# Generating Clustering-Based Interval Fuzzy Type-2 Triangular and Trapezoidal Membership Functions: A Structured Literature Review

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## Abstract

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## 1. Introduction

_{A}(y) = 0.6. This means that the degree of membership of y in fuzzy set A is 0.6, and this kind of structure is known as Fuzzy type-1 (FT1) [4]. However, it is shown that even though the membership degree is within the 0 and 1 range, the value is still considered a crisp value—which can represent a limited amount of uncertainties and fuzziness. It limits the capability of FT1 to address complex uncertainty problems and has become the main argument in the research community. This has led to the introduction of Fuzzy type-2 (FT2) set whereby a membership degree is also represented in the form of fuzzy value. This is made possible because FT2 contains a third dimension known as the footprint of uncertainty (FOU) that provides more degrees of freedom, hence making it capable of handling complex uncertainties present in a system. For instance, the previous FT1 set may be represented in FT2 as µ$\tilde{A}$(y) = [0.5, 0.7] whereby the membership degree is in the range of 0.5–0.7. This range represents FOU, while 0.5 and 0.7 are, respectively, known as Lower MF (LMF) and Upper MF (UMF).

## 2. Background

#### 2.1. Existing Works on the Construction of IT2 MF

- The blurring method is applied on the readily constructed FT1 MF as shown in Figure 3 [5]. The original FT1 MF in Figure 3a is blurred to the left and right which produces the shape as shown in Figure 3b. Then, the blurred MF is improved so that the FOU of IT2 MF is properly formed as shown in Figure 3c. To relate this construction of IT2 MF with FCM, the original FT1 MF in Figure 3a can be constructed using the outputs of FCM and a heuristic method. As discussed above, this approach only produces Gaussian FT1 MF, hence imposing a blurring method on it will generate Gaussian IT2 MF. Existing research proposed that the triangular and trapezoidal MFs construction is carried out using the grid partitioning method. Although the grid partitioning method can fulfill those objectives, its generated MFs lack representation of the actual composition of the underlying data set because the width of all clusters is equally distributed or spread out. Hence, an alternative approach on how to construct triangular and trapezoidal IT2 MFs from FCM should be investigated.
- Another possible method to form the MF FOU is through adaptive network-based fuzzy inference system (ANFIS). Unfortunately, applying ANFIS directly to IT2 FIS is not possible, and optimization of FT1 needs to be performed in order to generate the MF [26]. Moreover, the MF type produced is Gaussian type [26]. Hence, there remains a need for a study to extend the capability of FCM in generating more than a single type of MF.
- Most research proposed a double fuzzifier FCM method to form IT2 FOU [7,27]. This was achieved by applying FCM clustering using two different fuzzifier values upon a single data set [28]. This means that an MF constructed with one fuzzifier value will represent the LMF and another MF constructed with another fuzzifier value will generate the UMF of an FOU. However, this method is also based on FCM, hence it generates Gaussian IT2 MF type only.

#### 2.2. Interval Fuzzy Type-2

#### 2.3. Fuzzy Membership Function

#### 2.4. Fuzzy C-Means

_{1}, X

_{2}, …, X

_{n}} and c number of clusters. Firstly, FCM guesses the center of each cluster, c

_{i}, i = 1, 2, …, c. Next, FCM computes and assigns each data point with a membership degree for each cluster [43]. This is computed based on Equation (4), and the generated membership degrees are stored in matrix U.

_{ij}= ||c

_{i}– x

_{j}|| is the Euclidean distance from jth data point to the ith cluster center, while m is the fuzzy weighting exponent [43].

## 3. Materials and Methods

#### 3.1. Methodology

#### 3.1.1. Identification (Pi)

_{1}= 104.

_{2}= 6. Hence, the total number of manuscripts collected from this stage, Pi, was Pi = Pi

_{1}+ Pi

_{2}= 110 manuscripts.

#### 3.1.2. Screening (Ps)

_{1}= 94. Next, we assessed the collected 94 manuscripts by reviewing their title and abstract. As mentioned above, the manuscripts were excluded from being reviewed if they did not meet the selection criteria: (1) the manuscripts must describe the triangular MF construction using clustering or FCM method, and (2) the manuscripts must describe the trapezoidal MF construction using clustering or FCM method. The result from this screening was that the number of excluded manuscripts was Ps

_{2}= 73. Among the reasons for the exclusion of the manuscripts were the topic that does not focus on FCM or clustering in MF construction, and the use of nonfuzzy solution such as Neural Network and bio-inspired algorithms. Therefore, the number of manuscripts was reduced in this stage, Ps = Ps

_{1}– Ps

_{2}= 21 manuscripts.

#### 3.1.3. Eligibility (Pe) and Included (Pn)

#### 3.2. Results

_{1}– c

_{3}, in the triangular MF where c

_{1}is adjacent to c

_{2}, and c

_{3}is adjacent to c

_{2}. Based on Equation (4) and the vicinity principle, a value of c

_{2}is equal to b value of c

_{1}, and a value of c

_{3}is equal to b value of c

_{2}. Through these steps, a type-1 MF will be generated. This work, however, did not enhance FCM algorithm in generating triangular and trapezoidal IT2 MFs as being focused on by the present paper. The work also did not focus on generating IT2 MF.

#### 3.3. Fuzzy Calculation and Operation

#### 3.3.1. Fuzzy Calculation

#### 3.3.2. Fuzzy Operation

#### 3.4. Contributions and Limitations of the Existing Works

## 4. Challenges and Future Research

- Introducing a method that can heuristically produce the correct triangular and trapezoidal shapes given any types of data. The improvement in terms of accuracy or precision, for example, can be further explored.
- Optimizing IT2 FIS results through the application of optimization algorithms in MF generation processes. The existing works mainly focused on finding the effective range of LMF and UMF based on fuzzier values of FCM.
- Exploring the application of the triangular and trapezoidal IT2 MFs in various applications such as prediction, classification, multiattribute decision making system, and case-based reasoning system.
- Investigating the IT2 MFs generated by FCM being applied in the general fuzzy Type-2 FIS. The performance effects in various general fuzzy Type-2 applications can be further explored.

- FCM faces the challenges in managing different uncertainties associated with data and getting trapped in local optima and fails to find optimal cluster centers when dealing with large data. Qaiyum et al. (2019) [120] described this challenge and proposed an optimized Interval Type-2 Fuzzy C-Means (IT2FCM) using Ant Colony-based Optimization (ACO). Therefore, while exploring the construction of triangular and trapezoidal IT2 MF using FCM, the optimization issue needs to be of concern too.
- Despite its simplicity and effectiveness in clustering, FCM suffers from being sensitive to initial values and susceptible to noise. Many studies proposed the utilization of bio-inspired algorithms such as particle swarm optimization (PSO) Dhanachandra and Chanu (2020) [121] to tackle the issues. The study on IT2 FIS developed with trapezoidal and triangular MF-based FCM may require further consideration in terms of noise reduction, in order to ensure the results are accurate and acceptable.
- FCM is thought to have a slight defect when dealing with large datasets [122]. Hence, exploring the construction and application of IT2 triangular and trapezoidal MF based on FCM will be a challenging issue when dealing with large datasets.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Guillaume, S. Designing fuzzy inference systems from data: An interpretability-oriented review. IEEE Trans. Fuzzy Syst.
**2001**, 9, 426–443. [Google Scholar] [CrossRef] [Green Version] - Bolos, M.I.; Bradea, I.A.; Delcea, C. The Development of a Fuzzy Logic System in a Stochastic Environment with Normal Distribution Variables for Cash Flow Deficit Detection in Corporate Loan Policy. Symmetry
**2019**, 11, 548. [Google Scholar] [CrossRef] [Green Version] - Ma, X.; Liu, Q.; Zhan, J. A survey of decision making methods based on certain hybrid soft set models. Artif. Intell. Rev.
**2017**, 47, 507–530. [Google Scholar] [CrossRef] [Green Version] - Mohd Adnan, M.R.H.; Sarkheyli, A.; Mohd Zain, A.; Haron, H. Fuzzy logic for modeling machining process: A review. Artif. Intell. Rev.
**2015**, 43, 345–379. [Google Scholar] [CrossRef] - Castillo, O.; Melin, P. Design of Intelligent Systems with Interval Type-2 Fuzzy Logic. In Handbook of Granular Computing; Springer: Berlin, Germany, 2008; pp. 575–601. [Google Scholar]
- Jang, J.-R. ANFIS: Adaptive-network-based fuzzy inference system. IEEE Trans. Syst. Man, Cybern.
**1993**, 23, 665–685. [Google Scholar] [CrossRef] - Rubio, E.; Castillo, O. Interval type-2 fuzzy clustering for membership function generation. In Proceedings of the 2013 IEEE Workshop on Hybrid Intelligent Models and Applications (HIMA), Singapore, 16–19 April 2013; pp. 13–18. [Google Scholar]
- Rodríguez-Sánchez, J.E.; Orozco-del-Castillo, M.G.; Rodríguez-Castellanos, A.; Ávila-Carrera, R.; Valle-Molina, C. A fuzzy inference system applied to estimate the error in the recovery of the Green’s function by means of seismic noise correlations. J. Geophys. Eng.
**2018**, 15, 2110–2123. [Google Scholar] [CrossRef] [Green Version] - Ahmad, K.A.; Abdullah, S.L.S.; Mahmod, O.; A Bakar, M. Induction of Membership Function and Fuzzy Rules for Harumanis Classification. J. Fundam. Appl. Sci.
**2018**, 10, 1202–1215. [Google Scholar] - Ghani, U.; Bajwa, I.; Ashfaq, A. A Fuzzy Logic Based Intelligent System for Measuring Customer Loyalty and Decision Making. Symmetry
**2018**, 10, 761. [Google Scholar] [CrossRef] [Green Version] - Pancardo, P.; Hernández-Nolasco, J.A.; Acosta-Escalante, F. A Fuzzy Logic-Based Personalized Method to Classify Perceived Exertion in Workplaces Using a Wearable Heart Rate Sensor. Mob. Inf. Syst.
**2018**, 2018, 1–17. [Google Scholar] [CrossRef] [Green Version] - Subbotin, I. Trapezoidal Fuzzy Logic Model for Learning Assessment. 2014. Available online: https://www.researchgate.net/publication/263582488_Trapezoidal_Fuzzy_Logic_Model_for_Learning_Assessment (accessed on 17 November 2020).
- Hasan, M.H.; Jaafar, J.; Hassan, M.F. Fuzzy C-Means and two clusters’ centers method for generating interval type-2 membership function. In Proceedings of the 2016 3rd International Conference on Computer and Information Sciences (ICCOINS), Kuala Lumpur, Malaysia, 15–17 August 2016; pp. 627–632. [Google Scholar]
- Dunn, J.C. A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact Well-Separated Clusters. J. Cybern.
**1973**, 3, 32–57. [Google Scholar] [CrossRef] - Kalist, V.; Ganesan, P.; Sathish, B.S.; Jenitha, J.M.M.; Basha.shaik, K. Possiblistic-Fuzzy C-Means Clustering Approach for the Segmentation of Satellite Images in HSL Color Space. Procedia Comput. Sci.
**2015**, 57, 49–56. [Google Scholar] [CrossRef] [Green Version] - Sridevi, P. Identification of suitable membership and kernel function for FCM based FSVM classifier model. Clust. Comput.
**2018**, 22, 11965–11974. [Google Scholar] [CrossRef] - Rhee, F.; Choi, B.-I. Interval Type-2 Fuzzy Membership Function Generation Methods for Representing Sample Data. In Studies in Fuzziness and Soft Computing; Springer: Berlin, Germany, 2013; Volume 301, pp. 165–184. [Google Scholar]
- Faustino, C.P.; Novaes, C.P.; Pinheiro, C.A.M.; Carpinteiro, O.A. Improving the performance of fuzzy rules-based forecasters through application of FCM algorithm. Artif. Intell. Rev.
**2012**, 41, 287–300. [Google Scholar] [CrossRef] - Krishnapuram, R.; Keller, J.M. A possibilistic approach to clustering. IEEE Trans. Fuzzy Syst.
**1993**, 1, 98–110. [Google Scholar] [CrossRef] - Wang, L.; Wang, J. Feature Weighting Fuzzy Clustering Integrating Rough Sets and Shadowed Sets. Int. J. Pattern Recognit. Artif. Intell.
**2012**, 26, 26. [Google Scholar] [CrossRef] - Saha, S.; Pal, M.; Konar, A. Triangular membership function based real-time gesture monitoring system for physical disorder detection. Comput. Vis. Sci.
**2019**, 22, 1–14. [Google Scholar] [CrossRef] - Felix, G.; Nápoles, G.; Falcon, R.; Froelich, W.; Vanhoof, K.; Bello, R. A review on methods and software for fuzzy cognitive maps. Artif. Intell. Rev.
**2019**, 52, 1707–1737. [Google Scholar] [CrossRef] - Liu, F.; Peng, Y.; Chen, Z.; Shi, Y. Modeling of Characteristics on Artificial Intelligence IQ Test: A Fuzzy Cognitive Map-Based Dynamic Scenario Analysis. Int. J. Comput. Commun. Control.
**2019**, 14, 653–669. [Google Scholar] [CrossRef] [Green Version] - Raj, R.; Mohan, B.M. General structure of Interval Type-2 fuzzy PI/PD controller of Takagi–Sugeno type. Eng. Appl. Artif. Intell.
**2020**, 87, 103273. [Google Scholar] [CrossRef] - Ali, O.A.M.; Ali, A.Y.; Sumait, B.S. Comparison between the Effects of Different Types of Membership Functions on Fuzzy Logic Controller Performance. Int. J. Emerg. Eng. Res. Technol.
**2015**, 3, 76–83. [Google Scholar] - Umoh, U.; Udoh, S.; Isong, E.; Asuquo, R.; Nyoho, E. PSO Optimized Interval Type-2 Fuzzy Design for Elections Results Prediction. Int. J. Fuzzy Log. Syst.
**2019**, 9, 1–19. [Google Scholar] [CrossRef] - Mai, D.S.; Ngo, L.T. Interval Type-2 Fuzzy C-Means Clustering with Spatial Information for Land-Cover Classification. In Proceedings of the Asian Conference on Intelligent Information and Database Systems 2015, Dong Hoi City, Vietnam, 19–21 March 2018; pp. 387–397. [Google Scholar]
- Choi, B.-I.; Rhee, F.C.-H. Interval type-2 fuzzy membership function generation methods for pattern recognition. Inf. Sci.
**2009**, 179, 2102–2122. [Google Scholar] [CrossRef] - Mendel, J.M. x Type-2 Fuzzy Sets and Systems: A Retrospective. Inform. Spektrum
**2015**, 38, 523–532. [Google Scholar] [CrossRef] - Zhang, Z. Trapezoidal interval type-2 fuzzy aggregation operators and their application to multiple attribute group decision making. Neural Comput. Appl.
**2016**, 29, 1039–1054. [Google Scholar] [CrossRef] - Dan, S.; Kar, M.B.; Majumder, S.; Roy, B.; Kar, S.; Pamucar, D. Intuitionistic Type-2 Fuzzy Set and Its Properties. Symmetry
**2019**, 11, 808. [Google Scholar] [CrossRef] [Green Version] - Memon, K.H. A histogram approach for determining fuzzifier values of interval type-2 fuzzy c-means. Expert Syst. Appl.
**2018**, 91, 27–35. [Google Scholar] [CrossRef] - Melin, P.; Castillo, O. A review on type-2 fuzzy logic applications in clustering, classification and pattern recognition. Appl. Soft Comput.
**2014**, 21, 568–577. [Google Scholar] [CrossRef] - Zadeh, L.A. Fuzzy sets. Inf. Control.
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Casillas, J.; Moreno, D. Analyzing Strengths and Weaknesses of Fuzzy Association Rules Algorithms; Universidad de Granada: Granada, Spain, 2011; Available online: http://repositorio.conicit.go.cr:8080/xmlui/bitstream/handle/123456789/89/paper_Daniel_Moreno_vFinal%20%2812-12-2011%29.pdf?sequence=1&isAllowed=y (accessed on 17 November 2020).
- Chen, C.; John, R.; Twycross, J.; Garibaldi, J.M. Type-1 and interval type-2 ANFIS: A comparison. In Proceedings of the 2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Naples, Italy, 9–12 July 2017; pp. 1–6. [Google Scholar]
- Wu, K.-L. Analysis of parameter selections for fuzzy c-means. Pattern Recognit.
**2012**, 45, 407–415. [Google Scholar] [CrossRef] - Garima, G.H.; Singh, P.K. Clustering techniques in data mining: A comparison. In Proceedings of the 2015 2nd International Conference on Computing for Sustainable Global Development (INDIACom), New Delhi, India, 11–13 March 2015; pp. 410–415. [Google Scholar]
- Bezdek, J.C.; Ehrlich, R.; Full, W. FCM: The fuzzy c-means clustering algorithm. Comput. Geosci.
**1984**, 10, 191–203. [Google Scholar] [CrossRef] - Sadollah, A. Introductory Chapter: Which Membership Function is Appropriate in Fuzzy System? In Fuzzy Logic Based in Optimization Methods and Control Systems and Its Applications; Sadollah, A., Ed.; IntechOpen, 2018; Available online: https://www.researchgate.net/publication/328643706_Introductory_Chapter_Which_Membership_Function_is_Appropriate_in_Fuzzy_System (accessed on 17 November 2020).
- Reyna Vargas, M.E. Fuzzy Analytical Hierarchy Process Approach for Multicriteria Decision-Making with an Application to developing an ‘Urban Greenness Index’; University of Toronto: Toronto, ON, Canada, 2018; Available online: https://www.semanticscholar.org/paper/Fuzzy-Analytical-Hierarchy-Process-Approach-for-an-Vargas-Elvia/e2b9376e5b2f222a6dd5cc9a5b1095fb4bf4879e (accessed on 17 November 2020).
- Kreinovich, V.; Kosheleva, O.; Shahbazova, S. Why Triangular and Trapezoid Membership Functions: A Simple Explanation. Recent Dev. Fuzzy Log. Fuzzy Sets
**2020**, 25–31. [Google Scholar] [CrossRef] [Green Version] - Guldemır, H.; Sengur, A. Comparison of clustering algorithms for analog modulation classification. Expert Syst. Appl.
**2006**, 30, 642–649. [Google Scholar] [CrossRef] - Egrioglu, E.; Aladag, C.H.; Yolcu, U. Fuzzy time series forecasting with a novel hybrid approach combining fuzzy c-means and neural networks. Expert Syst. Appl.
**2013**, 40, 854–857. [Google Scholar] [CrossRef] - Zhao, F.; Fan, J.; Liu, H. Optimal-selection-based suppressed fuzzy c-means clustering algorithm with self-tuning non local spatial information for image segmentation. Expert Syst. Appl.
**2014**, 41, 4083–4093. [Google Scholar] [CrossRef] - Babaei, H.; Karimpour, J.; Oroji, H. Using fuzzy c-means clustering algorithm for common lecturers timetabling among departments. In Proceedings of the 2016 6th International Conference on Computer and Knowledge Engineering (ICCKE), Mashhad, Iran, 26–27 October 2016; pp. 243–250. [Google Scholar]
- Yazdani-Chamzini, A.; Razani, M.; Yakhchali, S.H.; Zavadskas, E.K.; Turskis, Z. Developing a fuzzy model based on subtractive clustering for road header performance prediction. Autom. Constr.
**2013**, 35, 111–120. [Google Scholar] [CrossRef] - Dhanachandra, N.; Manglem, K.; Chanu, Y.J. Image Segmentation Using K -means Clustering Algorithm and Subtractive Clustering Algorithm. Procedia Comput. Sci.
**2015**, 54, 764–771. [Google Scholar] [CrossRef] [Green Version] - Chu, F.; Ma, X.; Wang, F.; Jia, R. Novel robust approach for constructing Mamdani-type fuzzy system based on PRM and subtractive clustering algorithm. J. Central South Univ.
**2015**, 22, 2620–2628. [Google Scholar] [CrossRef] - Shukla, A.K.; Muhuri, P.K. Big-data clustering with interval type-2 fuzzy uncertainty modeling in gene expression datasets. Eng. Appl. Artif. Intell.
**2019**, 77, 268–282. [Google Scholar] [CrossRef] - Cao, H.; Jia, L.; Si, G.; Zhang, Y. A Clustering-analysis-based membership functions formation method for fuzzy controller of ball mill pulverizing system. J. Process. Control.
**2013**, 23, 34–43. [Google Scholar] [CrossRef] - Lv, Z.; Zhao, J.; Liu, Y.; Wang, W.; Han, M. A multi-objective clustering-based membership functions formation method for fuzzy modeling of gas pipeline pressure. IFAC-PapersOnLine
**2017**, 50, 12823–12828. [Google Scholar] [CrossRef] - Bulutsuz, A.G.; Yetilmezsoy, K.; Durakbasa, N. Application of fuzzy logic methodology for predicting dynamic measurement errors related to process parameters of coordinate measuring machines. IFAC-PapersOnLine
**2017**, 50, 12823–12828. [Google Scholar] [CrossRef] - Kowalczyk, A.; Pelikant, A. Implementation of automatically generated membership functions based on grouping algorithms. In Proceedings of the EUROCON 2007—The International Conference on “Computer as a Tool”, Warsaw, Poland, 9–12 September 2007; pp. 835–840. [Google Scholar]
- Kumar, A.; Quek, C.; Cho, S. DCT-Yager FNN: A Novel Yager-Based Fuzzy Neural Network With the Discrete Clustering Technique. IEEE Trans. Neural Netw.
**2008**, 19, 625–644. [Google Scholar] - Heng, Z.; Jie, W. Determination Method of Piecewise Linear Membership Function Based on the Interval Density Cluster. In Proceedings of the 2012 International Conference on Industrial Control and Electronics Engineering, Xi’an, China, 23–25 August 2012; pp. 1134–1137. [Google Scholar]
- Alemu, M.N. A fuzzy model for chaotic time series prediction. Int. J. Innov. Comput. Inf. Control.
**2018**, 14, 1767–1786. [Google Scholar] - Moewes, C.; Kruse, R. Evolutionary Fuzzy Rules for Ordinal Binary Classification with Monotonicity Constraints. Stud. Fuzziness Soft Comput.
**2013**, 291, 105–112. [Google Scholar] - Khayatzadeh, R.; Yelten, M.B. A Novel Multiple Membership Function Generator for Fuzzy Logic Systems. In Proceedings of the 2018 15th International Conference on Synthesis, Modeling, Analysis and Simulation Methods and Applications to Circuit Design (SMACD), Prague, Czech, 2–5 July 2018; pp. 101–104. [Google Scholar]
- Ruanpeng, C.; Auephanwiriyakul, S.; Theera-Umpon, N. Human and dog movement recognition using fuzzy inference system with automatically generated membership functions. In Proceedings of the 2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Naples, Italy, 9–12 July 2017; pp. 1–5. [Google Scholar]
- Viattchenin, D. A Heuristic Approach to Possibilistic Clustering for Fuzzy Data. J. Inf. Organ. Sci.
**2008**, 32, 149–163. [Google Scholar] - Viattchenin, D.; Tati, R.; Damaratski, A. Designing Gaussian Membership Functions for Fuzzy Classifier Generated by Heuristic Possibilistic Clustering. J. Inf. Organ. Sci.
**2013**, 37, 127–139. [Google Scholar] - Bhatt, R.B.; Narayanan, S.J.; Paramasivam, I.; Khalid, M. Approximating fuzzy membership functions from clustered raw data. In Proceedings of the 2012 Annual IEEE India Conference (INDICON), Kerala, India, 7–9 December 2012; pp. 487–492. [Google Scholar]
- Liao, T.W. A procedure for the generation of interval type-2 membership functions from data. Appl. Soft Comput.
**2017**, 52, 925–936. [Google Scholar] [CrossRef] - Liao, T.W.; Celmins, A.K.; Hammell, R.J. A fuzzy c-means variant for the generation of fuzzy term sets. Fuzzy Sets Syst.
**2003**, 135, 241–257. [Google Scholar] [CrossRef] - Koduru, G.K.; Nageswararao, K.; Namburu, A. T1 Weighted MR Brain Image Segmentation with Triangular Intuitionistic Fuzzy Set. Int. J. Innov. Technol. Explor. Eng.
**2020**, 9, 762–768. [Google Scholar] - Mahdipour, H.; Khademi, M.; Sadoghi Yazdi, H. Vector fuzzy C-means. J. Intell. Fuzzy Syst.
**2013**, 24, 363–381. [Google Scholar] - Rajendran, V. FCM Scheduled Multiple Model Controller for the Simulated Model of Spherical Tank Process. In Proceedings of the 2019 IEEE International Conference on Electrical, Computer and Communication Technologies (ICECCT), Coimbatore, India, 20–22 February 2019; pp. 1–6. [Google Scholar]
- Amsini, P.; Rani, R.U. Enhanced Type 2 Triangular Intuitionistic Fuzzy C Means Clustering Algorithm for Breast Cancer Histopathology Images. In Proceedings of the 2020 Fourth International Conference on Computing Methodologies and Communication (ICCMC), 11–13 March 2020; pp. 589–594. Available online: https://ieeexplore.ieee.org/document/9076454 (accessed on 17 November 2020).
- Shi, H.F.; Li, T.; Zhang, C. The Transformer Condition Assessment Model is Based on The Fuzzy Calculation. In Proceedings of the 3rd International Conference on Computer Science and Service System, Bangkok, Thailand, 13–15 June 2014; pp. 224–228. [Google Scholar]
- Yu, X.H.; Xiang, L.B. Classifying Cervical Spondylosis Based on Fuzzy Calculation. Abstr. Appl. Anal.
**2014**, 2014, 1–7. [Google Scholar] [CrossRef] - Ghazinoory, S.; Esmail Zadeh, A.; Kheirkhah, A.S. Application of fuzzy calculations for improving portfolio matrices. Inf. Sci.
**2010**, 180, 1582–1590. [Google Scholar] [CrossRef] - Vostroknutov, I.; Kaneda, Y. The possibilities of using modern casio cg-50 graphing calculators for volumetric and complex calculations, including fuzzy calculations. In Advances in Intelligent Systems and Computing; Springer: Berlin, Germany, 2018; pp. 702–708. [Google Scholar]
- Encheva, S. Selecting processes supported by fuzzy calculations. In Lecture Notes in Electrical Engineering; Springer: Berlin, Germany, 2014; pp. 39–43. [Google Scholar]
- Ghazinoory, S.; Esmail Zadeh, A.; Kheirkhah, A.S. Development of new evaluation methods for qualitative alternatives, using fuzzy calculations. Eur. J. Sci. Res.
**2011**, 51, 305–314. [Google Scholar] - Wei, X.; Zhang, H.; Wang, W.; Mo, H.; Li, B. Fingerprint chromatogram and fuzzy calculation for quality control of shenrong tonic wine. In Proceedings of the 5th International Conference on Natural Computation, Sendai, Japan, 12–13 December 2009; pp. 519–522. [Google Scholar]
- Degrauwe, D.; Arman, E.O.; Reynders, E.; De Roeck, G.; Lombaert, G. An efficient fuzzy calculation algorithm with application to finite element model updating. In Proceedings of the International Conference on Noise and Vibration Engineering, Heverlee, The Netherlands, 18–20 September 2006; pp. 4105–4116. [Google Scholar]
- Na, R.S.; Liu, Y.; Li, Y. Semantic Fuzzy Calculation and Product Recommendation Based on Online Reviews. J. Guangxi Norm. Univ.
**2010**, 1. Available online: https://en.cnki.com.cn/Article_en/CJFDTotal-GXSF201001038.htm (accessed on 17 November 2020). - Wang, C.; Wei, Y. Simulation of financial risk spillover effect based on ARMA-GARCH and fuzzy calculation model. J. Intell. Fuzzy Syst.
**2020**, 1–12. [Google Scholar] [CrossRef] - Ramos, S.; Khodr, H.M.; Azevedo, F.; Vale, Z. Power systems reliability calculation based on fuzzy data mining. In Proceedings of the 2009 IEEE Power & Energy Society General Meeting, Calgary, AB, Canada, 26-30 July 2009; pp. 1–7. [Google Scholar]
- Yu, X.; Meng, W.; Xiang, L. Comprehensive evaluation chronic pelvic pain based on fuzzy matrix calculation. Neurocomputing
**2016**, 173, 2097–2101. [Google Scholar] [CrossRef] - Yang, M.S.; Ba, L.; Zheng, H.Y.; Liu, Y.; Wang, X.F.; He, J.Z.; Li, Y. An integrated system for scheduling of processing and assembly operations with fuzzy operation time and fuzzy delivery time. Adv. Prod. Eng. Manag.
**2019**, 14, 367–378. [Google Scholar] [CrossRef] [Green Version] - Bocewicz, G.; Banaszak, Z.; Nielsen, I. Multimodal processes prototyping subject to grid-like network and fuzzy operation time constraints. Ann. Oper. Res.
**2017**, 273, 561–585. [Google Scholar] [CrossRef] [Green Version] - Bocewicz, G.; Banaszak, Z.; Nielsen, I. Multimodal processes prototyping subject to fuzzy operation time constraints. IFAC-PapersOnLine
**2015**, 48, 2103–2108. [Google Scholar] [CrossRef] - Nielsen, I.; Wójcik, R.; Bocewicz, G. Multimodal processes optimization subject to fuzzy operation time constraints: Declarative modeling approach. Front. Inf. Technol. Electron. Eng.
**2016**, 17, 338–347. [Google Scholar] [CrossRef] - Wójcik, R.; Nielsen, I.; Bocewicz, G.; Banaszak, Z. Multimodal Processes Optimization Subject to Fuzzy Operation Time Constraints. In Proceedings of the 12th International Conference. Advances in Intelligent Systems and Computing, Seville, Spain, 13–15 May 2019; p. 373. [Google Scholar]
- Chen, D.-J.; Wang, P. Research on Weighing Strategy of Vehicle Entering Plant Based on Fuzzy Operation Time. In Proceedings of the 2017 International Conference on Applied Mechanics and Mechanical Automation (AMMA 2017), Phuket, Thailand, 6–7 August 2017; pp. 132–138. [Google Scholar]
- Chen, C.-T.; Huang, S.-F. Order-fulfillment ability analysis in the supply-chain system with fuzzy operation times. Int. J. Prod. Econ.
**2006**, 101, 185–193. [Google Scholar] [CrossRef] - Reiser, R.; Lemke, A.; Avila, A.; Vieira, J.; Pilla, M.; Bois, A.D. Interpretations on Quantum Fuzzy Computing: Intuitionistic Fuzzy Operations×Quantum Operators. Electron. Notes Theor. Comput. Sci.
**2016**, 324, 135–150. [Google Scholar] [CrossRef] [Green Version] - Ledeneva, T. Additive generators of fuzzy operations in the form of linear fractional functions. Fuzzy Sets Syst.
**2020**, 386, 1–24. [Google Scholar] [CrossRef] - Han, S.; Liu, G.; Zhang, T. Mean almost periodicity and moment exponential stability of semi-discrete random cellular neural networks with fuzzy operations. PLoS ONE
**2019**, 14, e0220861. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bo, H.; Lvqing, B.; Songsong, D.; Sizhao, L. Distances of Complex Fuzzy Sets and Continuity of Complex Fuzzy Operations. J. Intell. Fuzzy Syst.
**2018**, 35, 2247–2255. [Google Scholar] - Zernov, M.M.; Mladov, V.V. Associative methods of fuzzy operations implementation. In Proceedings of the 2017 Second Russia and Pacific Conference on Computer Technology and Applications (RPC), Vladivostok, Russia, 25–29 September 2017; pp. 199–204. [Google Scholar]
- Ökmen, Ö.; Öztaş, A. A CPM-based scheduling method for construction projects with fuzzy sets and fuzzy operations. J. S. Afr. Inst. Civ. Eng.
**2014**, 56, 2–8. [Google Scholar] - Pietraszek, J. The modified sequential-binary approach for fuzzy operations on correlated assessments. In Proceedings of the 12th International Conference on Artificial Intelligence and Soft Computing, Zakopane, Poland, 9–13 June 2013. [Google Scholar]
- Zhu, X.-P.; Zhang, L.-B.; Liang, W.; Shi, G.-N. A quantitative comprehensive safety evaluation method for centrifugal compressors using FMEA-fuzzy operations. In Proceedings of the 2nd International Symposium on Instrumentation and Measurement, Sensor Network and Automation, Toronto, ON, Canada, 23–24 December 2013. [Google Scholar]
- Yoon, J.H.; Choi, S.H. Fuzzy Least Squares Estimation with New Fuzzy Operations. In Advances in Intelligent Systems and Computing; Kruse, R., Berthold, M., Moewes, C., Gil, M., Grzegorzewski, P., Hryniewicz, O., Eds.; Synergies of Soft Computing and Statistics for Intelligent Data Analysis; Springer: Berlin/Heidelberg, Germany, 2013; p. 190. [Google Scholar]
- Wang, A.; Jeong, J. ShiAdaptive bilateral filter with local intensity histogram combine generalized fuzzy operation (GFO) for intra-frame deinterlacing. In Proceedings of the 2012 International Conference on Systems and Informatics, Jiaxing, China, 18–20 December 2012. [Google Scholar]
- Zhang, L.; Duan, W.; Guo, H. Fuzzy Operation Forensics Research Based on Mathematical Morphology. In Lecture Notes in Electrical Engineering; Zhu, R., Ma, Y., Eds.; Information Engineering and Applications; Springer: London, UK, 2012; p. 154. [Google Scholar]
- Huang, S.; Huang, Y. Monitor and control the desk-top illumination based on fuzzy operation. In Proceedings of the 2011 International Conference on Machine Learning and Cybernetics, Guilin, China, 10–13 July 2011; pp. 248–253. [Google Scholar]
- Saneifard, R. A new algorithm for selecting equip system based on fuzzy operations. Int. J. Phys. Sci.
**2011**, 6, 3279–3287. [Google Scholar] - Gál, L.; Lovassy, R.; Kóczy, L.T. Function approximation performance of Fuzzy Neural Networks based on frequently used fuzzy operations and a pair of new trigonometric norms. In Proceedings of the International Conference on Fuzzy Systems, Barcelona, Spain, 18–23 July 2010; pp. 1–8. [Google Scholar]
- Maturo, A. Alternative fuzzy operations and applications to social sciences. Int. J. Intell. Syst.
**2009**. [Google Scholar] [CrossRef] - Rudas, I.J.; Batyrshin, I.Z.; Zavala, A.H.; Nieto, O.C.; Horváth, L.; Vargas, L.V. Generators of Fuzzy Operations for Hardware Implementation of Fuzzy Systems. In MICAI 2008: Advances in Artificial Intelligence. MICAI 2008. Lecture Notes in Computer Science; Gelbukh, A., Morales, E.F., Eds.; Springer: Berlin/Heidelberg, Germany, 2008; p. 5317. [Google Scholar]
- Xu, W.; Xiao, T. Mixed model assembly line balancing problem with fuzzy operation times and drifting operations. In Proceedings of the 2008 Winter Simulation Conference, Miami, FL, USA, 7–10 December 2008; pp. 1752–1760. [Google Scholar]
- Cai, S.; Chen, X.; Wang, Q.; Yin, M. FPGA Implementation of Generalized Fuzzy Operations. In Proceedings of the 2008 Fifth International Conference on Fuzzy Systems and Knowledge Discovery, Shandong, China, 18–20 October 2008; pp. 560–564. [Google Scholar]
- Batyrshin, I.; Zavala, A.; Camacho, O.; Vargas, L. Generalized Fuzzy Operations for Digital Hardware Implementation. In Proceedings of the Mexican International Conference on Artificial Intelligence, Aguascalientes, Mexico, 4–10 November 2007; pp. 9–18. [Google Scholar]
- Su, Z.-X.; Guo, S.-C. Fuzzy operation with equality constraint. J. Liaoning Tech. Univ.
**2005**, 24, 299–302. [Google Scholar] - Koprinkova-Hristova, P.D. Fuzzy operations’ parameters versus membership functions’ parameters influence on fuzzy control systems properties. In Proceedings of the 2004 2nd International IEEE Conference on ‘Intelligent Systems’, Varna, Bulgaria, 22–24 June 2004; pp. 219–224. [Google Scholar]
- Tang, L.; Xie, W.-X.; Huang, J.; Huang, J.-X. Fuzzy operation based multitarget-multisensor tracking algorithm. Xi Tong Gong Cheng Yu Dian Zi Ji Shu/Syst. Eng. Electron.
**2004**, 26, 1573–1577. [Google Scholar] - Turksen, I.B.; Esper, A.; Patel, K.; Starks, S.A.; Kreinovich, V. Selecting a fuzzy logic operation from the DNF-CNF interval: How practical are the resulting operations? In Proceedings of the 2002 Annual Meeting of the North American Fuzzy Information Processing Society Proceedings, New Orleans, LA, USA, 27–29 June 2002; pp. 28–33. [Google Scholar]
- Li, F.; Rao, Y. Weighted fuzzy operations based on Vague sets. Huazhong Ligong Daxue Xuebao/J. Huazhong Univ. Sci. Technol.
**2001**, 29, 12–14. [Google Scholar] - Czogala, E.; Kowalczyk, R. Investigation of selected fuzzy operations and implications for engineering. In Proceedings of the IEEE 5th International Fuzzy Systems, New Orleans, LA, USA, 11 September 1996; pp. 879–885. Available online: https://ieeexplore.ieee.org/abstract/document/552295 (accessed on 17 November 2020).
- Han, J.; Singh, S. Fast digital fuzzy operation units using comparison look-ahead. In Proceedings of the 33rd Midwest Symposium on Circuits and Systems, Calgary, AB, Canada, 12–15 August 1990; pp. 870–873. [Google Scholar]
- Han, J.; Singh, S. Comparison look-ahead and design of fast fuzzy operation units. In Proceedings of the Twentieth International Symposium on Multiple-Valued Logic, Charlotte, NC, USA, 23–25 May 1990; pp. 121–125. [Google Scholar]
- Wygralak, M. Fuzzy inclusion and fuzzy equality of two fuzzy subsets, fuzzy operations for fuzzy subsets. Fuzzy Sets Syst.
**1983**, 10, 157–168. [Google Scholar] [CrossRef] - Arakawa, M.; Yamakawa, H.; Ishikawa, H. Robust design using fuzzy numbers (consideration of correlation of design variables in fuzzy operation). In Proceedings of theDETC2000: ASME 2000 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Anaheim, CA, USA, 18–21 August 2000. [Google Scholar]
- Tabata, T.; Ueno, F.; Eguchi, K.; Zhu, H. CMOS-Based Fuzzy Operation Circuit Using Binary-Coded Redundantly-Represented Positive-Digit Numbers. In Proceedings of the IEEK Conference, The Institute of Electronics and Information Engineers, Atlanta, GA, USA, 2000; Available online: https://www.koreascience.or.kr/article/CFKO200011921291336.page (accessed on 17 November 2020).
- Swathi, J.N.; Bhatt, R.; Paramasivam, I.; Khalid, M. A study on the approximation of clustered data to parameterized family of Fuzzy membership functions for the induction of Fuzzy Decision Trees. Cybern. Inf. Technol.
**2015**, 15, 75–96. [Google Scholar] - Qaiyum, S.; Izzatdin, A.; Jaafar, J.; Kai, A. Ant Colony Optimization of Interval Type-2 Fuzzy C-Means with Subtractive Clustering and Multi-Round Sampling for Large Data. Int. J. Adv. Comput. Sci. Appl.
**2019**, 10, 10. [Google Scholar] [CrossRef] - Dhanachandra, N.; Chanu, Y.J. An image segmentation approach based on fuzzy c-means and dynamic particle swarm optimization algorithm. Multimedia Tools Appl.
**2020**, 79, 18839–18858. [Google Scholar] [CrossRef] - Venkat, R.; Reddy, K.S. Dealing Big Data using Fuzzy C-Means (FCM) Clustering and Optimizing with Gravitational Search Algorithm (GSA). In Proceedings of the 2019 3rd International Conference on Trends in Electronics and Informatics (ICOEI), Tirunelveli, India, 23–25 April 2019; pp. 465–467. [Google Scholar]

**Figure 3.**IT2 MF generation from the blurring method [5].

MF. | Source | Description |
---|---|---|

Gaussian | Rubio and Castillo, 2013 [7] | This paper shows that Fuzzy C-Means (FCM) can handle uncertainty in data clustering, and the MF generated by FCM presented a significant FOU. |

Rodríguez-Sánchez et al., 2018 [8] | Equations for seismic noise correlation were applied and fuzzy logic was used to estimate the error in the recovery of the Green’s function. | |

Triangular | Ahmad et al., 2018 [9] | Fuzzy classification system was applied as a decision making technique to classify the Harumanis fruit quality based on fruit contour. An algorithm for generating MFs and If-Else()fuzzy rules for fruit classification were proposed. |

Ghani et al., 2018 [10] | Fuzzy logic was applied to find customers’ loyalty and their decision making. A set of MFs and rule-based system of fuzzy sets were used to classify data in types of loyalties. | |

Trapezoidal | Pancardo et al., 2018 [11] | A heart rate-based personalized method to assess perceived exertion in workplaces was proposed and fuzzy logic was used as a method to manage imprecision and uncertainty in the used variables. |

Subbotin, 2014 [12] | Fuzzy logic was applied for the assessment in learning process such as grading and assessing students’ work. |

Manuscript | Generate MF Using FCM? | Generate Triangular/Trapezoidal MF Using FCM? | Generate IT2 MF? |
---|---|---|---|

Shukla and Muhuri (2019) [50] | No | Yes | Yes |

Cao et al. (2013) [51] | No | Yes | No |

Lv et al. (2017) [52] | No | Yes | No |

Bulutsuz et al. (2015) [53] | No | No | No |

Kowalczyk and Pelikant (2007) [54] | No | No | No |

Kumar et al. (2008) [55] | No | Yes | No |

Heng and Jie (2012) [56] | No | No | No |

Rubio and Castillo (2013) [7] | Yes | No | Yes |

Alemu (2018) [57] | Yes | No | No |

Moewes and Kruse (2013) [58] | No | No | No |

Jang (1993) [6] | No | No | No |

Chen et al. (2017) [36] | No | No | No |

Khayatzadeh and Yelten (2018) [59] | No | No | No |

Ruanpeng et al. (2017) [60] | No | No | No |

Viattchenin et al. (2013) [61] | No | No | No |

Bhatt et al. (2012) [63] | Yes | Yes | No |

Liao (2017) [64] | Yes | Yes | Yes |

Koduru et al. (2020) [66] | Yes | Yes | No |

Mahdipour et al. (2013) [67] | Yes | No | No |

Rajendran (2019) [68] | Yes | No | No |

Amsini and Rani (2020) [69] | Yes | No | No |

Manuscript | Generate MF Using FCM? | Generate Triangular/Trapezoidal MF Using FCM? | Generate IT2 MF? |
---|---|---|---|

Shi et al. (2014) [70] | No | No | No |

Yu and Xiang (2014) [71] | No | No | No |

Ghazinoory et al. (2010) [72] | No | No | No |

Vostroknutov and Kaneda (2018) [73] | No | No | No |

Encheva (2014) [74] | No | No | No |

Hamidreza et al. (2011) [75] | No | No | No |

Wei et al. (2009) [76] | No | No | No |

Degrauwe et al. (2006) [77] | No | No | No |

Na et al. (2010) [78] | No | No | No |

Wang and Wei (2020) [79] | No | No | No |

Ramos et al. (2009) [80] | No | No | No |

Yu et al. (2015) [81] | No | No | No |

Yang et al. (2019) [82] | No | No | No |

Bocewicz et al. (2019) | No | No | No |

Bocewicz et al. (2015) | No | No | No |

Wojcik et al. (2015) | No | No | No |

Nielsen et al. (2016) | No | No | No |

Chen and Wang (2017) [87] | No | No | No |

Cheng and Huang (2006) [88] | No | No | No |

Reiser et al. (2016) [89] | No | No | No |

Ledeneva (2020) [90] | No | No | No |

Han et al. (2019) [91] | No | No | No |

Hu et al. (2018) [92] | No | No | No |

Zernov and Mladov (2017) [93] | No | No | No |

Okmen and Oztas (2014) [94] | No | No | No |

Pietraszek (2013) [95] | No | No | No |

Zhu et al. (2013) [96] | No | No | No |

Yoon and Choi (2013) [97] | No | No | No |

Wang and Jeong (2012) [98] | No | No | No |

Zhang et al. (2012) [99] | No | No | No |

Huang and Huang (2011) [100] | No | No | No |

Saneifard (2011) [101] | No | No | No |

Gal et al. (2010) [102] | No | No | No |

Maturo (2009) [103] | No | No | No |

Rudas et al. (2008) [104] | No | No | No |

Xu and Xiao (2008) [105] | No | No | No |

Cai et al. (2008) [106] | No | No | No |

Batyrshin et al. (2007) [107] | No | No | No |

Su and Guo (2005) [108] | No | No | No |

Koprinkova-Hristova (2004) [109] | No | No | No |

Tang et al. (2004) [110] | No | No | No |

Turken et al. (2002) [111] | No | No | No |

Li and Rao (2001) [112] | No | No | No |

Czogala and Kowalczyk (1996) [113] | No | No | No |

Han and Singh (1990) [114] | No | No | No |

Han and Singh (1990) [115] | No | No | No |

Wygralak (1983) [116] | No | No | No |

Arakawa et al. (2000) [117] | No | No | No |

Tabata et al. (2000) [118] | No | No | No |

Contribution | Description |
---|---|

Alternative membership functions | The triangular and trapezoidal MF provide alternatives for IT2-based applications especially in the condition whereby Gaussian IT2 MF cannot produce good results. |

Application | By having more options of MFs, IT2 FIS can be applied to a wider range of applications. |

Performance | IT2 FIS generated from FCM can leverage on the benefit of learning from data, which may help in producing good and representative results especially in data-based applications such as forecasting and prediction systems. |

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## Share and Cite

**MDPI and ACS Style**

Khairuddin, S.H.; Hasan, M.H.; Hashmani, M.A.; Azam, M.H.
Generating Clustering-Based Interval Fuzzy Type-2 Triangular and Trapezoidal Membership Functions: A Structured Literature Review. *Symmetry* **2021**, *13*, 239.
https://doi.org/10.3390/sym13020239

**AMA Style**

Khairuddin SH, Hasan MH, Hashmani MA, Azam MH.
Generating Clustering-Based Interval Fuzzy Type-2 Triangular and Trapezoidal Membership Functions: A Structured Literature Review. *Symmetry*. 2021; 13(2):239.
https://doi.org/10.3390/sym13020239

**Chicago/Turabian Style**

Khairuddin, Siti Hajar, Mohd Hilmi Hasan, Manzoor Ahmed Hashmani, and Muhammad Hamza Azam.
2021. "Generating Clustering-Based Interval Fuzzy Type-2 Triangular and Trapezoidal Membership Functions: A Structured Literature Review" *Symmetry* 13, no. 2: 239.
https://doi.org/10.3390/sym13020239