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On entropy function and reliability indicator for D numbers

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Abstract

As an extension of Dempster-Shafer evidence theory, D numbers theory is developed with two more generalized properties that the elements in its frame of discernment do not require to be mutually exclusive strictly and the completeness constraint is released in its framework, which make it an effective tool for expressing and handling uncertain information. As a form of information representation and processing, there exists a certain amount of uncertainty in D numbers. How to reasonably measure the uncertainty of D numbers is still an open issue, and it is crucial to its further development and application. To address this issue, in this paper, a novel entropy function is defined to measure the uncertainty of D numbers, and based on which, the reliability indicator of D numbers is proposed motivated by TOPSIS method. Some numerical examples are given to illustrate the process of the proposed entropy function and reliability indicator, and in order to further demonstrate their effectiveness, a real-life application is conducted for bridge condition assessment problem using the developed D numbers approach.

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References

  1. Bian T, Zheng H, Yin L, Deng Y (2018) Failure mode and effects analysis based on D numbers and topsis. Qual Reliab Eng Int 34(4):501–515

    Article  Google Scholar 

  2. Cobb BR, Shenoy PP (2006) On the plausibility transformation method for translating belief function models to probability models. Int J Approx Reason 41(3):314–330

    Article  MathSciNet  MATH  Google Scholar 

  3. Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 38 (2):325–339

    Article  MathSciNet  MATH  Google Scholar 

  4. Deng X, Hu Y, Deng Y (2014) Bridge condition assessment using D numbers. Sci World J 2014:1–11

    Google Scholar 

  5. Deng X, Hu Y, Deng Y, Mahadevan S (2014) Environmental impact assessment based on D numbers. Expert Syst Appl 41(2):635–643

    Article  Google Scholar 

  6. Deng X, Hu Y, Deng Y, Mahadevan S (2014) Supplier selection using ahp methodology extended by D numbers. Expert Syst Appl 41(1):156–167

    Article  Google Scholar 

  7. Deng X, Jiang W (2017) Fuzzy risk evaluation in failure mode and effects analysis using a D numbers based multi-sensor information fusion method. Sensors 17(9):2086

    Article  Google Scholar 

  8. Deng X, Jiang W (2018) An evidential axiomatic design approach for decision making using the evaluation of belief structure satisfaction to uncertain target values. Int J Intell Syst 33(1):15– 32

    Article  Google Scholar 

  9. Deng X, Lu X, Chan FTS, Sadiq R, Mahadevan S, Deng Y (2015) D-CFPR: D numbers extended consistent fuzzy preference relations. Knowl-Based Syst 73:61–68

    Article  Google Scholar 

  10. Deng Y (2012) D numbers: theory and applications. Int J Inf Comput Sci 9(9):2421–2428

    Google Scholar 

  11. Deng Y (2016) Deng entropy. Chaos, Solitons Fractals 91:549–553

    Article  MATH  Google Scholar 

  12. Deng Y, Shi W, Zhu Z, Liu Q (2004) Combining belief functions based on distance of evidence. Decis Support Syst 38(3):489–493

    Article  Google Scholar 

  13. Dubois D, Prade H (1985) A note on measures of specificity for fuzzy sets. Int J Gen Syst 10(4):279–283

    Article  MathSciNet  MATH  Google Scholar 

  14. Dubois D, Prade H (1987) Properties of measures of information in evidence and possibility theories. Fuzzy Sets Syst 24(2):161–182

    Article  MathSciNet  MATH  Google Scholar 

  15. Dunker KF, Rabbat BG (1995) Assessing infrastructure deficiencies: the case of highway bridges. J Infrastruct Syst 1(2):100–119

    Article  Google Scholar 

  16. Fan G, Zhong D, Yan F, Yue P (2016) A hybrid fuzzy evaluation method for curtain grouting efficiency assessment based on an ahp method extended by D numbers. Expert Syst Appl 44:289–303

    Article  Google Scholar 

  17. Fei L, Deng Y (2018) A new divergence measure for basic probability assignment and its applications in extremely uncertain environments. International Journal of Intelligent Systems, https://doi.org/10.1002/int.22066

  18. Fei L, Deng Y, Hu Y (2018) DS-VIKOR: A New Multi-criteria Decision-Making Method for Supplier Selection. International Journal of Fuzzy Systems. https://doi.org/10.1007/s40815-018-0543-y https://doi.org/10.1007/s40815-018-0543-y

  19. Han D, Liu W, Dezert J, Yang Y (2016) A novel approach to pre-extracting support vectors based on the theory of belief functions. Knowl-Based Syst 110:210–223

    Article  Google Scholar 

  20. Hohle U (1982) Entropy with respect to plausibility measures. In: Proc of 12th IEEE Int Symp on Multiple Valued Logic, Paris, 1982s

  21. Hwang CL, Yoon K (1981) Multiple Attribute Decision Making. Springer, Berlin

    Book  MATH  Google Scholar 

  22. Jiang W, Hu W (2018) An improved soft likelihood function for Dempster-Shafer belief structures. Int J Intell Syst 33(6):1264–1282

    Article  Google Scholar 

  23. Jiang W, Wei B, Liu X, Li X, Zheng H (2018) Intuitionistic fuzzy power aggregation operator based on entropy and its application in decision making. Int J Intell Syst 33(1):49–67

    Article  Google Scholar 

  24. Jirousek R, Shenoy PP (2018) A new definition of entropy of belief functions in the Dempster-Shafer theory. Int J Approx Reason 92:49–65

    Article  MathSciNet  MATH  Google Scholar 

  25. Jousselme A, Liu C, Grenier D, Bosse E (2006) Measuring ambiguity in the evidence theory. IEEE Trans Syst Man Cybern Part A Syst Hum 36(5):890–903

    Article  Google Scholar 

  26. Jousselme AL, Grenier D (2001) ÉLoi bossé: a new distance between two bodies of evidence. Information Fusion 2(2):91–101

    Article  Google Scholar 

  27. Klein J, Colot O (2010) Automatic discounting rate computation using a dissent criterion. In: Workshop on the theory of belief functions, Brest, 1–6

  28. Klir George J, Ramer A (1990) Uncertainty in the Dempster-Shafer theory: a critical re-examination. Int J Gen Syst 18(2):155–166

    Article  MATH  Google Scholar 

  29. Klir GJ, Parviz B (1992) A note on the measure of discord. In: Eighth International conference on uncertainty in artificial intelligence, pp 138–141

  30. Klir GJ, Wierman MJ (1999) Uncertainty-Based Information. Physica-Verlag HD

  31. Lamata MT, Moral S (1988) Measures of entropy in the theory of evidence. Int J Gen Syst 14(4):297–305

    Article  MathSciNet  MATH  Google Scholar 

  32. Li M, Hu Y, Zhang Q, Deng Y (2016) A novel distance function of D numbers and its application in product engineering. Eng Appl Artif Intell 47:61–67

    Article  Google Scholar 

  33. Li X, Chen X (2018) D-intuitionistic hesitant fuzzy sets and their application in multiple attribute decision making. Cogn Comput 10(3):496–505

    Article  Google Scholar 

  34. Liang MT, Wu JH, Liang CH (2001) Multiple layer fuzzy evaluation for existing reinforced concrete bridges. J Infrastruct Syst 7(4):144–159

    Article  MathSciNet  Google Scholar 

  35. Lin S, Li C, Xu F, Liu D, Liu J (2018) Risk identification and analysis for new energy power system in China based on D numbers and decision-making trial and evaluation laboratory (dematel). J Clean Prod 180:81–96

    Article  Google Scholar 

  36. Liu HC, You JX, Fan XJ, Lin QL (2014) Failure mode and effects analysis using D numbers and grey relational projection method. Expert Syst Appl 41(10):4670–4679

    Article  Google Scholar 

  37. Liu ZG, Pan Q, Dezert J (2013) Evidential classifier for imprecise data based on belief functions. Knowl-Based Syst 52(6):246–257

    Article  Google Scholar 

  38. Martin A, Jousselme A, Osswald C (2008) Conflict measure for the discounting operation on belief functions. In: 2008 11th International conference on information fusion, pp 1–8

  39. Mo H, Deng Y (2016) A new aggregating operator for linguistic information based on D numbers. Int J Uncertainty Fuzziness Knowledge Based Syst 24(6):831–846

    Article  MathSciNet  MATH  Google Scholar 

  40. Nguyen HT (1985) On entropy of random sets and possibility distributions. In: Bezdek JC (ed) The analysis of fuzzy information, CRC Press, pp 145–156

  41. Pal NR, Bezdek JC, Hemasinha R (1992) Uncertainty measures for evidential reasoning i: a review. Int J Approx Reason 7(3):165–183

    Article  MathSciNet  MATH  Google Scholar 

  42. Pal NR, Bezdek JC, Hemasinha R (1993) Uncertainty measures for evidential reasoning ii: a new measure of total uncertainty. Int J Approx Reason 8(1):1–16

    Article  MathSciNet  MATH  Google Scholar 

  43. Ramer A (1987) Uniqueness of information measure in the theory of evidence. Fuzzy Sets Syst 24(2):183–196

    Article  MathSciNet  MATH  Google Scholar 

  44. Samet A, Lefèvre E, Yahia SB (2014) Integration of extra-information for belief function theory conflict management problem through generic association rules. Int J Uncertainty Fuzziness Knowledge Based Syst 22 (4):531–551

    Article  MathSciNet  MATH  Google Scholar 

  45. Shafer G (1976) A Mathematical Theory of Evidence, vol 1. Princeton University Press, Princeton

    MATH  Google Scholar 

  46. Smets P (1983) Information content of an evidence. Int J Man Mach Stud 19(1):33–43

    Article  Google Scholar 

  47. Sun L, Liu Y, Zhang B, Shang Y, Yuan H, Ma Z (2016) An integrated decision-making model for transformer condition assessment using game theory and modified evidence combination extended by D numbers. Energies 9(9):1–22

    Google Scholar 

  48. Tang Y, Zhou D, Xu S, He Z (2017) A weighted belief entropy-based uncertainty measure for multi-sensor data fusion. Sensors 17(4):1–15

    Article  Google Scholar 

  49. Wang N, Liu F, Wei D (2016) A modified combination rule for D numbers theory. Math Probl Eng 2016(2):1–10

    MathSciNet  MATH  Google Scholar 

  50. Wang N, Liu X, Wei D (2018) A modified D numbers’ integration for multiple attributes decision making. Int J Fuzzy Syst 20(1):104–115

    Article  MathSciNet  Google Scholar 

  51. Wang YM, Elhag TM (2008) Evidential reasoning approach for bridge condition assessment. Expert Systems with Applications 34(1):689–699

    Article  Google Scholar 

  52. Wang YM, Elhag TMS (2007) A fuzzy group decision making approach for bridge risk assessment. Comput Ind Eng 53(1):137–148

    Article  Google Scholar 

  53. Xia J, Feng Y, Liu L, Liu D (2018) Information fusion model of innovation alliances based on bayesian networks. Tsinghua Sci Technol 23(3):347–356

    Article  MathSciNet  Google Scholar 

  54. Xia J, Feng Y, Liu L, Liu D, Fei L (2018) An evidential reliability indicator-based fusion rule for dempster-shafer theory and its applications in classification. IEEE Access 6:24,912– 24,924

    Article  Google Scholar 

  55. Xiao F (2016) An intelligent complex event processing with D numbers under fuzzy environment. Math Probl Eng 2016(1):1– 10

    Google Scholar 

  56. Xiao F (2017) A novel evidence theory and fuzzy preference approach-based multi-sensor data fusion technique for fault diagnosis. Sensors 17(11):2504

    Article  Google Scholar 

  57. Xiao F (2018) An improved method for combining conflicting evidences based on the similarity measure and belief function entropy. Int J Fuzzy Syst 20(4):1256–1266

    Article  MathSciNet  Google Scholar 

  58. Yager RR (1983) Entropy and specificity in a mathematical theory of evidence. Int J Gen Syst 9(4):249–260

    Article  MathSciNet  MATH  Google Scholar 

  59. Zhou X, Deng X, Deng Y, Mahadevan S (2017) Dependence assessment in human reliability analysis based on D numbers and AHP. Nucl Eng Des 313:243–252

    Article  Google Scholar 

Download references

Acknowledgments

The authors greatly appreciate the reviews’ suggestions and the editor’s encouragement. This research was funded by the grants from the National Natural Science Foundation of China (#71472053, #71429001, #91646105).

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Authors and Affiliations

  1. School of Management, Harbin Institute of Technology, Harbin, 150001, China

    Jun Xia, Yuqiang Feng, Luning Liu & Liguo Fei

  2. Shenzhen Institute of Electronics, Shenzhen, 518055, China

    Jun Xia & Dongjun Liu

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  1. Jun Xia
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  2. Yuqiang Feng
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  3. Luning Liu
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  4. Dongjun Liu
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  5. Liguo Fei
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Correspondence to Yuqiang Feng or Luning Liu.

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Xia, J., Feng, Y., Liu, L. et al. On entropy function and reliability indicator for D numbers. Appl Intell 49, 3248–3266 (2019). https://doi.org/10.1007/s10489-019-01442-3

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