Abstract
The hesitant fuzzy linguistic term set (HFLTS) and its extensions have been intensively investigated as a useful tool for the qualitative information problems in recent years. In the present paper, the concept of hesitant fuzzy linguistic term soft sets (HFLTSSs), a combination of HFLTSs and soft sets, are developed. First of all, relationships between any two HFLTSSs are provided, including inclusion, equivalence, and complementation. Then, several basic set operations for HFLTSSs are studied, such as AND, OR, union, and intersection. Meanwhile, corresponding properties of these operations are also discussed. In view of decision-making (DM) problem, we propose two algorithms for HFLTSSs, using the level soft sets and the hesitant fuzzy linguistic weighted distance (HFLWD) operator. Finally, a numerical “third-party evaluation”-related example in green development is used to present the utility and effectiveness of our algorithms. We also make comparisons between proposed method and some existing ones to confirm its feasibility and rationality. The main contribution of this paper possesses three points: (1) Enriching soft set theory by proposing the HFLTSSs, a soft set that can reflect the evaluation on objects more reasonably. (2) Redefining the HFLWD operator based on a novel distance of any two hesitant fuzzy linguistic term elements (HFLTEs), which can make the operations among HFLTEs much easier. (3) Applying the level soft set and new distance operator to develop two algorithms for decision-making problem with the information of HFLTSs.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs40815-018-0506-3/MediaObjects/40815_2018_506_Fig1_HTML.png)
Similar content being viewed by others
References
Zadeh, L.A.: The concept of a linguistic variable and its applications to approximate reasoning-Part I. Inform. Sciences. 8, 199–249 (1975)
Herrera, F., Martinez, L.: A 2-tuple linguistic representation model for computing with words. IEEE T. Fuzzy Syst. 8, 746–752 (2000)
Xu, Z.S.: Deviation measures of linguistic preference relations in group decision making. Omega. 33, 249–254 (2005)
Rodríguez, R.M., Martínez, L., Herrera, F.: Hesitant fuzzy linguistic term sets for decision making. IEEE T. Fuzzy Syst. 20(1), 109–119 (2012)
Torra, V., Narukawa, Y.: On hesitant fuzzy sets and decision. In: Proceedings in the 18th IEEE International Conference On Fuzzy Systems, Korea, pp. 1378–1382 (2009)
Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25(6), 529–539 (2010)
Wei, C.P., Zhao, N., Tang, X.J.: Operators and comparisons of hesitant fuzzy linguistic term sets. IEEE Trans. Fuzzy Syst. 22(3), 575–585 (2014)
Zhang, Z.M., Wu, C.: Hesitant fuzzy linguistic aggregation operators and their applications to multiple attribute group decision making. J. Intell. Fuzzy Syst. 26, 2185–2202 (2014)
Wang, J.Q., Wu, J.T., Wang, J., Zhang, H.Y., Chen, X.H.: Interval-valued hesitant fuzzy linguistic sets and their applications in multi-criteria decision-making problems Inform. Sciences. 288, 55–72 (2014)
Durand, M., Truck, I.: A new proposal to deal with hesitant linguistic expressions on preference assessments. Inform. Fusion. 41, 176–181 (2018)
Liao, H.C., Xu, Z.S., Zeng, X.J.: Distance and similarity measures for hesitant fuzzy linguistic term sets and their application in multi-criteria decision making. Inform. Sci. 271, 125–142 (2014)
Liao, H.C., Xu, Z.S.: Approaches to manage hesitant fuzzy linguistic information based on the cosine distance and similarity measures for HFLTSs and their application in qualitative decision making. Expert Syst. Appl. 42, 5328–5336 (2015)
Liao, H.C., Xu, Z.S., Zeng, X.J., Merigó, J.M.: Qualitative decision making with correlation coefficients of hesitant fuzz y linguistic term sets. Knowl.-Based Syst. 76, 127–138 (2015)
Xu, Y.J., Xu, A.W., Merigó, J.M., Wang, H.M.: Hesitant fuzzy linguistic ordered weighted distance operators for group decision making. J. Appl. Math. Comput. 49, 285–308 (2015)
Gou, X.J., Xu, Z.S., Liao, H.C.: Hesitant fuzzy linguistic entropy and cross-entropy measures and alternative queuing method for multiple criteria decision making. Inform. Sciences, pp. 225–246 (2017)
Meng, F.Y., Chen, X.H.: A hesitant fuzzy linguistic multi-granularity decision making model based on distance measures. J. Intell. Fuzzy Syst. 28, 1519–1531 (2015)
Zeng, W.Y., Li, D.Q.: Distance and similarity measures between hesitant fuzzy sets and their application in pattern recognition. Pattern Recog. Lett. 84, 267–271 (2016)
Peng, D.H., Wang, T.D., Gao, C.Y., Wang, H.: Enhancing relative ratio method for MCDM via attitudinal distance measures of interval-valued hesitant fuzzy sets. Int. J. Mach. Learn. & Cyber. 8, 1347–1368 (2017)
Farhadinia, B.: Distance and similarity measures for higher order hesitant fuzzy sets. Knowl.-Based Syst. 55, 43–48 (2014)
Peng, D.H., Gao, C.Y., Gao, Z.F.: Generalized hesitant fuzzy synergetic weighted distance measures and their application to multiple criteria decision-making. Appl. Math. Model. 37(8), 5837–5850 (2013)
Zhu, B., Xu, Z.S.: Consistency measures for hesitant fuzzy linguistic preference relations. IEEE T. Fuzzy Syst. 22(1), 35–45 (2014)
Liu, H.B., Cai, J.F., Jiang, L.: On improving the additive consistency of the fuzzy preference relations based on comparative linguistic expressions. Int. J. Intell. Syst. 29, 544–559 (2014)
Zhang, Z.M., Wu, C.: On the use of multiplicative consistency in hesitant fuzzy linguistic preference relations. Knowl.-Based Syst. 72, 13–27 (2014)
Song, Y.M., Hu, J.: A group decision-making model based on incomplete comparative expressions with hesitant linguistic terms. Appl. Soft Comput. 59, 174–181 (2017)
Beg, I., Rashid, T.: TOPSIS for hesitant fuzzy linguistic term sets. Int. J. Intell. Syst. 28, 1162–1171 (2013)
Wei, C.P., Ren, Z.L., Rodriguez, R.M.: A hesitant fuzzy linguistic TODIM method based on a score function. Int. J. Comput. Intell. Syst. 8(4), 701–712 (2015)
Liao, H.C., Xu, Z.S., Zeng, X.J.: Hesitant fuzzy linguistic VIKOR method and its application in qualitative multiple criteria decision making. IEEE T. Fuzzy Syst. 23(5), 1343–1355 (2015)
Wang, J.Q., Wang, J., Chen, Q.H., Zhang, H.Y., Chen, X.H.: An outranking approach for multi-criteria decision-making with hesitant fuzzy linguistic term sets. Inform. Sciences. 280, 338–351 (2014)
Wang, H., Xu, Z.S., Zeng, X.J.: Hesitant fuzzy linguistic term sets for linguistic decision making: current developments, issues and challenges. Inform. Fusion. 43, 1–12 (2018)
Molodtsov, D.: Soft set theory—first results. Comput. Math Appl. 37(4–5), 19–31 (1999)
Maji, P.K., Biswas, R., Roy, A.R.: Soft set theory. Comput. Math Appl. 45, 555–562 (2003)
Ali, M.I., Feng, F., Liu, X.: On some new operations in soft set theory. Comput. Math Appl. 57(9), 1547–1553 (2009)
Sezgin, A., Atagun, A.O.: On operations of soft sets. Comput. Math Appl. 61, 1457–1467 (2011)
Majumdar, P., Samanta, S.K.: Similarity measures of soft sets. New Math. Nat. Compu. 4(1), 1–12 (2008)
Wang, C., Qu, A.J.: Entropy, similarity measure and distance measure of vague soft sets and their relations. Inform. Sciences. 244, 92–106 (2013)
Şahin, R., Kucuk, A.: On similarity and entropy of neutrosophic soft sets. J. Intell. Fuzzy Syst. 27, 2417–2430 (2014)
Roy, A.R., Maji, P.K.: A fuzzy soft set theoretic approach to decision making problems. J. Comput. Appl. Math. 203, 412–418 (2007)
Feng, F., Jun, Y.B., Liu, X.Y., et al.: An adjustable approach to fuzzy soft set based decision making. J. Comput. Appl. Math. 234, 10–20 (2010)
Çağman, N., Enginoğlu, S.: Soft set theory and uni-int decision making. Eur. J. Oper. Res. 207, 848–855 (2010)
Li, M.Y., Fan, Z.P., You, T.H.: Screening alternatives considering different evaluation index sets: a method based on soft set theory. Appl. Soft Comput. 64, 614–626 (2018)
Zhan, J.M., Alib, M.I., Mehmood, N.: On a novel uncertain soft set model: z-soft fuzzy rough set model and corresponding decision making methods. Appl. Soft Comput. 56, 446–457 (2017)
Xiao, Z., Yang, X.L., Niu, Q., Dong, Y.X., Gong, K., Xia, S.S., Pang, Y.: A new evaluation method based on D-S generalized fuzzy soft sets and its application in medical diagnosis problem. Appl. Math. Model. 36, 4592–4604 (2012)
Maji, P.K., Biswas, R., Roy, A.R.: Fuzzy soft sets. J. Fuzzy Math. 9, 589–602 (2001)
Yang, W.B., Lin, T.Y., Yang, J.Y., Li, Y., Yu, D.J.: Combination of interval-valued fuzzy set and soft set. Comput. Math Appl. 58, 521–527 (2009)
Maji, P.K., Biswas, R., Roy, A.R.: Intuitionistic fuzzy soft sets. J. Fuzzy Math. 9(3), 677–692 (2001)
Xiao, Z., Xia, S., Gong, K., Li, D.: The trapezoidal fuzzy soft set and its application in MCDM. Appl. Math. Model. 36, 5844–5855 (2012)
Yang, Y., Tan, X., Meng, C.: The multi-fuzzy soft set and its application in decision making. Appl. Math. Model. 37, 4915–4923 (2013)
Feng, F., Li, C.X., Davvaz, B., Ali, M.I.: Soft sets combined with fuzzy sets and rough sets: a tentative approach. Soft. Comput. 14, 899–911 (2010)
Feng, F., Liu, X.Y., Fotea, V., Jun, Y.B.: Soft sets and soft rough sets. Inform. Sci. 181, 1125–1137 (2011)
Tao, Z.F., Chen, H.Y., Song, X., Zhou, L.G., Liu, J.P.: Uncertain linguistic fuzzy soft sets and their applications in group decision making. Appl. Soft Comput. 34, 587–605 (2015)
Tao, Z.F., Chen, H.Y., Zhou, L.G., Liu, J.P.: 2-Tuple linguistic soft set and its application to group decision making. Soft. Comput. 19, 1201–1213 (2015)
Wang, F.Q., Li, X.H., Chen, X.H.: Hesitant fuzzy soft set and its applications in multicriteria decision making. J. Appli. Math. 3, 1–10 (2014)
Das, S., Kar, S.: The hesitant fuzzy soft set and its application in decision-making. Proc. Math. Stat. 125, 235–247 (2015)
Wang, J.Q., Li, X.E., Chen, X.H.: Hesitant fuzzy soft sets with application in multicriteria group decision making problems. Sci. World. J. Article ID 806983. (2015)
Peng, X.D., Yang, Y.: Interval-valued hesitant fuzzy soft sets and their application in decision making. Fund. Inform. 141, 71–93 (2015)
Zhang, H.D., Xiong, L.L., Ma, W.Y.: On interval-valued hesitant fuzzy soft set. Math. Probl. Eng. 3, 1–17 (2015)
Peng, X.D., Yang, Y.: Approaches to interval-valued intuitionistic hesitant fuzzy soft sets based decision making. Annals of Fuzzy Mathematics and Informatics. 10(4), 657–680 (2015)
Malakar, D., Gope, S., Das, S.: Correlation measure of hesitant fuzzy linguistic term soft set and its application in decision making. In: The 4th International Conference on Frontiers in Intelligent Computing: Theory and Applications (FICTA), Advances in Intelligent Systems and Computing, Springer, New Delhi, pp. 413–423(2016)
Gou, X.J., Xu, Z.S., Liao, H.C.: Multi-criteria decision making based on Bonferroni means with hesitant fuzzy linguistic information. Soft. Comput. (2016). https://doi.org/10.1007/s00500-016-2211-1
Gou, X.J., Liao, H.C., Xu, Z.S., Herrera, F.: Double hierarchy hesitant fuzzy linguistic term set and MULTIMOORA method: a case of study to evaluate the implementation status of haze controlling measures. Inform. Fusion. 38, 22–34 (2017)
Xu, Z.S.: Uncertain Multiple Attribute Decision Making: Methods And Applications. Tsinghua University Press, Beijing (2004)
Xia, M.M., Xu, Z.S.: Hesitant fuzzy information aggregation in decision making. Int. J. Approx. Reason. 52, 395–407 (2011)
Gou, X.J., Xu, Z.S.: Novel basic operational laws for linguistic, hesitant fuzzy linguistic term sets and probabilistic linguistic term sets. Inform. Sci. 372, 407–427 (2016)
Hwang, C.L., Yoon, K.: Multiple Attribute Decision Making Methods and Applications. Springer, Berlin (1981)
Wang, Y.M.: Using the method of maximizing deviations to make decision for multiindices. Syst. Eng. Electron. 20(7), 24–26 (1998)
Acknowledgements
The authors are deeply grateful to the editor and referees for their careful reading and insightful comments. We acknowledge the support of NSF of Anhui Province (No. 1808085QG211), SSRF of China (No. 2017LZ11), Doctoral Research Start-up Funds Projects of Hefei Normal University (No.2017rcjj03), and Open Project of Anhui University.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, X., Chen, H. & Zhou, L. Hesitant Fuzzy Linguistic Term Soft Sets and Their Applications in Decision Making. Int. J. Fuzzy Syst. 20, 2322–2336 (2018). https://doi.org/10.1007/s40815-018-0506-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40815-018-0506-3