Abstract
Granular computing which focuses on everyday and commonly used concepts and notions is a new field of multi-disciplinary study in dealing with theories, methodologies and techniques. As an important role in granular computing, hierarchy has attracted considerable attention. We are usually hesitant and irresolute for one thing when making decisions, which leads to a set of possible membership degrees. However, the existing hierarchies focus on crisp environment or fuzzy environment in which each element of the set has only one membership degree. To fill this gap, we research the hierarchies on hesitant fuzzy information granulations whose information granule has at least one membership degree of one object to the reference set. Firstly, we put forward new orders on hesitant fuzzy sets to characterize the hierarchies on hesitant fuzzy sets, the relationships of these orders are also researched. Moreover, we characterize the hierarchies on hesitant fuzzy information granulations from the viewpoint of granular computing. And then, new orders are presented to characterize the hierarchies on hesitant fuzzy information granulations. The order based hierarchies on hesitant fuzzy approximation space provide us with a more comprehensible perspective for the study of granular computing. Finally, two examples are given. One example is employed to compare the differences among the proposed orders on hesitant fuzzy sets, the other example is illustrated to show the orders on hesitant fuzzy sets that can be applied to hesitant fuzzy multi-attribute decision making. The results show that the orders proposed in this paper are effective to characterize the hierarchies on hesitant fuzzy approximation space.
Similar content being viewed by others
References
Chen N, Xu ZS, Xia MM (2013) Correlation coefficients of hesiant fuzzy sets and their applications to clustering analysis. Appl Math Model 37:2197–2211
Chiaselotti G, Gentile T, Infusino F (2017) Knowledge pairing systems in granular computing. Knowl Based Syst 124:144–163
Deepak D, Sunil JJ (2014) Hesitant fuzzy rough sets through hesitant fuzzy relations. Ann Fuzzy Math Inf 8(1):33–46
Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17:191–209
Farhadinia B (2014) Correlation for dual hesitant fuzzy sets and dual interval-valued hesitant fuzzy sets. Int J Intell Syst 29:184–205
Feng XQ, Tan QY, Wei CP (2017) Hesitant fuzzy linguisitc multi-criteria decision making based on possibility theory. Int J Mach Learn Cybern. https://doi.org/10.1007/s1304201706597
Han ZY, Zhao J, Liu QL, Wang W (2016) Granular-computing based hybrid collaborative fuzzy clustering for long-term prediction of multiple gas holders levels. Inf Sci 330:175–185
Hobbs JR (1985) Granularity. In: Proceedings of the 9th Intational Joint Conference on Artificial Intelligence, pp 432–435
Huang B, Guo CX, Li HX, Feng GF, Zhang XZ (2016) Hierarchical structures and uncertainty measures for intuitionistic fuzzy approximation space. Inf Sci 336:92–114
Kahraman C, Kaya I (2010) A fuzzy multicriteria methodology for selection among alternatives. Expert Syst Appl 37:6270–6281
Kang XP, Miao DQ, Lin GP, Liu Y (2017) Relation granulation and algebraic structure based on concept lattice in complex information systems. Int J Mach Learn Cybern. https://doi.org/10.1007/s1304201706980
Kuo RJ, Lin L, Zulvia FE, Lin CC (2017) Integration of cluster analysis and granular computing for imbalanced data classification: a case study on prostate cancer prognosis in Taiwan. J Intell Fuzzy Syst 32(3):2251–2267
Li JH, Mei CL, Xu WH, Qian YH (2015) Concept learning via granular computing: a cognitive view point. Inf Sci 298:447–467
Li WT, Xu WH (2015) Double-quantitative decision-theoretic rough set. Inf Sci 316:54–67
Liao HC, Xu ZS, Xia MM (2014) Multiplicative consistency of hesitant fuzzy preference relation and its application in group decision making. Int J Inf Technol Decis Mak 13(1):47–76
Liao HC, Xu ZS (2017) Hesitant fuzzy decision making methodologies and applications. Springer, New York
Liu HB, Li WH, Li R (2017) A comparative analysis of granular computing clustering from the view of set. J Intell Fuzzy Syst 32(1):509–519
Priestley HA (2002) Ordered sets and complete lattices: a primer for computer science. In: Backhouse R et al (eds) Algebraic and coalgebraic methods in the mathematics of program construction, vol 2297. LNCS, pp 21–78
Qian YH, Liang JY, Dang CY (2009) Knowledge structure, knowledge granulation and knowledge distance in a knowledge base. Int J Approx Reason 50:174–188
Qian YH, Liang JY, Wu WZ, Dang CY (2011) Information granularity in fuzzy binary GrC model. IEEE Trans Fuzzy Syst 19(2):253–264
Qian YH, Dang CY, Liang JY, Wu WZ (2012) Partial ordering of information granulations: a further investigation. Expert Syst 29(1):3–24
Qian YH, Li YB, Liang JY, Lin GP, Dang CY (2015) Fuzzy granular structure distance. IEEE Trans Fuzzy Syst 23(6):2245–2259
Rodríguez RM, Martínez L, Torra V, Xu ZS, Herrera F (2014) Hesitant fuzzy sets: State of the art and future directions. Int J Intell Syst 29:495–524
Sang BB, Guo YT, Shi DR, Xu WH (2017) Decision-theoretic rough set model of multi-source decision systems. Int J Mach Learn Cybern. https://doi.org/10.1007/s13042-017-0729-x
Ślȩzak D, Skowron A (2015) Preface. Nat Comput 14:567–568
Song JJ, Yang XB, Song XN, Yu HL, Yang JY (2014) Hierarchies on fuzzy information granulations: a knowledge distantce based lattice approach. J Intell Fuzzy Syst 27:1107–1117
Song JJ, Yang XB, Qi Y, Yu HL, Song XN, Yang JY (2014) Characterizing hierarchies on covering-based multigranulation spaces. In: Miao D et al (eds) The 9th International Conference on Rough Sets and Knowledge Technology, vol 8818. LNAI, pp 467–478
Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25:529–539
Verma R (2017) Hesitant interval-valued fuzzy sets: some new results. Int J Mach Learn Cybern 8(3):865–876
Wang R, Wang XZ, Kwong S, Xu C (2017) Incorporating diversity and informativeness in multiple-instance active learning. IEEEE Trans Fuzzy Syst 25(6):1460–1475
Wang XZ, Xing HJ, Li Y, Hua Q, Dong CR, Pedrycz W (2015) A study on relationship between generalization abilities and fuzziness of base classifiers in ensemble learning. IEEE Trans Fuzzy Syst 23(5):1638–1654
Wang XZ, Wang R, Xu C (2018) Discovering the relationship between generalization and uncertainty by incorporating complexity of classification. IEEE Trans Cybern 48(2):703–715
Xia MM, Xu ZS (2011) Hesitant fuzzy information aggregation in decision making. Int J Approx Reason 52:395–407
Xia MM, Xu ZS (2017) Some studies on properties of hesitant fuzzy sets. Int J Mach Learn Cyben 8(2):489–495
Xu SP, Yang XB, Yu HL, Yu DJ, Yang JY, Tsang Eric CC (2016) Multi-label learning with label-specific feature reduction. Knowl Based Syst 104:52–61
Xu WH, Guo YT (2016) Generalized multigranulation double-quantitative decision-theoretic rough set. Knowl Based Syst 105:190–205
Xu WH, Li WT (2016) Granular computing approach to two-way learning based on formal concept analysis in fuzzy datasets. IEEE Trans Cybern 46(2):366–379
Xu WH, Li WT, Zhang XT (2017) Generalized multigranulation rough sets and optimal granularity selection. Granul Comput 2:271–288
Xu WH, Wang QR, Zhang XT (2011) Multi-granulation fuzzy rough sets in a fuzzy tolerance approximation space. Int J Fuzzy Syst 13(4):246–259
Xu WH, Yu JH (2017) A novel approach to information fusion in multi-source datasets: a granular computing viewpoint. Inf Sci 378:410–423
Xu YJ, Rui D, Wang HM (2015) Dual hesitant fuzzy interaction operators and their application to group decision making. J Ind Prod Eng 32(4):273–290
Xu ZS, Xia MM (2011) Distance and similarity measures for hesitant fuzzy sets. Inf Sci 181:2128–2138
Xu ZS, Xia MM (2012) Hesitant fuzzy entropy and cross-enropy and their use in multiattribute decision-making. Int J Intell Syst 27:799–822
Xu ZS (2014) Hesitant fuzzy sets theory. Springer, New York
Yan L, Yan S (2016) Granular computing and attribute reduction based on a new discernibility function. Int J Simul Syst Sci Technol 17(33):1–10
Yang XB, Qian YH, Yang JY (2012) Hierarchical structures on multigranulation spaces. J Comput Sci Technol 27(6):1169–1183
Yang XB, Qian YH, Yang JY (2012) On characterizing hierarchies of granulation structures via distances. Fundam Inf 122:1–16
Yang XB, Song XN, Qi YS (2014) Constuctive and axiomatic approaches to hesitant fuzzy rough set. Soft Comput 18:1067–1077
Yao YY (2008) A unified framework of granular computing. In: Pedrycz W, Skowron A, Kreinovich V (eds) Handbook of Granular Computing. Wiley, New York, pp 401–410
Yao YY, Zhang N, Miao DQ, Xu FF (2012) Set-theoretic approaches to granular computing. Fundam Inf 115:247–264
Yao YY (2016) A triarchic theory of granular computing. Granul Comput 1(2):145–157
Zadeh LA (1996) Fuzzy logic equals computing with words. IEEE Trans Fuzzy Syst 4:103–111
Zadeh LA (1997) Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. Fuzzy Sets Syst 90(2):111–127
Zadeh LA (1998) Some reflections on soft computing, granular computing and their roles in the conception, design and utilization of information/intelligent systems. Soft Comput 2:23–25
Zhang HY, Yang SY (2015) Inclusion measure for typical hesitant fuzzy sets, the relative similarity measure and fuzzy entropy. Soft Comput 20:1–11
Zhang HY, Yang SY (2016) Representations of typical hesitant fuzzy rough sets. J Intell Fuzzy Syst 31:457–468
Zhu B, Xu ZS, Xia MM (2012) Dual hesitant fuzzy sets. J Appl Math 2012:1–13
Zhu B, Xu ZS (2013) Regression methods for hesitant fuzzy preference relations. Technol Econ Dev Econ 19(Supplement 1):S214–S227
Zhu B, Xu ZS, Xu JP (2014) Deriving a ranking from hesitant fuzzy preference relations under group decision making. IEEE Trans Cybern 44:1328–1337
Zhu H, Wang XZ, Tsang ECC (2018) Training an extreme learning machine by localized generalization error model. Soft Comput. https://doi.org/10.1007/s00500-018-3012-5
Acknowledgements
This work is supported by the Macau Science and Technology Development Fund (No. 081/2015/A3), National Natural Science Foundation of China (Nos. 71471060 and 61572242).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Tsang, E.C.C., Song, J., Chen, D. et al. Order based hierarchies on hesitant fuzzy approximation space. Int. J. Mach. Learn. & Cyber. 10, 1407–1422 (2019). https://doi.org/10.1007/s13042-018-0822-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13042-018-0822-9