Farhadinia, B. Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. (English) Zbl 1320.68187 Inf. Sci. 240, 129-144 (2013). Summary: The main purpose of this paper is to investigate the relationship between the entropy, the similarity measure and the distance measure for hesitant fuzzy sets (HFSs) and interval-valued hesitant fuzzy sets (IVHFSs). The primary goal of the study is to suggest the systematic transformation of the entropy into the similarity measure for HFSs and vice versa. Achieving this goal is important to the task of introducing new formulas for the entropy and the similarity measure of HFSs. With results having been obtained for HFSs, similar results are also obtainable for IVHFSs. This paper also discusses the need for proposing a new entropy for HFSs and subsequently a new similarity measure for HFSs. Finally, two clustering algorithms are developed under a hesitant fuzzy environment in which indices of similarity measures of HFSs and IVHFSs are applied in data analysis and classification. Moreover, two practical examples are examined to compare the proposed methods with the existing ones. Cited in 83 Documents MSC: 68T37 Reasoning under uncertainty in the context of artificial intelligence 94A17 Measures of information, entropy Keywords:hesitant fuzzy set; interval-valued hesitant fuzzy set; entropy; similarity measure; distance measure; clustering algorithm PDFBibTeX XMLCite \textit{B. Farhadinia}, Inf. Sci. 240, 129--144 (2013; Zbl 1320.68187) Full Text: DOI References: [1] Atanassov, K., Intuitionistic Fuzzy Sets: Theory and Applications (1999), Physica-Verlag: Physica-Verlag Heidelberg, NewYork · Zbl 0939.03057 [2] Burillo, P.; Bustince, H., Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets, Fuzzy Sets and Systems, 118, 305-316 (1996) · Zbl 0872.94061 [3] De Luca, A.; Termini, S., A definition of nonprobabilistic entropy in the setting of fuzzy sets theory, Information and Control, 20, 301-312 (1972) · Zbl 0239.94028 [4] Chaudhuri, B. B.; Rosenfeld, A., A modified Hausdorff distance between fuzzy sets, Information Sciences, 118, 159-171 (1999) · Zbl 0946.68126 [5] Chen, T. Y.; Li, C. H., Determining objective weights with intuitionistic fuzzy entropy measures: A comparative analysis, Information Sciences, 180, 4207-4222 (2010) [6] Chen, N.; Xu, Z.; Xia, M., Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis, Applied Mathematical Modelling, 37, 2197-2211 (2013) · Zbl 1349.62293 [7] Grzegorzewski, P., Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric, Fuzzy Sets and Systems, 148, 319-328 (2004) · Zbl 1056.03031 [8] Hong, D. H.; Kim, C., A note on similarity measures between vague sets and between elements, Information Science, 115, 83-96 (1995) · Zbl 0936.03052 [9] Hung, W. L.; Yang, M. S., Similarity measures of intuitionistic fuzzy sets based on Huasdorff distance, Pattern Recognition Letter, 25, 1603-1611 (2004) [10] Hung, W. L.; Yang, M. S., Fuzzy entropy on intuitionistic fuzzy sets, International Journal of Intelligent Systems, 21, 443-451 (2006) · Zbl 1091.94012 [11] Li, D. F.; Cheng, C. T., New similarity measure of intuitionistic fuzzy sets and application to pattern recognitions, Pattern Recognition Letter, 23, 221-225 (2002) · Zbl 0996.68171 [12] Liu, X. C., Entropy, distance measure and similarity measure of fuzzy sets and their relations, Fuzzy Sets and Systems, 52, 305-318 (1992) · Zbl 0782.94026 [13] Mitchell, H. B., On the Dengfeng-Chuntian similarity measure and its application to pattern recognition, Pattern Recognition letters, 24, 3101-3104 (2003) [14] Pal, S. K.; King, R. A., Image enhancement using smoothing with fuzzy sets, IEEE Transactions on Systems, Man and Cybernetics, 11, 495-501 (1981) [15] Qian, G.; Wang, H.; Feng, X., Generalized hesitant fuzzy sets and their application in decision support system, Knowledge Based Systems, 37, 357-365 (2013) [16] Rodriguez, R. M.; Martinez, L.; Herrera, F., Hesitant fuzzy linguistic term sets for decision making, IEEE Transactions on Systems, 20, 109-119 (2012) [17] Szmidt, E.; Kacprzyk, J., Entropy for intuitionistic fuzzy sets, Fuzzy Sets and Systems, 118, 467-477 (2001) · Zbl 1045.94007 [19] Szmidt, E.; Kacprzyk, J.; Torra, V.; Narukawa, Y.; Miyamoto, S., A new concept of a similarity measure for intuitionistic fuzzy sets and its use in group decision making, Modelling Decision for Artificial Intelligence, LNAI 3558. Modelling Decision for Artificial Intelligence, LNAI 3558, Springer, 272-282 (2005) · Zbl 1121.68428 [20] Torra, V., Hesitant fuzzy sets, International Journal of Intelligent Systems, 25, 529-539 (2010) · Zbl 1198.03076 [21] Vlochos, I. K.; Sergiadis, G. D., Intuitionistic fuzzy information-applications to pattern recognition, Pattern Recognition Letter, 28, 197-206 (2007) [22] Wang, Y.; Lei, Y. J., A technique for constructing intuitionistic fuzzy entropy, Control and Decision, 22, 1390-1394 (2007) · Zbl 1150.94317 [23] Wei, G., Hesitant fuzzy prioritized operators and their application to multiple attribute decision making, Knowledge Based Systems, 31, 176-182 (2012) [26] Wu, J. Z.; Zhang, Q., Multicriteria decision making method based on intuitionistic fuzzy weighted entropy, Expert Systems with Applications, 38, 916-922 (2011) [27] Xia, M.; Xu, Z., Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning, 52, 395-407 (2011) · Zbl 1217.68216 [28] Xu, Z. S., A method based on distance measure for interval-valued intuitionistic fuzzy group decision making, Information Sciences, 180, 181-190 (2010) · Zbl 1183.91039 [29] Xu, Z. S., Deviation measures of linguistic preference relations in group decision making, Omega, 17, 432-445 (2005) [30] Xu, Z. S.; Chen, J., An overview of distance and similarity measures of intuitionistic fuzzy sets, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 16, 529-555 (2008) · Zbl 1154.03317 [31] Xu, Z. S.; Chen, J., Approach to group decision making based on interval-valued intuitionistic judgement matrices, System Engineering Theory and Practice, 27, 126-133 (2007) [32] Xu, Z. S.; Chen, J.; Wu, J. J., Clustering algorithm for intuitionistic fuzzy sets, Information Sciences, 178, 3775-3790 (2008) · Zbl 1256.62040 [33] Xu, Z.; Xia, M., Distance and similarity measures for hesitant fuzzy sets, Information Sciences, 181, 2128-2138 (2011) · Zbl 1219.03064 [34] Xu, Z.; Xia, M., Hesitant fuzzy entropy and cross-entropy and their use in multiattribute decision-making, International Journal of Intelligent Systems, 27, 799-822 (2012) [35] Yao, J.; Dash, M., Fuzzy clustering and fuzzy modeling, Fuzzy Sets and Systems, 113, 381-388 (2000) · Zbl 1147.62348 [36] Zadeh, L. A., Fuzzy sets, Information and Computation, 8, 338-353 (1965) · Zbl 0139.24606 [37] Zeng, W. Y.; Guo, P., Normalized distance, similarity measure, inclusion measure and entropy of interval-valued fuzzy sets and their relationship, Information Sciences, 178, 1334-1342 (2008) · Zbl 1134.68059 [38] Zeng, W. Y.; Li, H. X., Relationship between similarity measure and entropy of interval-valued fuzzy sets, Fuzzy Sets and Systems, 157, 1477-1484 (2006) · Zbl 1093.94038 [39] Zeng, W. Y.; Yu, F. S.; Yu, X. C.; Cui, B. Z., Entropy of intuitionistic fuzzy set based on similarity measure, (Proceedings of the 2008 3rd International Conference on Innovative Computing Information and Control, vol. 254 (2008), IEEE Computer Society: IEEE Computer Society Washington, DC, USA), 398 [40] Zhang, H. Y.; Zhang, W. X.; Mei, C. L., Entropy of interval-valued fuzzy sets based on distance and its relationship with similarity measure, Knowledge Based Systems, 22, 449-454 (2009) [41] Zhu, B.; Xu, Z.; Xia, M., Dual hesitant fuzzy sets, Journal of Applied Mathematics, 2012, 1-13 (2012) · Zbl 1244.03152 [42] Zhu, B.; Xu, Z.; Xia, M., Hesitant fuzzy geometric Bonferroni means, Information Sciences, 205, 72-85 (2012) · Zbl 1250.91035 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.