{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T17:02:05Z","timestamp":1779296525532,"version":"3.51.4"},"reference-count":57,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,14]],"date-time":"2023-02-14T00:00:00Z","timestamp":1676332800000},"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":["61902221"],"award-info":[{"award-number":["61902221"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Intuitionistic fuzzy sets (IFSs), as a representative variant of fuzzy sets, has substantial advantages in managing and modeling uncertain information, so it has been widely studied and applied. Nevertheless, how to perfectly measure the similarities or differences between IFSs is still an open question. The distance metric offers an elegant and desirable solution to such a question. Hence, in this paper, we propose a new distance measure, named DIFS, inspired by the Hellinger distance in probability distribution space. First, we provide the formal definition of the new distance measure of IFSs, and analyze the outstanding properties and axioms satisfied by DIFS, which means it can measure the difference between IFSs well. Besides, on the basis of DIFS, we further present a normalized distance measure of IFSs, denoted DIFS\u02dc. Moreover, numerical examples verify that DIFS\u02dc can obtain more reasonable and superior results. Finally, we further develop a new decision-making method on top of DIFS\u02dc and evaluate its performance in two applications.<\/jats:p>","DOI":"10.3390\/sym15020500","type":"journal-article","created":{"date-parts":[[2023,2,14]],"date-time":"2023-02-14T02:41:56Z","timestamp":1676342516000},"page":"500","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["An Intuitionistic Fuzzy Version of Hellinger Distance Measure and Its Application to Decision-Making Process"],"prefix":"10.3390","volume":"15","author":[{"given":"Xiang","family":"Li","sequence":"first","affiliation":[{"name":"School of Computer Science and Technology, Shandong Jianzhu University, Jinan 250101, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8580-9655","authenticated-orcid":false,"given":"Zhe","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Hainan University, Haikou 570228, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xue","family":"Han","sequence":"additional","affiliation":[{"name":"School of Management, Xi\u2019an University of Architecture and Technology, Xi\u2019an 710055, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nan","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Shandong Jianzhu University, Jinan 250101, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Weihua","family":"Yuan","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Shandong Jianzhu University, Jinan 250101, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1007\/s10479-019-03431-8","article-title":"A survey of decision making and optimization under uncertainty","volume":"300","author":"Keith","year":"2021","journal-title":"Ann. Oper. Res."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1186\/s40537-019-0206-3","article-title":"Uncertainty in big data analytics: Survey, opportunities, and challenges","volume":"6","author":"Hariri","year":"2019","journal-title":"J. Big Data"},{"key":"ref_3","first-page":"1","article-title":"Handling of uncertainty\u2014A survey","volume":"3","author":"Dutt","year":"2013","journal-title":"Int. J. Sci. Res. Publ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2463","DOI":"10.1109\/TKDE.2012.179","article-title":"Dealing with uncertainty: A survey of theories and practices","volume":"25","author":"Li","year":"2013","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1672","DOI":"10.1007\/s10489-017-1024-y","article-title":"Uncertainty measure in evidence theory with its applications","volume":"48","author":"Wang","year":"2018","journal-title":"Appl. Intell."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1303","DOI":"10.1109\/TFUZZ.2017.2718966","article-title":"On using the Shapley value to approximate the Choquet integral in cases of uncertain arguments","volume":"26","author":"Yager","year":"2018","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/j.eswa.2017.03.072","article-title":"Integrated group fuzzy multi-criteria model: Case of facilities management strategy selection","volume":"82","author":"Zavadskas","year":"2017","journal-title":"Expert Syst. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_9","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_10","doi-asserted-by":"crossref","first-page":"3775","DOI":"10.1016\/j.ins.2008.06.008","article-title":"Clustering algorithm for intuitionistic fuzzy sets","volume":"178","author":"Xu","year":"2008","journal-title":"Inf. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s10700-009-9052-7","article-title":"Some aspects of intuitionistic fuzzy sets","volume":"8","author":"Yager","year":"2009","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1109\/TFUZZ.2017.2670551","article-title":"Rough information set and its applications in decision making","volume":"25","author":"Aggarwal","year":"2017","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Ayub, S., Shabir, M., Riaz, M., Mahmood, W., Bozanic, D., and Marinkovic, D. (2022). Linear diophantine fuzzy rough sets: A new rough set approach with decision making. Symmetry, 14.","DOI":"10.3390\/sym14030525"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/j.inffus.2018.08.007","article-title":"Information fusion in rough set theory: An overview","volume":"48","author":"Wei","year":"2019","journal-title":"Inf. Fusion"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1109\/TFUZZ.2018.2859899","article-title":"Generalized dempster\u2013shafer structures","volume":"27","author":"Yager","year":"2019","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11432-020-3006-9","article-title":"Uncertainty measure in evidence theory","volume":"63","author":"Deng","year":"2020","journal-title":"Sci. China Inf. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"7101","DOI":"10.3233\/JIFS-210991","article-title":"Evidential classification of incomplete instance based on K-nearest centroid neighbor","volume":"41","author":"Ma","year":"2021","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1109\/TFUZZ.2019.2955061","article-title":"R-sets, comprehensive fuzzy sets risk modeling for risk-based information fusion and decision-making","volume":"29","author":"Seiti","year":"2021","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1016\/j.ins.2019.01.006","article-title":"R-numbers, a new risk modeling associated with fuzzy numbers and its application to decision making","volume":"483","author":"Seiti","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.isatra.2020.01.015","article-title":"Developing the modified R-numbers for risk-based fuzzy information fusion and its application to failure modes, effects, and system resilience analysis (FMESRA)","volume":"113","author":"Seiti","year":"2021","journal-title":"ISA Trans."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2792","DOI":"10.1007\/s10489-017-1091-0","article-title":"Intuitive distance for intuitionistic fuzzy sets with applications in pattern recognition","volume":"48","author":"Luo","year":"2018","journal-title":"Appl. Intell."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3980","DOI":"10.1109\/TSMC.2019.2958635","article-title":"A distance measure for intuitionistic fuzzy sets and its application to pattern classification problems","volume":"51","author":"Xiao","year":"2021","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Ejegwa, P.A., and Ahemen, S. (2022). Enhanced intuitionistic fuzzy similarity operators with applications in emergency management and pattern recognition. Granul. Comput., 1\u201312.","DOI":"10.1007\/s41066-022-00334-1"},{"key":"ref_24","first-page":"127","article-title":"Novel distance measure between intuitionistic fuzzy sets and its application in pattern recognition","volume":"19","author":"Zeng","year":"2022","journal-title":"Iran. J. Fuzzy Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.artmed.2009.03.002","article-title":"Intuitionistic fuzzy set vs. fuzzy set application in medical pattern recognition","volume":"47","author":"Khatibi","year":"2009","journal-title":"Artif. Intell. Med."},{"key":"ref_26","first-page":"51","article-title":"An algorithm for an improved intuitionistic fuzzy correlation measure with medical diagnostic application","volume":"3","author":"Ejegwa","year":"2020","journal-title":"Ann. Optim. Theory Pract."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/s41066-018-0140-3","article-title":"Novel distance measures for cubic intuitionistic fuzzy sets and their applications to pattern recognitions and medical diagnosis","volume":"5","author":"Garg","year":"2020","journal-title":"Granul. Comput."},{"key":"ref_28","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_29","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.engappai.2017.02.008","article-title":"Novel intuitionistic fuzzy decision making method based on an improved operation laws and its application","volume":"60","author":"Garg","year":"2017","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1002\/int.21583","article-title":"Prioritized information fusion method for triangular intuitionistic fuzzy set and its application to teaching quality evaluation","volume":"28","author":"Yu","year":"2013","journal-title":"Int. J. Intell. Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"103276","DOI":"10.1016\/j.engappai.2019.103276","article-title":"A novel target threat assessment method based on three-way decisions under intuitionistic fuzzy multi-attribute decision making environment","volume":"87","author":"Gao","year":"2020","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1002\/int.22312","article-title":"A novel distance measure for intuitionistic fuzzy sets with diverse applications","volume":"36","author":"Mahanta","year":"2021","journal-title":"Int. J. Intell. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"361","DOI":"10.3233\/JIFS-171372","article-title":"Some new contractive conditions and related fixed point theorems in intuitionistic fuzzy n-Banach spaces","volume":"34","author":"Konwar","year":"2018","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"867","DOI":"10.1007\/s41066-020-00235-1","article-title":"Generalized intuitionistic fuzzy aggregation operators based on confidence levels for group decision making","volume":"6","author":"Rahman","year":"2021","journal-title":"Granul. Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1109\/TFUZZ.2018.2860967","article-title":"Another view on generalized intuitionistic fuzzy soft sets and related multiattribute decision making methods","volume":"27","author":"Feng","year":"2019","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1142\/S0218488508005406","article-title":"An overview of distance and similarity measures of intuitionistic fuzzy sets","volume":"16","author":"Xu","year":"2008","journal-title":"Int. J. Uncertain. Fuzziness Knowl.-Based Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"37","DOI":"10.3233\/IFS-2012-0527","article-title":"Similarity measures for intuitionistic fuzzy sets: State of the art","volume":"24","author":"Baccour","year":"2013","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1609","DOI":"10.1016\/j.patrec.2013.05.015","article-title":"Distance and similarity measures between intuitionistic fuzzy sets: A comparative analysis from a pattern recognition point of view","volume":"34","author":"Papakostas","year":"2013","journal-title":"Pattern Recognit. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Szmidt, E. (2014). Distances and Similarities in Intuitionistic Fuzzy Sets, Springer.","DOI":"10.1007\/978-3-319-01640-5"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1007\/s00500-019-03932-5","article-title":"Distance measures on intuitionistic fuzzy sets based on intuitionistic fuzzy dissimilarity functions","volume":"24","author":"He","year":"2020","journal-title":"Soft Comput."},{"key":"ref_41","unstructured":"Wu, X., Chen, G., Wang, T., Zhu, Z., and Liu, P. (2022). Strict intuitionistic fuzzy distance\/similarity measures based on Jensen-Shannon divergence. arXiv, arXiv:2207.06980."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Atanassov, K.T. (1999). Intuitionistic Fuzzy Sets, Springer.","DOI":"10.1007\/978-3-7908-1870-3"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/j.fss.2003.08.005","article-title":"Distances between intuitionistic fuzzy sets and\/or interval-valued fuzzy sets based on the Hausdorff metric","volume":"148","author":"Grzegorzewski","year":"2004","journal-title":"Fuzzy Sets Syst."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2063","DOI":"10.1016\/j.patrec.2005.03.018","article-title":"Distance measure between intuitionistic fuzzy sets","volume":"26","author":"Wang","year":"2005","journal-title":"Pattern Recognit. Lett."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"7805","DOI":"10.1002\/int.22608","article-title":"Construction and generation of distance and similarity measures for intuitionistic fuzzy sets and various applications","volume":"36","author":"Gohain","year":"2021","journal-title":"Int. J. Intell. Syst."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1016\/S0165-0114(98)00244-9","article-title":"Distances between intuitionistic fuzzy sets","volume":"114","author":"Szmidt","year":"2000","journal-title":"Fuzzy Sets Syst."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"8665","DOI":"10.1016\/j.eswa.2012.01.199","article-title":"Consistency of 2D and 3D distances of intuitionistic fuzzy sets","volume":"39","author":"Yang","year":"2012","journal-title":"Expert Syst. Appl."},{"key":"ref_48","first-page":"556","article-title":"Distance measure between intuitionistic fuzzy sets and its application to pattern recognition","volume":"19","author":"Park","year":"2009","journal-title":"J. Korean Inst. Intell. Syst."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.patrec.2006.07.004","article-title":"Intuitionistic fuzzy information\u2013applications to pattern recognition","volume":"28","author":"Vlachos","year":"2007","journal-title":"Pattern Recognit. Lett."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1985","DOI":"10.1007\/s00500-017-2912-0","article-title":"A new approach to construct similarity measure for intuitionistic fuzzy sets","volume":"23","author":"Song","year":"2019","journal-title":"Soft Comput."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.ins.2017.10.045","article-title":"An extended intuitionistic fuzzy TOPSIS method based on a new distance measure with an application to credit risk evaluation","volume":"428","author":"Shen","year":"2018","journal-title":"Inf. Sci."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1016\/j.eswa.2018.08.046","article-title":"A new similarity\/distance measure between intuitionistic fuzzy sets based on the transformed isosceles triangles and its applications to pattern recognition","volume":"116","author":"Jiang","year":"2019","journal-title":"Expert Syst. Appl."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"116228","DOI":"10.1016\/j.eswa.2021.116228","article-title":"Novel distance measures for intuitionistic fuzzy sets based on various triangle centers of isosceles triangular fuzzy numbers and their applications","volume":"191","author":"Garg","year":"2022","journal-title":"Expert Syst. Appl."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1515\/crll.1909.136.210","article-title":"Neue begr\u00fcndung der theorie quadratischer formen von unendlichvielen ver\u00e4nderlichen","volume":"1909","author":"Hellinger","year":"1909","journal-title":"J. F\u00dcR Die Reine Und Angew. Math."},{"key":"ref_55","first-page":"147","article-title":"Several new results based on the study of distance measures of intuitionistic fuzzy sets","volume":"17","author":"Chen","year":"2020","journal-title":"Iran. J. Fuzzy Syst."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1002\/int.21529","article-title":"A novel distance measure of intuitionistic fuzzy sets and its application to pattern recognition problems","volume":"27","author":"Hatzimichailidis","year":"2012","journal-title":"Int. J. Intell. Syst."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1007\/s12559-020-09809-2","article-title":"Novel similarity measure based on the transformed right-angled triangles between intuitionistic fuzzy sets and its applications","volume":"13","author":"Garg","year":"2021","journal-title":"Cogn. Comput."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/500\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:34:24Z","timestamp":1760121264000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/500"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,14]]},"references-count":57,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["sym15020500"],"URL":"https:\/\/doi.org\/10.3390\/sym15020500","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,14]]}}}