# Information Volume of Fuzzy Membership Function

• Jixiang Deng
• Yong Deng

## Keywords:

fuzzy sets, membership function, information volume, higher-order information volume, entropy

## Abstract

Fuzzy membership function plays an important role in fuzzy set theory. However, how to measure the information volume of fuzzy membership function is still an open issue. The existing methods to determine the uncertainty of fuzzy membership function only measure the first-order information volume, but do not take higher-order information volume into consideration. To address this issue, a new information volume of fuzzy membership function is presented in this paper, which includes the first-order and the higher-order information volume. By continuously separating the hesitancy degree until convergence, the information volume of the fuzzy membership function can be calculated. In addition, when the hesitancy degree of a fuzzy membership function equals to zero, the information volume of this special fuzzy membership function is identical to Shannon entropy. Two typical fuzzy sets, namely classic fuzzy sets and intuitiontistic fuzzy sets, are studied. Several examples are illustrated to show the efficiency of the proposed information volume of fuzzy membership function.

## References

[1] Anjaria, K. (2020). Negation and Entropy: Effectual Knowledge Management Equipment for Learning Organizations, Expert Systems with Applications, 113497, 2020. https://doi.org/10.1016/j.eswa.2020.113497

[2] Atanassov, K. T. (1999). Intuitionistic fuzzy sets, In Intuitionistic fuzzy sets, Springer, 1-137, 1999. https://doi.org/10.1007/978-3-7908-1870-3_1

[3] Buono, F.; Longobardi, M. (2020). A dual measure of uncertainty: The Deng extropy, Entropy, 22(5), 582, 2020. https://doi.org/10.3390/e22050582

[4] Cao, Z.; Lin, C.T. (2018). Inherent fuzzy entropy for the improvement of EEG complexity evaluation, IEEE Transactions on Fuzzy Systems, 26(2), 1032-1035, 2018. https://doi.org/10.1109/TFUZZ.2017.2666789

[5] Cao, Z.; Lin, C.T.; Lai, K.L.; Ko, L.W.; King, J.T.; Liao, K.K.; Fuh, J.L.; Wang, S.J. (2019). Extraction of SSVEPs-based inherent fuzzy entropy using a wearable headband EEG in migraine patients, IEEE Transactions on Fuzzy Systems, 28(1), 14-27, 2019. https://doi.org/10.1109/TFUZZ.2019.2905823

[6] Dempster, A. P. (1967). Upper and Lower Probabilities Induced by a Multivalued Mapping, The Annals of Mathematical Statistics, 38(2), 325-339, 1967. https://doi.org/10.1214/aoms/1177698950

[7] Deng, Y. (2020). Information Volume of Mass Function, International Journal of Computers Communications & Control, 15(6), 3983, 2020. https://doi.org/10.15837/ijccc.2020.6.3983

[8] Deng, Y. (2020). Uncertainty measure in evidence theory, Science China Information Sciences, 63(11), 1-19, 2020. https://doi.org/10.1007/s11432-020-3006-9

[9] Dzitac, I.; Filip, F.G.; Manolescu, M.J. (2017). Fuzzy logic is not fuzzy: World-renowned computer scientist Lotfi A. Zadeh, International Journal of Computers Communications & Control, 12(6), 748-789, 2017. https://doi.org/10.15837/ijccc.2017.6.3111

[10] Fei, L.; Feng, Y.; Liu, L. (2019). On Pythagorean fuzzy decision making using soft likelihood functions, International Journal of Intelligent Systems, 34(12), 3317-3335, 2019. https://doi.org/10.1002/int.22199

[11] Fei, L.; Lu, J.; Feng, Y. (2020). An extended best-worst multi-criteria decision-making method by belief functions and its applications in hospital service evaluation, Computers & Industrial Engineering, 142, 106355, 2020. https://doi.org/10.1016/j.cie.2020.106355

[12] Feng, F.; Xu, Z.; Fujita, H.; Liang, M. (2020). Enhancing PROMETHEE method with intuitionistic fuzzy soft sets, International Journal of Intelligent Systems, 35(7), 1071-1104, 2020. https://doi.org/10.1002/int.22235

[13] Gao, X.; Deng, Y. (2020). The Pseudo-Pascal Triangle of Maximum Deng Entropy, International Journal of Computers Communications & Control, 15(1), 1006, 2020. https://doi.org/10.15837/ijccc.2020.1.3735

[14] Gou, X.; Liao, H.; Xu, Z.; Min, R.; Herrera, F. (2019). Group decision making with double hierarchy hesitant fuzzy linguistic preference relations: Consistency based measures, index and repairing algorithms and decision model, Information Sciences, 489, 93-112, 2019. https://doi.org/10.1016/j.ins.2019.03.037

[15] Gou, X.; Xu, Z.; Liao, H.; Herrera, F. (2018). Multiple criteria decision making based on distance and similarity measures under double hierarchy hesitant fuzzy linguistic environment, Computers & Industrial Engineering, 126, 516-530, 2018. https://doi.org/10.1016/j.cie.2018.10.020

[16] Greenfield, S. (2016). Uncertainty Measurement for the Interval Type-2 Fuzzy Set, Lecture Notes in Artificial Intelligence, 9692, 183-194, 2016. https://doi.org/10.1007/978-3-319-39378-0_17

[17] Jiang, W.; Zhang, Z.; Deng, X. (2019). A novel failure mode and effects analysis method based on fuzzy evidential reasoning rules, IEEE Access, 7, 113605-113615, 2019. https://doi.org/10.1109/ACCESS.2019.2934495

[18] Lai, J.W.; Cheong, K.H. (2020). Parrondo's paradox from classical to quantum: A review, Nonlinear Dynamics, 1-13, 2020. https://doi.org/10.1007/s11071-020-05496-8

[19] Lee, P. (1980). Probability theory, Bulletin of the London Mathematical Society, 12(4), 318-319, 1980. https://doi.org/10.1112/blms/12.4.318

[20] Li, M.; Huang, S.; De Bock, J.; De Cooman, G.; PiÅ¾urica, A. (2020). A Robust Dynamic Classifier Selection Approach for Hyperspectral Images with Imprecise Label Information, Sensors, 20(18), 5262, 2020. https://doi.org/10.3390/s20185262

[21] Li, Z.; Zhang, P.; Ge, X.; Xie, N.; Zhang, G.; Wen, C.F. (2019). Uncertainty measurement for a fuzzy relation information system, IEEE Transactions on Fuzzy Systems, 27(12), 2338-2352, 2019.

[22] Liao, H.; Ren, Z.; Fang, R. (2020). A Deng-Entropy-Based Evidential Reasoning Approach for Multi-expert Multi-criterion Decision-Making with Uncertainty, International Journal of Computational Intelligence Systems, 13(1), 1281-1294, 2020. https://doi.org/10.2991/ijcis.d.200814.001

[23] Liu, B.; Deng, Y. (2019). Risk Evaluation in Failure Mode and Effects Analysis Based on D Numbers Theory, International Journal of Computers Communications & Control, 14(5), 672- 691, 2019.

[24] Liu, P.; Zhang, X. (2020). A novel approach to multi-criteria group decision-making problems based on linguistic D numbers, Computational and Applied Mathematics, 39, 132, 2020. https://doi.org/10.1007/s40314-020-1132-x

[25] Liu, P.; Zhang, X.; Wang, Z. (2020). An extended VIKOR method for multiple attribute decision making with linguistic d numbers based on fuzzy entropy, International Journal of Information Technology & Decision Making, 19(01), 143-167, 2020. https://doi.org/10.1142/S0219622019500433

[26] Pan, L.; Deng, Y. (2020). Probability transform based on the ordered weighted averaging and entropy difference, International Journal of Computers Communications & Control, 15(4), 3743, 2020. https://doi.org/10.15837/ijccc.2020.4.3743

[27] Pan, Y.; Zhang, L.; Li, Z.; Ding, L. (2019). Improved fuzzy Bayesian network-based risk analysis with interval-valued fuzzy sets and DS evidence theory. IEEE Transactions on Fuzzy Systems, 2019.

[28] Shafer, G. (1976). A mathematical theory of evidence, Princeton university press, 1976.

[29] Shannon, C.E. (1948). A mathematical theory of communication, Bell System Technical Journal, 27(4), 379-423, 1948. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x

[30] Stanojevic, B.; Dzitac, S.; Dzitac, I. (2020). Fuzzy Numbers and Fractional Programming in Making Decisions, International Journal of Information Technology & Decision Making, 19(04), 1123-1147, 2020. https://doi.org/10.1142/S0219622020300037

[31] Thao, N. X.; Smarandache, F. (2019). A new fuzzy entropy on Pythagorean fuzzy sets, Journal of Intelligent & Fuzzy Systems, 37(1), 1065-1074, 2019. https://doi.org/10.3233/JIFS-182540

[32] Wang, C.; Tan, Z.X.; Ye, Y.; Wang, L.; Cheong, K.H.; Xie, N.g. (2017). A rumor spreading model based on information entropy, Scientific reports , 7(1), 1-14, 2017. https://doi.org/10.1038/s41598-017-09171-8

[33] Wang, H.; Fang, Y. P.; Zio, E. (2019). Risk assessment of an electrical power system considering the influence of traffic congestion on a hypothetical scenario of electrified transportation system in New York state, IEEE Transactions on Intelligent Transportation Systems, 2019. https://doi.org/10.1109/TITS.2019.2955359

[34] Wei, B.; Xiao, F.; Shi, Y. (2019). Fully distributed synchronization of dynamic networked systems with adaptive nonlinear couplings, IEEE Transactions on Cybernetics, 1-9, 2019.

[35] Wei, B.; Xiao, F.; Shi, Y. (2019). Synchronization in kuramoto oscillator networks with sampleddata updating law, IEEE Transactions on Cybernetics, 50(6), 2380-2388, 2019. https://doi.org/10.1109/TCYB.2019.2940987

[36] Xiao, F. (2019). A distance measure for intuitionistic fuzzy sets and its application to pattern classification problems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2019. https://doi.org/10.1109/TSMC.2019.2958635

[37] Xiao, F. (2019). Generalization of Dempster-Shafer theory: A complex mass function, Applied Intelligence, 50(10), 3266-3275, 2019. https://doi.org/10.1007/s10489-019-01617-y

[38] Xiao, F. (2020). CED: A distance for complex mass functions, IEEE Transactions on Neural Networks and Learning Systems, 2020. https://doi.org/10.1109/TNNLS.2020.2984918

[39] Xiao, F. (2020). CEQD: A complex mass function to predict interference effects, IEEE Transactions on Cybernetics, 2020. https://doi.org/10.1109/TCYB.2020.3040770

[40] Xiao, F. (2020). EFMCDM: Evidential fuzzy multicriteria decision making based on belief entropy, IEEE Transactions on Fuzzy Systems, 28(7), 1477-1491, 2020.

[41] Xiao, F. (2020). On the maximum entropy negation of a complex-valued distribution, IEEE Transactions on Fuzzy Systems, 2020. https://doi.org/10.1109/TFUZZ.2020.3016723

[42] Xue, Y.; Deng, Y. (2020). Entailment for intuitionistic fuzzy sets based on generalized belief structures, International Journal of Intelligent Systems, 35(6), 963-982, 2020. https://doi.org/10.1002/int.22232

[43] Xue, Y.; Deng, Y. (2020). On the conjunction of possibility measures under intuitionistic evidence sets, Journal of Ambient Intelligence and Humanized Computing, 1-10, 2020. https://doi.org/10.1007/s12652-020-02508-8

[44] Xue, Y.; Deng, Y. (2020). Refined Expected Value Decision Rules under Orthopair Fuzzy Environment, Mathematics, 8(3), 442, 2020. https://doi.org/10.3390/math8030442

[45] Xue, Y.; Deng, Y.; Garg, H. (2020). Uncertain database retrieval with measure-Based belief function attribute values under intuitionistic fuzzy set, Information Sciences, 546, 436-447, 2020. https://doi.org/10.1016/j.ins.2020.08.096

[46] Yager, R. R. (2013). Pythagorean fuzzy subsets, In 2013 joint IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS), IEEE, 57-61, 2013. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375

[47] Yager, R. R. (2018). Fuzzy rule bases with generalized belief structure inputs, Engineering Applications of Artificial Intelligence, 72, 93-98, 2018. https://doi.org/10.1016/j.engappai.2018.03.005

[48] Yager, R. R.; Alajlan, N. (2017). Approximate reasoning with generalized orthopair fuzzy sets, Information Fusion, 38, 65-73, 2017. https://doi.org/10.1016/j.inffus.2017.02.005

[49] Yager, R. R.; Alajlan, N. (2016). Maxitive belief structures and imprecise possibility distributions, IEEE Transactions on Fuzzy Systems, 25(4), 768-774, 2016. https://doi.org/10.1109/TFUZZ.2016.2574930

[50] Yao, D.; Wang, C. (2018). Hesitant intuitionistic fuzzy entropy/cross-entropy and their applications, Soft Computing, 22(9), 2809-2824, 2018. https://doi.org/10.1007/s00500-017-2753-x

[51] Zadeh, L.A. (1965). Fuzzy sets, Information and control, 8(3), 338-353, 1965. https://doi.org/10.1016/S0019-9958(65)90241-X

[52] Zadeh, L.A. (1975). The concept of a linguistic variable and its application to approximate reasoning-I, Information Sciences, 8(3), 199-249, 1975. https://doi.org/10.1016/0020-0255(75)90036-5

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

[54] Zadeh, L. A.; Abbasov, A. M.; Shahbazova, S. N. (2015). Fuzzy-based techniques in human-like processing of social network data, International Journal of Uncertainty, Fuzziness and Knowledge- Based Systems, 23, 1-14, 2015. https://doi.org/10.1142/S0218488515400012

[55] Zhang, J.; Liu, R.; Zhang, J.; Kang, B. (2020). Extension of Yager's negation of a probability distribution based on Tsallis entropy, International Journal of Intelligent Systems, 35(1), 72-84, 2020. https://doi.org/10.1002/int.22198

2021-01-17

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