Consensus based on multiplicative consistent double hierarchy linguistic preferences: venture capital in real estate market
Abstract
Based on the Computing with Words (CW), double hierarchy hesitant fuzzy linguistic term set (DHHFLTS) can be used to express complex linguistic information accurately with two simple linguistic hierarchies. This paper proposes a group decision making (GDM) model based on multiplicative consistency and consensus with double hierarchy hesitant fuzzy linguistic preference relations (DHHFLPRs). Firstly, a correlation coefficient of DHHFLTSs is defined based on the distance measures of double hierarchy hesitant fuzzy linguistic elements (DHHFLEs). Then, a multiplicative consistency property of DHHFLPRs is investigated, and a consistency checking method and a feedback mechanism-based repairing algorithm are developed to ensure all DHHFLPRs with acceptable multiplicative consistency. Furthermore, a correlation measure for DHHFLPRs based on the correlation coefficient of DHHFLTSs is proposed, and a new consensus reaching method on the basis of the correlation measure is developed, which can be used to fully obtain the consensus degree from both positive and negative angles. Finally, we make some comparative analyses with other existing consistency checking and repairing method as well as the consensus reaching approach to illustrate the effectiveness of the proposed method by a case study concerning the assessment of the venture capital project about real estate market in some cities of China.
First published online 8 October 2019
Keyword : double hierarchy hesitant fuzzy linguistic preference relations, group decision making, multiplicative consistency, consensus, venture capital, real estate market
How to Cite
Gou, X., Liao, H., Wang, X., Xu, Z., & Herrera, F. (2020). Consensus based on multiplicative consistent double hierarchy linguistic preferences: venture capital in real estate market. International Journal of Strategic Property Management, 24(1), 1-23. https://doi.org/10.3846/ijspm.2019.10431
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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Dubois, D. (2011). The role of fuzzy sets indecision sciences: old techniques and new directions. Fuzzy Sets and Systems, 184, 3-28. https://doi.org/10.1016/j.fss.2011.06.003
Dutta, S., & Folta, T. B. (2016). A comparison of the effect of angels and venture capitalists on innovation and value creation. Journal of Business Venturing, 31(1), 39-54. https://doi.org/10.1016/j.jbusvent.2015.08.003
Gomes, L. F. A. M., & González, X. I. (2012). Behavioral multicriteria decision analysis: further elaborations on the TODIM method. Foundations of Computing and Decision Sciences, 37(1), 3-8. https://doi.org/10.2478/v10209-011-0001-1
Gou, X. J., Liao, H. C., Xu, Z. S., & Herrera, F. (2017). Double hierarchy hesitant fuzzy linguistic term set and MULTIMOORA method: a case of study to evaluate the implementation status of haze controlling measures. Information Fusion, 38, 22-34. https://doi.org/10.1016/j.inffus.2017.02.008
Gou, X. J., Liao, H. C., Xu, Z. S., Min. R, & Herrera, F. (2019). Group decision making with double hierarchy hesitant fuzzy linguistic preference relations: consistency based measures, index and repairing algorithms and decision model. Information Sciences, 489, 93-112. https://doi.org/10.1016/j.ins.2019.03.037
Gou, X. J., Xu, Z. S., & Herrera, F. (2018a). Consensus reaching process for large-scale group decision making with double hierarchy hesitant fuzzy linguistic preference relations. Knowledge-Based Systems, 157, 20-33. https://doi.org/10.1016/j.knosys.2018.05.008
Gou, X. J., Xu, Z. S., Liao, H. C., & Herrera, F. (2018b). Multiple criteria decision making based on distance and similarity measures with double hierarchy hesitant fuzzy linguistic term sets. Computers & Industrial Engineering, 126, 516-530. https://doi.org/10.1016/j.cie.2018.10.020
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Herrera, F., Herrera-Viedma, E., & Verdegay, J. L. (1995). A sequential selection process in group decision making with a linguistic assessment approach. Information Sciences, 85(4), 223-239. https://doi.org/10.1016/0020-0255(95)00025-K
Herrera, F., & Martínez, L. (2000). A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Transactions on Fuzzy Systems, 8, 746-752. https://doi.org/10.1109/91.890332
Herrera-Viedma, E., Alonso, S., Chiclana, F., & Herrera, F. (2007). A consensus model for group decision making with incomplete fuzzy preference relations. IEEE Transactions on Fuzzy Systems, 15(5), 863-877. https://doi.org/10.1109/TFUZZ.2006.889952
Karplus, K. P., & Diederichs, K. (2012). Linking crystallographic model and data quality. Science, 336(6084), 1030-1033. https://doi.org/10.1126/science.1218231
Krishankumar, R., Subrajaa, L. S., Ravichandran, K. S., Kar, S., & Saeid, A. B. (2019). A framework for multi-attribute group decision-making using double hierarchy hesitant fuzzy linguistic term set. International Journal of Fuzzy Systems. https://doi.org/10.1007/s40815-019-00618-w
Li, C. C., Dong, Y. C., Xu, Y. J., Chiclana, F., Herrera-Viedma, E., & Herrera, F. (2019). An overview on managing additive consistency of reciprocal preference relations for consistency-driven decision making and fusion: taxonomy and future directions. Information Fusion, 52, 143-156. https://doi.org/10.1016/j.inffus.2018.12.004
Li, J., & Wang, J. Q. (2018). Multi-criteria decision making with probabilistic hesitant fuzzy information based on expected multiplicative consistency. Neural Computing and Applications. https://doi.org/10.1007/s00521-018-3753-1
Li, J., Wang, J. Q., & Hu, J. H. (2019). Consensus building for hesitant fuzzy preference relations with multiplicative consistency. Computers & Industrial Engineering, 128, 387-400. https://doi.org/10.1016/j.cie.2018.12.051
Liao, H. C., Xu, Z. S., Zeng, X. J., & Xu, D. L. (2015). An enhanced consensus reaching process in group decision making with intuitionistic fuzzy preference relations. Information Sciences, 329(C), 274-286. https://doi.org/10.1016/j.ins.2015.09.024
Liu, C., & Xiong, W. (2018). China’s real estate market (Working paper). National Bureau of Economic Research. https://doi.org/10.3386/w25297
Luo, S. Z., Zhang, H. Y., Wang, J. Q., & Li, L. (2019) Group decision-making approach for evaluating the sustainability of constructed wetlands with probabilistic linguistic preference relations. Journal of the Operational Research Society. https://doi.org/10.1080/01605682.2018.151080
Ma, L. C. (2016). A new group ranking approach for ordinal preferences based on group maximum consensus sequences. European Journal of Operational Research, 251(1), 171-181. https://doi.org/10.1016/j.ejor.2015.10.042
Mendel, J. M. (2007). Type-2 fuzzy sets and systems: an overview. IEEE Computational Intelligence Magazine, 2(2), 20-29. https://doi.org/10.1109/MCI.2007.380672
Millet, I. (1997). The effectiveness of alternative preference elicitation methods in the analytic hierarchy process. Journal of Multi‐Criteria Decision Analysis, 6, 41-51. https://doi.org/10.1002/(SICI)1099-1360(199701)6:1<41::AID-MCDA122>3.0.CO;2-D
Montserrat-Adell, J., Xu, Z. S., Gou, X. J., & Agell, N. (2019). Free double hierarchy hesitant fuzzy linguistic term sets: an application on ranking alternatives in GDM. Information Fusion, 47, 45-59. https://doi.org/10.1016/j.inffus.2018.07.002
Orlorski, S. A. (1978). Decision-making with a fuzzy preference relation. Fuzzy Sets and Systems, 3, 155-167. http://doi.org/10.1016/0165-0114(78)90001-5
Qin, J. D., Liu, X. W., & Pedrycz, W. (2017). An extended TODIM multi-criteria group decision making method for green supplier selection in interval type-2 fuzzy environment. European Journal of Operational Research, 258(2), 626-638. https://doi.org/10.1016/j.ejor.2016.09.059
Rodríguez, R. M., Martínez, L., & Herrera, F. (2012). Hesitant fuzzy linguistic terms sets for decision making. IEEE Transactions on Fuzzy Systems, 20, 109-119. https://doi.org/10.1109/TFUZZ.2011.2170076
Saaty, T. L. (1980). The analytical hierarchy process. New York: McGraw-Hill. https://doi.org/10.21236/ADA214804
Schumpeter, J. A. (1934). The theory of economic development: an inquiry into profits, capital, credit, interest, and the business cycle. Social Science Electronic Publishing, 25(1), 90-91.
Wang, H., Xu, Z. S., & Zeng, X. J. (2018). Linguistic terms with weakened hedges: a model for qualitative decision making under uncertainty. Information Sciences, 433, 37-45. https://doi.org/10.1016/j.ins.2017.12.036
Wu, X. L., & Liao, H. C. (2019). A consensus-based probabilistic linguistic gained and lost dominance score method. European Journal of Operational Research, 272(3), 1017-1027. https://doi.org/10.1016/j.ejor.2018.07.044
Wu, Z. B., & Xu, J. P. (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
Xia, M. M., & Xu, Z. S. (2011). Some issues on multiplicative consistency of interval reciprocal relations. International Journal of Information Technology & Decision Making, 10(6), 1043-1065. https://doi.org/10.1142/S0219622011004701
Xia, M. M., Xu, Z. S., & Chen, J. (2013). Algorithms for improving consistency or consensus of reciprocal [0,1]-valued preference relations. Fuzzy Sets and Systems, 216, 108-133. https://doi.org/10.1016/j.fss.2012.09.016
Xia, M. M., Xu, Z. S., & Wang, Z. (2014). Multiplicative consistency-based decision support system for incomplete linguistic preference relations. International Journal of Systems Science, 45(3), 625-636. https://doi.org/10.1080/00207721.2012.724112
Xu, Z. S. (2005). Deviation measures of linguistic preference relations in group decision making. Omega, 33(3), 249-254. https://doi.org/10.1016/j.omega.2004.04.008
Xu, Z. S. (2007). Intuitionistic preference relations and their application in group decision making. Information Sciences, 177, 2363-2379. https://doi.org/10.1016/j.ins.2006.12.019
Xu, Z. S. (2008). Group decision making based on multiple types of linguistic preference relations. Information Sciences, 178, 452-467. https://doi.org/10.1016/j.ins.2007.05.018
Xu, Z. S., Cai, X. Q., & Szmidt, E. (2011). Algorithms for estimating missing elements of incomplete intuitionistic preference relations. International Journal of Intelligent Systems, 26(9), 787-813. https://doi.org/10.1002/int.20494
Yazdipour, R. (2011). Advances in entrepreneurial finance. New York: Springer. https://doi.org/10.1007/978-1-4419-7527-0
Yu, D. J., Xu, Z. S., & Wang, W. (2018). Bibliometric analysis of fuzzy theory research in China: a 30-year perspective. Knowledge-Based Systems, 141, 188-199. https://doi.org/10.1016/j.knosys.2017.11.018
Zacharakis, A. L., Mcmullen, J. S., & Shepherd, D. A. (2007). Venture capitalists’ decision making across three countries: an institutional theory perspective. Journal of International Business Studies, 38(5), 691-708. https://doi.org/10.1057/palgrave.jibs.8400291
Zadeh, L. A. (1975). The concept of a linguistic variable and its application to approximate reasoning I. Information Sciences, 8, 199-249. https://doi.org/10.1016/0020-0255(75)90036-5
Zadeh, L. A. (2012). Computing with words: what is computing with words (CWW)? (pp. 1-40). Springer. https://doi.org/10.1007/978-3-642-27473-2
Zhang, W. K., Du, J., & Tian, X. L. (2018). Finding a promising venture capital project with todim under probabilistic hesitant fuzzy circumstance. Technological and Economic Development of Economy, 24(5), 2026-2044. https://doi.org/10.3846/tede.2018.5494
Zhang, Y. X., Xu, Z. S., Wang, H., & Liao, H. C. (2016). Consistency-based risk assessment with probabilistic linguistic preference relation. Applied Soft Computing, 49, 817-833. https://doi.org/10.1016/j.asoc.2016.08.045
Zhang, X. Y., Zhang H. Y., & Wang, J. Q. (2019). Discussing incomplete 2-tuple fuzzy linguistic preference relations in multi-granular linguistic MCGDM with unknown weight information. Soft Computing, 23(6), 2015-2032. https://doi.org/10.1007/s00500-017-2915-x
Zhou, W., & Xu, Z. S. (2018). Probability calculation and element optimization of probabilistic hesitant fuzzy preference relations based on expected consistency. IEEE Transactions on Fuzzy Systems, 26(3), 1367-1378. https://doi.org/10.1109/TFUZZ.2017.2723349
Zhu, B., & Xu, Z. S. (2014). Consistency measures for hesitant fuzzy linguistic preference relations. IEEE Transactions on Fuzzy Systems, 22(1), 35-45. https://doi.org/10.1109/TFUZZ.2013.2245136
Zhu, B., Xu, Z. S., & Xu, J. P. (2014). Deriving a ranking from hesitant fuzzy preference relations under group decision making. IEEE Transactions on Cybernetics, 44(8), 1328-1337. https://doi.org/10.1109/TCYB.2013.2283021
Chen, T., Liu, L., Xiong, W., & Zhou, L. A. (2017). Real estate boom and misallocation of capital in China (Working paper). Princeton University.
Chiclana, F., Herrera-Viedma, E., Alonso, S., & Herrera F. (2009). Cardinal consistency of reciprocal preference relations: a characterization of multiplicative transitivity. IEEE Transactions on Fuzzy Systems, 17(1), 14-23. https://doi.org/10.1109/TFUZZ.2008.2008028
Chiclana, F., Herrera-Viedma, E., Herrera, F., & Alonso, S. (2007). Some induced ordered weighted averaging operators and their use for solving group decision-making problems based on fuzzy preference relations. European Journal of Operational Research, 182, 383-399. https://doi.org/10.1016/j.ejor.2006.08.032
Dong, Q. X., & Cooper, O. (2016). A peer-to-peer dynamic adaptive consensus reaching model for the group AHP decision making. European Journal of Operational Research, 250(2), 521-530. https://doi.org/10.1016/j.ejor.2015.09.016
Dong, Y. C., Xu, Y. F., Li, H. Y., & Feng, B. (2010). The OWAbased consensus operator under linguistic representation models using position indexes. European Journal of Operational Research, 203(2), 455-463. https://doi.org/10.1016/j.ejor.2009.08.013
Dubois, D. (2011). The role of fuzzy sets indecision sciences: old techniques and new directions. Fuzzy Sets and Systems, 184, 3-28. https://doi.org/10.1016/j.fss.2011.06.003
Dutta, S., & Folta, T. B. (2016). A comparison of the effect of angels and venture capitalists on innovation and value creation. Journal of Business Venturing, 31(1), 39-54. https://doi.org/10.1016/j.jbusvent.2015.08.003
Gomes, L. F. A. M., & González, X. I. (2012). Behavioral multicriteria decision analysis: further elaborations on the TODIM method. Foundations of Computing and Decision Sciences, 37(1), 3-8. https://doi.org/10.2478/v10209-011-0001-1
Gou, X. J., Liao, H. C., Xu, Z. S., & Herrera, F. (2017). Double hierarchy hesitant fuzzy linguistic term set and MULTIMOORA method: a case of study to evaluate the implementation status of haze controlling measures. Information Fusion, 38, 22-34. https://doi.org/10.1016/j.inffus.2017.02.008
Gou, X. J., Liao, H. C., Xu, Z. S., Min. R, & Herrera, F. (2019). Group decision making with double hierarchy hesitant fuzzy linguistic preference relations: consistency based measures, index and repairing algorithms and decision model. Information Sciences, 489, 93-112. https://doi.org/10.1016/j.ins.2019.03.037
Gou, X. J., Xu, Z. S., & Herrera, F. (2018a). Consensus reaching process for large-scale group decision making with double hierarchy hesitant fuzzy linguistic preference relations. Knowledge-Based Systems, 157, 20-33. https://doi.org/10.1016/j.knosys.2018.05.008
Gou, X. J., Xu, Z. S., Liao, H. C., & Herrera, F. (2018b). Multiple criteria decision making based on distance and similarity measures with double hierarchy hesitant fuzzy linguistic term sets. Computers & Industrial Engineering, 126, 516-530. https://doi.org/10.1016/j.cie.2018.10.020
Herrera, F., Herrera-Viedma, E., & Verdegay, J. L. (1996). A model of consensus in group decision making under linguistic assessments. Fuzzy Sets and Systems, 78(1), 73-87. https://doi.org/10.1016/0165-0114(95)00107-7
Herrera, F., Herrera-Viedma, E., & Verdegay, J. L. (1995). A sequential selection process in group decision making with a linguistic assessment approach. Information Sciences, 85(4), 223-239. https://doi.org/10.1016/0020-0255(95)00025-K
Herrera, F., & Martínez, L. (2000). A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Transactions on Fuzzy Systems, 8, 746-752. https://doi.org/10.1109/91.890332
Herrera-Viedma, E., Alonso, S., Chiclana, F., & Herrera, F. (2007). A consensus model for group decision making with incomplete fuzzy preference relations. IEEE Transactions on Fuzzy Systems, 15(5), 863-877. https://doi.org/10.1109/TFUZZ.2006.889952
Karplus, K. P., & Diederichs, K. (2012). Linking crystallographic model and data quality. Science, 336(6084), 1030-1033. https://doi.org/10.1126/science.1218231
Krishankumar, R., Subrajaa, L. S., Ravichandran, K. S., Kar, S., & Saeid, A. B. (2019). A framework for multi-attribute group decision-making using double hierarchy hesitant fuzzy linguistic term set. International Journal of Fuzzy Systems. https://doi.org/10.1007/s40815-019-00618-w
Li, C. C., Dong, Y. C., Xu, Y. J., Chiclana, F., Herrera-Viedma, E., & Herrera, F. (2019). An overview on managing additive consistency of reciprocal preference relations for consistency-driven decision making and fusion: taxonomy and future directions. Information Fusion, 52, 143-156. https://doi.org/10.1016/j.inffus.2018.12.004
Li, J., & Wang, J. Q. (2018). Multi-criteria decision making with probabilistic hesitant fuzzy information based on expected multiplicative consistency. Neural Computing and Applications. https://doi.org/10.1007/s00521-018-3753-1
Li, J., Wang, J. Q., & Hu, J. H. (2019). Consensus building for hesitant fuzzy preference relations with multiplicative consistency. Computers & Industrial Engineering, 128, 387-400. https://doi.org/10.1016/j.cie.2018.12.051
Liao, H. C., Xu, Z. S., Zeng, X. J., & Xu, D. L. (2015). An enhanced consensus reaching process in group decision making with intuitionistic fuzzy preference relations. Information Sciences, 329(C), 274-286. https://doi.org/10.1016/j.ins.2015.09.024
Liu, C., & Xiong, W. (2018). China’s real estate market (Working paper). National Bureau of Economic Research. https://doi.org/10.3386/w25297
Luo, S. Z., Zhang, H. Y., Wang, J. Q., & Li, L. (2019) Group decision-making approach for evaluating the sustainability of constructed wetlands with probabilistic linguistic preference relations. Journal of the Operational Research Society. https://doi.org/10.1080/01605682.2018.151080
Ma, L. C. (2016). A new group ranking approach for ordinal preferences based on group maximum consensus sequences. European Journal of Operational Research, 251(1), 171-181. https://doi.org/10.1016/j.ejor.2015.10.042
Mendel, J. M. (2007). Type-2 fuzzy sets and systems: an overview. IEEE Computational Intelligence Magazine, 2(2), 20-29. https://doi.org/10.1109/MCI.2007.380672
Millet, I. (1997). The effectiveness of alternative preference elicitation methods in the analytic hierarchy process. Journal of Multi‐Criteria Decision Analysis, 6, 41-51. https://doi.org/10.1002/(SICI)1099-1360(199701)6:1<41::AID-MCDA122>3.0.CO;2-D
Montserrat-Adell, J., Xu, Z. S., Gou, X. J., & Agell, N. (2019). Free double hierarchy hesitant fuzzy linguistic term sets: an application on ranking alternatives in GDM. Information Fusion, 47, 45-59. https://doi.org/10.1016/j.inffus.2018.07.002
Orlorski, S. A. (1978). Decision-making with a fuzzy preference relation. Fuzzy Sets and Systems, 3, 155-167. http://doi.org/10.1016/0165-0114(78)90001-5
Qin, J. D., Liu, X. W., & Pedrycz, W. (2017). An extended TODIM multi-criteria group decision making method for green supplier selection in interval type-2 fuzzy environment. European Journal of Operational Research, 258(2), 626-638. https://doi.org/10.1016/j.ejor.2016.09.059
Rodríguez, R. M., Martínez, L., & Herrera, F. (2012). Hesitant fuzzy linguistic terms sets for decision making. IEEE Transactions on Fuzzy Systems, 20, 109-119. https://doi.org/10.1109/TFUZZ.2011.2170076
Saaty, T. L. (1980). The analytical hierarchy process. New York: McGraw-Hill. https://doi.org/10.21236/ADA214804
Schumpeter, J. A. (1934). The theory of economic development: an inquiry into profits, capital, credit, interest, and the business cycle. Social Science Electronic Publishing, 25(1), 90-91.
Wang, H., Xu, Z. S., & Zeng, X. J. (2018). Linguistic terms with weakened hedges: a model for qualitative decision making under uncertainty. Information Sciences, 433, 37-45. https://doi.org/10.1016/j.ins.2017.12.036
Wu, X. L., & Liao, H. C. (2019). A consensus-based probabilistic linguistic gained and lost dominance score method. European Journal of Operational Research, 272(3), 1017-1027. https://doi.org/10.1016/j.ejor.2018.07.044
Wu, Z. B., & Xu, J. P. (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
Xia, M. M., & Xu, Z. S. (2011). Some issues on multiplicative consistency of interval reciprocal relations. International Journal of Information Technology & Decision Making, 10(6), 1043-1065. https://doi.org/10.1142/S0219622011004701
Xia, M. M., Xu, Z. S., & Chen, J. (2013). Algorithms for improving consistency or consensus of reciprocal [0,1]-valued preference relations. Fuzzy Sets and Systems, 216, 108-133. https://doi.org/10.1016/j.fss.2012.09.016
Xia, M. M., Xu, Z. S., & Wang, Z. (2014). Multiplicative consistency-based decision support system for incomplete linguistic preference relations. International Journal of Systems Science, 45(3), 625-636. https://doi.org/10.1080/00207721.2012.724112
Xu, Z. S. (2005). Deviation measures of linguistic preference relations in group decision making. Omega, 33(3), 249-254. https://doi.org/10.1016/j.omega.2004.04.008
Xu, Z. S. (2007). Intuitionistic preference relations and their application in group decision making. Information Sciences, 177, 2363-2379. https://doi.org/10.1016/j.ins.2006.12.019
Xu, Z. S. (2008). Group decision making based on multiple types of linguistic preference relations. Information Sciences, 178, 452-467. https://doi.org/10.1016/j.ins.2007.05.018
Xu, Z. S., Cai, X. Q., & Szmidt, E. (2011). Algorithms for estimating missing elements of incomplete intuitionistic preference relations. International Journal of Intelligent Systems, 26(9), 787-813. https://doi.org/10.1002/int.20494
Yazdipour, R. (2011). Advances in entrepreneurial finance. New York: Springer. https://doi.org/10.1007/978-1-4419-7527-0
Yu, D. J., Xu, Z. S., & Wang, W. (2018). Bibliometric analysis of fuzzy theory research in China: a 30-year perspective. Knowledge-Based Systems, 141, 188-199. https://doi.org/10.1016/j.knosys.2017.11.018
Zacharakis, A. L., Mcmullen, J. S., & Shepherd, D. A. (2007). Venture capitalists’ decision making across three countries: an institutional theory perspective. Journal of International Business Studies, 38(5), 691-708. https://doi.org/10.1057/palgrave.jibs.8400291
Zadeh, L. A. (1975). The concept of a linguistic variable and its application to approximate reasoning I. Information Sciences, 8, 199-249. https://doi.org/10.1016/0020-0255(75)90036-5
Zadeh, L. A. (2012). Computing with words: what is computing with words (CWW)? (pp. 1-40). Springer. https://doi.org/10.1007/978-3-642-27473-2
Zhang, W. K., Du, J., & Tian, X. L. (2018). Finding a promising venture capital project with todim under probabilistic hesitant fuzzy circumstance. Technological and Economic Development of Economy, 24(5), 2026-2044. https://doi.org/10.3846/tede.2018.5494
Zhang, Y. X., Xu, Z. S., Wang, H., & Liao, H. C. (2016). Consistency-based risk assessment with probabilistic linguistic preference relation. Applied Soft Computing, 49, 817-833. https://doi.org/10.1016/j.asoc.2016.08.045
Zhang, X. Y., Zhang H. Y., & Wang, J. Q. (2019). Discussing incomplete 2-tuple fuzzy linguistic preference relations in multi-granular linguistic MCGDM with unknown weight information. Soft Computing, 23(6), 2015-2032. https://doi.org/10.1007/s00500-017-2915-x
Zhou, W., & Xu, Z. S. (2018). Probability calculation and element optimization of probabilistic hesitant fuzzy preference relations based on expected consistency. IEEE Transactions on Fuzzy Systems, 26(3), 1367-1378. https://doi.org/10.1109/TFUZZ.2017.2723349
Zhu, B., & Xu, Z. S. (2014). Consistency measures for hesitant fuzzy linguistic preference relations. IEEE Transactions on Fuzzy Systems, 22(1), 35-45. https://doi.org/10.1109/TFUZZ.2013.2245136
Zhu, B., Xu, Z. S., & Xu, J. P. (2014). Deriving a ranking from hesitant fuzzy preference relations under group decision making. IEEE Transactions on Cybernetics, 44(8), 1328-1337. https://doi.org/10.1109/TCYB.2013.2283021