Multi-attribute group decision making based on hesitant fuzzy sets, TOPSIS method and fuzzy preference relations
Abstract
Hesitant fuzzy sets (HFSs) are widely applied in pattern recognition, classification, clustering, and multiple attribute decision making. In order to get more accurate decision results, the order relation of HFSs is particularly important. In this paper, some defects of the existing order relations for HFSs are discussed. In order to solve these problems, by employing a distance measure and the TOPSIS method, we propose a new order relation extraction method based on a new additive consistency fuzzy preference relation for hesitant fuzzy elements (HFEs). Then, the proposed additive consistency fuzzy preference relation is applied to integrate group decision information. In multi-attribute group decision making (MAGDM), it is particularly important to ensure the consensus of the decision-makers (DMs), and the consistency of the decision process is the precondition for DMs to reach consensus. The proposed method can maintain the consistency of the decision process for MAGDM under hesitant fuzzy environments, so as to get the consensus of DMs, besides, it can overcome the limitations of the existing order relations for HFSs. At the end of this paper, a numerical example is used to illustrate the effectiveness and feasibility of the new approach, and some comparative analyses are given. The obtained results confirm the theoretical and numerical analyses and emphasize the advantages, which can ensure the consistency of the whole decision process and avoid the original decision information change and loss of the proposed method, so as to be more in line with the actual situation.
Keyword : multi-attribute group decision making, hesitant fuzzy sets, TOPSIS, distance measure, fuzzy preference relation, additive consistency
How to Cite
Lan, J., Yang, M., Hu, M., & Liu, F. (2018). Multi-attribute group decision making based on hesitant fuzzy sets, TOPSIS method and fuzzy preference relations. Technological and Economic Development of Economy, 24(6), 2295-2317. https://doi.org/10.3846/tede.2018.6768
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
Atanassov, K. T. (1989). Gargov, g.: Interval valued intuitionistic fuzzy sets. Fuzzy Sets & Systems, 31(3), 343-349. https://doi.org/10.1016/0165-0114(89)90205-4
Atanassov, K. T., & Rangasamy, P. (1986). Intuitionistic fuzzy sets. Fuzzy Sets & Systems, 20(1), 87-96. https://doi.org/10.1016/S0165-0114(86)80034-3
Ballesteros-Pérez, P., Campo-Hitschfeld, M. L. D., Mora-Melià, D., & Domínguez, D. (2015). Modeling bidding competitiveness and position performance in multi-attribute construction auctions. Operations Research Perspectives, 2(C), 24-35. https://doi.org/10.1016/j.orp.2015.02.001
Bedregal, B., Reiser, R., Bustince, H., & Lopez-Molina, C. (2014). Aggregation functions for typical hesitant fuzzy elements and the action of automorphisms. Information Sciences an International Journal, 255(1), 82-99. https://doi.org/10.1016/j.ins.2013.08.024
Chen, N., Xu, Z., & Xia, M. (2013). Interval-valued hesitant preference relations and their applications to group decision making. Knowledge-Based Systems, 37(2), 528-540. https://doi.org/10.1016/j.knosys.2012.09.009
Chen, N., Xu, Z., & Xia, M. (2013a). Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis. Applied Mathematical Modelling, 37(4), 2197-2211. https://doi.org/10.1016/j.apm.2012.04.031
Deschrijver, G., & Kerre, E. E. (2003). On the relationship between some extensions of fuzzy set theory. Fuzzy Sets & Systems, 133(2), 227-235. https://doi.org/10.1016/S0165-0114(02)00127-6
Dong, Y., Chen, X., & Herrera, F. (2015). Minimizing adjusted simple terms in the consensus reaching process with hesitant linguistic assessments in group decision making. Information Sciences, 297(C), 95-117. https://doi.org/10.1016/j.ins.2014.11.011
Erceg, M. A. (1979). Metric spaces in fuzzy set theory. Journal of Mathematical Analysis and Applications, 69(1), 205-230. https://doi.org/10.1016/0022-247X(79)90189-6
Farhadinia, B. (2013). Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Information Sciences, 240(10), 129-144. https://doi.org/10.1016/j.ins.2013.03.034
Farhadinia, B. (2014). Distance and similarity measures for higher order hesitant fuzzy sets. KnowledgeBased Systems, 55, 43-48. https://doi.org/10.1016/j.knosys.2013.10.008
Farhadinia, B. (2016). Hesitant fuzzy set lexicographical ordering and its application to multi-attribute decision making. Information Sciences, 327(C), 233-245. https://doi.org/10.1016/j.ins.2015.07.057
Feng, X., Zuo, W., Wang, J., & Feng, L. (2014). Topsis method for hesitant fuzzy multiple attribute decision making. Journal of Intelligent & Fuzzy Systems Applications in Engineering & Technology, 26(5), 2263-2269.
Hesamian, G., & Shams, M. (2015). Measuring similarity and ordering based on hesitant fuzzy linguistic term sets. Journal of Intelligent & Fuzzy Systems, 28(2), 983-990.
Huo, L. A., Lan, J., & Wang, Z. (2011). New parametric prioritization methods for an analytical hierarchy process based on a pairwise comparison matrix. Mathematical & Computer Modelling, 54(11-12), 2736-2749. https://doi.org/10.1016/j.mcm.2011.06.062
Hwang, C., & Yoon, K. (1981). Multiple attribute decision making methods and applications: a state-of-the-art survey. In Multiple attribute decision making (pp. 58-191). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-48318-9_3
Jin, F., Ni, Z., Chen, H., Li, Y., & Zhou, L. (2016). Multiple attribute group decision making based on interval-valued hesitant fuzzy information measures. Computers & Industrial Engineering, 101, 103-115. https://doi.org/10.1016/j.cie.2016.08.019
Kacprzyk, J., & Orlovski, S. A. (Eds.). (1987). Optimization models using fuzzy sets and possibility theory. Theory and Decision Library book series (TDLB, vol. 4). Springer. https://doi.org/10.1007/978-94-009-3869-4
Kessler, M., Hiesse, C., Hestin, D., Mayeux, D., Boubenider, K., & Charpentier, B. (2011). A novel hybrid decision-making model for selecting locations in a fuzzy environment. Mathematical & Computer Modelling, 54(1), 88-104.
Khalid, A., & Beg, I. (2016). Incomplete hesitant fuzzy preference relations in group decision making. International Journal of Fuzzy Systems, 19(3), 1-9.
Lan, J., Jin, R., Zheng, Z., & Hu, M. (2017). Priority degrees for hesitant fuzzy sets: application to multiple attribute decision making. Operations Research Perspectives, 4, 67-73. https://doi.org/10.1016/j.orp.2017.05.001
Li, D., Zeng, W., & Li, J. (2015). New distance and similarity measures on hesitant fuzzy sets and their applications in multiple criteria decision making. Engineering Applications of Artificial Intelligence, 40, 11-16. https://doi.org/10.1016/j.engappai.2014.12.012
Meng, F. Y., & An, Q. X. (2017). A new approach for group decision making method with hesitant fuzzy preference relations. Knowledge-Based Systems, 127, 1-15. https://doi.org/10.1016/j.knosys.2017.03.010
Papakostas, G. A., Hatzimichailidis, A. G., & Kaburlasos, V. G. (2013). Distance and similarity measures between intuitionistic fuzzy sets: A comparative analysis from a pattern recognition point of view. Pattern Recognition Letters, 34(14), 1609-1622. https://doi.org/10.1016/j.patrec.2013.05.015
Peng, D. H., Gao, C. Y., & Gao, Z. F. (2013). Generalized hesitant fuzzy synergetic weighted distance measures and their application to multiple criteria decision-making. Applied Mathematical Modelling, 37(8), 5837-5850. https://doi.org/10.1016/j.apm.2012.11.016
Pramanik, S., Pramanik, S., & Giri, B. C. (2016). TOPSIS method for multi-attribute group decisionmaking under single-valued neutrosophic environment. Neural Computing and Applications, 27(3), 727-737. https://doi.org/10.1007/s00521-015-1891-2
Qian, G., Wang, H., & Feng, X. (2013). Generalized hesitant fuzzy sets and their application in decision support system. Knowledge-Based Systems, 37(4), 357-365. https://doi.org/10.1016/j.knosys.2012.08.019
Qin, J., Liu, X., & Pedrycz, W. (2016). Frank aggregation operators and their application to hesitant fuzzy multiple attribute decision making. Applied Soft Computing, 41, 428-452. https://doi.org/10.1016/j.asoc.2015.12.030
Rodriguez, R. M., Martinez, L., & Herrera, F. (2012). Hesitant fuzzy linguistic term sets for decision making. IEEE Transactions on Fuzzy Systems, 20(1), 109-119. https://doi.org/10.1109/TFUZZ.2011.2170076
Roy, A. R., & Maji, P. K. (2007). A fuzzy soft set theoretic approach to decision making problems. Journal of Computational and Applied Mathematics, 203(2), 412-418. https://doi.org/10.1016/j.cam.2006.04.008
Schubert, J., Moradi, F., Asadi, H., Luotsinen, L., Sjöberg, E., Hörling, P., Linderhed, A., & Oskarsson, D. (2015). Simulation-based decision support for evaluating operational plans. Operations Research Perspectives, 2, 36-56. https://doi.org/10.1016/j.orp.2015.02.002
Singh, P. (2015). Distance and similarity measures for multiple-attribute decision making with dual hesitant fuzzy sets. Computational & Applied Mathematics, 36(1), 1-16.
Su, Z., Xu, Z., Liu, H., & Liu, S. (2015). Distance and similarity measures for dual hesitant fuzzy sets and their applications in pattern recognition. Journal of Intelligent & Fuzzy Systems, 29(2), 731-745. https://doi.org/10.3233/IFS-141474
Tang, X., Fu, C., Xu, D. L., & Yang, S. (2017). Analysis of fuzzy hamacher aggregation functions for uncertain multiple attribute decision making. Information Sciences, 387, 19-33. https://doi.org/10.1016/j.ins.2016.12.045
Tanino, T. (1984). Fuzzy preference orderings in group decision making. Fuzzy Sets & Systems, 12(2), 117-131. https://doi.org/10.1016/0165-0114(84)90032-0
Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25(6), 529-539. https://doi.org/10.1002/int.20418
Torra, V., & Narukawa, Y. (2007). Modeling decisions – information fusion and aggregation operators. Cognitive Technologies, 61(Pt12), 1090-1093.
Torra, V., & Narukawa, Y. (2009, August). On hesitant fuzzy sets and decision. In IEEE International Conference on Fuzzy Systems (pp. 1378-1382). Jeju Island, South Korea. https://doi.org/10.1109/FUZZY.2009.5276884
Torra, V., Xu, Z. S., & Herrera, F. (2014). Hesitant fuzzy sets: state of the art and future directions. International Journal of Intelligent Systems, 29(6), 495-524. https://doi.org/10.1002/int.21654
Turksen. (1986). Interval valued fuzzy sets based on normal forms. Fuzzy Sets & Systems, 20(2), 191-210. https://doi.org/10.1016/0165-0114(86)90077-1
Wang, Y. (2012). Using the method of maximizing deviation to make decision for multiindices. Journal of Systems Engineering and Electronics, 8(3), 21-26.
Wei, G. (2012). Hesitant fuzzy prioritized operators and their application to multiple attribute decision making. Knowledge-Based Systems, 31(7), 176-182. https://doi.org/10.1016/j.knosys.2012.03.011
Williams, J., & Steele, N. (2002). Difference, distance and similarity as a basis for fuzzy decision support based on prototypical decision classes. Fuzzy Sets & Systems, 131(1), 35-46. https://doi.org/10.1016/S0165-0114(01)00253-6
Wu, Z., & Xu, J. (2016). Managing consistency and consensus in group decision making with hesitant fuzzy linguistic preference relations. Omega, 65(3), 28-40. https://doi.org/10.1016/j.omega.2015.12.005
Wu, Z., Jin, B., & Xu, J. (2018). Local feedback strategy for consensus building with probability-hesitant fuzzy preference relations. Applied Soft Computing, 67, 691-705. https://doi.org/10.1016/j.asoc.2017.06.011
Xia, M., & Xu, Z. (2011). Hesitant fuzzy information aggregation in decision making. International Journal of Approximate Reasoning, 52(3), 395-407. https://doi.org/10.1016/j.ijar.2010.09.002
Xia, M., Xu, Z., & Chen, N. (2013). Some hesitant fuzzy aggregation operators with their application in group decision making. Group Decision & Negotiation, 22(2), 259-279. https://doi.org/10.1007/s10726-011-9261-7
Xu, Y., Cabrerizo, F. J., & Herrera-Viedma, E. (2017). A consensus model for hesitant fuzzy preference relations and its application in water allocation management. Applied Soft Computing, 58, 265-284. https://doi.org/10.1016/j.asoc.2017.04.068
Xu, Y., Chen, L., Herrera, F., & Wang, H. (2016). Deriving the priority weights from incomplete hesitant fuzzy preference relations in group decision making. Knowledge-Based Systems, 99(C), 71-78. https://doi.org/10.1016/j.knosys.2016.01.047
Xu, Z. (2015). Uncertain Multi-attribute decision making: Methods and applications. Springer, Berlin, Heidelberg, 50-55. https://doi.org/10.1007/978-3-662-45640-8
Xu, Z., & Xia, M. (2011). Distance and similarity measures for hesitant fuzzy sets. Information Sciences, 181(11), 2128-2138. https://doi.org/10.1016/j.ins.2011.01.028
Xu, Z., & Xia, M. (2011a). On distance and correlation measures of hesitant fuzzy information. International Journal of Intelligent Systems, 26(5), 410-425. https://doi.org/10.1002/int.20474
Xu, Z., & Xia, M. (2012). Hesitant fuzzy entropy and cross-entropy and their use in multiattribute decision-making. International Journal of Intelligent Systems, 27(9), 799-822. https://doi.org/10.1002/int.21548
Xu, Z., & Zhang, X. (2013). Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information. Knowledge-Based Systems, 52(6), 53-64. https://doi.org/10.1016/j.knosys.2013.05.011
Yager. R. R. (1986). On the theory of bags. International Journal of General Systems, 13(1), 23-37. https://doi.org/10.1080/03081078608934952
Yu, D., Wu, Y., & Zhou, W. (2011). Multi-criteria decision making based on Choquet integral under hesitant fuzzy environment. Journal of Computational Information Systems, 7(12), 4506-4513.
Yue, C. (2016). A geometric approach for ranking interval-valued intuitionistic fuzzy numbers with an application to group decision-making. Computers & Industrial Engineering, 102, 233-245. https://doi.org/10.1016/j.cie.2016.10.027
Zadeh, L. A. (1965). Fuzzy sets. Information & Control, 8(3), 338-356. https://doi.org/10.1016/S0019-9958(65)90241-X
Zeng, W., Li, D., & Yin, Q. (2016). Distance and similarity measures between hesitant fuzzy sets and their application in pattern recognition. Pattern Recognition Letters, 84(C), 267-271. https://doi.org/10.1016/j.patrec.2016.11.001
Zhang, F., Li, J., Chen, J., Sun, J., & Attey, A. (2017). Hesitant distance set on hesitant fuzzy sets and its application in urban road traffic state identification. Engineering Applications of Artificial Intelligence, 61(C), 57-64. https://doi.org/10.1016/j.engappai.2017.02.004
Zhang, X., & Xu, Z. (2015). Novel distance and similarity measures on hesitant fuzzy sets with applications to clustering analysis. Journal of Intelligent & Fuzzy Systems, 28(5), 2279-2296.
Zhang, Y., Xie, A., & Wu, Y. (2015a). A hesitant fuzzy multiple attribute decision making method based on linear programming and TOPSIS. IFAC-PapersOnLine, 48(28), 427-431. https://doi.org/10.1016/j.ifacol.2015.12.165
Zhang, Z., Wang, C., & Tian, X. (2015b). A decision support model for group decision making with hesitant fuzzy preference relations. Knowledge-Based Systems, 86(C), 77-101. https://doi.org/10.1016/j.knosys.2015.05.023
Zhang, Z., Wang, C., Tian, D., & Li, K. (2014). Induced generalized hesitant fuzzy operators and their application to multiple attribute group decision making. Computers & Industrial Engineering, 67(1), 116-138. https://doi.org/10.1016/j.cie.2013.10.011
Zhou, W., Xu, Z., & Chen, M. (2015). Preference relations based on hesitant-intuitionistic fuzzy information and their application in group decision making. Computers & Industrial Engineering, 87, 163-175. https://doi.org/10.1016/j.cie.2015.04.020
Zhu, B., Xu, Z., & Xia, M. (2012). Hesitant fuzzy geometric Bonferroni means. Information Sciences, 205, 72-85. https://doi.org/10.1016/j.ins.2012.01.048
Zhu, C., Zhu, L., & Zhang, X. (2016). Linguistic hesitant fuzzy power aggregation operators and their applications in multiple attribute decision-making. Information Sciences, 367-308, 809-826. https://doi.org/10.1016/j.ins.2016.07.011
Atanassov, K. T., & Rangasamy, P. (1986). Intuitionistic fuzzy sets. Fuzzy Sets & Systems, 20(1), 87-96. https://doi.org/10.1016/S0165-0114(86)80034-3
Ballesteros-Pérez, P., Campo-Hitschfeld, M. L. D., Mora-Melià, D., & Domínguez, D. (2015). Modeling bidding competitiveness and position performance in multi-attribute construction auctions. Operations Research Perspectives, 2(C), 24-35. https://doi.org/10.1016/j.orp.2015.02.001
Bedregal, B., Reiser, R., Bustince, H., & Lopez-Molina, C. (2014). Aggregation functions for typical hesitant fuzzy elements and the action of automorphisms. Information Sciences an International Journal, 255(1), 82-99. https://doi.org/10.1016/j.ins.2013.08.024
Chen, N., Xu, Z., & Xia, M. (2013). Interval-valued hesitant preference relations and their applications to group decision making. Knowledge-Based Systems, 37(2), 528-540. https://doi.org/10.1016/j.knosys.2012.09.009
Chen, N., Xu, Z., & Xia, M. (2013a). Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis. Applied Mathematical Modelling, 37(4), 2197-2211. https://doi.org/10.1016/j.apm.2012.04.031
Deschrijver, G., & Kerre, E. E. (2003). On the relationship between some extensions of fuzzy set theory. Fuzzy Sets & Systems, 133(2), 227-235. https://doi.org/10.1016/S0165-0114(02)00127-6
Dong, Y., Chen, X., & Herrera, F. (2015). Minimizing adjusted simple terms in the consensus reaching process with hesitant linguistic assessments in group decision making. Information Sciences, 297(C), 95-117. https://doi.org/10.1016/j.ins.2014.11.011
Erceg, M. A. (1979). Metric spaces in fuzzy set theory. Journal of Mathematical Analysis and Applications, 69(1), 205-230. https://doi.org/10.1016/0022-247X(79)90189-6
Farhadinia, B. (2013). Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Information Sciences, 240(10), 129-144. https://doi.org/10.1016/j.ins.2013.03.034
Farhadinia, B. (2014). Distance and similarity measures for higher order hesitant fuzzy sets. KnowledgeBased Systems, 55, 43-48. https://doi.org/10.1016/j.knosys.2013.10.008
Farhadinia, B. (2016). Hesitant fuzzy set lexicographical ordering and its application to multi-attribute decision making. Information Sciences, 327(C), 233-245. https://doi.org/10.1016/j.ins.2015.07.057
Feng, X., Zuo, W., Wang, J., & Feng, L. (2014). Topsis method for hesitant fuzzy multiple attribute decision making. Journal of Intelligent & Fuzzy Systems Applications in Engineering & Technology, 26(5), 2263-2269.
Hesamian, G., & Shams, M. (2015). Measuring similarity and ordering based on hesitant fuzzy linguistic term sets. Journal of Intelligent & Fuzzy Systems, 28(2), 983-990.
Huo, L. A., Lan, J., & Wang, Z. (2011). New parametric prioritization methods for an analytical hierarchy process based on a pairwise comparison matrix. Mathematical & Computer Modelling, 54(11-12), 2736-2749. https://doi.org/10.1016/j.mcm.2011.06.062
Hwang, C., & Yoon, K. (1981). Multiple attribute decision making methods and applications: a state-of-the-art survey. In Multiple attribute decision making (pp. 58-191). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-48318-9_3
Jin, F., Ni, Z., Chen, H., Li, Y., & Zhou, L. (2016). Multiple attribute group decision making based on interval-valued hesitant fuzzy information measures. Computers & Industrial Engineering, 101, 103-115. https://doi.org/10.1016/j.cie.2016.08.019
Kacprzyk, J., & Orlovski, S. A. (Eds.). (1987). Optimization models using fuzzy sets and possibility theory. Theory and Decision Library book series (TDLB, vol. 4). Springer. https://doi.org/10.1007/978-94-009-3869-4
Kessler, M., Hiesse, C., Hestin, D., Mayeux, D., Boubenider, K., & Charpentier, B. (2011). A novel hybrid decision-making model for selecting locations in a fuzzy environment. Mathematical & Computer Modelling, 54(1), 88-104.
Khalid, A., & Beg, I. (2016). Incomplete hesitant fuzzy preference relations in group decision making. International Journal of Fuzzy Systems, 19(3), 1-9.
Lan, J., Jin, R., Zheng, Z., & Hu, M. (2017). Priority degrees for hesitant fuzzy sets: application to multiple attribute decision making. Operations Research Perspectives, 4, 67-73. https://doi.org/10.1016/j.orp.2017.05.001
Li, D., Zeng, W., & Li, J. (2015). New distance and similarity measures on hesitant fuzzy sets and their applications in multiple criteria decision making. Engineering Applications of Artificial Intelligence, 40, 11-16. https://doi.org/10.1016/j.engappai.2014.12.012
Meng, F. Y., & An, Q. X. (2017). A new approach for group decision making method with hesitant fuzzy preference relations. Knowledge-Based Systems, 127, 1-15. https://doi.org/10.1016/j.knosys.2017.03.010
Papakostas, G. A., Hatzimichailidis, A. G., & Kaburlasos, V. G. (2013). Distance and similarity measures between intuitionistic fuzzy sets: A comparative analysis from a pattern recognition point of view. Pattern Recognition Letters, 34(14), 1609-1622. https://doi.org/10.1016/j.patrec.2013.05.015
Peng, D. H., Gao, C. Y., & Gao, Z. F. (2013). Generalized hesitant fuzzy synergetic weighted distance measures and their application to multiple criteria decision-making. Applied Mathematical Modelling, 37(8), 5837-5850. https://doi.org/10.1016/j.apm.2012.11.016
Pramanik, S., Pramanik, S., & Giri, B. C. (2016). TOPSIS method for multi-attribute group decisionmaking under single-valued neutrosophic environment. Neural Computing and Applications, 27(3), 727-737. https://doi.org/10.1007/s00521-015-1891-2
Qian, G., Wang, H., & Feng, X. (2013). Generalized hesitant fuzzy sets and their application in decision support system. Knowledge-Based Systems, 37(4), 357-365. https://doi.org/10.1016/j.knosys.2012.08.019
Qin, J., Liu, X., & Pedrycz, W. (2016). Frank aggregation operators and their application to hesitant fuzzy multiple attribute decision making. Applied Soft Computing, 41, 428-452. https://doi.org/10.1016/j.asoc.2015.12.030
Rodriguez, R. M., Martinez, L., & Herrera, F. (2012). Hesitant fuzzy linguistic term sets for decision making. IEEE Transactions on Fuzzy Systems, 20(1), 109-119. https://doi.org/10.1109/TFUZZ.2011.2170076
Roy, A. R., & Maji, P. K. (2007). A fuzzy soft set theoretic approach to decision making problems. Journal of Computational and Applied Mathematics, 203(2), 412-418. https://doi.org/10.1016/j.cam.2006.04.008
Schubert, J., Moradi, F., Asadi, H., Luotsinen, L., Sjöberg, E., Hörling, P., Linderhed, A., & Oskarsson, D. (2015). Simulation-based decision support for evaluating operational plans. Operations Research Perspectives, 2, 36-56. https://doi.org/10.1016/j.orp.2015.02.002
Singh, P. (2015). Distance and similarity measures for multiple-attribute decision making with dual hesitant fuzzy sets. Computational & Applied Mathematics, 36(1), 1-16.
Su, Z., Xu, Z., Liu, H., & Liu, S. (2015). Distance and similarity measures for dual hesitant fuzzy sets and their applications in pattern recognition. Journal of Intelligent & Fuzzy Systems, 29(2), 731-745. https://doi.org/10.3233/IFS-141474
Tang, X., Fu, C., Xu, D. L., & Yang, S. (2017). Analysis of fuzzy hamacher aggregation functions for uncertain multiple attribute decision making. Information Sciences, 387, 19-33. https://doi.org/10.1016/j.ins.2016.12.045
Tanino, T. (1984). Fuzzy preference orderings in group decision making. Fuzzy Sets & Systems, 12(2), 117-131. https://doi.org/10.1016/0165-0114(84)90032-0
Torra, V. (2010). Hesitant fuzzy sets. International Journal of Intelligent Systems, 25(6), 529-539. https://doi.org/10.1002/int.20418
Torra, V., & Narukawa, Y. (2007). Modeling decisions – information fusion and aggregation operators. Cognitive Technologies, 61(Pt12), 1090-1093.
Torra, V., & Narukawa, Y. (2009, August). On hesitant fuzzy sets and decision. In IEEE International Conference on Fuzzy Systems (pp. 1378-1382). Jeju Island, South Korea. https://doi.org/10.1109/FUZZY.2009.5276884
Torra, V., Xu, Z. S., & Herrera, F. (2014). Hesitant fuzzy sets: state of the art and future directions. International Journal of Intelligent Systems, 29(6), 495-524. https://doi.org/10.1002/int.21654
Turksen. (1986). Interval valued fuzzy sets based on normal forms. Fuzzy Sets & Systems, 20(2), 191-210. https://doi.org/10.1016/0165-0114(86)90077-1
Wang, Y. (2012). Using the method of maximizing deviation to make decision for multiindices. Journal of Systems Engineering and Electronics, 8(3), 21-26.
Wei, G. (2012). Hesitant fuzzy prioritized operators and their application to multiple attribute decision making. Knowledge-Based Systems, 31(7), 176-182. https://doi.org/10.1016/j.knosys.2012.03.011
Williams, J., & Steele, N. (2002). Difference, distance and similarity as a basis for fuzzy decision support based on prototypical decision classes. Fuzzy Sets & Systems, 131(1), 35-46. https://doi.org/10.1016/S0165-0114(01)00253-6
Wu, Z., & Xu, J. (2016). Managing consistency and consensus in group decision making with hesitant fuzzy linguistic preference relations. Omega, 65(3), 28-40. https://doi.org/10.1016/j.omega.2015.12.005
Wu, Z., Jin, B., & Xu, J. (2018). Local feedback strategy for consensus building with probability-hesitant fuzzy preference relations. Applied Soft Computing, 67, 691-705. https://doi.org/10.1016/j.asoc.2017.06.011
Xia, M., & Xu, Z. (2011). Hesitant fuzzy information aggregation in decision making. International Journal of Approximate Reasoning, 52(3), 395-407. https://doi.org/10.1016/j.ijar.2010.09.002
Xia, M., Xu, Z., & Chen, N. (2013). Some hesitant fuzzy aggregation operators with their application in group decision making. Group Decision & Negotiation, 22(2), 259-279. https://doi.org/10.1007/s10726-011-9261-7
Xu, Y., Cabrerizo, F. J., & Herrera-Viedma, E. (2017). A consensus model for hesitant fuzzy preference relations and its application in water allocation management. Applied Soft Computing, 58, 265-284. https://doi.org/10.1016/j.asoc.2017.04.068
Xu, Y., Chen, L., Herrera, F., & Wang, H. (2016). Deriving the priority weights from incomplete hesitant fuzzy preference relations in group decision making. Knowledge-Based Systems, 99(C), 71-78. https://doi.org/10.1016/j.knosys.2016.01.047
Xu, Z. (2015). Uncertain Multi-attribute decision making: Methods and applications. Springer, Berlin, Heidelberg, 50-55. https://doi.org/10.1007/978-3-662-45640-8
Xu, Z., & Xia, M. (2011). Distance and similarity measures for hesitant fuzzy sets. Information Sciences, 181(11), 2128-2138. https://doi.org/10.1016/j.ins.2011.01.028
Xu, Z., & Xia, M. (2011a). On distance and correlation measures of hesitant fuzzy information. International Journal of Intelligent Systems, 26(5), 410-425. https://doi.org/10.1002/int.20474
Xu, Z., & Xia, M. (2012). Hesitant fuzzy entropy and cross-entropy and their use in multiattribute decision-making. International Journal of Intelligent Systems, 27(9), 799-822. https://doi.org/10.1002/int.21548
Xu, Z., & Zhang, X. (2013). Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information. Knowledge-Based Systems, 52(6), 53-64. https://doi.org/10.1016/j.knosys.2013.05.011
Yager. R. R. (1986). On the theory of bags. International Journal of General Systems, 13(1), 23-37. https://doi.org/10.1080/03081078608934952
Yu, D., Wu, Y., & Zhou, W. (2011). Multi-criteria decision making based on Choquet integral under hesitant fuzzy environment. Journal of Computational Information Systems, 7(12), 4506-4513.
Yue, C. (2016). A geometric approach for ranking interval-valued intuitionistic fuzzy numbers with an application to group decision-making. Computers & Industrial Engineering, 102, 233-245. https://doi.org/10.1016/j.cie.2016.10.027
Zadeh, L. A. (1965). Fuzzy sets. Information & Control, 8(3), 338-356. https://doi.org/10.1016/S0019-9958(65)90241-X
Zeng, W., Li, D., & Yin, Q. (2016). Distance and similarity measures between hesitant fuzzy sets and their application in pattern recognition. Pattern Recognition Letters, 84(C), 267-271. https://doi.org/10.1016/j.patrec.2016.11.001
Zhang, F., Li, J., Chen, J., Sun, J., & Attey, A. (2017). Hesitant distance set on hesitant fuzzy sets and its application in urban road traffic state identification. Engineering Applications of Artificial Intelligence, 61(C), 57-64. https://doi.org/10.1016/j.engappai.2017.02.004
Zhang, X., & Xu, Z. (2015). Novel distance and similarity measures on hesitant fuzzy sets with applications to clustering analysis. Journal of Intelligent & Fuzzy Systems, 28(5), 2279-2296.
Zhang, Y., Xie, A., & Wu, Y. (2015a). A hesitant fuzzy multiple attribute decision making method based on linear programming and TOPSIS. IFAC-PapersOnLine, 48(28), 427-431. https://doi.org/10.1016/j.ifacol.2015.12.165
Zhang, Z., Wang, C., & Tian, X. (2015b). A decision support model for group decision making with hesitant fuzzy preference relations. Knowledge-Based Systems, 86(C), 77-101. https://doi.org/10.1016/j.knosys.2015.05.023
Zhang, Z., Wang, C., Tian, D., & Li, K. (2014). Induced generalized hesitant fuzzy operators and their application to multiple attribute group decision making. Computers & Industrial Engineering, 67(1), 116-138. https://doi.org/10.1016/j.cie.2013.10.011
Zhou, W., Xu, Z., & Chen, M. (2015). Preference relations based on hesitant-intuitionistic fuzzy information and their application in group decision making. Computers & Industrial Engineering, 87, 163-175. https://doi.org/10.1016/j.cie.2015.04.020
Zhu, B., Xu, Z., & Xia, M. (2012). Hesitant fuzzy geometric Bonferroni means. Information Sciences, 205, 72-85. https://doi.org/10.1016/j.ins.2012.01.048
Zhu, C., Zhu, L., & Zhang, X. (2016). Linguistic hesitant fuzzy power aggregation operators and their applications in multiple attribute decision-making. Information Sciences, 367-308, 809-826. https://doi.org/10.1016/j.ins.2016.07.011