: Implementation, Informatics and Emerging Technologies
How to Design Optimization Algorithms by Applying Natural Behavioral Patterns
Human-Computer Interaction and Beyond: Advances Towards Smart and Interconnected Environments (Part I)
Article Metrics
3