General Type-2 Fuzzy Logic Systems Made Simple: A Tutorial
Abstract
The purpose of this tutorial paper is to make general type-2 fuzzy logic systems (GT2 FLSs) more accessible to fuzzy logic researchers and practitioners, and to expedite their research, designs, and use. To accomplish this, the paper 1) explains four different mathematical representations for general type-2 fuzzy sets (GT2 FSs); 2) demonstrates that for the optimal design of a GT2 FLS, one should use the vertical-slice representation of its GT2 FSs because it is the only one of the four mathematical representations that is parsimonious; 3) shows how to obtain set theoretic and other operations for GT2 FSs using type-1 (T1) FS mathematics (α- cuts play a central role); 4) reviews Mamdani and TSK interval type-2 (IT2) FLSs so that their mathematical operations can be easily used in a GT2 FLS; 5) provides all of the formulas that describe both Mamdani and TSK GT2 FLSs; 6) explains why center-of sets type-reduction should be favored for a GT2 FLS over centroid type-reduction; 7) provides three simplified GT2 FLSs (two are for Mamdani GT2 FLSs and one is for a TSK GT2 FLS), all of which bypass type reduction and are generalizations from their IT2 FLS counterparts to GT2 FLSs; 8) explains why gradient-based optimization should not be used to optimally design a GT2 FLS; 9) explains how derivative-free optimization algorithms can be used to optimally design a GT2 FLS; and 10) provides a three-step approach for optimally designing FLSs in a progressive manner, from T1 to IT2 to GT2, e…
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Traditional fuzzy logic systems rely on manually designed rule bases, where the number of rules grows exponentially with input dimensions, leading to significant computational burdens.
“Traditional fuzzy logic systems rely on manually designed rule bases, where the number of rules grows exponentially with the input dimensions, leading to significant computational burdens21.”
From Intelligent evaluation of coal mine solid filling effect using fuzzy logic and improved D-S evidence theory (2025) · cites this paper as [21]
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“EC algorithms are recommended for the optimization of IT2 fuzzy systems, because derivatives are difficult to compute in an IT2 fuzzy system (especially when the LMF and/or UMF formulas include tests about the location of their independent variable), and such algorithms are globally convergent (Mendel, 2014; Cervantes…”
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“olynomial functions of the same order. A crispactivation degreeis produced by performing the type reductionmechanism using the q type-reduction mechanism of the interval-valued activation degree Ge i [16] as follows: Ge i = (1−q)Gi +qGi, (4) where q ∈ ℜ1×m is a design coefficient which adjusts the influence of upper and…”
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“In addition, GT2FLS has more design parameters (degree of freedom) than IT2FLS [10].”
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Output Feedback Predictive Control of Interval Type-2 T–S Fuzzy Systems With Markovian Packet Loss
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“Compared with type-1 T–S fuzzy models, the IT2 T– S fuzzy models are proposed to handle the parameter uncertainties of nonlinear systems captured by the IT2 T–S membership functions [36]– [38].”
Chen, Y., Wang, D., & Tong, S. (2016) · Neurocomputing · doi:10.1016/j.neucom.2015.10.032 · Cited by 58
“There is much interesting work that lies ahead, including the study of Mamdani or TSK IT2 FLSs with triangular type, and trapezoid type MFs of IT2 FSs, Mamdani or TSK general type-2 FLSs [42-43] and seeking other global optimization algorithms, for example, genetic algorithms [45], quantum particle swarm optimization…”
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“degree of freedom in designing varieties of systems with uncertainties [1, 2, 8].”
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“The mathematical expressions as provided in [5] and [7] are also repeated for comprehensiveness.”
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“…engine for the IT2-RBFNN is either of Mamdani or TSK type, the matrix representation of the sth output in the MIMO IT2-RBFNN can be written as [16, 43]: in which y sl = Y sl w Ts and y sr = Y sr w Ts and In which w s = ( w 1 s , . . . , w Ms ) is the set of original rule-ordered consequent…”
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Chen, Y., Wang, D., & Ning, W. (2018) · doi:10.1002/oca.2353 · Cited by 38
“In dealing with the uncertain problems, GT2 FLSs 18 have the capability to outperform the IT2 FLSs. Generally speaking, a T2 FLS is composed of 5 blocks as follows: fuzzifier, inference, rules, type-reducer, and defuzzifier.”
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“More recent works [5]–[7] studied the geometrical properties of some GT2FSs to find closed formulas or”
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“2(a), the membership function of T2 FS is three-dimensional [61].”
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“Liu and Liao [37] performed an extensive survey of the FDM approaches.”
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“Therefore, a general type-2 fuzzy inference system is described using more parameters than a type-1 and an interval type-2 fuzzy inference system (Melin et al. 2014; Mendel 2014, 2016).”
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“IT2 FLSs composed of IT2 FSs are much simpler than general T2 FLSs [7, 8] in mathematics.”
BackgroundInterval-valued fuzzy morphological associative memories: Some theoretical aspects and applications
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Hybrid Learning for General Type-2 TSK Fuzzy Logic Systems
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“Information granules which intrinsically support uncertainty can be represented by general type-2 fuzzy sets (GT2 FSs), and in turn these GT2 FSs can be inferred by a general type-2 fuzzy logic system (GT2 FLS) [11].”
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“A type2 fuzzy system contains a set of type-2 fuzzy rules in which variables are expressed in terms of type-2 fuzzy sets [1], [15]– [17].”
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“A type-reducer and a defuzzifier are included in the output r cessi g l c , ic produces a type-1 fuzzy set output (fro the type-reducer) or a recise alue (fro t e defuzzification block) [35,36].”
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Ochoa, P., Castillo, O., & Soria, J. (2019) · International Journal of Fuzzy Systems · doi:10.1007/s40815-019-00723-w · Cited by 18
Type-3 fuzzy dynamic adaptation of Bee colony optimization applied to mathematical functions
Amador-Angulo, L., Castillo, O., Melin, P., & Geem, Z. (2024) · Fuzzy Sets Syst. · doi:10.1016/j.fss.2024.109014 · Cited by 17
Restricted crossing U-turn traffic control by interval Type-2 fuzzy logic
Jovanović, A., Kukic, K., Stevanovic, A., & Teodorovic, D. (2022) · Expert systems with applications · doi:10.1016/j.eswa.2022.118613 · Cited by 17
Ma, B., & Wang, Y.-F. (2021) · Proceedings of the Institution of mechanical engineers. Part D, journal of automobile engineering · doi:10.1177/0954407021995390 · Cited by 17
“D and includes a FOU with the new third dimension of T2 FS. 26 A FOU provides additional degrees of freedom that make it possible to provide the capability to model high levels of uncertainties.”
BackgroundMethodLiu, J., Zhao, T., & Dian, S. (2020) · Soft Computing - A Fusion of Foundations, Methodologies and Applications · doi:10.1007/s00500-020-05202-1 · Cited by 17
“(14)–(15) (Mendel 2014).”
BackgroundChen, Y. (2020) · Soft Computing - A Fusion of Foundations, Methodologies and Applications · doi:10.1007/s00500-020-04998-2 · Cited by 17
“In early times, Professor Mendel developed the most popular Karnik–Mendel (KM) algorithms (Karnik and Mendel 2001; Mendel 2013, 2014) to compute the centroids of IT2 FSs or perform the centroid TR of IT2 FLSs.”
MethodAzar, A., Serrano, F. E., & Koubâa, A. (2020) · 2020 IEEE International Conference on Autonomous Robot Systems and Competitions (ICARSC) · doi:10.1109/icarsc49921.2020.9096163 · Cited by 17
“It is important to define the fuzzy type-2 system process as fuzzificator, inference rule engine, type reduction and deffuzzificator [15,17,18].”
MethodVD-IT2, Virtual Disk Cloning on Disk Arrays Using a Type-2 Fuzzy Controller
Navarro, G., Umberger, D. K., & Manic, M. (2017) · IEEE transactions on fuzzy systems · doi:10.1109/tfuzz.2016.2633371 · Cited by 17
“There are other methods [21], but the center-of-sets is the most common”
BackgroundMethodDesign and Application of Interval Type-2 TSK Fuzzy Logic System Based on QPSO Algorithm
Fan, Q.-F., Wang, T., Chen, Y., & Zhang, Z.-F. (2017) · International Journal of Fuzzy Systems · doi:10.1007/s40815-017-0357-3 · Cited by 17
“In 2014, Mendel et al. [43] proposed using the QPSO algorithm to train the fuzzy logic system.”
MethodType-2 fuzzy logic based transit priority strategy
Jovanović, A., & Teodorovic, D. (2022) · Expert systems with applications · doi:10.1016/j.eswa.2021.115875 · Cited by 16
Feng, J.-C., Huang, H.-A., Yin, Y., & Zhang, K. (2019) · doi:10.1016/j.wse.2019.12.009 · Cited by 16
Rubio-Solis, A., Musah, A., dos Santos, W. P., Massoni, T., Birjovanu, G., & Kostkova, P. (2019) · PDH · doi:10.1145/3357729.3357738 · Cited by 16 · see citing works
“| M L) (M L) ⇥ 1 [24, 30], ej 2 RL (j = 1, .”
Wang, Y., Liu, Y., Ding, J., & Wang, D. (2024) · Engineering applications of artificial intelligence · doi:10.1016/j.engappai.2023.107833 · Cited by 15
Decoding emotional changes of android-gamers using a fused Type-2 fuzzy deep neural network
Ghosh, L., Saha, S., & Konar, A. (2021) · Computers in Human Behavior · doi:10.1016/j.chb.2020.106640 · Cited by 15 · see citing works
Luo, G., Ma, B., Wang, Z.-Z., Yin, L., & Wang, Y.-F. (2020) · doi:10.1063/5.0014788 · Cited by 15
Chen, Y., Zhao, T., Dian, S., Zeng, X., & Wang, H.-P. (2020) · Symmetry · doi:10.3390/sym12030479 · Cited by 15
- Authors
- Mendel, J. M.
- Affiliations
- University of Southern California (US)
- Corresponding author
- Mendel, Jerry M.
- ORCID
- Mendel, Jerry M.
- Venue
- IEEE Transactions on Fuzzy Systems
- Published
- 2014-10
- Volume
- 22
- Issue
- 5
- Pages
- 1162-1182
- ISSN
- 1063-6706 (ISSN-L)
- 1941-0034
- OpenAlex
- W2031176346
- Cited by
- 311
- References
- 0
- Citations by year
- 2026: 13 · 2025: 16 · 2024: 27 · 2023: 29 · 2022: 24 · 2021: 33 · 2020: 38 · 2019: 29 · 2018: 26 · 2017: 32 · 2016: 20 · 2015: 20 · 2014: 4
- FWCI
- 16.22
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- IEEE Transactions on Fuzzy Systems (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Fuzzy Logic and Control Systems
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Advanced Algebra and Logic
- Physical Sciences › Computer Science › Artificial Intelligence › Mathematical Control Systems and Analysis
- Keywords
- general type-2 fuzzy sets, type reduction, interval type-2 fuzzy systems, alpha-cuts, derivative-free optimization, quantum-behaved particle swarm optimization, fuzzy system design
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