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CODAS Method for Multi-Attribute Decision-Making Based on Some Novel Distance and Entropy Measures Under Probabilistic Dual Hesitant Fuzzy Sets

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Abstract

Probabilistic dual hesitant fuzzy set (PDHFS) as an extension of the generalization of hesitant fuzzy set (HFS) and dual hesitant fuzzy set (DHFS). It not only reflects the hesitant attitude of decision-makers (DMs), but also reflects probabilistic information. It is a more powerful and important tool to express uncertain information. As we all know, the distance and entropy measures are very useful tool in the MADM problems. In many fuzzy environments, their distance and entropy measures are proposed, and the MADM methods depend on the distance and entropy measures are given. First, to overcome the disadvantages of destroying the original information caused by artificially adding elements, we defined the membership degree (MD) mean, non-membership degree (NMD) mean and standard deviation for PDHFE, based on above definitions, the mean and standard deviation distance were proposed. Secondly, without the aid of other auxiliary functions, we built some novel PDHF entropy measures. Third, depend on the distance and entropy measures built, we integrate the classical CODAS method with the PDHF setting, and build a novel MADM technique to solving the MADM problem. Finally, the built MADM technique is used to the evaluation of enterprise credit risk to testify the practicability and feasibility of the built MADM technique. Meanwhile, the MADM technique built in this study is compared with some existing methods, and the advantages of the MADM technique proposed in this study are put forward, which has a better effect in solving MADM problems.

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References

  1. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MATH  Google Scholar 

  2. William-West, T.O., Ciucci, D.: Decision-theoretic five-way approximation of fuzzy sets. Inf. Sci. 572, 200–222 (2021)

    Article  MathSciNet  Google Scholar 

  3. Mockor, J., Hynar, D.: On unification of methods in theories of fuzzy sets, hesitant fuzzy set, fuzzy soft sets and intuitionistic fuzzy sets. Mathematics 9, 447 (2021)

    Article  Google Scholar 

  4. Su, Y., Zhao, M., Wei, G., Wei, C., Chen, X.: Probabilistic uncertain linguistic EDAS method based on prospect theory for multiple attribute group decision-making and its application to green finance. Int. J. Fuzzy Syst. 24, 1318–1331 (2022)

    Article  Google Scholar 

  5. Zhao, M., Gao, H., Wei, G., Wei, C., Guo, Y.: Model for network security service provider selection with probabilistic uncertain linguistic TODIM method based on prospect theory. Technol. Econ. Dev. Econ. 28, 638–654 (2022)

    Article  Google Scholar 

  6. Pramanik, R., Baidya, D.K., Dhang, N.: Reliability assessment of three-dimensional bearing capacity of shallow foundation using fuzzy set theory, Frontiers of Structural and Civil. Engineering 15, 478–489 (2021)

    Google Scholar 

  7. Lima, A., Palmeira, E.S., Bedregal, B., Bustince, H.: Multidimensional Fuzzy Sets. IEEE Trans. Fuzzy Syst. 29, 2195–2208 (2021)

    Article  Google Scholar 

  8. Shang, Y.G., Yuan, X.H., Lee, E.S.: The n-dimensional fuzzy sets and Zadeh fuzzy sets based on the finite valued fuzzy sets. Comput. Math. Appl. 60, 442–463 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986)

    Article  MATH  Google Scholar 

  10. Atanassov, K., Gargov, G.: Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31, 343–349 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  11. Zulqarnain, R.M., Siddique, I., Ali, R., Pamucar, D., Marinkovic, D., Bozanic, D.: Robust aggregation operators for intuitionistic fuzzy hypersoft set with their application to solve MCDM problem. Entropy 23, 688 (2021)

    Article  MathSciNet  Google Scholar 

  12. Zhao, R.J., Yang, F.B., Ji, L.N., Bai, Y.Q.: Dynamic air target threat assessment based on interval-valued intuitionistic fuzzy sets, game theory, and evidential reasoning methodology. Math. Probl. Eng. 2021, 6652706 (2021)

    Article  Google Scholar 

  13. Mishra, A.R., Mardani, A., Rani, P., Zavadskas, E.K.: A novel EDAS approach on intuitionistic fuzzy set for assessment of health-care waste disposal technology using new parametric divergence measures. J. Cleaner Prod. (2020). https://doi.org/10.1016/j.jclepro.2020.122807

    Article  Google Scholar 

  14. Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25, 529–539 (2010)

    MATH  Google Scholar 

  15. Narayanamoorthy, S., Ramya, L., Kang, D., Baleanu, D., Kureethara, J.V., Annapoorani, V.: A new extension of hesitant fuzzy set: An application to an offshore wind turbine technology selection process. IET Renew. Power Gener. 15, 2340–2355 (2021)

    Article  Google Scholar 

  16. Zhu, B., Xu, Z., Xia, M.: Dual hesitant fuzzy sets. J. Appl. Math. 2012, 2607–2645 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhang, C., Li, D.Y., Liang, J.Y., Wang, B.L.: MAGDM-oriented dual hesitant fuzzy multigranulation probabilistic models based on MULTIMOORA. Int. J. Mach. Learn. Cybern. 12, 1219–1241 (2021)

    Article  Google Scholar 

  18. Du, S.B., Yang, F., Tian, X.D.: Ancient chinese character image retrieval based on dual hesitant fuzzy sets. Sci. Program. 2021, 6621037 (2021)

    Google Scholar 

  19. Hao, Z.N., Xu, Z.S., Zhao, H., Su, Z.: Probabilistic dual hesitant fuzzy set and its application in risk evaluation. Knowl.-Based Syst. 127, 16–28 (2017)

    Article  Google Scholar 

  20. Zhao, Q., Ju, Y.B., Pedrycz, W.: A method based on bivariate almost stochastic dominance for multiple criteria group decision making with probabilistic dual hesitant fuzzy information, Ieee. Access 8, 203769–203786 (2020)

    Article  Google Scholar 

  21. Garg, H., Kaur, G.: A robust correlation coefficient for probabilistic dual hesitant fuzzy sets and its applications. Neural Comput. Appl. 32, 8847–8866 (2020)

    Article  Google Scholar 

  22. Z.L. Ren, Z.S. Xu, H. Wang (2017) An extended TODIM method under probabilistic dual hesitant fuzzy information and its application on enterprise strategic assessment, 2017 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM) 1464–1468.

  23. Garg, H., Kaur, G.: Quantifying gesture information in brain hemorrhage patients using probabilistic dual hesitant fuzzy sets with unknown probability information. Comput. Ind. Eng. 140, 106211 (2020)

    Article  Google Scholar 

  24. Ning, B., Wei, G., Lin, R., Guo, Y.: A novel MADM technique based on extended power generalized Maclaurin symmetric mean operators under probabilistic dual hesitant fuzzy setting and its application to sustainable suppliers selection. Expert Syst. Appl. 204, 117419 (2022)

    Article  Google Scholar 

  25. Gong, C.J., Jiang, L.W., Hou, L.: Group decision-making with distance induced fuzzy operators. Int. J. Fuzzy Syst. 24, 440–456 (2022)

    Article  Google Scholar 

  26. Zhan, Q.S., Fu, C., Xue, M.: Distance-based large-scale group decision-making method with group influence. Int. J. Fuzzy Syst. 23, 535–554 (2021)

    Article  Google Scholar 

  27. Surono, S., Putri, R.D.A.: Optimization of fuzzy C-Means clustering algorithm with combination of minkowski and chebyshev distance using principal component analysis. Int. J. Fuzzy Syst. 23, 139–144 (2021)

    Article  Google Scholar 

  28. Ding, Q.Y., Wang, Y.M., Goh, M.: TODIM dynamic emergency decision-making method based on hybrid weighted distance under probabilistic hesitant fuzzy information. Int. J. Fuzzy Syst. 23, 474–491 (2021)

    Article  Google Scholar 

  29. Liu, X.: Entropy, distance measure and similarity measure of fuzzy sets and their relations. Fuzzy Sets Syst. 52, 305–318 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  30. Du, W.S.: Subtraction and division operations on intuitionistic fuzzy sets derived from the Hamming distance. Inf. Sci. 571, 206–224 (2021)

    Article  MathSciNet  Google Scholar 

  31. Xiao, F.Y.: A distance measure for intuitionistic fuzzy sets and its application to pattern classification problems. Ieee Transactions on Systems Man Cybernetics-Systems 51, 3980–3992 (2021)

    Article  Google Scholar 

  32. Che, R.Q., Suo, C.F., Li, Y.M.: An approach to construct entropies on interval-valued intuitionistic fuzzy sets by their distance functions. Soft. Comput. 25, 6879–6889 (2021)

    Article  Google Scholar 

  33. Liu, Y.N., Jiang, W.: A new distance measure of interval-valued intuitionistic fuzzy sets and its application in decision making. Soft. Comput. 24, 6987–7003 (2020)

    Article  MATH  Google Scholar 

  34. He, X.X., Li, Y.F., Qin, K.Y., Meng, D.: Distance measures on intuitionistic fuzzy sets based on intuitionistic fuzzy dissimilarity functions. Soft. Comput. 24, 523–541 (2020)

    Article  MATH  Google Scholar 

  35. Li, C.Q., Zhao, H., Xu, Z.S.: Hesitant fuzzy psychological distance measure. Int. J. Mach. Learn. Cybern. 11, 2089–2100 (2020)

    Article  Google Scholar 

  36. Farhadinia, B., Xu, Z.S.: A novel distance-based multiple attribute decision-making with hesitant fuzzy sets. Soft. Comput. 24, 5005–5017 (2020)

    Article  MATH  Google Scholar 

  37. Xu, Z.S., Xia, M.M.: Distance and similarity measures for hesitant fuzzy sets. Inf. Sci. 181, 2128–2138 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Liu, J.B., Malik, M.A., Ayub, N., Siddiqui, H.M.A.: Distance measures for multiple-attributes decision-making based on connection numbers of set pair analysis with dual hesitant fuzzy sets, Ieee. Access 8, 9172–9184 (2020)

    Article  Google Scholar 

  39. Garg, H., Kumar, K.: A novel correlation coefficient of intuitionistic fuzzy sets based on the connection number of set pair analysis and its application. Scientia Iranica 25, 2373–2388 (2018)

    Google Scholar 

  40. Singh, P.: Distance and similarity measures for multiple-attribute decision making with dual hesitant fuzzy sets. Comput. Appl. Math. 36, 111–126 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  41. Su, Z., Xu, Z.S., Liu, H.F., Liu, S.S.: Distance and similarity measures for dual hesitant fuzzy sets and their applications in pattern recognition. J. Intell. Fuzzy Sys. 29, 731–745 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. Garg, H., Kaur, G.: Algorithm for probabilistic dual hesitant fuzzy multi-criteria decision-making based on aggregation operators with new distance measures. Mathematics 6, 280 (2018)

    Article  MATH  Google Scholar 

  43. Liu, X.D., Wu, J., Zhang, S.T., Wang, Z.W., Garg, H.: Extended cumulative residual entropy for emergency group decision-making under probabilistic hesitant fuzzy environment. Int. J. Fuzzy Syst. 24, 159–179 (2022)

    Article  Google Scholar 

  44. Versaci, M., Morabito, F.C.: image edge detection: a new approach based on fuzzy entropy and fuzzy divergence. Int. J. Fuzzy Syst. 23, 918–936 (2021)

    Article  Google Scholar 

  45. Wu, C.M., Liu, N.: Robust suppressed competitive picture fuzzy clustering driven by entropy. Int. J. Fuzzy Syst. 22, 2466–2492 (2020)

    Article  Google Scholar 

  46. Rahimi, M., Kumar, P., Moomivand, B., Yari, G.: An intuitionistic fuzzy entropy approach for supplier selection. Complex & Intelligent Systems 7, 1869–1876 (2021)

    Article  Google Scholar 

  47. Xu, L., Tang, Q.: Cold chain vulnerability assessment through two-stage grey comprehensive measurement of intuitionistic fuzzy entropy. Kybernetes (2021). https://doi.org/10.1108/K-1102-2021-0161

    Article  Google Scholar 

  48. Meng, F.Y., Xu, Y.W., Wang, N.: Correlation coefficients of dual hesitant fuzzy sets and their application in engineering management. J. Ambient Intell. Humanized Comput. 11, 2943–2961 (2020)

    Article  Google Scholar 

  49. Thao, N.X., Smarandache, F.: A new fuzzy entropy on Pythagorean fuzzy sets. J. Intell. Fuzzy Sys. 37, 1065–1074 (2019)

    Article  Google Scholar 

  50. Anees, J., Zhang, H.C., Baig, S., Lougou, B.G., Bona, T.G.R.: Hesitant fuzzy entropy-based opportunistic clustering and data fusion algorithm for heterogeneous wireless sensor networks. Sensors 20, 913 (2020)

    Article  Google Scholar 

  51. Zhang, H.M.: Distance and entropy measures for dual hesitant fuzzy sets. Comput. Appl. Math. 39, 91 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  52. Hwang, C.L., Yoon, K.P.: Multiple attribute decision making. A state-of- the-art survey, in, Springer, Berlin, Methods and applications (1981)

    Book  MATH  Google Scholar 

  53. Zhang, H.Y., Wei, G.W., Wei, C.: TOPSIS method for spherical fuzzy MAGDM based on cumulative prospect theory and combined weights and its application to residential location. Journal of Intelligent & Fuzzy Systems 42, 1367–1380 (2022)

    Article  Google Scholar 

  54. Opricovic, S., Tzeng, G.H.: Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS. Eur. J. Oper. Res. 156, 445–455 (2004)

    Article  MATH  Google Scholar 

  55. Roy, B.: Classement et choix en présence de points de vue multiples (la methode ELECTRE). Rairo 2, 57–75 (1968)

    Google Scholar 

  56. Brans, J.P., Vincke, P.: A preference ranking organization method: The PROMETHEE method for MCDM. Manage. Sci. 31, 641–656 (1985)

    Google Scholar 

  57. SrinivasanAllan, V., Shocker, D.: Linear programming techniques for multidimensional analysis of preferences. Psychometrika 38, 337–369 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  58. Paelinck, J.: Qualiflex: A flexible multiple-criteria method. Econ. Lett. 1, 193–197 (1978)

    Article  Google Scholar 

  59. Gomes, L., Rangel, L.A.D.: An application of the TODIM method to the multicriteria rental evaluation of residential properties. Eur. J. Oper. Res. 193, 204–211 (2009)

    Article  MATH  Google Scholar 

  60. Zhang, D., Su, Y., Zhao, M., Chen, X.: CPT-TODIM method for interval neutrosophic MAGDM and its application to third-party logistics service providers selection. Technol. Econ. Dev. Econ. 28, 201–219 (2022)

    Article  Google Scholar 

  61. Brauers, W.K.M., Zavadskas, E.K.: Project management by MULTIMOORA as an instrument for transition economies. Ukio Technologinis Ir Ekonominis Vystymas 16, 5–24 (2010)

    Google Scholar 

  62. Zavadskas, E.K., Kaklauskas, A., Sarka, V.: The new method of multicriteria complex proportional assessment of projects. Technol. Econ. Dev. Econ. 1, 131–139 (1994)

    Google Scholar 

  63. Ghorabaee, M.K., Zavadskas, E.K., Olfat, L., Turskis, Z.: Multi-criteria inventory classification using a new method of evaluation based on distance from average solution (EDAS). Informatica 26, 435–451 (2015)

    Article  Google Scholar 

  64. Zhang, H., Wei, G., Chen, X.: SF-GRA method based on cumulative prospect theory for multiple attribute group decision making and its application to emergency supplies supplier selection. Eng. Appl. Artif. Intell. 110, 104679 (2022)

    Article  Google Scholar 

  65. Rezaei, J.: Best-worst multi-criteria decision-making method: Some properties and a linear model, Omega-International Journal of. Manage. Sci. 64, 126–130 (2016)

    Google Scholar 

  66. Zavadskas, E.: Kazimieras, Turskis, Zenonas, A new additive ratio assessment (ARAS) method in multicriteria decision-making. Technol. Economic Develop. Econ. 16, 159–172 (2010)

    Article  Google Scholar 

  67. Zavadskas, E.K., Turskis, Z., Antucheviciene, J., Zakarevicius, A.: Optimization of weighted aggregated sum product assessment, Elektronika Ir. Elektrotechnika 122, 3–6 (2012)

    Google Scholar 

  68. Keshavarz Ghorabaee, M., Zavadskas, E.K., Turskis, Z., Antucheviciene, J.: A new combinative distance-based assessment (CODAS) method for multi-criteria decision-making. Econ. Comput. Econ. Cybernetics Studies Res. 50, 25–44 (2016)

    Google Scholar 

  69. Simic, V., Karagoz, S., Deveci, M., Aydin, N.: Picture fuzzy extension of the CODAS method for multi-criteria vehicle shredding facility location. Expert Syst. Appl. 175, 114644 (2021)

    Article  Google Scholar 

  70. Perez-Dominguez, L., Duran, S.N.A., Lopez, R.R., Perez-Olguin, I.J.C., Luviano-Cruz, D., Gomez, J.A.H.: Assessment urban transport service and pythagorean fuzzy sets CODAS method: A case of study of Ciudad Juarez. Sustainability 13, 1281 (2021)

    Article  Google Scholar 

  71. Seker, S.: A novel interval-valued intuitionistic trapezoidal fuzzy combinative distance-based assessment (CODAS) method. Soft. Comput. 24, 2287–2300 (2020)

    Article  Google Scholar 

  72. Dahooie, J.H., Vanaki, A.S., Mohammadi, N.: Choosing the appropriate system for cloud computing implementation by using the interval-valued intuitionistic fuzzy codas multiattribute decision-making method (case study: Faculty of new sciences and technologies of tehran university). IEEE Trans. Eng. Manage. 67, 855–868 (2020)

    Article  Google Scholar 

  73. Xu, Z.S., Zhou, W.: Consensus building with a group of decision makers under the hesitant probabilistic fuzzy environment. Fuzzy Optim. Decis. Making 16, 481–503 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  74. Farhadinia, B.: Correlation for dual hesitant fuzzy sets and dual interval-valued hesitant fuzzy sets. Int. J. Intell. Syst. 29, 184–205 (2014)

    Article  Google Scholar 

  75. Diamond, P., Kloeden, P.E.: Metric spaces of fuzzy sets: Theory and applications. World Scientific Publishing, Singapore (1994)

    Book  MATH  Google Scholar 

  76. Tcvetkov, R., Szmidt, E., Kacprzyk, J., Atanassov, K.: A modified hausdorff distance between intuitionistic fuzzy sets. Comptes Rendus De L Academie Bulgare Des Sciences 65, 1035–1042 (2012)

    MathSciNet  MATH  Google Scholar 

  77. Janusz, K.: Multistage fuzzy control. Wiley, Chichester (1997)

    MATH  Google Scholar 

  78. Janusz, K.: Applied Inequalities, in. Shandong Science and Technology Press, Jinan (in Chinese) (2004)

    Google Scholar 

  79. Vlachos, I.K., Sergiadis, G.D.: Inner product based entropy in the intuitionistic fuzzy setting. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 14, 351–366 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  80. Fan, P., Liang, J.R., Li, T.Z.: The fuzzy entropy of vague sets and its measure method, dynamics of continuous discrete and impulsive systems-series a-mathematical. Analysis 13, 722–724 (2006)

    Google Scholar 

  81. Wei, C.P., Gao, Z.H., Guo, T.T.: An intuitionistic fuzzy entropy measure based on trigonometric function, Journal of. Control and Decision 11, 3–14 (2012)

    Google Scholar 

  82. Ning, B.Q., Xie, J., Shan, Z.P.: Sorting method for multi-attribute decision-making based on relative entropy and VIKOR. Math. Practice Theory 49, 35–45 (2019)

    MATH  Google Scholar 

  83. Dombi, J.: A general class of fuzzy operators, the DeMorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators. Fuzzy Sets Systems 8, 149–163 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  84. Bonferroni, C.: Sulle medie multiple di potenze. Bollettino Matematica Italiana 5, 267–270 (1950)

    MathSciNet  MATH  Google Scholar 

  85. Muirhead, R.F.: Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters. Proc. Edinb. Math. Soc. 21, 144–162 (1902)

    Article  Google Scholar 

  86. Garg, H.: A new generalized pythagorean fuzzy information aggregation using einstein operations and its application to decision making. Int. J. Intell. Syst. 31, 886–920 (2016)

    Article  Google Scholar 

  87. Ren, Y., Yuan, X., Lin, R.: A novel MADM algorithm for landfill site selection based on q-rung orthopair probabilistic hesitant fuzzy power Muirhead mean operator. PLoS ONE (2021). https://doi.org/10.1371/journal.pone.0258448

    Article  Google Scholar 

  88. Xue, J., Yip, T.L., Wu, B., Wu, C., van Gelder, P.H.: A novel fuzzy Bayesian network-based MADM model for offshore wind turbine selection in busy waterways: An application to a case in China. Renewable Energy 172, 897–917 (2021)

    Article  Google Scholar 

  89. Kaneesamkandi, Z., Almujahid, A., Salim, B.: Selection of an appropriate solar thermal technology for solar vapor absorption cooling-an MADM approach. Energies 15, 1882 (2022)

    Article  Google Scholar 

  90. Mahesh, V., Mahesh, V., Nagaraj, S.M., Subhashaya, P., Singh, G.S.T.S.: Physio-mechanical and thermal characterization of jute/rubber crumb hybrid composites and selection of optimal configuration using the MADM approach. Proc. Ins. Mech. Eng. (2022). https://doi.org/10.1177/09544062221079166

    Article  Google Scholar 

  91. Mao, Q., Chen, J., Lv, J., Chen, S.: Emergency plan selection for epidemic prevention and control based on cumulative prospect theory and hybrid-information MADM. Kybernetes (2022). https://doi.org/10.1108/k-1108-2021-0736

    Article  Google Scholar 

  92. Zolfani, S.H., Hasheminasab, H., Torkayesh, A.E., Zavadskas, E.K., Derakhti, A.: A literature review of MADM applications for site selection problems - one decade review from to 2020. Int. J. Inf. Technol. Decis. Mak. 21(2022), 7–57 (2011)

    Google Scholar 

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Acknowledgements

The work was supported by the Young scientific and technological talents growth project of Guizhou Provincial Department of Education(Qian jiao he KY[2018]369, Qian jiao he KY[2018]385 and Qian jiao he KY[2017]274)and Science and technology innovation team of Liupanshui Normal University(LPSSYKJTD201702).

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  1. School of Mathematical Sciences, Sichuan Normal University, Chengdu, 610101, People’s Republic of China

    Baoquan Ning, Fan Lei & Guiwu Wei

  2. School of Mathematics and Statistics, Liupanshui Normal University, 553004, Liupanshui, People’s Republic of China

    Baoquan Ning

  3. School of Business, Sichuan Normal University, Chengdu, 610101, People’s Republic of China

    Guiwu Wei

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  1. Baoquan Ning
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Ning, B., Lei, F. & Wei, G. CODAS Method for Multi-Attribute Decision-Making Based on Some Novel Distance and Entropy Measures Under Probabilistic Dual Hesitant Fuzzy Sets. Int. J. Fuzzy Syst. 24, 3626–3649 (2022). https://doi.org/10.1007/s40815-022-01350-8

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