Probability Transform Based on the Ordered Weighted Averaging and Entropy Difference


  • Lipeng Pan Institute of Fundamental and Frontier Science, University of Electronic Science and Technology of China
  • Yong Deng Institute of Fundamental and Frontier Science, University of Electronic Science and Technology of China


Dempster-Shafer evidence theory, probability transform, mass function, ordered weighted averaging, entropy difference


Dempster-Shafer evidence theory can handle imprecise and unknown information, which has attracted many people. In most cases, the mass function can be translated into the probability, which is useful to expand the applications of the D-S evidence theory. However, how to reasonably transfer the mass function to the probability distribution is still an open issue. Hence, the paper proposed a new probability transform method based on the ordered weighted averaging and entropy difference. The new method calculates weights by ordered weighted averaging, and adds entropy difference as one of the measurement indicators. Then achieved the transformation of the minimum entropy difference by adjusting the parameter r of the weight function. Finally, some numerical examples are given to prove that new method is more reasonable and effective.


Abellán, J. (2011). Combining nonspecificity measures in Dempster-Shafer theory of evidence, International journal of general systems, 40(6), 611-622, 2011.

Abellán, J. (2017). Analyzing properties of Deng entropy in the theory of evidence, Chaos Solitons & Fractals, 95, 195-199, 2017.

Abellán, J.; Mantas, C.J; Bossé, É. (2019). Basic Properties for Total Uncertainty Measures in the Theory of Evidence, Information Quality in Information Fusion and Decision Making, 99-108, 2019.

Atanassov, K.T. (1999). Intuitionistic fuzzy sets, Intuitionistic fuzzy sets, 1-137, 1999.

Cai, Q.; Gao, X.; Deng, Y. (2020). Pignistic belief transform: A new method of conflict measurement, IEEE Access, 8(1), 15265-15272, 2020.

Cobb, B.R; Shenoy, P.P. (2006). On the plausibility transformation method for translating belief function models to probability models, International journal of approximate reasoning, 41(3), 314-330, 2006.

Cheong, K.H.; Koh, J.M. (2019). A hybrid genetic-Levenberg Marquardt algorithm for automated spectrometer design optimization, Ultramicroscopy, 202, 100-106, 2019.

Deng, Y. (2016). Deng entropy, Chaos Solitons & Fractals, 91, 549-55, 2016.

Deng, W.; Deng, Y. (2018). Entropic methodology for entanglement measures, Physica A: Statistical Mechanics and its Applications, 512, 693-697, 2018.

Deng, X.; Jiang, W. (2019). Evaluating green supply chain management practices under fuzzy environment: a novel method based on D number theory, International Journal of Fuzzy Systems, 21, 1389-1402, 2019.

Deng, Y.; Shi, W.; Zhu, Z.; Liu, Q. (2004). Combining belief functions based on distance of evidence, Decision support systems, 38(3), 489-493, 2004.

Dempster, A.P. (1968). Upper and lower probabilities generated by a random closed interval, The Annals of Mathematical Statistics, 39(3), 957-966, 1968.

Dezert, J.; Smarandache, F. (2008). A new probabilistic transformation of belief mass assignment, 2008 11th International Conference on Information Fusion, 1-8, 2008.

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.

Fei, L.; Deng, Y. (2020). Multi-criteria decision making in Pythagorean fuzzy environment, Applied Intelligence, 50(2), 537-561, 2020.

Fei, L.; Zhang, Q.; Deng, Y. (2018). Identifying influential nodes in complex networks based on the inverse-square law, Physica A: Statistical Mechanics and its Applications, 512, 1044-1059, 2018.

Gao, Q.; Xu, D. (2019). An empirical study on the application of the Evidential Reasoning rule to decision making in financial investment, Knowledge-Based Systems, 164, 226-234, 2019.

Gao, S.; Deng, Y. (2019). An evidential evaluation of nuclear safeguards, International Journal of Distributed Sensor Networks, 15(12), 2019.

Ho, A. F. W.; To, B. Z. Y. S.; Koh, J.M.; Cheong, K. H. (2019). Forecasting Hospital Emergency Department Patient Volume Using Internet Search Data, IEEE Access, 7, 93387-93395, 2019.

Jaunzemis, A.D.; Holzinger, M.J.; Chan, M. W.; Shenoy, P.P. (2019). Evidence gathering for hypothesis resolution using judicial evidential reasoning, Information Fusion, 49, 26-45, 2019.

Jiang, W.; Cao, Y.; Deng, X. (2019). A Novel Z-network Model Based on Bayesian Network and Z-number, IEEE Transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2019.2918999.

Jiang, W.; Zhang, A.; Deng, Y. (2011). Proposing Interval Probability Transform(IPT) Method for Decision Making and Its Application, Journal of Northwestern Polytechnical University, 29(1), 44-48, 2011.

Jiroušek, R.; Shenoy, P. P. (2018). A new definition of entropy of belief functions in the Dempster- Shafer theory, International Journal of Approximate Reasoning, 92, 49-65, 2018.

Kang, B.; Zhang, P.; Gao, Z. et al. (2019). Environmental assessment under uncertainty using Dempster-Shafer theory and Z-numbers, Journal of Ambient Intelligence and Humanized Computing, DOI: 10.1007/s12652-019-01228-y.

Li, H.; Yuan, R.; Fu, J. (2019). A reliability modeling for multi-component systems considering random shocks and multistate degradation, IEEE Access, 7(1), 168805-168814, 2019.

Li, M.; Xu, H.; Deng, Y. (2019). Evidential Decision Tree Based on Belief Entropy, Entropy, 21(9), 897, 2019.

Li, Y.; Garg, H.; Deng, Y. (2020). A New Uncertainty Measure of Discrete Z-numbers, International Journal of Fuzzy Systems, 22(3), 760-776, 2020.

Liao, H.; Wu, X.; Mi, X.; Herrera, F. (2019). An integrated method for cognitive complex multiple experts multiple criteria decision making based on ELECTRE III with weighted Borda rule, Omega, DOI: 10.1016/

Liu, F.; Gao, X.; Zhao, J.; Deng, Y. (2019). Generalized Belief Entropy and Its Application in Identifying Conflict Evidence, IEEE Access, 7(1), 126625-126633, 2019.

Liu, Q.; Tian, Y.; Kang, B. (2019). Derive knowledge of Z-number from the perspective of Dempster-Shafer evidence theory, Engineering Applications of Artificial Intelligence, 85, 754-764, 2019.

Liu, W.; Wang, T.; Zang, T. et al. (2020). A fault diagnosis method for power transmission networks based on spiking neural P systems with self-updating rules considering biological apoptosis mechanism, Complexity, DOI: 10.1155/2020/2462647.

Liu, Y.; Jiang, W. (2019). A new distance measure of interval-valued intuitionistic fuzzy sets and its application in decision making, Soft Computing, 23, 2019.

Liu, Y.-T.; Pal, N.R; Marathe, A.R; Lin, C.-T. (2018). Weighted Fuzzy Dempster-Shafer Framework for Multimodal Information Integration, IEEE Transactions on Fuzzy Systems, 26(1), 338- 352, 2018.

Liu, Z.; Pan, Q.; Dezert, J.; Martin, A. (2018). Combination of classifiers with optimal weight based on evidential reasoning, IEEE Transactions on Fuzzy Systems, 26(3), 1217-1230, 2018.

Liu, Z.; Liu, Y.; Dezert, J.; Cuzzolin, F. (2019). Evidence combination based on credal belief redistribution for pattern classification, IEEE Transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2019.2911915.

Luo, Z.; Deng, Y. (2019). A matrix method of basic belief assignment's negation in Dempster- Shafer theory, IEEE Transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2019.2930027.

Luo, Z.; Deng, Y. (2020). A vector and geometry interpretation of basic probability assignment in Dempster-Shafer theory, International Journal of Intelligent Systems, 35(6), 944-962, 2020.

Marra, M.; Emrouznejad, A.; Ho, W.; Edwards, J.S. (2015). The value of indirect ties in citation networks: SNA analysis with OWA operator weights, Information Sciences, 314, 135-151, 2015.

Mo, H.; Deng, Y. (2019). Identifying node importance based on evidence theory in complex networks, Physica A: Statistical Mechanics & Its Applications, DOI: 10.1016/j.physa.2019.121538.

Murphy, C.K. (2000). Combining belief functions when evidence conflicts, Decision support systems, 29(1), 1-9, 2000.

Pan, Y.; Zhang, L.; Li, Z.W.; Ding, L. (2019). Improved Fuzzy Bayesian Network-Based Risk Analysis With Interval-Valued Fuzzy Sets and D-S Evidence Theory, IEEE Transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2019.2929024.

Seiti, H.; Hafezalkotob, A.; Najafi, S.E.; Khalaj, M. (2018). A risk-based fuzzy evidential framework for FMEA analysis under uncertainty: An interval-valued DS approach, Journal of Intelligent & Fuzzy Systems, 1-12, 2018.

Shafer, G. (1976). A mathematical theory of evidence, Princeton university press, 42, 1976.

Smets, P. (2005). Decision making in the TBM: the necessity of the pignistic transformation, International Journal of Approximate Reasoning, 38(2), 133-147, 2005.

Song, Y.; Deng, Y. (2019). A new soft likelihood function based on power ordered weighted average operator, International Journal of Intelligent Systems, 34(11), 2988-2999, 2019.

Song, Y.; Deng, Y. (2019). Divergence measure of belief function and its application in data fusion, IEEE Access, 71(1), 107465-107472, 2019.

Song, Y.F.; Wang, X.D.; Lei, L.; Xue, A.J. (2014). Measurement of evidence conflict based on correlation coefficient, Journal on Communications, 35(5),95-100, 2014.

Tang, M.; Liao, H.; Li, Z.; Xu, Z.S. (2018). Nature disaster risk evaluation with a group decision making method based on incomplete hesitant fuzzy linguistic preference relations, International journal of environmental research and public health, 15(4), 751, 2018.

Wang, C.; Tan, Z.; Ye, Y,. et al. (2017). A rumor spreading model based on information entropy, Scientific reports, 7(1), 1-14, 2017.

Wang, H.; Fang, Y.; 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 Stat, IEEE Transactions on Intelligent Transportation Systems, DOI: 10.1109/TITS.2019.2955359.

Wang, T.; Liu, W.; Zhao, J. et al. (2020). A rough set-based bio-inspired fault diagnosis method for electrical substations, International Journal of Electrical Power & Energy Systems, 119, 105961, 2020.

Wang, T.; Wei, X.; Huang, T. et al. (2019). Cascading Failures Analysis Considering Extreme Virus Propagation of Cyber-Physical Systems in Smart Grids, Complexity, DOI: 10.1155/2019/7428458.

Wang, T.; Wei, X.; Huang, T. et al. (2019). Modeling fault propagation paths in power systems: A new framework based on event SNP systems with neurotransmitter concentration, IEEE Access, 7, 12798-12808, 2019.

Wang, T.; Wang, J.; Ming, J. et al. (2018). Application of neural-like P systems with state values for power coordination of photovoltaic/battery microgrids, IEEE Access, 6, 46630-4664, 2018.

Wang, T.; Wei, X.; Wang, J. et al. (2020). A weighted corrective fuzzy reasoning spiking neural P system for fault diagnosis in power systems with variable topologies, Engineering Applications of Artificial Intelligence, 92, 103680, 2020.

Wen, T.; Deng, Y. (2020). The vulnerability of communities in complex networks: An entropy approach, Reliability Engineering & System Safety, 196, 106782, 2020.

Wu, X.; Liao, H.; Xu, Z.S. et al. (2018). Probabilistic Linguistic MULTIMOORA: A Multicriteria Decision Making Method Based on the Probabilistic Linguistic Expectation Function and the Improved Borda Rule, IEEE Transactions on Fuzzy Systems, 26(6), 3688-3702, 2018.

Xiao, F. (2019). Generalization of Dempster-Shafer theory: A complex mass function, Applied Intelligence, DOI: 10.1007/s10489-019-01617-y.

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, DOI: 10.1109/TSMC.2019.2958635.

Xiao, F. (2019). EFMCDM: Evidential fuzzy multicriteria decision making based on belief entropy, IEEE Transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2019.2936368.

Xiao, F. (2020). A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion, Information Sciences, 514, 462-483, 2020.

Xiao, F. (2020). Generalized belief function in complex evidence theory, Journal of Intelligent & Fuzzy Systems, DOI: 10.3233/JIFS-179589.

Xiao, F. (2020). CED: A distance for complex mass functions, IEEE Transactions on Neural Networks and Learning Systems, DOI: 10.1109/TNNLS.2020.2984918.

Xiao, F.; Zhang, Z.; Abawajy, J. (2019). Workflow scheduling in distributed systems under fuzzy environment, Journal of Intelligent & Fuzzy Systems, 37(4), 5323-5333, 2019.

Xu, P.; Zhang, R.; Deng, Y. (2018). A Novel Visibility Graph Transformation of Time Series into Weighted Networks, Chaos, Solitons & Fractals, 117, 201-208, 2018.

Xue, Y.; Deng, Y. (2020). Refined Expected Value Decision Rules under Orthopair Fuzzy Environment, Mathematics, 8(3), 442, 2020.

Yager, R.R. (1988). On ordered weighted averaging aggregation operators in multicriteria decisionmaking, IEEE Transactions on systems, Man, and Cybernetics, 18(1), 183-190, 1988.

Yager, R.R. (2017). Generalized regret based decision making, Engineering Applications of Artificial Intelligence, 65, 400-405, 2017.

Yager, R.R.(2018). Interval valued entropies for dempster-shafer structures, Knowledge-Based Systems, 161, 390-397, 2018.

Yager, R.R. (2019). OWA aggregation with an uncertainty over the arguments, Information Fusion, 52, 206-212, 2019.

Yager, R.R. (2019). Generalized Dempster-Shafer Structures, IEEE Transactions on Fuzzy Systems, 27(3), 428-435, 2019.

Yager, R.R. (2019). Extending Set Measures to Pythagorean Fuzzy Sets, International Journal of Fuzzy Systems, 21(2), 343-354, 2019.

Yan, H.; Deng, Y. (2020). An Improved Belief Entropy in Evidence Theory, IEEE Access, 8(1), 57505-57516, 2020.

Yang, G.; Yang, J.; Xu, D.; Khoveyni, M. (2017). A three-stage hybrid approach for weight assignment in MADM, Omega, 71, 93-105, 2017.

Yuan, R.; Tang, M.; Wang, H.; Li, H. (2019). A Reliability Analysis Method of Accelerated Performance Degradation Based on Bayesian Strategy, IEEE Access, 7(1), 169047-169054, 2019.

Zadeh, L.A. (1965). Fuzzy sets, Information and control, 8(3), 338-353, 1965.

Zadeh, L.A. (2011). A note on Z-numbers, Information Sciences, 181(14), 2923-2932, 2011.

Zhang, H.; Deng, Y. (2020). Weighted belief function of sensor data fusion in engine fault diagnosis, Soft computing, 24(3), 2329-2339, 2020.

Zhou, M.; Liu, X.; Chen, al. (2019). Assignment of attribute weights with belief distributions for MADM under uncertainties, Knowledge-Based Systems, DOI: 10.1016/j.knosys.2019.105110.

Zhou, M.; Liu, X.; Yang, J. et al. (2019). Evidential reasoning approach with multiple kinds of attributes and entropy-based weight assignment, Knowledge-Based Systems, 163, 358-375, 2019.

Zhou, Q.; Mo, H.; Deng, Y. (2020). A new divergence measure of pythagorean fuzzy sets based on belief function and its application in medical diagnosis, Mathematics, 8(1), 142, 2020.



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