Review:

# On the Monotonicity of Fuzzy Inference Models

## Hirosato Seki^{*} and Kai Meng Tay^{**}

^{*}Department of Mathematical Sciences, Kwansei Gakuin University, 2-1 Gakuen, Sanda, Hyogo 669-1337, Japan

^{**}Faculty of Engineering, Universiti Malaysia Sarawak, 94300 Kota Samarahan, Sarawak, Malaysia

Monotonicity property is very important in real systems. The monotonicity may need to be satisfied in a variety of application domains, e.g., control, medical diagnosis, educational evaluation, etc. A search in the literature reveals that the importance of the monotonicity in fuzzy inference system has been highlighted. Therefore, this paper surveys the works relating the monotonicity for various fuzzy inference systems. It firstly focuses on the monotonicity of the Mamdani inference model. Themonotonicity ofMamdani model is shown by using a defuzzification method in cases of three t-norms. Secondly, the monotonicity conditions and applications of the T–S inference model are stated. Finally, the monotonicity of the single input type fuzzy inference models is surveyed.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.16, No.5, pp. 592-602, 2012.

- [1] E. H.Mamdani, “Application of fuzzy algorithms for control of simple dynamic plant,” in Proc. IEE, Vol.121, No.12, pp. 1585-1588, 1974.
- [2] L. P. Holmblad and J. J. Ostergard, “Control of a cement kiln by fuzzy logic,” Information and Decision Processes, M. M. Gupta and E. Sanchez (Eds.), Amsterdam, North Holland, pp. 386-389, 1982.
- [3] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications,” IEEE Trans. on Systems, Man, and Cybernetics, Vol.15, No.1, pp. 116-132, 1985.
- [4] J. S. R. Jang, C. T. Sun, and E. Mizutami, “Neural-Fuzzy and soft Computing,” Prentice-Hall, 1997.
- [5] N. Yubazaki, J. Yi, and K. Hirota, “SIRMs (Single Input Rule Modules) connected fuzzy inference model,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.1, No.1, pp. 23-30, 1997.
- [6] J. Yi, N. Yubazaki, and K. Hirota, “Upswing and stabilization control of inverted pendulum and cart system by the SIRMs dynamically connected fuzzy inference model,” in Proc. 1999 IEEE Int. Conf. Fuzzy Syst., Vol.1, pp. 400-405, 1999.
- [7] J. Yi, N. Yubazaki, and K. Hirota, “A proposal of SIRMs dynamically connected fuzzy inference model for plural input fuzzy control,” Fuzzy Sets Syst., Vol.125, No.1, pp. 79-92, 2002.
- [8] J. Yi, N. Yubazaki, and K. Hirota, “A new fuzzy controller for stabilization of parallel-type double inverted pendulum system,” Fuzzy Sets Syst., Vol.126, No.1, pp. 105-119, 2002.
- [9] J. Yi, N. Yubazaki, and K. Hirota, “Anti-swing fuzzy control of overhead traveling crane,” in Proc. 2002 IEEE Int. Conf. Fuzzy Syst., Vol.2, pp. 1298-1303, 2002.
- [10] S. Khwan-on, T. Kulworawanichpong, A. Srikaew, and S. Sujitjorn, “Neuro-tabu-fuzzy controller to stabilize an inverted pendulum system,” in Proc. 2004 IEEE Region 10 Conf. TENCON 2004, Vol.D, pp. 562-565, 2004.
- [11] H. Seki and M. Mizumoto, “On the Equivalence Conditions of Fuzzy Inference Methods – Part 1: Basic Concept and Definition,” IEEE Trans. Fuzzy Syst., Vol.19, No.6, pp. 1097-1106, 2011.
- [12] K. Hayashi, A. Otsubo, S. Murakami, and M. Maeda, “Realization of nonlinear and linear PID controls using simplified indirect fuzzy inference method,” Fuzzy Sets Syst., Vol.105, pp. 409-414, 1999.
- [13] K. Hayashi, A. Otsubo, and K. Shiranita, “Improvement of conventional method of PI fuzzy control,” IEICE Trans. Fundamentals, Vol.E84-A, No.6, pp. 1588-1592, June 2001.
- [14] A. Ben-David, “Monotonicity maintenance in information-theoretic machine learning algorithms,” Mach. Learn., Vol.19, pp. 29-43, 1995.
- [15] K. Cao-Van and B. De Baets, “Growing decision trees in an ordinal setting,” Int. J. Intell. Syst., Vol.18, pp. 733-750, 2003.
- [16] H. A. M. Daniels and M. V. Velikova, “Derivation of monotone decision models from noisy data,” IEEE Trans. Syst., Man, Cybern. C, Appl. Rev., Vol.36, pp. 705-710, 2006.
- [17] J. Sill, “Monotonic networks,” in Proc. NIPS Conf. 1997, Advances in Neural Information Processing Systems, Vol.10, M. Jordan, M. Kearno, and S. Solla (Eds.), The MIT Press, Denver, CO, USA, pp. 661-667, 1998.
- [18] S. Lievens, B. De Baets, and K. Cao-Van, “A probabilistic framework for the design of instance-based supervised ranking algorithms in an ordinal setting,” Annals of Operations Research, Vol.163, pp. 115-142, 2008.
- [19] C. J. Wu and A. H. Sung, “A general purpose fuzzy controller for monotone functions,” IEEE Trans. Syst., Man, Cybern. B, Cybern., Vol.26, No.5, pp. 803-808, 1996.
- [20] C. J. Wu, “Guaranteed accurate fuzzy controllers for monotone functions,” Fuzzy Sets Syst., Vol.92, pp. 71-82, 1997.
- [21] P. Lindskog and L. Ljung, “Ensuring monotonic gain characteristics in estimated models by fuzzy model structures,” Automatica, Vol.36, pp. 311-317, 2000.
- [22] H. Zhao and C. Zhu, “Monotone fuzzy control method and its control performance,” in Proc. 2000 IEEE Int. Conf. Syst., Man, Cybern., pp. 3740-3745, Nashville, TN, 2000.
- [23] H. Seki, H. Ishii, and M.Mizumoto, “On the monotonicity of single input type fuzzy reasoning methods,” IEICE Trans. on Fundamentals, Vol.E90-A, No.7, pp. 1462-1468, July 2007.
- [24] J. M. Won, S. Y. Park, and J. S. Lee “Parameter conditions for monotonic Takagi-Sugeno-Kang fuzzy system,” Fuzzy Sets Syst., Vol.132, pp. 135-146, 2002.
- [25] K. Koo, J. M. Won, and J. S. Lee, “Least squares identification of monotonic fuzzy systems,” in Proc. the Annual Meeting of the North American Fuzzy Information Processing Society, Vol.2, Banff, Canada, pp. 745-749, 2004.
- [26] V. S. Kouikoglou and Y. A. Phillis, “On the monotonicity of hierarchical sum–product fuzzy systems,” Fuzzy Sets Syst., Vol.160, pp. 3530-3538, 2009.
- [27] B. Schott and T. Whalen, “Nonmonotonicity and discretization error in fuzzy rule-based control COA and MOM defuzzification,” in Proc. 1996 IEEE Int. Conf. Fuzzy Syst., New Orleans, LA, USA, pp. 450-456, 1996.
- [28] E. Van Broekhoven and B. De Baets, “Monotone Mamdani–Assilian models under mean of maxima defuzzification,” Fuzzy Sets Syst., Vol.159, pp. 2819-2844, 2008.
- [29] E. Van Broekhoven and B. De Baets, “Only smooth rule bases can generate monotone Mamdani–Assilian models under COG defuzzification,” IEEE Trans. Fuzzy Syst., Vol.17, No.5, pp. 1157-1174, October 2009.
- [30] L. A. Zadeh, “Fuzzy sets,” Information and Control, Vol.8, pp. 338-353, 1965.
- [31] J. Ramik and J. Rimanek, “Inequality relation between fuzzy numbers and its use in fuzzy optimization,” Fuzzy Sets Syst., Vol.16, No.2, pp. 123-138, 1985.
- [32] E. H. Mamdani and S. Assilian, “An experiment in linguistic synthesis with a fuzzy logic controller,” Int. J. Man-Machine Studies, Vol.7, pp. 1-13, 1975.
- [33] E. H. Mamdani, “Advances in the linguistic synthesis of fuzzy controllers,” Int. J. Man-Machine Studies, Vol.8, pp. 669-679, 1976.
- [34] M. Sugeno and T. Yasukawa, “A fuzzy logic based approach to qualitative modeling,” IEEE Trans. Fuzzy Systems, Vol.1, No.1, pp. 7-31, 1993.
- [35] M. da Silva Peixoto, L. de Barros, and R. Bassanezi, “A model of cellular automata for the spatial and temporal analysis of citrus sudden death with the fuzzy parameter,” Ecological Modelling, Vol.214, pp. 45-52, 2008.
- [36] V. S. Kouikoglou and Y. A. Phillis, “On the monotonicity of hierarchical sum-product fuzzy systems,” Fuzzy Sets and Systems, Vol.160, No.24, pp. 3530-3538, 2009.
- [37] K. M. Tay and C. P. Lim, “On monotonic sufficient conditions of fuzzy inference systems and their applications,” Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol.19, No.5, pp. 731-757, 2011.
- [38] J. M. Won, S. Y. Park, and J. S. Lee, “Parameter conditions for monotonic Takagi-Sugeno-Kang fuzzy system,” Fuzzy Sets and Systems, Vol.132, No.2, pp. 135-146, 2002.
- [39] K. M. Tay and C. P. Lim, “On the use of fuzzy inference techniques in assessment models: Part I – theoretical properties,” Fuzzy Optimization and Decision Making, Vol.7, No.3, pp. 269-281, 2008.
- [40] J. Liu, L. Martinez, H. Wang, R. M. Rodriguez, and V. Novozhilov, “Computing with words in risk assessment,” Int. J. of Computational Intelligence Systems, Vol.3, No.4, pp. 396-419, 2010.
- [41] P. Lindskog and L. Ljung, “Ensuring monotonic gain characteristics in estimated models by fuzzy model structures,” Automatica, Vol.36, No.2, pp. 311-317, 2000.
- [42] K. M. Tay and C. P. Lim, “A fuzzy inference system-based criterionreferenced assessment model,” Expert Systems With Applications, Vol.38, No.9, pp. 11129-11136, 2011.
- [43] D. H. Stamatis, “Failure mode and effect analysis: FMEA from theory to execution,” ASQ Quality Press, 2003.
- [44] J. B. Bowles and C. E. Pelaez, “Application of fuzzy logic to reliability engineering,” Proc. of the IEEE, Vol.83, No.3, pp. 435-449, 1995.
- [45] T. Boongoen, Q. Shen, and C. Price. “Fuzzy qualitative link analysis for academic performance evaluation,” Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol.19, No.3, pp. 559-585, 2011.
- [46] S. Saliu, “Constrained Subjective Assessment of Student Learning,” J. of Science Education and Technology, Vol.14, No.3, pp. 271-284, 2005.
- [47] V. D. Kouloumpis, V. S. Kouikoglou, and Y. A. Phillis, “Sustainability Assessment of Nations and Related Decision Making Using Fuzzy Logic,” IEEE Systems J., Vol.2, No.2, pp. 224-235, 2008.
- [48] Y. A. Phillis, V. S. Kouikoglou, and X. Zhu, “Fuzzy Assessment of Material Recyclability and Its Applications,” J. of Intelligent and Robotic Systems, Vol.15, No.1, pp. 21-38, 2009.
- [49] H. Seki and M.Mizumoto, “Fuzzy singleton-type SIRMs connected fuzzy inference method: property and application to a medical diagnosis,” Proc. 2011 IFSA World Congress and AFSS Int. Conf., Surabaya, Indonesia, FI-004, June 2011.
- [50] H. Seki, H. Ishii, and M. Mizumoto, “On the Monotonicity of Fuzzy-Inference Methods Related to T–S Inference Method,” IEEE Trans. Fuzzy Syst., Vol.18, No.3, pp. 629-634, 2010.
- [51] H. Seki, H. Ishii, and M. Mizumoto, “On the generalization of single input rule modules connected type fuzzy reasoning method,” IEEE Trans. Fuzzy Syst., Vol.16, No.5, pp. 1180-1187, 2008.