Ordinal fuzzy entropy

Document Type : Research Paper


Institute of Fundamental and Frontier Science, University of Electronic Science and Technology of China, 610054 Chengdu, China


In real life, occurrences of a series of things are supposed to come in an order. Therefore, it is necessary to regard sequence as a crucial factor in managing different kinds of things in fuzzy environment. However, few related researches have been made to provide a reasonable solution to this demand. Therefore, how to measure degree of uncertainty of ordinal fuzzy sets is still an open issue. To address this issue, a novel ordinal fuzzy entropy is proposed in this paper taking orders of propositions into consideration in measuring level of uncertainty in fuzzy environment. Compared with previously proposed entropies, effects on degrees of fuzzy uncertainty brought by sequences of sequential propositions are embodied in values of measurement using proposed method in this article. Moreover, some numerical examples are offered to verify the correctness and validity of the proposed entropy.


[1] M. Akram, S. Shumaiza, Multi-criteria decision making based on q-rung orthopair fuzzy promethee approach, Iranian Journal of Fuzzy Systems, 18(5) (2021), 107-127.
[2] K. T. Atanassov, Intuitionistic fuzzy sets, Springer, 1999, 1-137.
[3] F. Buono, M. Longobardi, A dual measure of uncertainty: The Deng extropy, Entropy, 22(5) (2020), DOI:10.3390/e22050582.
[4] P. Burillo, H. Sola, Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets, Fuzzy Sets and Systems, 78 (1996), 305-316.
[5] Z. Cao, W. Ding, Y. Wang, F. K. Hussain, A. Al-Jumaily, C. Lin, Effects of repetitive ssveps on EEG complexity using multiscale inherent fuzzy entropy, Neurocomputing, 389 (2020), 198-206.
[6] Z. Cao, C. T. Lin, Inherent fuzzy entropy for the improvement of EEG complexity evaluation, IEEE Transactions on Fuzzy Systems, 26(2) (2018), 1032-1035.
[7] Z. Cao, C. Lin, K. Lai, L. Ko, J. King, K. Liao, J. Fuh, S. Wang, Extraction of ssveps-based inherent fuzzy entropy using a wearable headband EEG in migraine patients, IEEE Transactions on Fuzzy Systems, 28(1) (2020), 14-27.
[8] D. Chakraborty, S. Pal, Journal pre-proofs rough video conceptualization for real-time event precognition with motion entropy, Information Sciences, 543 (2020), DOI:10.1016/j.ins.2020.09.021.
[9] C. Cheng, F. Xiao, A distance for belief functions of orderable set, Pattern Recognition Letters, 145 (2021), 165-170.
[10] H. Cui, L. Zhou, Y. Li, B. Kang, Belief entropy-of-entropy and its application in the cardiac interbeat interval time series analysis, Chaos, Solitons and Fractals, 155 (2022), 111736, DOI:10.1016/j.chaos.
[11] A. P. Dempster, Upper and lower probabilities induced by a multi-valued mapping, Annals of Mathematical Statistics, 38(2) (1967), 325-339.
[12] Y. Deng, Random permutation set, International Journal of Computers Communications and Control, 17(1) (2022), 4542, DOI:10.15837/ijccc.2022.1.4542.
[13] L. Fei, Y. Feng, A dynamic framework of multi-attribute decision making under Pythagorean fuzzy environment by using dempster-shafer theory, Engineering Applications of Artificial Intelligence, 101 (2021), 104213.
[14] L. Fei, Y. Feng, Modeling interactive multiattribute decision-making via probabilistic linguistic term set extended by dempstershafer theory, International Journal of Fuzzy Systems, 23(2) (2021), 599-613.
[15] L. Fei, Y. Feng, L. Liu, Evidence combination using OWA-based soft likelihood functions, International Journal of Intelligent Systems, 34(9) (2019), 2269-2290.
[16] L. Fei, J. Lu, Y. Feng, An extended best-worst multi-criteria decision-making method by belief functions and its applications in hospital service evaluation, Computers and Industrial Engineering, 142 (2020), 106355.
[17] X. Gao, L. Pan, Y. Deng, A generalized divergence of information volume and its applications, Engineering Applications of Artificial Intelligence, 108 (2021), 104584, DOI:10.1016/j.engappai.2021.
[18] X. Gao, X. Su, H. Qian, X. Pan, Dependence assessment in human reliability analysis under uncertain and dynamic situations, Nuclear Engineering and Technology, 54 (2021), DOI:10.1016/j.net.2021.09.045.
[19] Q. Gao, T. Wen, Y. Deng, Information volume fractal dimension, Fractals, 29(8) (2021), 2150263, DOI:10.1142/ S0218348X21502637.
[20] G. Hesamian, M. G. Akbar, Fuzzy time series model using weighted least square estimation, Iranian Journal of Fuzzy Systems, 19(2) (2022), 63-81.
[21] S. M. Hosseini, M. Manthouri, Type 2 adaptive fuzzy control approach applied to variable speed dfig based wind turbines with mppt algorithm, Iranian Journal of Fuzzy Systems, 19(1) (2022), 31-45.
[22] J. Huang, J. Xin, D. Fang, S. J. Lee, Q. Jiang, S. Yao, New entropy and distance measures of intuitionistic fuzzy sets, EEE International Conference on Fuzzy Systems, (2020), 1-8.
[23] W. L. Hung, M. S. Yang, Fuzzy entropy on intuitionistic fuzzy sets, International Journal of Intelligent Systems, 21 (2006), 443-451.
[24] G. Isik, I. Kaya, Design and analysis of acceptance sampling plans based on intuitionistic fuzzy linguistic terms, Iranian Journal of Fuzzy Systems, 18(6) (2021), 101-118.
[25] G. Jumarie, From entropy of fuzzy sets to fuzzy set of entropies: A critical review and new results, Kybernetes, 21 (1992), 33-51.
[26] M. R. Kazemi, S. Tahmasebi, F. Buono, M. Longobardi, Fractional Deng entropy and extropy and some applications, Entropy, 23(5) (2021), DOI:10.3390/e23050623.
[27] S. Kullback, R. Leibler, On information and sufficiency, Annals of Mathematical Statistics, 22 (1951), 79-86.
[28] Q. Liu, H. Cui, Y. Tian, B. Kang, On the negation of discrete z-numbers, Information Sciences, 537 (2020), 18-29.
[29] A. Mesiarova-Zemankova, R. Mesiar, Y. Su, Ordinal sum constructions for aggregation functions on the real unit interval, Iranian Journal of Fuzzy Systems, 19(1) (2022), 83-96.
[30] N. Pal, S. Pal, Object background segmentation using new definition of entropy, Computers and Digital Techniques, IEE Proceedings E, 136 (1989), 284-295, DOI:10.1049/ip-e.1989.0039.
[31] T. Senapati, R. Yager, Fermatean fuzzy sets, Journal of Ambient Intelligence and Humanized Computing, 11 (2020), DOI:10.1007/s12652-019-01377-0.
[32] G. Shafer, A mathematical theory of evidence, 1, 1976, DOI:10.2307/j.ctv10vm1qb.
[33] C. Shannon, A mathematical theory of communication, The Bell System Technical Journal, 27 (1948), 379-423.
[34] M. Song, C. Sun, D. Cai, S. Hong, H. Li, Classifying vaguely labeled data based on evidential fusion, Information Sciences, 583 (2022), 159-173.
[35] C. Tsallis, Nonadditive entropy: The concept and its use, European Physical Journal A, 40 (2008), 257-266.
[36] H. Wang, Y. Fang, E. Zio, Resilience-oriented optimal post-disruption reconfiguration for coupled traffic-power systems, Reliability Engineering and System Safety, 222 (2022), 108408, DOI:10.1016/j.ress.
[37] B. Wei, F. Xiao, F. Fang, Y. Shi, Velocity-free event-triggered control for multiple euler-lagrange systems with communication time delays, IEEE Transactions on Automatic Control, 66(11) (2021), 5599-5605.
[38] T. Wen, K. H. Cheong, The fractal dimension of complex networks: A review, Information Fusion, 73 (2021), 87-102.
[39] Q. Wu, Y. Deng, N. N. Xiong, Exponential negation of a probability distribution, Soft Computing, 26(5) (2022), 2147-2156.
[40] F. Xiao, Ceqd: A complex mass function to predict interference effects, IEEE Transactions on Cybernetics, PP (2021), 1-13.
[41] F. Xiao, A distance measure for intuitionistic fuzzy sets and its application to pattern classification problems, IEEE Transactions on Systems, Man, and Cybernetics, 51(6) (2021), 3980-3992.
[42] F. Xiao, On the maximum entropy negation of a complex-valued distribution, IEEE Transactions on Fuzzy Systems, 29(11) (2021), 3259-3269.
[43] D. Xie, F. Xiao, W. Pedrycz, Information quality for intuitionistic fuzzy values with its application in decision making, Engineering Applications of Artificial Intelligence, 109 (2022), 104568, DOI:10.1016/j.
[44] L. Xiong, X. Su, H. Qian, Conflicting evidence combination from the perspective of networks, Information Sciences, 580 (2021), 408-418.
[45] T. T. Xu, H. Zhang, B. Q. Li, Pythagorean fuzzy entropy and its application in multiple-criteria decision-making, International Journal of Fuzzy Systems, 22 (2020), DOI:10.1007/s40815-020-00877-y.
[46] N. Xuan Thao, F. Smarandache, A new fuzzy entropy of Pythagorean fuzzy sets, Journal of Intelligent and Fuzzy Systems, 37 (2019), 1065-1074.
[47] R. Yager, Pythagorean membership grades in multicriteria decision making, IEEE Transactions on Fuzzy Systems, 22 (2014), 958-965.
[48] R. Yager, Generalized orthopair fuzzy sets, IEEE Transactions on Fuzzy Systems, PP (2016), DOI:10.
1109/TFUZZ. 2016.2604005.
[49] R. Yager, Interval valued entropies for dempster-shafer structures, Knowledge-Based Systems, 161 (2018), DOI:10.1016/j.knosys.2018.08.001.
[50] R. Yager, A. Abbasov, Pythagorean membership grades, complex numbers, and decision making, International Journal of Intelligent Systems, 28 (2013), DOI:10.1002/int.21584.
[51] M. S. Yang, Z. Hussain, Fuzzy entropy for Pythagorean fuzzy sets with application to multicriterion decision making, Complexity, 2018 (2018), 1-14.
[52] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338-353.
[53] Q. S. Zhang, S. Jiang, A note on information entropy measures for vague sets and its applications, Information Sciences, 178 (2008), 4184-4191.
[54] Q. Zhou, Y. Deng, Belief extropy: Measure uncertainty from negation, Communications in Statistics-Theory and Methods, (2021), 1-23, DOI:10.1080/03610926.2021.1980049.
[55] M. Zhou, S. S. Zhu, J. Wu, E. Herrera-Viedma, A generalized belief entropy with non-specificity and structural conflict, IEEE Transactions on Systems, Man, and Cybernetics: Systems, PP (2021), 1-14.