A Review on Ranking of Z-numbers


  • Fırat Bilgin Graduate School of Natural and Applied Science, Ege University, Izmir, 35040, Turkey
  • Musa Alci Electrical and Electronics Engineering Department, Ege University, Izmir, 35040, Turkey




There are numerous studies about Z-numbers since its inception in 2011. Because Z-number concept reflects human ability to make rational decisions, Z-number based multi-criteria decision making problems are one of these studies. When the problem is translated from linguistic information into Z-number domain, the important question occurs that which Z-number should be selected. To answer this question, several ranking methods have been proposed. To compare the performances of these methods, benchmark set of fuzzy Z-numbers has been created in time. There are relatively new methods that their performances are not examined yet on this benchmark problem. In this paper, we worked on these studies which are relative entropy based Z-number ranking method and a method for ranking discrete Z-numbers. The authors tried to examine their performances on the benchmark problem and compared the results with the other ranking algorithms. The results are consistent with the literature, mostly. The advantages and the drawbacks of the methods are presented which can be useful for the researchers who are interested in this area.


Fuzzy Z-numbers; Ranking discrete Z-numbers; Ranking of Z-numbers; Relative entropy based ranking


[1] Zadeh, L.A., 2011. A note on Z-numbers. Information Sciences. 181(14), 2923-2932. DOI: https://doi.org/10.1016/j.ins.2011.02.022

[2] Zadeh, L.A., 1968. Probability measures of fuzzy events. Journal of Mathematical Analysis and Applications. 23(2), 421-427. DOI: https://doi.org/10.1016/0022-247X(68)90078-4

[3] Puri, M.L., Ralescu, D.A., Zadeh, L.A., 1993. Fuzzy random variables. Readings in Fuzzy Sets for Intelligent Systems. pp. 265-271. DOI: https://doi.org/10.1016/B978-1-4832-1450-4.50029-8

[4] Zadeh, L.A., 1979. Fuzzy sets and information granularity. Advances in Fuzzy Set Theory and Applications. 11, 3-18. DOI: https://doi.org/10.1142/9789814261302_0022

[5] Tian, Y., Kang, B., 2020. A modified method of generating Z-number based on OWA weights and maximum entropy. Soft Computing. 24(20), 15841-15852. DOI: https://doi.org/10.1007/s00500-020-04914-8

[6] Bilgin, F., Alcı, M., 2021. Generating Z-number by logistic regression. 7th International Symposium on Electrical and Electronics Engineering (ISEEE). pp. 1-6. DOI: https://doi.org/10.1109/ISEEE53383.2021.9628654

[7] Azadeh, A., Saberi, M., Atashbar, N.Z., et al., 2013. Z-AHP: A Z-number extension of fuzzy analytical hierarchy process. 2013 7th IEEE International Conference on Digital Ecosystems and Technologies (DEST). pp. 141-147. DOI: https://doi.org/10.1109/DEST.2013.6611344

[8] Khalif, K.M.N., Gegov, A., Abu Bakar, A.S., 2017. Z-TOPSIS approach for performance assessment using fuzzy similarity. 2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE). pp. 1-6. DOI: https://doi.org/10.1109/FUZZ-IEEE.2017.8015458

[9] Abiyev, R.H., Akkaya, N., Gunsel, I., 2019. Control of omnidirectional robot using Z-Number-based fuzzy system. IEEE Transactions on Systems, Man, and Cybernetics: Systems. 49(1), 238-252. DOI: https://doi.org/10.1109/TSMC.2018.2834728

[10] Peide, L., Fei, T., 2015. An extended TODIM method for multiple attribute group decision making based on intuitionistic uncertain linguistic variables. 29(2), 701-711. DOI: https://doi.org/10.3233/IFS-141441

[11] Aliev, R.A., Huseynov, O.H., Aliyev, R.R., et al., 2015. The arithmetic of Z-numbers: theory and applications. Singapore: World Scientific.DOI: https://doi.org/10.1142/9575

[12] Patel, P., Khorosani, E.S., Rahimi, S., 2015. Modeling and implementation of Z-number. Soft Computing. 20(4), 1341-1364. DOI: https://doi.org/10.1007/s00500-015-1591-y

[13] Shalabi, M.E., Hussieny, H., Abouelsoud, A.A., 2019. Control of automotive air-spring suspension system using Z-number based fuzzy system. IEEE International Conference on Robotics and Biomimetics (ROBIO). pp. 1306-1311. DOI: https://doi.org/10.1109/ROBIO49542.2019.8961492

[14] Abdelwahab, M., Parque, V., Elbab, A.M.F., et al., 2020. Trajectory tracking of wheeled mobile robots using z-number based fuzzy logic. IEEE Access. 8, 18426-18441. DOI: https://doi.org/10.1109/ACCESS.2020.2968421

[15] Jiang, W., Xie, C., Zhuang, M., et al., 2016. Sensor data fusion with Z-Numbers and its application in fault diagnosis. Sensors. 16(9), 1509. DOI: https://doi.org/10.3390/s16091509

[16] Aliev, R.A., Pedrycz, W., Guirimov, B.G., et al., 2020. Acquisition of Z-number-valued clusters by using a new compound function, IEEE Transactions on Fuzzy Systems. 30(1), 279-286. DOI: https://doi.org/10.1109/TFUZZ.2020.3037969

[17] Tian, Y., Mi, X., Cui, H., et al., 2021, Using Z-number to measure the reliability of new information fusion method and its application in pattern recognition. Applied Soft Computing. 111, 107658. DOI: https://doi.org/10.1016/j.asoc.2021.107658

[18] Kang, B., Wei, D., Li, Y., et al., 2012. A method of converting Z-number to classical fuzzy number. Journal of Information & Computational Science. 9(3), 703-709.

[19] Banerjee, R., Pal, S., Pal, J.K., 2021. A decade of the Z-numbers, IEEE Transactions on Fuzzy Systems. DOI: https://doi.org/10.1109/TFUZZ.2021.3094657

[20] K. Shen and J. Wang, 2018, Z-VIKOR method based on a new comprehensive weighted distance measure of Z-Number and its application, IEEE Transactions on Fuzzy Systems, 26(6), 3232-3245. DOI: https://doi.org/10.1109/TFUZZ.2018.2816581

[21] Qiao, D., Shen, K., Wang, J., 2020. Multi-criteria PROMETHEE method based on possibility degree with Z-numbers under uncertain linguistic environment. Journal of Ambient Intelligence and Humanized Computing. 11, 2187-2201. DOI: https://doi.org/10.1007/s12652-019-01251-z

[22] Mohamad, D., Shaharani, S.A., Kamis, N.H., 2017. Ordering of Z-numbers. AIP Conference Proceedings, AIP Publishing LLC. 1870(1). DOI: https://doi.org/10.1063/1.4995881

[23] Bakar, A.S.A., Gegov, A., 2015. Multi-layer decision methodology for ranking Z-numbers. International Journal of Computational Intelligence Systems. 8(2), 395-406. DOI: https://doi.org/10.1080/18756891.2015.1017371

[24] Ezadi, S., Allahviranloo, T., 2017. New multi-layer method for Z-number ranking using hyperbolic tangent function and convex combination. Intelligent Automation & Soft Computing. pp. 1-7. DOI: https://doi.org/10.1080/10798587.2017.1367146

[25] Ezadi, S., Allahviranloo, T., Mohammadi, S., 2018. Two new methods for ranking of Z‐numbers based on sigmoid function and sign method. International Journal of Intelligent Systems. 33(7), 1476-1487. DOI: https://doi.org/10.1002/int.21987

[26] Jiang, W., Xie, Ch.H., Luo, Y., et al., 2017. Ranking Z-numbers with an improved ranking method for generalized fuzzy numbers. Journal of Intelligent & Fuzzy Systems. 32(3), 1931-1943. DOI: https://doi.org/10.3233/JIFS-16139

[27] Chutia, R., 2021, Ranking of Z‐numbers based on value and ambiguity at levels of decision making. International Journal of Intelligent Systems. 36(1), 313-331. DOI: https://doi.org/10.1002/int.22301

[28] Li, Y.X., Pelusi, D., Deng, Y., et al., 2021, Relative entropy of Z-numbers. Information Sciences. pp. 1-17. DOI: https://doi.org/10.1016/j.ins.2021.08.077

[29] Gong, Y., Li, X., Jiang, W., 2020. A new method for ranking discrete Z‐number. 2020 Chinese Control and Decision Conference (CCDC). pp. 3591‐3596. DOI: https://doi.org/10.1109/CCDC49329.2020.9164654

[30] Basirzadeh, H., Farnam, M., Hakimi, E., 2012. An approach for ranking discrete fuzzy sets. Journal of Mathematics and Computer Science. 2(3), 584-592. https://scik.org/index.php/jmcs/article/view/90

[31] Chen, M.S., Wang, S.W., 1999, Fuzzy clustering analysis for optimizing fuzzy membership functions. Fuzzy Sets and Systems. 103(2), 239-254. DOI: https://doi.org/10.1016/S0165-0114(98)00224-3


How to Cite

Bilgin, F., & Alci, M. (2022). A Review on Ranking of Z-numbers. Journal of Computer Science Research, 4(2), 1–12. https://doi.org/10.30564/jcsr.v4i2.4499


Article Type