{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,26]],"date-time":"2025-10-26T14:58:13Z","timestamp":1761490693533,"version":"build-2065373602"},"reference-count":53,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2018,11,12]],"date-time":"2018-11-12T00:00:00Z","timestamp":1541980800000},"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>An aggregation operator performs the task of fusing multiple sources of information, which plays a pivotal part in realizing a collective opinion in most decision-making activities. Considering the increasing complexity of decision-making situations, it is imperative to extend aggregation operators for fusing uncertain information with the different forms of attribute values. This study focuses on the development of picture fuzzy sets and aims to design a managerial decision-making solving method. Some operational principles of hesitant picture 2-tuple linguistic variables on account of the Archimedean t-norm and t-conorm are initiated, on which two hesitant picture 2-tuple linguistic weighted operators are established by taking various weight forms. Moreover, we explore the aggregation operators\u2019 idempotency, boundedness, and monotonicity, as well as analyze some particular forms of these operators. Furthermore, these aggregation operators are employed to design a method of deriving an overall performance from evaluation of experts with hesitant picture 2-tuple linguistic terms. An example of selecting service outsourcing supplier is carried out to show the procedures of decision-making with a detailed comparative analysis.<\/jats:p>","DOI":"10.3390\/sym10110629","type":"journal-article","created":{"date-parts":[[2018,11,14]],"date-time":"2018-11-14T10:58:22Z","timestamp":1542193102000},"page":"629","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Hesitant Picture 2-Tuple Linguistic Aggregation Operators Based on Archimedean T-Norm and T-Conorm and Their Use in Decision-Making"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3702-5693","authenticated-orcid":false,"given":"Yanjun","family":"Wang","sequence":"first","affiliation":[{"name":"School of Science, Dalian Maritime University, Dalian 116026, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1462-9443","authenticated-orcid":false,"given":"Lidong","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Science, Dalian Maritime University, Dalian 116026, China"}]},{"given":"Huijuan","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Science, Dalian Maritime University, Dalian 116026, China"}]},{"given":"Xinghua","family":"Feng","sequence":"additional","affiliation":[{"name":"School of Information Engineering, Southwest University of Science and Technology, Mianyang 621010, China"}]}],"member":"1968","published-online":{"date-parts":[[2018,11,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"71","DOI":"10.5391\/IJFIS.2011.11.2.071","article-title":"A generalized intuitionistic fuzzy soft set theoretic approach to decision making problems","volume":"11","author":"Park","year":"2011","journal-title":"Int. J. Fuzzy Logic Intell. Syst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/j.ins.2017.04.028","article-title":"Reporting\/most influential objects in uncertain databases based on probabilistic reverse top-k queries","volume":"405","author":"Xiao","year":"2017","journal-title":"Inf. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.jcss.2016.05.010","article-title":"Efficient monochromatic and bichromatic probabilistic reverse top-k query processing for uncertain big data","volume":"89","author":"Xiao","year":"2017","journal-title":"J. Comput. Syst. Sci."},{"key":"ref_4","first-page":"4","article-title":"A study on the customer satisfaction towards online shopping","volume":"4","author":"Shenbaga","year":"2015","journal-title":"Sumedha J. Manag."},{"key":"ref_5","first-page":"67","article-title":"Evaluating factors influencing consumer satisfaction towards online shopping in viet nam","volume":"5","author":"Trong","year":"2014","journal-title":"J. Emerg. Trends Comput. Inf. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1007\/s12559-014-9253-4","article-title":"A two-stage methodology using K-NN and false-positive minimizing ELM for nominal data classification","volume":"6","author":"Akusok","year":"2014","journal-title":"Cognit. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1007\/s12559-015-9320-5","article-title":"Autonomous driver based on an intelligent system of decision-making","volume":"7","author":"Czubenko","year":"2015","journal-title":"Cognit. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1007\/s12559-014-9313-9","article-title":"Correlation coefficients of hesitant fuzzy sets and their application based on fuzzy measures","volume":"7","author":"Meng","year":"2015","journal-title":"Cognit. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"582","DOI":"10.1007\/s12559-015-9323-2","article-title":"An improved SVM-based cognitive diagnosis algorithm for operation states of distribution grid","volume":"7","author":"Yang","year":"2015","journal-title":"Cognit. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1016\/S0377-2217(98)00383-X","article-title":"Choice functions and mechanisms for linguistic preference relations","volume":"120","author":"Herrera","year":"2000","journal-title":"Eur. J. Oper. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/S0165-0114(99)00024-X","article-title":"Linguistic decision analysis: Steps for solving decision problems under linguistic information","volume":"115","author":"Herrera","year":"2000","journal-title":"Fuzzy Sets Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/0165-0114(95)00107-7","article-title":"A model of consensus in group decision making under linguistic assessments","volume":"78","author":"Herrera","year":"1996","journal-title":"Fuzzy Sets Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1880","DOI":"10.1007\/s40815-016-0291-9","article-title":"Method of multiple attribute group decision making based on 2-dimension interval type-2 fuzzy aggregation operators with multi-granularity linguistic information","volume":"19","author":"Wu","year":"2017","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.ins.2003.10.006","article-title":"A method based on linguistic aggregation operators for group decision making with linguistic preference relations","volume":"166","author":"Xu","year":"2014","journal-title":"Inf. Sci."},{"key":"ref_15","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":"2012","journal-title":"Fuzzy Sets Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"190","DOI":"10.5391\/IJFIS.2018.18.3.190","article-title":"Interval valued intuitionistic fuzzy evaluations for analysis of a student\u2019s knowledge in university e-learning courses","volume":"18","author":"Kim","year":"2018","journal-title":"Int. J. Fuzzy Logic Intell. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3207","DOI":"10.1007\/s00500-015-2004-y","article-title":"A multiple attribute interval type-2 fuzzy group decision making and its application to supplier selection with extended LINMAP method","volume":"21","author":"Qin","year":"2017","journal-title":"Soft Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/j.ijar.2010.09.002","article-title":"Hesitant fuzzy information aggregation in decision making","volume":"52","author":"Xia","year":"2011","journal-title":"Int. J. Approx. Reason."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/s10726-011-9261-7","article-title":"Some hesitant fuzzy aggregation operators with their application in group decision making","volume":"22","author":"Xia","year":"2013","journal-title":"Group Decis. Negot."},{"key":"ref_20","first-page":"394","article-title":"Fuzzy Sets, Fuzzy logic, & Fuzzy Systems","volume":"8","author":"Zadeh","year":"1996","journal-title":"Fuzzy Sets"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Mart\u00ednez, L., Rodriguez, R.M., and Herrera, F. (2015). The 2-Tuple Linguistic Model-Computing with Words in Decision Making, Springer Publishing Company.","DOI":"10.1007\/978-3-319-24714-4_2"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"747","DOI":"10.1080\/18756891.2015.1061394","article-title":"An approach to multiple attribute group decision making based on linguistic intuitionistic fuzzy numbers","volume":"8","author":"Chen","year":"2015","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_23","first-page":"1571","article-title":"Multi-criteria decision-making method based on aggregation operators for intuitionistic linguistic fuzzy numbers","volume":"25","author":"Wang","year":"2010","journal-title":"Control Decis."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"746","DOI":"10.1109\/91.890332","article-title":"A 2-tuple fuzzy linguistic representation model for computing with words","volume":"8","author":"Herrera","year":"2000","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1109\/3477.915345","article-title":"A model based on linguistic 2-tuples for dealing with multigranular hierarchical linguistic contexts in multi-expert decision-making","volume":"31","author":"Herrera","year":"2000","journal-title":"IEEE Trans. Syst. Man Cybern. Part B Cybern."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Wang, R., and Li, Y.L. (2018). Picture hesitant fuzzy set and its application to multiple criteria decision-making. Symmetry, 10.","DOI":"10.3390\/sym10070295"},{"key":"ref_27","first-page":"409","article-title":"Picture fuzzy sets-first results: Part 1","volume":"30","author":"Cuong","year":"2014","journal-title":"J. Comput. Sci. Cybern."},{"key":"ref_28","first-page":"80","article-title":"Smarandache. Standard neutrosophic soft theory: Some first results","volume":"12","author":"Cuong","year":"2016","journal-title":"Neutrosophic Sets Syst."},{"key":"ref_29","first-page":"245","article-title":"Rough picture fuzzy set and picture fuzzy topologies","volume":"31","author":"Dinh","year":"2015","journal-title":"J. Comput. Sci. Cybern."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"547","DOI":"10.15388\/Informatica.2017.144","article-title":"Some cosine similarity measures for picture fuzzy sets and their applications to strategic decision making","volume":"28","author":"Wei","year":"2017","journal-title":"Informatica"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"652","DOI":"10.1007\/s10489-016-0856-1","article-title":"Picture inference system: A new fuzzy inference system on picture fuzzy set","volume":"46","author":"Le","year":"2017","journal-title":"Appl. Intell."},{"key":"ref_32","unstructured":"Dutta, P., and Ganju, S. (2017, January 7\u20138). Application of interval valued picture fuzzy set in medical diagnosis. Proceedings of the 19th International Conference on Knowledge Discovery and Fuzzy Systems, Mumbai, India."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"997","DOI":"10.1007\/s40815-016-0266-x","article-title":"Picture 2-tuple linguistic Bonferroni mean operators and their application to multiple attribute decision making","volume":"19","author":"Wei","year":"2017","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Nie, R.X., Wang, J.Q., and Wang, T.L. (2018). A hybrid outranking method for greenhouse gas emissions\u2019 institution selection with picture 2-tuple linguistic information. Comput. Appl. Math.","DOI":"10.1007\/s40314-018-0708-1"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1007\/s00500-016-2403-8","article-title":"Picture 2-tuple linguistic aggregation operators in multiple attribute decision making","volume":"22","author":"Wei","year":"2018","journal-title":"Soft Comput."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"958","DOI":"10.1007\/s40815-017-0441-8","article-title":"On novel operational laws and aggregation operators of picture 2-tuple linguistic information for MCDM problems","volume":"20","author":"Zhang","year":"2018","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"181","DOI":"10.5391\/IJFIS.2014.14.3.181","article-title":"Group decision making using intuitionistic hesitant fuzzy sets","volume":"14","author":"Beg","year":"2014","journal-title":"Int. J. Fuzzy Logic Intell. Syst."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"7895","DOI":"10.1016\/j.eswa.2010.04.047","article-title":"A method for multiple attribute group decision making based on the ET-WG and ET-OWG operators with 2-tuple linguistic information","volume":"37","author":"Wei","year":"2010","journal-title":"Exp. Syst. Appl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"3988","DOI":"10.1016\/j.asoc.2011.02.027","article-title":"Approaches based on 2-tuple linguistic power aggregation operators for multiple attribute group decision making under linguistic environment","volume":"11","author":"Xu","year":"2011","journal-title":"Soft Comput."},{"key":"ref_40","first-page":"108","article-title":"A hesitant fuzzy language decision making method based on 2-tuple","volume":"26","author":"Ge","year":"2017","journal-title":"Oper. Res. Manag. Sci."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Wang, L.D., Wang, Y.J., and Liu, X.D. (2018). Prioritized aggregation operators and correlated aggregation operators for hesitant 2-tuple linguistic variables. Symmetry, 10.","DOI":"10.3390\/sym10020039"},{"key":"ref_42","first-page":"283","article-title":"Fuzzy sets and fuzzy logic: Theory and applications","volume":"1","author":"Klir","year":"1995","journal-title":"Prent. Hall"},{"key":"ref_43","unstructured":"Nguyen, H.T., and Walker, E.A. (1997). A First Course in Fuzzy Logic, CRC Press."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1007\/s40815-016-0195-8","article-title":"The aggregation operators based on Archimedean t-conorm and t-norm for single-valued neutrosophic numbers and their application to decision making","volume":"18","author":"Liu","year":"2016","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1016\/j.knosys.2012.02.004","article-title":"Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm","volume":"31","author":"Xia","year":"2012","journal-title":"Knowl. Based Syst."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Liu, P.D., and Chen, S.M. (2016, January 10\u201313). Heronian aggregation operators of intuitionistic fuzzy numbers based on the Archimedean t-norm and t-conorm. Proceedings of the 2016 International Conference on Machine Learning & Cybernetics, Jeju, Korea.","DOI":"10.1109\/ICMLC.2016.7872971"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1002\/int.21710","article-title":"Group decision making under interval-valued multiplicative intuitionistic fuzzy environment based on Archimedean t-conorm and t-norm","volume":"30","author":"Yu","year":"2015","journal-title":"Int. J. Intell. Syst."},{"key":"ref_48","first-page":"2737","article-title":"Some interval-valued hesitant fuzzy aggregation operators based on Archimedean t-norm and t-conorm with their application in multi-criteria decision making","volume":"27","author":"Zhang","year":"2014","journal-title":"Int. J. Intell. Syst."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1016\/j.knosys.2014.04.037","article-title":"On new operational laws of 2-tuple linguistic information using Archimedean t-norm and s-norm","volume":"66","author":"Tao","year":"2014","journal-title":"Knowl. Based Syst."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1142\/S0218488500000381","article-title":"An approach for combining linguistic and numerical information based on the 2-tuple fuzzy linguistic representation model in decision-making","volume":"8","author":"Herrera","year":"2000","journal-title":"Int. J. Uncertain. Fuzziness Knowl. Based Syst."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"2662","DOI":"10.1016\/j.eswa.2011.08.121","article-title":"New aggregation operators based on the Choquet integral and 2-tuple linguistic information","volume":"39","author":"Yang","year":"2012","journal-title":"Expert Syst. Appl."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1080\/00207721.2015.1074761","article-title":"A consensus reaching model for 2-tuple linguistic multiple attribute group decision making with incomplete weight information","volume":"47","author":"Zhang","year":"2016","journal-title":"Int. J. Syst. Sci."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1","DOI":"10.9734\/BJMCS\/2016\/27172","article-title":"Archimedean operations-based interval-valued hesitant fuzzy ordered weighted operators and their application to multi-criteria decision making","volume":"17","author":"Zhang","year":"2016","journal-title":"Br. J. Math. Comput. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/11\/629\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:29:15Z","timestamp":1760196555000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/11\/629"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,12]]},"references-count":53,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2018,11]]}},"alternative-id":["sym10110629"],"URL":"https:\/\/doi.org\/10.3390\/sym10110629","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2018,11,12]]}}}