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Complexity
Volume 2017, Article ID 4359195, 15 pages
https://doi.org/10.1155/2017/4359195
Research Article

An Improved Belief Entropy and Its Application in Decision-Making

School of Electronics and Information, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

Correspondence should be addressed to Yongchuan Tang; nc.ude.upwn.liam@nauhcgnoygnat and Wen Jiang; nc.ude.upwn@newgnaij

Received 22 December 2016; Accepted 23 January 2017; Published 16 March 2017

Academic Editor: Jurgita Antucheviciene

Copyright © 2017 Deyun Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. E. Šiožinyte, J. Antuchevičiene, and V. Kutut, “Upgrading the old vernacular building to contemporary norms: multiple criteria approach,” Journal of Civil Engineering & Management, vol. 20, no. 2, pp. 291–298, 2014. View at Publisher · View at Google Scholar · View at Scopus
  2. J. Antucheviciene, Z. Kala, M. Marzouk, and E. R. Vaidogas, “Solving civil engineering problems by means of fuzzy and stochastic mcdm methods: current state and future research,” Mathematical Problems in Engineering, vol. 2015, Article ID 362579, 16 pages, 2015. View at Publisher · View at Google Scholar · View at Scopus
  3. E. K. Zavadskas, J. Antucheviciene, S. H. Razavi Hajiagha, and S. S. Hashemi, “The interval-valued intuitionistic fuzzy MULTIMOORA method for group decision making in engineering,” Mathematical Problems in Engineering, vol. 2015, Article ID 560690, 13 pages, 2015. View at Publisher · View at Google Scholar · View at Scopus
  4. M. A. Sodenkamp, M. Tavana, and D. Di Caprio, “Modeling synergies in multi-criteria supplier selection and order allocation: an application to commodity trading,” European Journal of Operational Research, vol. 254, no. 3, pp. 859–874, 2016. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. M. K. Ghorabaee, E. K. Zavadskas, M. Amiri, and J. Antucheviciene, “A new method of assessment based on fuzzy ranking and aggregated weights (AFRAW) for MCDM problems under type-2 fuzzy environment,” Economic Computation & Economic Cybernetics Studies & Research, vol. 50, pp. 39–68, 2016. View at Google Scholar
  6. T. Aven, “Supplementing quantitative risk assessments with a stage addressing the risk understanding of the decision maker,” Reliability Engineering and System Safety, vol. 152, pp. 51–57, 2016. View at Publisher · View at Google Scholar · View at Scopus
  7. T. Aven and E. Zio, “Some considerations on the treatment of uncertainties in risk assessment for practical decision making,” Reliability Engineering and System Safety, vol. 96, no. 1, pp. 64–74, 2011. View at Publisher · View at Google Scholar · View at Scopus
  8. H. Ahmadi, M. Nilashi, and O. Ibrahim, “Organizational decision to adopt hospital information system: an empirical investigation in the case of Malaysian public hospitals,” International Journal of Medical Informatics, vol. 84, no. 3, pp. 166–188, 2015. View at Publisher · View at Google Scholar · View at Scopus
  9. M. Nilashi, H. Ahmadi, A. Ahani, R. Ravangard, and O. B. Ibrahim, “Determining the importance of hospital information system adoption factors using fuzzy Analytic Network Process (ANP),” Technological Forecasting & Social Change, vol. 111, pp. 244–264, 2016. View at Publisher · View at Google Scholar · View at Scopus
  10. M. J. North, C. M. Macal, J. St. Aubin et al., “Multiscale agent-based consumer market modeling,” Complexity, vol. 15, no. 5, pp. 37–47, 2010. View at Publisher · View at Google Scholar · View at Scopus
  11. E. K. Zavadskas, R. Baušys, D. Stanujkic, and M. Magdalinovic-Kalinovic, “Selection of lead-zinc flotation circuit design by applying WASPAS method with single-valued neutrosophic set,” Acta Montanistica Slovaca, vol. 21, no. 2, pp. 85–92, 2016. View at Google Scholar · View at Scopus
  12. E. K. Zavadskas, R. Baušys, and M. Lazauskas, “Sustainable assessment of alternative sites for the construction of a waste incineration plant by applying WASPAS method with single-valued neutrosophic set,” Sustainability, vol. 7, no. 12, pp. 15923–15936, 2015. View at Publisher · View at Google Scholar · View at Scopus
  13. W. Feller, An Introduction to Probability Theory and Its Applications, John Wiley & Sons, New York, NY, USA, 2nd edition, 1957.
  14. Z.-S. Chen, K.-S. Chin, Y.-L. Li, and Y. Yang, “Proportional hesitant fuzzy linguistic term set for multiple criteria group decision making,” Information Sciences, vol. 357, pp. 61–87, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. K. Khalili-Damghani, M. Tavana, and M. Amirkhan, “A fuzzy bi-objective mixed-integer programming method for solving supply chain network design problems under ambiguous and vague conditions,” International Journal of Advanced Manufacturing Technology, vol. 73, no. 9-12, pp. 1567–1595, 2014. View at Publisher · View at Google Scholar · View at Scopus
  17. F. Sabahi and M.-R. Akbarzadeh-T, “Introducing validity in fuzzy probability for judicial decision–making,” International Journal of Approximate Reasoning, vol. 55, no. 6, pp. 1383–1403, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. X. Wang, J. Zhu, Y. Song, and L. Lei, “Combination of unreliable evidence sources in intuitionistic fuzzy MCDM framework,” Knowledge-Based Systems, vol. 97, pp. 24–39, 2016. View at Publisher · View at Google Scholar · View at Scopus
  19. A. P. Dempster, “Upper and lower probabilities induced by a multivalued mapping,” Annals of Mathematical Statistics, vol. 38, pp. 325–339, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. G. Shafer, A Mathematical Theory of Evidence, Princeton University Press, Princeton, NJ, USA, 1976. View at MathSciNet
  21. R. R. Yager, “Decision making using minimization of regret,” International Journal of Approximate Reasoning, vol. 36, no. 2, pp. 109–128, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. C. Fu, J.-B. Yang, and S.-L. Yang, “A group evidential reasoning approach based on expert reliability,” European Journal of Operational Research, vol. 246, no. 3, pp. 886–893, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. Z. Pawlak, “Rough sets,” International Journal of Computer & Information Sciences, vol. 11, no. 5, pp. 341–356, 1982. View at Publisher · View at Google Scholar · View at Scopus
  24. B. Sun, W. Ma, and H. Zhao, “An approach to emergency decision making based on decision-theoretic rough set over two universes,” Soft Computing, vol. 20, no. 9, pp. 3617–3628, 2016. View at Publisher · View at Google Scholar · View at Scopus
  25. C. Luo, T. Li, Z. Yi, and H. Fujita, “Matrix approach to decision-theoretic rough sets for evolving data,” Knowledge-Based Systems, vol. 99, pp. 123–134, 2016. View at Publisher · View at Google Scholar · View at Scopus
  26. C. Fu and D. L. Xu, “Determining attribute weights to improve solution reliability and its application to selecting leading industries,” Annals of Operations Research, vol. 245, pp. 401–426, 2014. View at Google Scholar
  27. J. Wang, Y. Hu, F. Xiao, X. Deng, and Y. Deng, “A novel method to use fuzzy soft sets in decision making based on ambiguity measure and Dempster-Shafer theory of evidence: an application in medical diagnosis,” Artificial Intelligence in Medicine, vol. 69, pp. 1–11, 2016. View at Publisher · View at Google Scholar · View at Scopus
  28. X. Deng, X. Zheng, X. Su et al., “An evidential game theory framework in multi-criteria decision making process,” Applied Mathematics and Computation, vol. 244, pp. 783–793, 2014. View at Publisher · View at Google Scholar · View at Scopus
  29. B. Kang, Y. Hu, Y. Deng, and D. Zhou, “A new methodology of multicriteria decision-making in supplier selection based on Z-numbers,” Mathematical Problems in Engineering, vol. 2016, Article ID 8475987, 17 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. Z.-G. Liu, Q. Pan, and J. Dezert, “A new belief-based K-nearest neighbor classification method,” Pattern Recognition, vol. 46, no. 3, pp. 834–844, 2013. View at Publisher · View at Google Scholar · View at Scopus
  31. J. Ma, W. Liu, P. Miller, and H. Zhou, “An evidential fusion approach for gender profiling,” Information Sciences, vol. 333, pp. 10–20, 2016. View at Publisher · View at Google Scholar · View at Scopus
  32. Z.-G. Liu, Q. Pan, J. Dezert, and G. Mercier, “Credal c-means clustering method based on belief functions,” Knowledge-Based Systems, vol. 74, pp. 119–132, 2015. View at Publisher · View at Google Scholar · View at Scopus
  33. D. Han, W. Liu, J. Dezert, and Y. Yang, “A novel approach to pre-extracting support vectors based on the theory of belief functions,” Knowledge-Based Systems, vol. 110, pp. 210–223, 2016. View at Publisher · View at Google Scholar
  34. Z.-G. Liu, Q. Pan, J. Dezert, and A. Martin, “Adaptive imputation of missing values for incomplete pattern classification,” Pattern Recognition, vol. 52, pp. 85–95, 2016. View at Publisher · View at Google Scholar · View at Scopus
  35. X. Su, Y. Deng, S. Mahadevan, and Q. Bao, “An improved method for risk evaluation in failure modes and effects analysis of aircraft engine rotor blades,” Engineering Failure Analysis, vol. 26, pp. 164–174, 2012. View at Publisher · View at Google Scholar · View at Scopus
  36. W. Jiang, C. Xie, M. Zhuang, Y. Shou, and Y. Tang, “Sensor data fusion with z-numbers and its application in fault diagnosis,” Sensors, vol. 16, no. 9, p. 1509, 2016. View at Publisher · View at Google Scholar
  37. K. Yuan, F. Xiao, L. Fei, B. Kang, and Y. Deng, “Modeling sensor reliability in fault diagnosis based on evidence theory,” Sensors, vol. 16, no. 1, article 113, 2016. View at Publisher · View at Google Scholar · View at Scopus
  38. W. Jiang, B. Wei, X. Qin, J. Zhan, and Y. Tang, “Sensor data fusion based on a new conflict measure,” Mathematical Problems in Engineering, vol. 2016, Article ID 5769061, 11 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  39. K.-S. Chin, C. Fu, and Y. Wang, “A method of determining attribute weights in evidential reasoning approach based on incompatibility among attributes,” Computers and Industrial Engineering, vol. 87, pp. 150–162, 2015. View at Publisher · View at Google Scholar · View at Scopus
  40. W. S. Du and B. Q. Hu, “Attribute reduction in ordered decision tables via evidence theory,” Information Sciences, vol. 364-365, pp. 91–110, 2016. View at Publisher · View at Google Scholar · View at Scopus
  41. C. Fu and Y. Wang, “An interval difference based evidential reasoning approach with unknown attribute weights and utilities of assessment grades,” Computers & Industrial Engineering, vol. 81, pp. 109–117, 2015. View at Publisher · View at Google Scholar · View at Scopus
  42. Y.-M. Wang and T. M. S. Elhag, “A comparison of neural network, evidential reasoning and multiple regression analysis in modelling bridge risks,” Expert Systems with Applications, vol. 32, no. 2, pp. 336–348, 2007. View at Publisher · View at Google Scholar · View at Scopus
  43. W. Jiang, C. Xie, B. Wei, and D. Zhou, “A modified method for risk evaluation in failure modes and effects analysis of aircraft turbine rotor blades,” Advances in Mechanical Engineering, vol. 8, no. 4, pp. 1–16, 2016. View at Publisher · View at Google Scholar · View at Scopus
  44. R. R. Yager and D. P. Filev, “Including probabilistic uncertainty in fuzzy logic controller modeling using dempster-shafer theory,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 25, no. 8, pp. 1221–1230, 1995. View at Publisher · View at Google Scholar · View at Scopus
  45. Y. Tang, D. Zhou, and W. Jiang, “A new fuzzy-evidential controller for stabilization of the planar inverted pendulum system,” PLoS ONE, vol. 11, no. 8, Article ID e0160416, 2016. View at Publisher · View at Google Scholar · View at Scopus
  46. Y.-M. Wang, J.-B. Yang, D.-L. Xu, and K.-S. Chin, “Consumer preference prediction by using a hybrid evidential reasoning and belief rule-based methodology,” Expert Systems with Applications, vol. 36, no. 4, pp. 8421–8430, 2009. View at Publisher · View at Google Scholar · View at Scopus
  47. J. Ma, W. Liu, and S. Benferhat, “A belief revision framework for revising epistemic states with partial epistemic states,” International Journal of Approximate Reasoning, vol. 59, pp. 20–40, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  48. K. Zhou, A. Martin, Q. Pan, and Z.-G. Liu, “Median evidential c-means algorithm and its application to community detection,” Knowledge-Based Systems, vol. 74, pp. 69–88, 2015. View at Publisher · View at Google Scholar · View at Scopus
  49. L. A. Zadeh, “Simple view of the dempster-shafer theory of evidence and its implication for the rule of combination,” AI Magazine, vol. 7, no. 2, pp. 85–90, 1986. View at Google Scholar · View at Scopus
  50. W. Liu, “Analyzing the degree of conflict among belief functions,” Artificial Intelligence, vol. 170, no. 11, pp. 909–924, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  51. J. Schubert, “Conflict management in Dempster-Shafer theory using the degree of falsity,” International Journal of Approximate Reasoning, vol. 52, no. 3, pp. 449–460, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  52. Y. Deng, “Generalized evidence theory,” Applied Intelligence, vol. 43, no. 3, pp. 530–543, 2015. View at Publisher · View at Google Scholar · View at Scopus
  53. X. Su, S. Mahadevan, W. Han, and Y. Deng, “Combining dependent bodies of evidence,” Applied Intelligence, vol. 44, no. 3, pp. 634–644, 2016. View at Publisher · View at Google Scholar · View at Scopus
  54. X. Su, S. Mahadevan, P. Xu, and Y. Deng, “Dependence assessment in human reliability analysis using evidence theory and AHP,” Risk Analysis, vol. 35, no. 7, pp. 1296–1316, 2015. View at Publisher · View at Google Scholar · View at Scopus
  55. X. Su, S. Mahadevan, P. Xu, and Y. Deng, “Handling of dependence in Dempster-Shafer theory,” International Journal of Intelligent Systems, vol. 30, no. 4, pp. 441–467, 2015. View at Publisher · View at Google Scholar · View at Scopus
  56. X. Deng, Q. Liu, Y. Deng, and S. Mahadevan, “An improved method to construct basic probability assignment based on the confusion matrix for classification problem,” Information Sciences, vol. 340-341, pp. 250–261, 2016. View at Publisher · View at Google Scholar · View at Scopus
  57. Y. Yi and Y. Liu, “Iterative approximation of basic belief assignment based on distance of evidence,” PLoS ONE, vol. 11, no. 2, Article ID e0147799, 2016. View at Publisher · View at Google Scholar · View at Scopus
  58. W. Jiang, J. Zhan, D. Zhou, and X. Li, “A method to determine generalized basic probability assignment in the open world,” Mathematical Problems in Engineering, vol. 2016, Article ID 3878634, 11 pages, 2016. View at Publisher · View at Google Scholar · View at Scopus
  59. P. Smets and R. Kennes, “The transferable belief model,” Artificial Intelligence, vol. 66, no. 2, pp. 191–234, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  60. P. Smets, “Belief functions on real numbers,” International Journal of Approximate Reasoning, vol. 40, no. 3, pp. 181–223, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  61. D. Zhou, Y. Tang, and W. Jiang, “A modified model of failure mode and effects analysis based on generalized evidence theory,” Mathematical Problems in Engineering, vol. 2016, Article ID 4512383, 11 pages, 2016. View at Publisher · View at Google Scholar
  62. C. E. Shannon, “A mathematical theory of communication,” ACM SIGMOBILE Mobile Computing and Communications Review, vol. 5, no. 1, pp. 3–55, 2001. View at Publisher · View at Google Scholar
  63. M. Tavana, M. A. Sodenkamp, and M. Pirdashti, “A fuzzy opportunity and threat aggregation approach in multicriteria decision analysis,” Fuzzy Optimization and Decision Making, vol. 9, no. 4, pp. 455–492, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  64. T. Dejus and J. Antuchevičiene, “Assessment of health and safety solutions at a construction site,” Journal of Civil Engineering and Management, vol. 19, no. 5, pp. 728–737, 2013. View at Publisher · View at Google Scholar · View at Scopus
  65. M. Tavana, K. Khalili-Damghani, and R. Rahmatian, “A hybrid fuzzy MCDM method for measuring the performance of publicly held pharmaceutical companies,” Annals of Operations Research, vol. 226, pp. 589–621, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  66. U. Hohle, “Entropy with respect to plausibility measures,” in Proceedings of the 12th IEEE International Symposium on Multiple-Valued Logic (MVL '82), pp. 167–169, Paris, France, 1982.
  67. R. R. Yager, “Entropy and specificity in a mathematical theory of evidence,” International Journal of General Systems, vol. 9, no. 4, pp. 249–260, 1983. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  68. D. Dubois and H. Prade, “A note on measures of specificity for fuzzy sets,” International Journal of General Systems, vol. 10, no. 4, pp. 279–283, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  69. G. J. Klir and A. Ramer, “Uncertainty in the Dempster-Shafer theory: a critical re-examination,” International Journal of General Systems, vol. 18, no. 2, pp. 155–166, 1990. View at Publisher · View at Google Scholar · View at Scopus
  70. G. J. Klir and B. Parviz, “A note on the measure of discord,” in Proceedings of the 8th International Conference on Uncertainty in Artificial Intelligence, pp. 138–141, 1992.
  71. T. George and N. R. Pal, “Quantification of conflict in Dempster-Shafer framework: a new approach,” International Journal of General Systems, vol. 24, no. 4, pp. 407–423, 1996. View at Publisher · View at Google Scholar · View at Scopus
  72. Y. Yang and D. Han, “A new distance-based total uncertainty measure in the theory of belief functions,” Knowledge-Based Systems, vol. 94, pp. 114–123, 2016. View at Publisher · View at Google Scholar · View at Scopus
  73. Y. Song, X. Wang, L. Lei, and S. Yue, “Uncertainty measure for interval-valued belief structures,” Measurement, vol. 80, pp. 241–250, 2016. View at Publisher · View at Google Scholar · View at Scopus
  74. Y. Song, X. Wang, and H. Zhang, “A distance measure between intuitionistic fuzzy belief functions,” Knowledge-Based Systems, vol. 86, pp. 288–298, 2015. View at Publisher · View at Google Scholar · View at Scopus
  75. Y. Deng, “Deng entropy,” Chaos, Solitons and Fractals, vol. 91, pp. 549–553, 2016. View at Publisher · View at Google Scholar · View at Scopus
  76. R. R. Yager, “On the Dempster-Shafer framework and new combination rules,” Information Sciences, vol. 41, no. 2, pp. 93–137, 1987. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  77. C. K. Murphy, “Combining belief functions when evidence conflicts,” Decision Support Systems, vol. 29, no. 1, pp. 1–9, 2000. View at Publisher · View at Google Scholar · View at Scopus
  78. D. Yong, S. WenKang, Z. ZhenFu, and L. Qi, “Combining belief functions based on distance of evidence,” Decision Support Systems, vol. 38, no. 3, pp. 489–493, 2004. View at Publisher · View at Google Scholar · View at Scopus
  79. Z. Zhang, T. Liu, D. Chen, and W. Zhang, “Novel algorithm for identifying and fusing conflicting data in wireless sensor networks,” Sensors, vol. 14, no. 6, pp. 9562–9581, 2014. View at Publisher · View at Google Scholar · View at Scopus
  80. K. Yuan, F. Xiao, L. Fei, B. Kang, and Y. Deng, “Conflict management based on belief function entropy in sensor fusion,” SpringerPlus, vol. 5, no. 1, article no. 638, 2016. View at Publisher · View at Google Scholar · View at Scopus
  81. W. Jiang, B. Wei, C. Xie, and D. Zhou, “An evidential sensor fusion method in fault diagnosis,” Advances in Mechanical Engineering, vol. 8, no. 3, pp. 1–7, 2016. View at Publisher · View at Google Scholar · View at Scopus