{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T02:31:59Z","timestamp":1771036319255,"version":"3.50.1"},"reference-count":42,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,1,23]],"date-time":"2020-01-23T00:00:00Z","timestamp":1579737600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61703426, 60975026, and 61273275."],"award-info":[{"award-number":["61703426, 60975026, and 61273275."]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Multiple attribute decision making (MADM) is full of uncertainty and vagueness due to intrinsic complexity, limited experience and individual cognition. Representative decision theories include fuzzy set (FS), intuitionistic fuzzy set (IFS), hesitant fuzzy set (HFS), dual hesitant fuzzy set (DHFS) and so on. Compared with IFS and HFS, DHFS has more advantages in dealing with uncertainties in real MADM problems and possesses good symmetry. The membership degrees and non-membership degrees in DHFS are simultaneously permitted to represent decision makers\u2019 preferences by a given set having diverse possibilities. In this paper, new distance measures for dual hesitant fuzzy sets (DHFSs) are developed in terms of the mean, variance and number of elements in the dual hesitant fuzzy elements (DHFEs), which overcomes some deficiencies of the existing distance measures for DHFSs. The proposed distance measures are effectively applicable to solve MADM problems where the attribute weights are completely unknown. With the help of the new distance measures, the attribute weights are objectively determined, and the closeness coefficients of each alternative can be objectively obtained to generate optimal solution. Finally, an evaluation problem of airline service quality is conducted by using the distance-based MADM method to demonstrate its validity and applicability.<\/jats:p>","DOI":"10.3390\/sym12020191","type":"journal-article","created":{"date-parts":[[2020,1,27]],"date-time":"2020-01-27T07:41:11Z","timestamp":1580110871000},"page":"191","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["New Distance Measures for Dual Hesitant Fuzzy Sets and Their Application to Multiple Attribute Decision Making"],"prefix":"10.3390","volume":"12","author":[{"given":"Rugen","family":"Wang","sequence":"first","affiliation":[{"name":"Air and Missile Defense College, Air Force Engineering University, Xi\u2019an 710051, China"}]},{"given":"Weimin","family":"Li","sequence":"additional","affiliation":[{"name":"Air and Missile Defense College, Air Force Engineering University, Xi\u2019an 710051, China"}]},{"given":"Tao","family":"Zhang","sequence":"additional","affiliation":[{"name":"Air and Missile Defense College, Air Force Engineering University, Xi\u2019an 710051, China"}]},{"given":"Qi","family":"Han","sequence":"additional","affiliation":[{"name":"Air and Missile Defense College, Air Force Engineering University, Xi\u2019an 710051, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1287\/mnsc.38.5.645","article-title":"Multiple criteria decision making, multiattribute utility theory:the next ten years","volume":"38","author":"Dyer","year":"1992","journal-title":"Manag. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets, Inform","volume":"8","author":"Zadeh","year":"1965","journal-title":"Control"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","article-title":"Intuitionistic fuzzy sets","volume":"20","author":"Atanassov","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Torra, V., and Narukawa, Y. (2009, January 20\u201324). On hesitant fuzzy sets and decision. Proceedings of the 18th IEEE International Conference on Fuzzy Systems, JejuIsland, Korea.","DOI":"10.1109\/FUZZY.2009.5276884"},{"key":"ref_5","first-page":"100","article-title":"Dual hesitant fuzzy sets","volume":"22","author":"Zhu","year":"2012","journal-title":"J. Appl. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.asoc.2015.09.012","article-title":"Dual hesitant fuzzy power aggregation operators based on Archimedean t-conorm and t-norm and their application to multiple attribute group decision making","volume":"38","author":"Wang","year":"2016","journal-title":"Appl. Soft. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"199","DOI":"10.3233\/IFS-141290","article-title":"Research on the computer network security evaluation based on the DHFHCG operator with dual hesitant fuzzy information","volume":"28","author":"Zhang","year":"2015","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"731","DOI":"10.3233\/IFS-141474","article-title":"Distance and similarity measure for dual hesitant fuzzy sets and their application in pattern recognition","volume":"29","author":"Su","year":"2015","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Wang, L., Xu, S., Wang, Q., and Ni, M. (2014, January 16\u201318). Distance and similarity measures of dual hesitant fuzzy sets with their applications to multiple attribute decision making. Proceedings of the International Conference on Progress in Informatics and Computing, Shanghai, China.","DOI":"10.1109\/PIC.2014.6972302"},{"key":"ref_10","first-page":"111","article-title":"Distance and similarity measures for multiple-attribute decision making with dual hesitant fuzzy sets","volume":"36","author":"Singh","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Zhao, N., and Xu, Z.S. (2015, January 4\u20136). Entropy measures for dual hesitant fuzzy information. Proceedings of the 2015 Fifth International Conference on Communication Systems and Network Technologies, Gwalior, India.","DOI":"10.1109\/CSNT.2015.266"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"7082","DOI":"10.1016\/j.apm.2015.02.046","article-title":"Correlation coefficient of dual hesitant fuzzy sets and its applications","volume":"39","author":"Tyagi","year":"2015","journal-title":"Appl. Math. Model."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1016\/j.apm.2013.07.010","article-title":"Correlation coefficient of dual hesitant fuzzy sets and its application to multiple attribute decision making","volume":"38","author":"Ye","year":"2014","journal-title":"Appl. Math. Model."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"20","DOI":"10.4018\/IJDSST.2016070102","article-title":"Cross-entropy of dual hesitant fuzzy sets for multiple attribute decision-making","volume":"8","author":"Ye","year":"2016","journal-title":"Int. J. Decis. Support Syst. Technol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1007\/s13042-015-0356-3","article-title":"A multi-attribute decision-making method with prioritization relationship and dual hesitant fuzzy decision information","volume":"8","author":"Ren","year":"2017","journal-title":"Int. J. Mach. Learn. Cybern."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2463","DOI":"10.3233\/JIFS-17649","article-title":"Some new Shapley dual hesitant fuzzy Choquet aggregation operators and their applications to multiple attribute group decision making-based TOPSIS","volume":"33","author":"Qu","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.ins.2017.02.038","article-title":"Three-way decisions based on decision-theoretic rough sets with dual hesitant fuzzy information","volume":"396","author":"Liang","year":"2017","journal-title":"Inf. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1784","DOI":"10.1007\/s40314-017-0423-3","article-title":"Distance measures for higher order dual hesitant fuzzy sets","volume":"37","author":"Chen","year":"2018","journal-title":"Comp. Appl. Math."},{"key":"ref_19","first-page":"155","article-title":"Dual hesitant bipolar fuzzy aggregation operators in multiple attribute decision making","volume":"21","author":"Xu","year":"2017","journal-title":"Int. J. Knowl.-Based Intell. Eng. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Zhang, F.W., Chen, J.H., Zhu, Y.H., Li, J.R., Li, Q., and Zhuang, Z.Y. (2017). A dual hesitant fuzzy rough pattern recognition approach based on deviation theories and its application in urban traffic modes recognition. Symmetry, 9.","DOI":"10.3390\/sym9110262"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/978-3-319-19105-8_17","article-title":"Dual Hesitant Fuzzy Soft Set and Its Properties","volume":"367","author":"Zhang","year":"2016","journal-title":"Adv. Intell. Syst. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.ins.2014.02.125","article-title":"Distance and similarity measures for hesitant fuzzy linguistic term sets and their application in multi-criteria decision making","volume":"271","author":"Liao","year":"2014","journal-title":"Inf. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/j.ins.2015.02.030","article-title":"An extension of ELECTRE to multi-criteria decision-making problem with multi-hesitant fuzzy sets","volume":"307","author":"Peng","year":"2015","journal-title":"Inf. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Su, Z., Xu, Z.S., Zhao, H., and Liu, S.S. (2019). Distribution-Based Approaches to Deriving Weights from Dual Hesitant Fuzzy Information. Symmetry, 11.","DOI":"10.3390\/sym11010085"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ins.2017.01.024","article-title":"Dual hesitant fuzzy VIKOR method for multi-criteria group decision making based on fuzzy measure and new comparison method","volume":"388","author":"Ren","year":"2017","journal-title":"Inf. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.engappai.2014.12.012","article-title":"New distance and similarity measures on hesitant fuzzy sets and their applications in multiple criteria decision making","volume":"40","author":"Li","year":"2015","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1016\/j.asoc.2014.08.008","article-title":"A new method for solving dual hesitant fuzzy assignment problems with restrictions based on similarity measure","volume":"24","author":"Singh","year":"2014","journal-title":"Appl. Soft Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1002\/int.21548","article-title":"Hesitant fuzzy entropy and cross-entropy and their use in multi-attribute decision-making","volume":"27","author":"Xu","year":"2012","journal-title":"Int. J. Intell. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2065","DOI":"10.1007\/s40815-018-0512-5","article-title":"Determining the Parameter of Distance Measure between Dual Hesitant Fuzzy Information in Multiple Attribute Decision Making","volume":"20","author":"Xue","year":"2018","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.ins.2015.03.076","article-title":"Note on distance measure of hesitant fuzzy sets","volume":"321","author":"Li","year":"2015","journal-title":"Inf. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1002\/int.21633","article-title":"Correlation for dual hesitant fuzzy sets and dual interval-valued hesitant fuzzy sets","volume":"29","author":"Farhadinia","year":"2014","journal-title":"Int. J. Intell. Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1016\/j.inffus.2015.07.001","article-title":"Information fusion for intuitionistic fuzzy decision making: An overview","volume":"28","author":"Xu","year":"2016","journal-title":"Inf. Fusion"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Zhang, H.B., Liu, X.M., Ji, H.H., Hou, Z.S., and Fan, L.L. (2019). Multi-Agent-Based Data-Driven Distributed Adaptive Cooperative Control in Urban Traffic Signal Timing. Energies, 12.","DOI":"10.3390\/en12071402"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1007\/s12559-017-9480-6","article-title":"A multiple criteria decision making model with entropy weight in an interval-transformed hesitant fuzzy environment","volume":"9","author":"Farhadinia","year":"2017","journal-title":"Cognit. Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/j.procs.2019.11.284","article-title":"Combined Model-Free Adaptive Control with Fuzzy Component by Virtual Reference Feedback Tuning for Tower Crane Systems","volume":"162","author":"Roman","year":"2019","journal-title":"Procedia Comput. Sci."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1002\/int.21841","article-title":"Entropy and cross-entropy for generalized hesitant fuzzy information and their use in multiple attribute decision making","volume":"32","author":"Park","year":"2017","journal-title":"Int. J. Intell. Syst."},{"key":"ref_37","first-page":"1","article-title":"Maximizing deviation method for neutrosophic multiple attribute decision making with incomplete weightinformation","volume":"7","author":"Sahin","year":"2016","journal-title":"Neural Comput. Appl."},{"key":"ref_38","unstructured":"Kuang, J. (2004). Applied Inequalities, Shandong Science and Technology Press. [3rd ed.]."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1016\/0305-0548(94)00059-H","article-title":"Determining objective weights in multiple criteria problems: The critic method","volume":"22","author":"Diakoulaki","year":"1995","journal-title":"Comput. Oper. Res."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.mcm.2009.07.016","article-title":"Integration of correlations with standard deviations for determining attribute weights in multiple attribute decision making","volume":"51","author":"Wang","year":"2010","journal-title":"Math. Comput. Model."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Alcantud, J., Santos-Garc\u00eda, G., Peng, X.D., and Zhan, J.M. (2019). Dual Extended Hesitant Fuzzy Sets. Symmetry, 11.","DOI":"10.3390\/sym11050714"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Zhang, R.C., Li, Z.M., and Liao, H.C. (2019). Multiple-attribute decision-making method based on the correlation coefficient between dual hesitant fuzzy linguistic term sets. Knowl.-Based. Syst.","DOI":"10.1016\/j.knosys.2018.07.014"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/2\/191\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:08:27Z","timestamp":1760364507000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/2\/191"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,23]]},"references-count":42,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,2]]}},"alternative-id":["sym12020191"],"URL":"https:\/\/doi.org\/10.3390\/sym12020191","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1,23]]}}}