Ranking Z-numbers Using the Optimal Clustering Method (CZ-number)

Subject Areas :

Saeed Jafari ¹ , Mojtaba Najafi ² , Naghi Moaddabi Pirkolahchahi ³ , Najmeh Cheraghi Shirazi ⁴

1 - Deputy of Planning and Engineering, Renewable Energy Office, Bushehr Power Distribution Company, Bushehr, Iran
2 - Department of Electrical Engineering, Bushehr Branch, Islamic Azad University, Bushehr, Iran
3 - Department of Electrical Engineering, Bushehr Branch, Islamic Azad University, Bushehr, Iran
4 - Department of Electrical Engineering, Bushehr Branch, Islamic Azad University, Bushehr, Iran

Received: 2023-06-06 Accepted : 2023-09-09 Published : 2025-10-12

Keywords: Probability-possibility, Unsupervised clustering, Z-number, CZ number, Fuzzy.,

Abstract :

In recent years, modeling uncertainties through natural language has attracted growing attention, with Z-numbers, introduced by Zadeh in 2011, being a key concept. A Z-number consists of two fuzzy components: the first represents the possibility of an event’s occurrence, while the second expresses the probability of that occurrence. A major challenge in applying Z-numbers lies in properly structuring these two components; inappropriate strategies can lead to inaccurate results, especially in group decision-making contexts with large data volumes. To address this, the paper proposes an optimal clustering approach for structuring the components of Z-numbers more systematically and purposefully. This method enhances the accuracy of Z-number modeling by reducing errors in component formation. The effectiveness of the proposed approach is validated through a comparison with conventional fuzzy methods, demonstrating improved performance in handling uncertainty.

Highlights:

• Using the unsupervised learning method in the optimal selection of expert opinions.
• Effectiveness of the proposed method in choosing the optimal opinions of experts in the conditions of large and complex data sets of expert opinions.
• Determining the optimal points to determine the points that make up the intervals of the Z-number.
• The use of any method or algorithm for unsupervised clustering of points forming the Z-number.

 

Citation: S. Jafari, M. Najafi, N. Moadabi Pirkolachahi, and N. Cheraghi Shirazi, “Ranking Z-numbers Using the Optimal Clustering Method,” Journal of Southern Communication Engineering, vol. 15, no. 57, pp. 56–75, 2025, doi:10.30495/jce.2023.1987707.1205 [in Persian].

References:

[1] A. L. Zadeh, "Fuzzy sets," Information and control, vol. 8, no. 3, pp. 338-353, 1965, doi: 10.1016/S0019-9958(65)90241-X.
[2] A. P. Dempster, "A generalization of Bayesian inference," Journal of the Royal Statistical Society, Series B (Methodological) , vol. 30, no. 2, pp. 205-232, 2018, doi: 10.1111/j.2517-6161.1968.tb00722.x.
[3] G. Shafer, “A mathematical theory of evidence turns 40,” International Journal of Approximate Reasoning, vol. 79, pp. 7-25, 2016, doi: 10.1016/j.ijar.2016.07.009.
[4] M. J. Mendel and J.R. Bob, "Type-2 fuzzy sets made simple," IEEE Transactions on fuzzy systems, vol. 10, no. 2, pp. 117-127, 2002, doi: 10.1109/91.995115.
[5] G. Ulutagay and V. Kreinovich, “Density-Based Fuzzy Clustering as a First Step to Learning Rules: Challenges and Solutions,” in Advance Trends in Soft Computing: Proceedings of WCSC , 2013, San Antonio, Texas, USA , pp. 357-372, doi: 10.1007/978-3-319-03674-8.
[6] Y. Wang et al., "Pairwise constraints-based semi-supervised fuzzy clustering with multi-manifold regularization," Information Sciences, vol. 638, p. 118994, 2023, doi: 10.1016/j.ins.2023.118994.
[7] L. Guo et al., "Pixel and region level information fusion in membership regularized fuzzy clustering for image segmentation," Information Fusion , vol. 92, pp. 479-497, 2023, doi: 10.1016/j.inffus.2022.12.008.
[8] F. Xiao, "A multiple-criteria decision-making method based on D numbers and belief entropy," International Journal of Fuzzy Systems , vol. 21, no. 4, pp. 1144-1153, 2019, doi: 10.1007/s40815-019-00620-2.
[9] J. Pérez-Ortega et al., "POFCM: A Parallel Fuzzy Clustering Algorithm for Large Datasets," Mathematics , vol. 11, no. 8, p. 1920, 2023, doi: 10.3390/math11081920.
[10] S. Jafari, M. Najafi, N. M. Pirkolachahi, and N. C. Shirazi, "Enhancing energy hub efficiency through advanced modelling and optimization techniques: A case study on micro-refinery output products and parking lot integration," Heliyon, vol. 10, no. 18, 2023, doi: 10.1016/j.heliyon.2024.e36233 .
[11] S. Jafari, M. Najafi, N. M. Pirkolachahi, and N. C. Shirazi, "Modelling multi-stage energy optimization in the smart hub energy system with hybrid demand management and local supply by Electric Vehicles," Research square, 2024, doi: 10.21203/rs.3.rs-3834421/v1.
[12] R. Cerqueti and R. Mattera, "Fuzzy clustering of time series with time-varying memory," International Journal of Approximate Reasoning, vol. 153, pp. 193-218, 2023, doi: 10.1016/j.ijar.2022.11.021.
[13] A. R. Rajeswari et al., "A trust-based secure neuro fuzzy clustering technique for mobile ad hoc networks," Electronics , vol. 12, no. 2, p. 274, 2023, doi: 10.3390/electronics12020274.
[14] R. Cheng, B. Kang and J. Zhang, "An Improved Method of Converting Z-number into Classical Fuzzy Number," 33rd Chinese Control and Decision Conference (CCDC), Kunming, China, 2021, pp. 3823-3828, doi: 10.1109/CCDC52312.2021.9601658.
[15] A. R. Aliev, A. V. Alizadeh and O. H. Huseynov, "The arithmetic of discrete Z-numbers," Information Sciences, vol. 290 , pp. 134-155, 2015, doi: 10.1016/j.ins.2014.08.024.
[16] F. Liu et al., "Evaluating Internet hospitals by a linguistic Z-number-based gained and lost dominance score method considering different risk preferences of experts," Information Sciences, vol. 630, pp. 647-668, 2023, doi: 10.1016/j.ins.2023.02.061.
[17] F. Teng et al., "Probabilistic linguistic Z number decision-making method for multiple attribute group decision-making problems with heterogeneous relationships and incomplete probability information," International Journal of Fuzzy Systems, vol. 24, no. 1, pp. 1-22, 2022, doi: 10.1007/s40815-021-01161-3.
[18] R. Jafari, W. Yu, and X. Li, "Numerical solution of fuzzy equations with Z-numbers using neural networks," Intelligent Automation & Soft Computing, vol. 24, no. 1, pp. 1-7, 2017, doi: 10.1080/10798587.2017.1327154.
[19] S. Massanet, J. V. Riera and J. Torrens, "A new vision of Zadeh’s Z-numbers," International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems. Cham: Springer International Publishing, vol. 47, 2016, pp. 581-592, doi: 10.1007/978-3-319-40581-0_47.
[20] A. R. Aliev et al., "Z‐number‐based linear programming," International Journal of Intelligent Systems , vol. 30, no. 5, pp. 563-589, 2015, doi: 10.1002/int.21709.
[21] D. Wu et al., "A new medical diagnosis method based on Z-numbers," Applied Intelligence , vol. 48, pp. 854-867, 2018, doi: 10.1007/s10489-017-1002-4.
[22] R. Cheng, J. Zhang and B. Kang, "Ranking of Z-Numbers Based on the Developed Golden Rule Representative Value," in IEEE Transactions on Fuzzy Systems, vol. 30, no. 12, pp. 5196-5210, Dec. 2022, doi: 10.1109/TFUZZ.2022.3170208.
[23] A. R. Aliev et al., "Eigensolutions of partially reliable decision preferences described by matrices of Z-numbers," International Journal of Information Technology & Decision Making, vol. 19, no. 06, pp. 1429-1450, 2020, doi: 10.1142/S0219622020010063.
[24] R. Banerjee and K. P. Sankar, "A machine-mind architecture and Z*-numbers for real-world comprehension," Pattern Recognition and Big Data, pp. 805-842, 2017, doi: 10.1142/9789813144552_0026.
[25] Q. Liu et al., "On the negation of discrete z-numbers," Information Sciences, vol. 537, pp. 18-29, 2020, doi: 10.1016/j.ins.2020.05.106.
[26] https://hourlypricing.comed.com/live-prices/?date=20230310.