, Volume 38, Issue 6, pp 523–532 | Cite as

Type-2 Fuzzy Sets and Systems: a Retrospective

  • Jerry M. MendelEmail author


This article provides a high-level retrospective of type-2 fuzzy sets and fuzzy logic systems. It explains how type-2 fuzzy sets can be used to model membership function uncertainties, and how by doing this smoother performance can be obtained than by using type-1 fuzzy sets. It also summarizes the notation that should be used for type-2 fuzzy sets, describes four important mathematical representations for these fuzzy sets, explains the differences between type-1 and type-2 fuzzy logic systems and which of the four representations is most useful when designing an optimal type-2 fuzzy logic system, provides a very useful strategy for optimal designs of fuzzy logic systems – one that guarantees performance improvement as one goes from a type-1 fuzzy logic system to a type-2 fuzzy logic system design – , and describes four methods for simplifying the designs of type-2 fuzzy logic systems. Finally, it explains why type-2 fuzzy sets can capture two kinds of linguistic uncertainties simultaneously (the uncertainty of an individual and the uncertainties of a group about a word), whereas type-1 fuzzy sets cannot, and that such type-2 fuzzy set word models are what should be used to implement Zadeh’s Computing With Words paradigm.


Membership Function Fuzzy Logic System Membership Function Parameter Computing With Word Secondary Membership 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aisbett J, Rickard JT, Morgenthaler DG (2010) Type-2 fuzzy sets as functions on spaces. IEEE T Fuzzy Syst 18:841–844CrossRefGoogle Scholar
  2. 2.
    Biglarbegian M, Melek WW, Mendel JM (2010) On the stability of interval type-2 TSK fuzzy logic control systems. IEEE Trans Syst Man Cybern – Part B: Cybern 40:798–818CrossRefGoogle Scholar
  3. 3.
    Biglarbegian M, Melek WW, Mendel JM (2011) On the robustness of type-1 and interval type-2 fuzzy logic systems in modeling. Inform Sci 181:1325–1347zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bustince H, Fernandez J, Hagras H, Herrera F, Pagola M, Barrenechea E (2015) Interval type-2 fuzzy sets are generalization of interval-valued fuzzy sets: towards a wide view on their relationship. IEEE T Fuzzy Syst, early access, doi:10.1109/TFUZZ.2014.2362149Google Scholar
  5. 5.
    Castillo O, Martinez-Marroquin R, Melin P, Valdez F, Soria J (2012) Comparative study of bio-inspired algorithms applied to the optimization of type-1 and type-2 fuzzy controllers for an autonomous mobile robot. Inform Sci 192:19–38CrossRefGoogle Scholar
  6. 6.
    Castillo O, Melin P, Alanis A, Montiel O, Sepulveda R (2011) Optimization of interval type-2 fuzzy logic controllers using evolutionary algorithms. J Soft Comp 15:1145–1160CrossRefGoogle Scholar
  7. 7.
    Derrac J, Garcia S, Hui S, Suganthan PN, Herrera F (2014) Analyzing convergence performance of evolutionary algorithms: a statistical approach. Inform Sci 289:41–58CrossRefGoogle Scholar
  8. 8.
    Karnik NN, Mendel JM (2001) Centroid of a type-2 fuzzy set. Inform Sci 132:195–220zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Karnik NN, Mendel JM, Liang Q (1999) Type-2 fuzzy logic systems. IEEE T Fuzzy Syst 7:643–658Google Scholar
  10. 10.
    Kayacan E, Ahmadieh M (2015) Fuzzy Neural Networks for Real Time Applications. Elsevier, AmsterdamGoogle Scholar
  11. 11.
    Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proc IEEE Int’l Conf on Neural Networks, pp 1942–1948Google Scholar
  12. 12.
    Liang Q, Mendel JM (2000) Interval type-2 fuzzy logic systems: theory and design. IEEE T Fuzzy Syst 8:535–550CrossRefGoogle Scholar
  13. 13.
    Liu F (2008) An efficient centroid type reduction strategy for general type-2 fuzzy logic system. Inform Sci 178:2224–2236CrossRefMathSciNetGoogle Scholar
  14. 14.
    Lynch C, Hagras H, Callaghan V (2006) Using uncertainty bounds in the design of embedded real-time type-2 neuro-fuzzy speed controller for marine diesel engines. In: Proc IEEE FUZZ Conf, Vancouver, BC, Canada, pp 7217–7224Google Scholar
  15. 15.
    Mendel JM (2001) Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions. Prentice-Hall, Upper Saddle River, NJGoogle Scholar
  16. 16.
    Mendel JM (2004) Computing derivatives in interval type-2 fuzzy logic systems. IEEE T Fuzzy Syst 12:84–98CrossRefGoogle Scholar
  17. 17.
    Mendel JM (2007a) Advances in type-2 fuzzy sets and systems. Inform Sci 177:84–110zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Mendel JM (2007b) Type-2 fuzzy sets and systems: an overview. IEEE Comput Intell Mag 2:20–29Google Scholar
  19. 19.
    Mendel JM (2009) On answering the question ‘Where do I start in order to solve a new problem involving type-2 fuzzy sets?’. Inform Sci 179:3418–3431zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Mendel JM (2013a) On KM algorithms for solving type-2 fuzzy set problems. IEEE T Fuzzy Syst 21:426–446CrossRefGoogle Scholar
  21. 21.
    Mendel JM (2013b) Type-2 fuzzy sets and beyond. In: Seising R, Trillas E, Moraga C, Termini S (eds) On Fuzziness: a Homage to Lotfi A. Zadeh, vol. 2, Ch 34. Springer, New YorkGoogle Scholar
  22. 22.
    Mendel JM (2014) General type-2 fuzzy logic systems made simple: a tutorial. IEEE T Fuzzy Syst 22:1162–1182CrossRefGoogle Scholar
  23. 23.
    Mendel JM, Hagras H, Bustince H, Herrera F (2015) Comments on ‘Interval type-2 fuzzy sets are generalization of interval-valued fuzzy sets: towards a wide view on their relationship’ accepted for publication in IEEE T Fuzzy SystGoogle Scholar
  24. 24.
    Mendel JM, Hagras H, Wan-Tan W, Melek W, Ying H (2014) Introduction to type-2 fuzzy logic control: theory and applications. Wiley and IEEE Press, Hoboken, NJCrossRefGoogle Scholar
  25. 25.
    Mendel JM, John RI (2002a) Type-2 fuzzy sets made simple. IEEE T Fuzzy Syst 10:117–127CrossRefGoogle Scholar
  26. 26.
    Mendel JM, John RI (2002b) Footprint of uncertainty and its importance to type-2 fuzzy sets. In: Proc 6th IASTED Int’l Conf on Artificial Intelligence and Soft Computing, Banff, CanadaGoogle Scholar
  27. 27.
    Mendel JM, John RI, Hagras H (2006) Standard background material about interval type-2 fuzzy logic systems that can be used by all authors, IEEE Computational Intelligence Society standard: can be accessed at, last access: 18.10.2015Google Scholar
  28. 28.
    Mendel JM, John RI, Liu F (2006) Interval type-2 fuzzy logic systems made simple. IEEE T Fuzzy Syst 14:808–821CrossRefGoogle Scholar
  29. 29.
    Mendel JM, Liu F, Zhai D (2009) Alpha-plane representation for type-2 fuzzy sets: theory and applications. IEEE T Fuzzy Syst 17:1189–1207CrossRefGoogle Scholar
  30. 30.
    Mendel JM, Liu X (2013) Simplified interval type-2 fuzzy logic systems. IEEE T Fuzzy Syst 21:1056–1069CrossRefGoogle Scholar
  31. 31.
    Mendel JM, Rajati MR (2015a) Advanced computing with words: status and challenges. In: Seising R, Trillas E, Kacprzyk J (eds) Fuzzy Logic: Towards the Future. Springer, New YorkGoogle Scholar
  32. 32.
    Mendel JM, Rajati MR (2015b) “On clarifying some notations used for type-2 fuzzy sets as well as some recommended notational changes,” revised submission sent to Information SciencesGoogle Scholar
  33. 33.
    Mendel JM, Wu D (2010) Perceptual Computing: Aiding People in Making Subjective Judgments. Wiley and IEEE Press, Hoboken, NJCrossRefGoogle Scholar
  34. 34.
    Mizumoto M, Tanaka K (1976) Some properties of fuzzy sets of type-2. Inform Control 31:312–340zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Mizumoto M, Tanaka K (1981) Fuzzy sets of type-2 under algebraic product and algebraic sum. Fuzzy Sets Syst 5:277–290zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Nie M, Tan WW (2008) Towards an efficient type-reduction method for interval type-2 fuzzy logic systems. In: Proc IEEE FUZZ Conf: Paper # FS0339, Hong Kong, ChinaGoogle Scholar
  37. 37.
    Trawinski B, Smetek M, Telec Z, Lasota T (2012) Nonparametric statistical analysis for multiple comparison of machine learning regression algorithms. Int J Appl Math Comput Sci 22:867–881zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Wagner C, Hagras H (2008) z slices – towards bridging the gap between interval and general type-2 fuzzy logic. In: Proc IEEE FUZZ Conf, Paper # FS0126, Hong Kong, China, pp 489–497Google Scholar
  39. 39.
    Wagner C, Hagras H (2010) Towards general type-2 fuzzy logic systems based on zslices. IEEE T Fuzzy Syst 18:637–660CrossRefGoogle Scholar
  40. 40.
    Wang L-X, Mendel JM (1992) Fuzzy basis function, universal approximation, and orthogonal least-squares learning. IEEE T Neural Netw 3:807–814CrossRefGoogle Scholar
  41. 41.
    Wei F, Jun S, Ping XZ, Xu WB (2010) Convergence analysis of quantum-behaved particle swarm optimization algorithm and study on its control parameter. Acta Phys Sin 59:3686–3694zbMATHGoogle Scholar
  42. 42.
    Wu D, Mendel JM (2011) On the continuity of type-1 and interval type-2 fuzzy logic systems. IEEE T Fuzzy Syst 19:179–192CrossRefGoogle Scholar
  43. 43.
    Wu H, Mendel JM (2002) Uncertainty bounds and their use in the design of interval type-2 fuzzy logic systems. IEEE T Fuzzy Syst 10:622–639CrossRefGoogle Scholar
  44. 44.
    Zadeh LA (1965) Fuzzy sets. Inform Control 8:338–353zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Zadeh LA (1996) Fuzzy logic = computing with words. IEEE T Fuzzy Syst 4:103–111CrossRefGoogle Scholar
  46. 46.
    Zadeh LA (1999) From computing with numbers to computing with words – from manipulation of measurements to manipulation of perceptions. IEEE T Circuits-1 4:105–119CrossRefMathSciNetGoogle Scholar
  47. 47.
    Zadeh LA (2012) Computing with Words: Principal Concepts and Ideas. Springer, New YorkCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Signal and Image Processing InstituteUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations