{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T19:07:03Z","timestamp":1783710423448,"version":"3.55.0"},"reference-count":46,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,21]],"date-time":"2022-11-21T00:00:00Z","timestamp":1668988800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Science and technology (MOST) of Taiwan","award":["MOST-110-2118-M-033-003-"],"award-info":[{"award-number":["MOST-110-2118-M-033-003-"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In recent years, q-rung orthopair fuzzy sets (q-ROFSs), a novel and rigorous generalization of the fuzzy set (FS) coined by Yager in 2017, have been used to manage inexplicit and indefinite information in daily life with a high precision and greater accuracy than intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFSs). The characterization of a measure of similarity between q-ROFSs is important, as they have applications in different areas, including pattern recognition, clustering, image segmentation and decision making. Therefore, this article is dedicated to the construction of a measure of similarity between q-ROFSs based on the Hausdorff metric. This is a very useful tool for establishing the similarity between two objects. Furthermore, some axiomatic definitions of the distances and similarity measures of q-ROFSs are also presented. In this article, we first present a novel method to calculate the distance between q-ROFSs based on the Hausdorff metric. We then utilize our proposed distance measure to construct the degree of similarity between q-ROFSs. We provide some properties for the proposed similarity measures. We offer several numerical examples related to pattern recognition and characterization linguistic variables to demonstrate the usefulness of the proposed similarity measures. We construct an algorithm for orthopair fuzzy TODIM (interactive and multi-criteria decision making, in Portuguese) based on our proposed methods. Finally, we use the constructed orthopair fuzzy TODIM method to address problems related to daily life settings involving multi-criteria decision making (MCDM). The numerical results show that the proposed similarity measures are suitable, applicable and well-suited to the contexts of pattern recognition, queries with fuzzy linguistic variables and MCDM.<\/jats:p>","DOI":"10.3390\/sym14112467","type":"journal-article","created":{"date-parts":[[2022,11,22]],"date-time":"2022-11-22T03:04:20Z","timestamp":1669086260000},"page":"2467","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["Distances and Similarity Measures of Q-Rung Orthopair Fuzzy Sets Based on the Hausdorff Metric with the Construction of Orthopair Fuzzy TODIM"],"prefix":"10.3390","volume":"14","author":[{"given":"Zahid","family":"Hussain","sequence":"first","affiliation":[{"name":"Department of Mathematical Science, Karakoram International University, Gilgit 15100, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8519-2264","authenticated-orcid":false,"given":"Sahar","family":"Abbas","sequence":"additional","affiliation":[{"name":"Department of Mathematical Science, Karakoram International University, Gilgit 15100, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4907-3548","authenticated-orcid":false,"given":"Miin-Shen","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taoyuan 32023, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,21]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","first-page":"2329","DOI":"10.1109\/TFUZZ.2015.2421071","article-title":"Analysis of parameter selection for Gustafson-Kessel fuzzy clustering using Jacobian matrix","volume":"23","author":"Chaomurilige","year":"2015","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.cie.2015.12.002","article-title":"Change-point detection for shifts in control charts using fuzzy shift change-point algorithms","volume":"93","author":"Lu","year":"2016","journal-title":"Comput. Ind. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"114686","DOI":"10.1016\/j.eswa.2021.114686","article-title":"Multi-criteria approach to stochastic and fuzzy uncertainty in the selection of electric vehicles with high social acceptance","volume":"173","author":"Ziemba","year":"2021","journal-title":"Expert Syst. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/0888-613X(87)90015-6","article-title":"Measures of similarity among fuzzy concepts: A comparative analysis","volume":"1","author":"Zwick","year":"1987","journal-title":"Int. J. Approx. Reason."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/0165-0114(93)90141-4","article-title":"A comparative assessment of measures of similarity of fuzzy values","volume":"56","author":"Pappis","year":"1993","journal-title":"Fuzzy Sets Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/0165-0114(95)00365-7","article-title":"New similarity measures on fuzzy sets and on elements","volume":"85","author":"Wang","year":"1997","journal-title":"Fuzzy Sets Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/S0169-023X(00)00025-2","article-title":"Similarity-based ranking and query processing in multimedia databases","volume":"35","author":"Candan","year":"2000","journal-title":"Data Knowl. Eng."},{"key":"ref_9","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_10","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_11","doi-asserted-by":"crossref","unstructured":"Atanassov, K.T. (1999). Intuitionistic Fuzzy Sets: Theory and Applications, Physica-Verlag.","DOI":"10.1007\/978-3-7908-1870-3"},{"key":"ref_12","first-page":"359","article-title":"New construction for similarity measures between intuitionistic fuzzy sets based on lower, upper and middle fuzzy sets","volume":"15","author":"Hwang","year":"2013","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Garg, H., and Kaur, J. (2018). A novel (R,S)-norm entropy measure of intuitionistic fuzzy sets and its applications in multi-attribute decision-making. Mathematics, 6.","DOI":"10.3390\/math6060092"},{"key":"ref_14","first-page":"7849686","article-title":"Belief and plausibility measures on intuitionistic fuzzy sets with construction of belief-plausibility TOPSIS","volume":"2020","author":"Yang","year":"2020","journal-title":"Complexity"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Gong, Z., and Fan, L. (2021). Three-way multi-attribute decision making based on outranking relations under intuitionistic fuzzy environments. Symmetry, 13.","DOI":"10.3390\/sym13081384"},{"key":"ref_16","first-page":"529","article-title":"Hesitant fuzzy sets","volume":"25","author":"Torra","year":"2010","journal-title":"Int. J. Intell. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1002\/int.21654","article-title":"Hesitant fuzzy sets: State of arts and future directions","volume":"29","author":"Torra","year":"2014","journal-title":"Int. J. Intell. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2128","DOI":"10.1016\/j.ins.2011.01.028","article-title":"Distance and similarity measures for hesitant fuzzy sets","volume":"181","author":"Xu","year":"2011","journal-title":"Inf. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2517","DOI":"10.1007\/s40815-018-0523-2","article-title":"Entropy for hesitant fuzzy sets based on Hausdorff metric with construction of hesitant fuzzy TOPSIS","volume":"20","author":"Hussian","year":"2018","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"5835","DOI":"10.1007\/s00500-018-3248-0","article-title":"Distance and similarity measures of hesitant fuzzy sets based on Hausdorff metric with applications to multi-criteria decision making and clustering","volume":"23","author":"Yang","year":"2019","journal-title":"Soft Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2981","DOI":"10.3233\/JIFS-190082","article-title":"Novel distance and similarity measures on Hesitant fuzzy linguistic term sets with application to pattern recognition","volume":"37","author":"Zhang","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1002\/int.21584","article-title":"Pythagorean membership grades, complex numbers, and decision making","volume":"28","author":"Yager","year":"2013","journal-title":"Int. J. Intell. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"958","DOI":"10.1109\/TFUZZ.2013.2278989","article-title":"Pythagorean membership grades in multicriteria decision making","volume":"22","author":"Yager","year":"2013","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1002\/int.21796","article-title":"A novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group decision making","volume":"31","author":"Zhang","year":"2016","journal-title":"Int. J. Intell. Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1731","DOI":"10.1002\/int.21994","article-title":"Pythagorean fuzzy multicriteria group decision making through similarity measure based on point operators","volume":"33","author":"Biswas","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Hussain, A., Ullah, K., Alshahrani, M.N., Yang, M.S., and Pamucar, D. (2022). Novel Aczel-Alsina operators for Pythagorean fuzzy sets with application in multi-attribute decision making. Symmetry, 14.","DOI":"10.3390\/sym14050940"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1222","DOI":"10.1109\/TFUZZ.2016.2604005","article-title":"Generalized orthopair fuzzy sets","volume":"25","author":"Yager","year":"2016","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2154","DOI":"10.1002\/int.22008","article-title":"Aspects of generalized orthopair fuzzy sets","volume":"33","author":"Yager","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1795","DOI":"10.1002\/int.22115","article-title":"Information measures for q-rung orthopair fuzzy sets","volume":"34","author":"Peng","year":"2019","journal-title":"Int. J. Intell. Syst."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Yang, M.S., Ali, Z., and Mahmood, T. (2021). Three-way decisions based on q-rung orthopair fuzzy 2-tuple linguistic sets with generalized Maclaurin symmetric mean operators. Mathematics, 9.","DOI":"10.3390\/math9121387"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"5225","DOI":"10.3233\/JIFS-169806","article-title":"Interval valued q-rung orthopair fuzzy sets and their properties","volume":"35","author":"Joshi","year":"2018","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1002\/int.22203","article-title":"Confidence levels q-rung orthopair fuzzy aggregation operators and its applications to MCDM problems","volume":"35","author":"Joshi","year":"2020","journal-title":"Int. J. Intell. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1749","DOI":"10.1007\/s13042-020-01070-1","article-title":"A q-rung orthopair fuzzy multi-criteria group decision making method for supplier selection based on a novel distance measure","volume":"11","author":"Pinar","year":"2020","journal-title":"Int. J. Mach. Learn. Cybern."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Wang, P., Wang, J., Wei, G., and Wei, C. (2019). Similarity measures of q-rung orthopair fuzzy sets based on cosine function and their applications. Mathematics, 7.","DOI":"10.3390\/math7040340"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1572","DOI":"10.1002\/int.22108","article-title":"Some cosine similarity measures and distance measures between q-rung orthopair fuzzy sets","volume":"34","author":"Liu","year":"2019","journal-title":"Int. J. Intell. Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1535","DOI":"10.1002\/int.22351","article-title":"A family of similarity measures for q-rung orthopair fuzzy sets and their applications to multiple criteria decision making","volume":"36","author":"Farhadinia","year":"2021","journal-title":"Int. J. Intell. Syst."},{"key":"ref_37","first-page":"635","article-title":"Hyperspaces of Sets","volume":"49","author":"Nalder","year":"1978","journal-title":"N. Y. Marcel Dekker"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"850","DOI":"10.1109\/34.232073","article-title":"Comparing images using the Hausdorff distance","volume":"15","author":"Huttenlocher","year":"1993","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_39","unstructured":"Olson, C.F. (1998, January 25). A probabilistic formulation for Hausdorff matching. Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Cat. No. 98CB36231), Santa Barbara, CA, USA."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2633","DOI":"10.1002\/int.22169","article-title":"Distance and similarity measures of Pythagorean fuzzy sets based on the Hausdorff metric with application to fuzzy TOPSIS","volume":"34","author":"Hussian","year":"2019","journal-title":"Int. J. Intell. Syst."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/0306-4573(77)90018-8","article-title":"A conceptual framework for fuzzy query processing\u2014A step toward very intelligent database systems","volume":"13","author":"Tahani","year":"1977","journal-title":"Inf. Process. Manag."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1109\/TSMC.1986.4308982","article-title":"Database queries with fuzzy linguistic quantifiers","volume":"16","author":"Kacprzyk","year":"1986","journal-title":"IEEE Trans. Syst. Man Cybern."},{"key":"ref_43","unstructured":"Petry, F.E. (1996). Fuzzy Database: Principles and Applications, Kluwer."},{"key":"ref_44","first-page":"113","article-title":"TODIM: Basics and application to multicriteria ranking of projects with environmental impacts","volume":"16","author":"Gomes","year":"1992","journal-title":"Found. Comput. Decis. Sci."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"3345","DOI":"10.1002\/(SICI)1097-0258(19991215)18:23<3345::AID-SIM321>3.0.CO;2-7","article-title":"Multi-criteria decision making-An approach to setting priorities in health care","volume":"18","author":"Trotta","year":"1999","journal-title":"Stat. Med."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.ejor.2007.10.046","article-title":"An application of the TODIM method to the multicriteria rental evaluation of residential properties","volume":"193","author":"Gomes","year":"2009","journal-title":"Eur. J. Oper. Res."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2467\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:22:50Z","timestamp":1760145770000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2467"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,21]]},"references-count":46,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["sym14112467"],"URL":"https:\/\/doi.org\/10.3390\/sym14112467","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,21]]}}}