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Multiple attribute decision-making method based on projection model for dual hesitant fuzzy set


In the decision-making process, retaining the original data information has become a most crucial step. Dual hesitant fuzzy sets (DHFS), which can reflect the original membership degree and non-membership degree information given by the DMs, is a kind of new tool for the DMs to provide the original information as much as possible. In this paper, we focus on the decision- making problem by a projection model (Algorithm I) whose attribute values are given in the forms of dual hesitant fuzzy elements (DHFEs). In order to reflect the information of the data more accurately, we first divide the dual hesitant fuzzy decision matrix into membership degree matrix and non-membership degree matrix. Then we gain the virtual positive ideal solution from the membership degree matrix and the negative positive ideal solution from the non-membership degree matrix. And then the projection values from every solution to the virtual positive ideal solution and the negative positive ideal solution are calculated. In combination with the two projection values, the relative consistent degree is further calculated to rank all the alternatives. At the same time, in order to guarantee the rationality of the decision-making result, a variation coefficient method is developed to determine the weights of the attributes under dual hesitant fuzzy environment objectively. Finally, the existing algorithms (Algorithm II and Algorithm III, Algorithm IV, Algorithm V) are compared with our algorithm (Algorithm I). The comparison result shows that Algorithm I is a valuable tool for decision making.

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  • Atanassov, K. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20, 87–96.

    MathSciNet  Article  Google Scholar 

  • Bland, J. M., & Altman, D. G. (1996). Statistics notes: Measurement error proportional to the mean. British Medical Journal, 313, 106.

    Article  Google Scholar 

  • Chen, N., & Xu, Z. S. (2015). Hesitant fuzzy ELECTRE II approach: A new way to handle multi-criteria decision making problems. Information Sciences, 292, 175–197.

    Article  Google Scholar 

  • Dubois, D., & Prade, H. (1980). Fuzzy sets and systems, theory and applications. Academic Press.

    MATH  Google Scholar 

  • Everitt, B. S., & Howell, D. C. (2005). Encyclopedia of statistics in behavioral science. Wiley.

    Book  Google Scholar 

  • Fan, Q. R., Ikejo, K., Nagamura, K., & Okada, K. (2016). Diagnosis of gear damage based on coefficient of variation method by analyzing vibration accelerations on one gear tooth. Journal of Advanced Mechanical Design, Systems, and Manufacturing, 10(2), 1–14.

    Article  Google Scholar 

  • Gao, H., Lu, M., & Wei, Y. (2019). Dual hesitant bipolar fuzzy hamacher aggregation operators and their applications to multiple attribute decision making. Journal of Intelligent & Fuzzy Systems, 37(4), 5755–5766.

    Article  Google Scholar 

  • Gou, X. J., Xu, Z. S., & Liao, H. C. (2017). Hesitant fuzzy linguistic entropy and cross-entropy measores and alternative queuing method for multiple criteria decision making. Information Sciences, 388, 225–246.

    Article  Google Scholar 

  • Guo, W. Q., An, Y. L., & Liu, S. X. (2011). Study on the driving forces of rocky desertification in Guizhou province based on variation coefficient method. Meteorological and Environmental Research, 2(2), 76–79.

    Google Scholar 

  • Miyamoto, S. (2005). Remarks on basics of fuzzy sets and fuzzy multisets. Fuzzy Sets and Systems, 156, 427–431.

    MathSciNet  Article  Google Scholar 

  • Opricovic, S., & Tzeng, G. H. (2004). Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS. European Journal of Operational Research, 156(2), 445–455.

    Article  Google Scholar 

  • Pearson, K. (1896). Mathematical contributions to the theory of evolution. III. Regression, heredity and panmixia. Philosophical Transactions of the Royal Society of London, A, 187, 253–318.

    Article  Google Scholar 

  • Su, Z., Xu, Z. S., Zhao, H., & Liu, S. S. (2019). Distribution-based approaches to deriving weights from dual hesitant fuzzy information. Symmetry-Basel, 11(1), 1–20.

    MATH  Google Scholar 

  • Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25, 529–539.

    MATH  Google Scholar 

  • Wang, R. G., & Li, W. M. (2019). Extended VIKOR method of multi-attribute decision making under intuitionistic fuzzy environment based on a new distance measure. System Engineering and Electronics, 11(41), 2529–2530.

    Google Scholar 

  • Wang, Z. B., & Qiu, B. Z. (2014). Fuzzy C-means clustering algorithm based on coefficient of variation. Advanced Materials Research, 998–999, 873–877.

    Article  Google Scholar 

  • Xu, R. N., & Li, C. L. (2001). Computation of dispersion coefficient of fuzzy valued data. Fuzzy System and Mathematics, 15(1), 62–66.

    Google Scholar 

  • Z.S. Xu, The uncertain multiple attribute decision making method and application, Beijing Qing Hua University, 2004.

  • Xu, Z. S. (2007). Multiple-attribute group decision making with different formats of preference information on attributes. IEEE Transactions on Systems, Man, and Cybernetics-Part B, 37(6), 1500–1511.

    Article  Google Scholar 

  • Xu, Z. S., & Da, Q. L. (2004). Projection method for uncertain multiple attribute decision making with preference information on alternatives. International Journal of Information Technology & Decision Making, 3(3), 429–434.

    Article  Google Scholar 

  • Yoon, K. (1980). Systems selection by multiple attribute decision making, Ph. D. dissertation. Kansas State University.

  • Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353.

    MathSciNet  Article  Google Scholar 

  • Zhao, H., Xu, Z. S., & Liu, S. S. (2017). Dual hesitant fuzzy information aggregation with Einstein t-conorm and t-norm. Journal of Systems Science and Systems Engineering, 26(2), 240–264.

    Article  Google Scholar 

  • Zhu, B., & Xu, Z. S. (2017). Hesitant fuzzy Bonferroni means for multi-criteria decision making. Journal of the Operational Research Society, 64(12), 1831–1840.

    Article  Google Scholar 

  • Zhu, B., Xu, Z. S., & Xia, M. M. (2012). Dual hesitant fuzzy sets. Journal of Applied Mathematics, 26(5), 410–425.

    MathSciNet  MATH  Google Scholar 

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This work was supported by the National Natural Science Foundation of China (Nos. 71771155, 72071135).

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Correspondence to Hua Zhao or Zeshui Xu.

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Ni, Y., Zhao, H., Xu, Z. et al. Multiple attribute decision-making method based on projection model for dual hesitant fuzzy set. Fuzzy Optim Decis Making 21, 263–289 (2022).

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  • Dual hesitant fuzzy set
  • Multiple attribute decision making
  • Projection
  • Relative consistent degree