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**Distance and similarity measures for hesitant fuzzy sets.**
*(English)*
Zbl 1219.03064

Summary: We propose a variety of distance measures for hesitant fuzzy sets, based on which the corresponding similarity measures can be obtained. We investigate the connections of the aforementioned distance measures and further develop a number of hesitant ordered weighted distance measures and hesitant ordered weighted similarity measures. They can alleviate the influence of unduly large (or small) deviations on the aggregation results by assigning them low (or high) weights. Several numerical examples are provided to illustrate these distance and similarity measures.

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\textit{Z. Xu} and \textit{M. Xia}, Inf. Sci. 181, No. 11, 2128--2138 (2011; Zbl 1219.03064)

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### References:

[1] | Atanassov, K., Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87-96 (1986) · Zbl 0631.03040 |

[2] | Beliakov, G., Learn weights in the generalized OWA operator operators, Fuzzy Optimization and Decision Making, 4, 119-130 (2005) · Zbl 1078.90541 |

[3] | Buckley, J. J.; Hayashi, Y., Fuzzy input-output controllers are universal approximates, Fuzzy Sets and Systems, 58, 273-278 (1993) · Zbl 0793.93078 |

[4] | Bustince, H., Indicator of inclusion grade for interval-valued fuzzy sets: application to approximate reasoning based on interval-valued fuzzy sets, International Journal of Approximate Reasoning, 23, 137-209 (2000) · Zbl 1046.68646 |

[5] | Candan, K. S.; Li, W. S.; Priya, M. L., Similarity-based ranking and query processing in multimedia databases, Data and Knowledge Engineering, 35, 259-298 (2000) · Zbl 0948.68058 |

[6] | Chaudhuri, B. B.; Rosenfeld, A., A modified Hausdorff distance between fuzzy sets, Information Sciences, 118, 159-171 (1999) · Zbl 0946.68126 |

[7] | Diamond, P.; Kloeden, P., Metric Spaces of Fuzzy Sets Theory and Applications (1994), World Scientific Publishing: World Scientific Publishing Singapore · Zbl 0873.54019 |

[8] | Dubois, D.; Prade, H., Fuzzy Sets and Systems: Theory and Applications (1980), Academic Press: Academic Press New York · Zbl 0444.94049 |

[9] | Gorzalczany, M. B., A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets and Systems, 21, 1-17 (1987) · Zbl 0635.68103 |

[10] | 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 |

[11] | Hung, W. L.; Yang, M. S., Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance, Pattern Recognition Letters, 25, 1603-1611 (2004) |

[12] | Hung, W. L.; Yang, M. S., Similarity measures of intuitionistic fuzzy sets based on \(L_p\) metric, International Journal of Approximate Reasoning, 46, 120-136 (2007) · Zbl 1141.68591 |

[13] | Kacprzyk, J., Multistage Fuzzy Control (1997), Wiley: Wiley Chichester |

[14] | Kahraman, C.; Kaya, I., A fuzzy multicriteria methodology for selection among energy alternatives, Expert Systems with Applications, 37, 6270-6281 (2010) |

[15] | Li, D. F.; Cheng, C. T., New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions, Pattern Recognition Letters, 23, 221-225 (2002) · Zbl 0996.68171 |

[16] | Li, Y. H.; Olson, D. L.; Qin, Z., Similarity measures between intuitionistic fuzzy (vague) sets: a comparative analysis, Pattern Recognition Letters, 28, 278-285 (2007) |

[17] | Liang, Z. Z.; Shi, P. F., Similarity measures on intuitionistic fuzzy sets, Pattern Recognition Letters, 24, 2687-2693 (2003) · Zbl 1091.68102 |

[18] | 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 |

[19] | Liu, H. W.; Wang, G. J., Multi-criteria decision making methods based on intuitionistic fuzzy sets, European Journal of Operational Research, 197, 220-233 (2007) · Zbl 1163.90558 |

[21] | Merigó, J. M.; Gil-Lafuente, A. M., The induced generalized OWA operator, Information Sciences, 179, 729-741 (2009) · Zbl 1156.91336 |

[22] | Merigó, J. M.; Casanovas, M., Induced aggregation operators in decision making with the Dempster-Shafer belief structure, International Journal of Intelligent Systems, 24, 934-954 (2009) · Zbl 1176.68202 |

[23] | Merigó, J. M.; Gil-Lafuente, A. M., New decision-making techniques and their application in the selection of financial products, Information Sciences, 180, 2085-2094 (2010) · Zbl 1194.91070 |

[24] | Mitchell, H. B., On the Dengfeng-Chuntian similarity measure and its application to pattern recognition, Pattern Recognition Letters, 24, 3101-3104 (2003) |

[25] | Mitchell, H. B., Pattern recognition using type-II fuzzy sets, Information Sciences, 170, 409-418 (2005) |

[26] | Miyamoto, S., Multisets and fuzzy multisets, (Liu, Z.-Q.; Miyamoto, S., Soft Computing and Human-centered Machines (2000), Springer: Springer Berlin), 9-33 · Zbl 0961.03048 |

[27] | Miyamoto, S., Remarks on basics of fuzzy sets and fuzzy multisets, Fuzzy Sets and Systems, 156, 427-431 (2005) · Zbl 1079.03548 |

[28] | Pal, S. K.; King, R. A., Image enhancement using smoothing with fuzzy sets, IEEE Transactions on Systems, Man and Cybernetics, 11, 495-501 (1981) |

[29] | Szmidt, E.; Kacprzyk, J., Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems, 114, 505-518 (2000) · Zbl 0961.03050 |

[30] | Szmidt, E.; Kacprzyk, J., Intuitionistic fuzzy sets in intelligent data analysis for medical diagnosis, (Alexandrov, V. A.; etal., CCS 2001. CCS 2001, LNCS, vol. 2074 (2001), Springer-Verlag: Springer-Verlag Berlin, Heidelberg), 263-271 · Zbl 0983.68658 |

[31] | Torra, V., Hesitant fuzzy sets, International Journal of Intelligent Systems, 25, 529-539 (2010) · Zbl 1198.03076 |

[33] | Turksen, I. B.; Zhong, Z., An approximate analogical reasoning approach based on similarity measures, IEEE Transactions on Systems, Man and Cybernetics, 18, 1049-1056 (1988) |

[34] | Wang, T. J.; Lu, Z. D.; Li, F., Bidirectional approximate reasoning based on weighted similarity measures of vague sets, Journal of Computer Engineering and Science, 24, 96-100 (2002) |

[35] | Wang, W. Q.; Xin, X. L., Distance measure between intuitionistic fuzzy sets, Pattern Recognition Letters, 26, 2063-2069 (2005) |

[36] | Wu, D.; Mendel, J. M., A vector similarity measure for linguistic approximation: interval type-2 and type-1 fuzzy sets, Information Sciences, 178, 381-402 (2008) · Zbl 1126.68612 |

[37] | Wu, D.; Mendel, J. M., A comparative study of ranking methods, similarity measures and uncertainty measures for interval type-2 fuzzy sets, Information Sciences, 179, 1169-1192 (2009) |

[38] | Xu, Z. S., Deviation measures of linguistic preference relations in group decision making, Omega, 17, 432-445 (2005) |

[39] | Xu, Z. S., Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making, Fuzzy Optimization and Decision Making, 6, 109-121 (2007) · Zbl 1158.03319 |

[40] | Xu, Z. S., Intuitionistic preference relations and their application in group decision making, Information Sciences, 177, 2363-2379 (2007) · Zbl 1286.91043 |

[41] | Xu, Z. S.; Chen, J.; Wu, J. J., Clustering algorithm for intuitionistic fuzzy sets, Information Sciences, 178, 3775-3790 (2008) · Zbl 1256.62040 |

[42] | 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 |

[43] | Xu, Z. S., Choquet integrals of weighted intuitionistic fuzzy information, Information Sciences, 180, 726-736 (2010) · Zbl 1186.68469 |

[44] | 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 |

[45] | Xu, Z. S.; Chen, J., Ordered weighted distance measure, Journal of Systems Science and Systems Engineering, 16, 529-555 (2008) · Zbl 1154.03317 |

[46] | S Xu, Z.; Xia, M. M., Induced generalized intuitionistic fuzzy operators, Knowledge-Based Systems, 24, 197-209 (2011) |

[47] | Yager, R. R., On the theory of bags, International Journal of General Systems, 13, 23-37 (1986) |

[48] | Yager, R. R., On ordered weighted averaging aggregation operators in multi-criteria decision making, IEEE Transactions on Systems, Man, and Cybernetics18, 183-190 (1988) · Zbl 0637.90057 |

[49] | Yager, R. R., Generalized OWA aggregation operators, Fuzzy Optimization and Decision Making, 3, 93-107 (2004) · Zbl 1057.90032 |

[50] | Yager, R. R., Norms induced from OWA operators, IEEE Transactions on Fuzzy Systems, 18, 57-66 (2010) |

[51] | Yang, M. S.; Lin, D. C., On similarity and inclusion measures between type-2 fuzzy sets with an application to clustering, Computers and Mathematics with Applications, 57, 896-907 (2009) · Zbl 1186.62091 |

[52] | Yang, M. S.; Shih, H. M., Cluster analysis based on fuzzy relations, Fuzzy Sets and Systems, 120, 197-212 (2001) · Zbl 1013.68187 |

[53] | Zadeh, L. A., Fuzzy sets, Information and Control, 8, 338-353 (1965) · Zbl 0139.24606 |

[54] | 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 |

[55] | Zhao, H.; Xu, Z. S.; Ni, M. F.; Liu, S. S., Generalized aggregation operators for intuitionistic fuzzy Sets, International Journal of Intelligent Systems, 25, 1-30 (2010) · Zbl 1185.68660 |

[56] | Zhou, L. G.; Chen, H. Y., Continuous generalized OWA operator and its application to decision making, Fuzzy Sets and Systems (2010) |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.