{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:00:52Z","timestamp":1760241652638,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2018,7,12]],"date-time":"2018-07-12T00:00:00Z","timestamp":1531353600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we demonstrate how interval-valued intuitionistic fuzzy sets (IVIFSs) can function as extended intuitionistic fuzzy sets (IFSs) using the interval-valued intuitionistic fuzzy numbers (IVIFNs) instead of precision numbers to describe the degree of membership and non-membership, which are more flexible and practical in dealing with ambiguity and uncertainty. By introducing IVIFSs into three-way decisions, we provide a new description of the loss function. Thus, we firstly propose a model of interval-valued intuitionistic fuzzy decision-theoretic rough sets (IVIFDTRSs). According to the basic framework of IVIFDTRSs, we design a strategy to address the IVIFNs and deduce three-way decisions. Then, we successfully extend the results of IVIFDTRSs from single-person decision-making to group decision-making. In this situation, we adopt a grey correlation accurate weighted determining method (GCAWD) to compute the weights of decision-makers, which integrates the advantages of the accurate weighted determining method and grey correlation analysis method. Moreover, we utilize the interval-valued intuitionistic fuzzy weighted averaging (IIFWA) operation to count the aggregated scores and the accuracies of the expected losses. By comparing these scores and accuracies, we design a simple and straightforward algorithm to deduce three-way decisions for group decision-making. Finally, we use an illustrative example to verify our results.<\/jats:p>","DOI":"10.3390\/sym10070281","type":"journal-article","created":{"date-parts":[[2018,7,12]],"date-time":"2018-07-12T11:19:24Z","timestamp":1531394364000},"page":"281","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Three-Way Decisions with Interval-Valued Intuitionistic Fuzzy Decision-Theoretic Rough Sets in Group Decision-Making"],"prefix":"10.3390","volume":"10","author":[{"given":"Dajun","family":"Ye","sequence":"first","affiliation":[{"name":"School of Management and Economics, Southwest Jiaotong University, Chengdu 610031, China"},{"name":"Department of Management and Economics, Sichuan College of Architectural Technology, Deyang 618000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0655-1255","authenticated-orcid":false,"given":"Decui","family":"Liang","sequence":"additional","affiliation":[{"name":"School of Management and Economics, University of Electronic Science and Technology of China, Chengdu 610054, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pei","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Management and Economics, Southwest Jiaotong University, Chengdu 610031, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,7,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Yao, Y. (2012, January 17\u201320). An outline of a theory of three-way decisions. Proceedings of the International Conference on Rough Sets and Current Trends in Computing, Chengdu, China.","DOI":"10.1007\/978-3-642-32115-3_1"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.ins.2014.05.015","article-title":"Three-way decisions space and three-way decisions","volume":"281","author":"Hu","year":"2014","journal-title":"Inform. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1007\/BF01001956","article-title":"Rough sets","volume":"11","author":"Pawlak","year":"1982","journal-title":"Int. J. Comput. Inf. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Yao, Y., and Deng, X. (2011, January 18\u201320). Sequential three-way decisions with probabilistic rough sets. Proceedings of the IEEE 10th International Conference on Cognitive Informatics and Cognitive Computing, Banff, AB, Canada.","DOI":"10.1109\/COGINF.2011.6016129"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1080","DOI":"10.1016\/j.ins.2010.11.019","article-title":"The superiority of three-way decisions in probabilistic rough set models","volume":"181","author":"Yao","year":"2011","journal-title":"Inform. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Yao, Y. (2012). Three-Way Decisions Using Rough Sets. Rough Sets: Selected Methods and Applications in Management and Engineering, Springer.","DOI":"10.1007\/978-1-4471-2760-4_5"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/j.ins.2009.09.021","article-title":"Three-way decisions with probabilistic rough sets","volume":"180","author":"Yao","year":"2010","journal-title":"Inform. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"614","DOI":"10.1016\/j.ins.2014.03.078","article-title":"Decision region distribution preservation reduction in decision-theoretic rough set model","volume":"278","author":"Ma","year":"2014","journal-title":"Inform. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/j.ins.2014.06.045","article-title":"Decision-theoretic rough fuzzy set model and application","volume":"283","author":"Sun","year":"2014","journal-title":"Inform. Sci."},{"key":"ref_10","unstructured":"Ras, Z.W., Zemankova, M., and Emrich, M.L. (1990). A decision-theoretic rough set model. Methodologies for Intelligent System, Elsevier North-Holland, Inc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"793","DOI":"10.1016\/0020-7373(92)90069-W","article-title":"A decision theoretic framework for approximating concepts","volume":"37","author":"Yao","year":"1992","journal-title":"Int. J. Man Mach. Stud."},{"key":"ref_12","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_13","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_14","doi-asserted-by":"crossref","first-page":"733","DOI":"10.1177\/0956797611407208","article-title":"In praise of vagueness: Malleability of vague information as a performance booster","volume":"22","author":"Mishra","year":"2011","journal-title":"Psychol. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1016\/j.ijar.2013.03.014","article-title":"Triangular fuzzy decision-theoretic rough sets","volume":"54","author":"Liang","year":"2013","journal-title":"Int. J. Approx. Reason."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1109\/TFUZZ.2014.2310495","article-title":"A novel risk decision making based on decision-theoretic rough sets under hesitant fuzzy information","volume":"23","author":"Liang","year":"2015","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.ins.2014.12.036","article-title":"Deriving three-way decisions from intuitionistic fuzzy decision-theoretic rough sets","volume":"300","author":"Liang","year":"2015","journal-title":"Inform. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1016\/j.ins.2014.02.054","article-title":"Systematic studies on three-way decisions with interval-valued decision-theoretic rough sets","volume":"276","author":"Liang","year":"2014","journal-title":"Inform. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Liu, D., Li, T., and Liang, D. (2012, January 17\u201320). Decision-Theoretic Rough Sets with Probabilistic Distribution. Proceedings of the International Conference on Rough Sets and Knowledge Technology, Chengdu, China.","DOI":"10.1007\/978-3-642-31900-6_48"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/0165-0114(89)90205-4","article-title":"Interval valued intuitionistic fuzzy sets","volume":"31","author":"Atanassov","year":"1989","journal-title":"Fuzzy Sets Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/0165-0114(94)90331-X","article-title":"Operators over interval valued intuitionistic fuzzy sets","volume":"64","author":"Atanassov","year":"1994","journal-title":"Fuzzy Sets Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/0165-0114(94)00343-6","article-title":"Correlation of interval-valued intuitionistic fuzzy sets","volume":"74","author":"Bustince","year":"1995","journal-title":"Fuzzy Sets Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/S0165-0114(02)00127-6","article-title":"On the relationship between some extensions of fuzzy set theory","volume":"133","author":"Deschrijver","year":"2003","journal-title":"Fuzzy Sets Syst."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Garg, H., Agarwal, N., and Tripathi, A. (2017). Choquet Integral-Based Information Aggregation Operators under the interval-valued intuitionistic fuzzy set and Its Applications to Decision-Making Process. Int. J. Uncertain. Quantif., 7.","DOI":"10.1615\/Int.J.UncertaintyQuantification.2017020076"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/S1874-8651(08)60026-5","article-title":"Approach to group decision making based on interval-valued intuitionistic judgment matrices","volume":"27","author":"Xu","year":"2007","journal-title":"Syst. Eng. Theory Pract."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/j.ins.2009.09.005","article-title":"A method based on distance measure for interval-valued intuitionistic fuzzy group decision making","volume":"180","author":"Xu","year":"2010","journal-title":"Inform. Sci."},{"key":"ref_27","first-page":"215","article-title":"Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making","volume":"22","author":"Xu","year":"2007","journal-title":"Control Decis."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Liu, X.D., Zheng, S.H., and Xiong, F.L. (2005, January 27\u201329). Entropy and Subsethood for General Interval-Valued Intuitionistic Fuzzy Sets. Proceedings of the International Conference on Fuzzy Systems and Knowledge Discovery, Changsha, China.","DOI":"10.1007\/11539506_6"},{"key":"ref_29","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":"Inform. Sci."},{"key":"ref_30","first-page":"716","article-title":"Accurate method of obtaining decision expert weights in intuitionistic fuzzy group decision making","volume":"28","author":"Zhou","year":"2013","journal-title":"Control Decis."},{"key":"ref_31","first-page":"158","article-title":"Method of determining weights of experts and attributes for interval-valued intuitionistic fuzzy decision making","volume":"18","author":"Li","year":"2016","journal-title":"Comput. Eng. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/S0165-0114(96)00311-9","article-title":"A note on correlation of interval-valued intuitionistic fuzzy sets","volume":"95","author":"Hong","year":"1998","journal-title":"Fuzzy Sets Syst."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Jia, X., Li, H., and Shang, L. (2016, January 10\u201313). Minimum cost attribute reduction in three-way decisions based Bayesian network. Proceedings of the 2016 International Conference on Machine Learning and Cybernetics, Jeju, Korea.","DOI":"10.1109\/ICMLC.2016.7860889"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Joshi, D.K., and Kumar, S. (2018). Entropy of interval-valued intuitionistic hesitant fuzzy set and its application to group decision making problems. Granul. Comput., 1\u201315.","DOI":"10.1080\/01969722.2017.1412890"},{"key":"ref_35","first-page":"319","article-title":"A Fuzzy Aras Method for Supply Chain Management Performance Measurement in SMEs under Uncertainty","volume":"16","author":"Rostamzadeh","year":"2017","journal-title":"Transform. Bus. Econ."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1002\/int.21953","article-title":"A novel aggregation method for Pythagorean fuzzy multiple attribute group decision making","volume":"33","author":"Zeng","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Zeng, S., Llopis-Albert, C., and Zhang, Y. (2018). A novel induced aggregation method for intuitionistic fuzzy set and its application in multiple attribute group decision making. Int. J. Intell. Syst.","DOI":"10.1002\/int.22009"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1016\/j.ijar.2013.03.015","article-title":"Analyzing uncertainties of probabilistic rough set regions with game-theoretic rough sets","volume":"55","author":"Azam","year":"2014","journal-title":"Int. J. Approx. Reason."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/7\/281\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:11:53Z","timestamp":1760195513000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/7\/281"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7,12]]},"references-count":38,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2018,7]]}},"alternative-id":["sym10070281"],"URL":"https:\/\/doi.org\/10.3390\/sym10070281","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2018,7,12]]}}}