{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,18]],"date-time":"2026-10-18T03:09:55Z","timestamp":1792292995522,"version":"4.4.0"},"reference-count":55,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,12,20]],"date-time":"2019-12-20T00:00:00Z","timestamp":1576800000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Information"],"abstract":"<jats:p>In this manuscript, the notions of q-rung orthopair fuzzy sets (q-ROFSs) and complex fuzzy sets (CFSs) are combined is to propose the complex q-rung orthopair fuzzy sets (Cq-ROFSs) and their fundamental laws. The Cq-ROFSs are an important way to express uncertain information, and they are superior to the complex intuitionistic fuzzy sets and the complex Pythagorean fuzzy sets. Their eminent characteristic is that the sum of the qth power of the real part (similarly for imaginary part) of complex-valued membership degree and the qth power of the real part (similarly for imaginary part) of complex-valued non\u2010membership degree is equal to or less than 1, so the space of uncertain information they can describe is broader. Under these environments, we develop the score function, accuracy function and comparison method for two Cq-ROFNs. Based on Cq-ROFSs, some new aggregation operators are called complex q-rung orthopair fuzzy weighted averaging (Cq-ROFWA) and complex q-rung orthopair fuzzy weighted geometric (Cq-ROFWG) operators are investigated, and their properties are described. Further, based on proposed operators, we present a new method to deal with the multi\u2010attribute group decision making (MAGDM) problems under the environment of fuzzy set theory. Finally, we use some practical examples to illustrate the validity and superiority of the proposed method by comparing with other existing methods.<\/jats:p>","DOI":"10.3390\/info11010005","type":"journal-article","created":{"date-parts":[[2019,12,20]],"date-time":"2019-12-20T09:50:33Z","timestamp":1576835433000},"page":"5","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":153,"title":["Complex q-Rung Orthopair Fuzzy Aggregation Operators and Their Applications in Multi-Attribute Group Decision Making"],"prefix":"10.3390","volume":"11","author":[{"given":"Peide","family":"Liu","sequence":"first","affiliation":[{"name":"School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan 250015, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tahir","family":"Mahmood","sequence":"additional","affiliation":[{"name":"Department of Mathematics & Statistics, International Islamic University Islamabad, Islamabad 44000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zeeshan","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics & Statistics, International Islamic University Islamabad, Islamabad 44000, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,20]]},"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":"199","DOI":"10.1016\/0020-0255(75)90036-5","article-title":"The concept of a linguistic variable and its application to approximate reasoning\u2014I","volume":"8","author":"Zadeh","year":"1975","journal-title":"Inf. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1109\/TFUZZ.2006.889764","article-title":"Geometric type-1 and type-2 fuzzy logic systems","volume":"15","author":"Coupland","year":"2007","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Atanassov, K.T. (1999). Intuitionistic fuzzy sets. Intuitionistic Fuzzy Sets, Physica.","DOI":"10.1007\/978-3-7908-1870-3_1"},{"key":"ref_5","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_6","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1016\/j.ijar.2007.08.008","article-title":"Dynamic intuitionistic fuzzy multi-attribute decision making","volume":"48","author":"Xu","year":"2008","journal-title":"Int. J. Approx. Reason."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1016\/j.asoc.2009.08.009","article-title":"Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making","volume":"10","author":"Wei","year":"2010","journal-title":"Appl. Soft Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/j.eswa.2009.05.005","article-title":"Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making","volume":"37","author":"Tan","year":"2010","journal-title":"Expert Syst. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1016\/j.knosys.2012.09.006","article-title":"Some intuitionistic fuzzy Einstein hybrid aggregation operators and their application to multiple attribute decision making","volume":"37","author":"Zhao","year":"2013","journal-title":"Knowl.-Based Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2026","DOI":"10.1016\/j.eswa.2011.08.031","article-title":"Some induced correlated aggregating operators with intuitionistic fuzzy information and their application to multiple attribute group decision making","volume":"39","author":"Wei","year":"2012","journal-title":"Expert Syst. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3846\/16486897.2017.1281139","article-title":"Intuitionistic fuzzy EDAS method: An application to solid waste disposal site selection","volume":"25","author":"Kahraman","year":"2017","journal-title":"J. Environ. Eng. Landsc. Manag."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4865","DOI":"10.3233\/JIFS-18979","article-title":"Interval-valued intuitionistic fuzzy CODAS method and its application to wave energy facility location selection problem","volume":"35","author":"Bolturk","year":"2018","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1002\/int.21939","article-title":"Intuitionistic fuzzy power aggregation operator based on entropy and its application in decision making","volume":"33","author":"Jiang","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1007\/s10726-018-9603-9","article-title":"Interval-valued Atanassov intuitionistic Fuzzy CODAS method for multi criteria group decision making problems","volume":"28","author":"Yeni","year":"2019","journal-title":"Group Decis. Negot."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1016\/j.patrec.2006.07.009","article-title":"Similarity measures between intuitionistic fuzzy (vague) sets: A comparative analysis","volume":"28","author":"Li","year":"2007","journal-title":"Pattern Recognit. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Yager, R.R. (2013, January 24). Pythagorean fuzzy subsets. Proceedings of the 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA\/NAFIPS), Edmonton, AB, Canada.","DOI":"10.1109\/IFSA-NAFIPS.2013.6608375"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1061","DOI":"10.1002\/int.21676","article-title":"Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets","volume":"29","author":"Zhang","year":"2014","journal-title":"Int. J. Intell. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1645","DOI":"10.1080\/10807039.2018.1546115","article-title":"Assessing occupational risks in pipeline construction using FMEA-based AHP-MOORA integrated approach under Pythagorean fuzzy environment","volume":"25","author":"Mete","year":"2018","journal-title":"Hum. Ecol. Risk Assess. Int. J."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1002\/int.21745","article-title":"A note on extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets","volume":"31","author":"Yang","year":"2016","journal-title":"Int. J. Intell. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"886","DOI":"10.1002\/int.21809","article-title":"A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision making","volume":"31","author":"Garg","year":"2016","journal-title":"Int. J. Intell. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1002\/int.21814","article-title":"Pythagorean fuzzy Choquet integral based MABAC method for multiple attribute group decision making","volume":"31","author":"Peng","year":"2016","journal-title":"Int. J. Intell. Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1002\/int.21860","article-title":"Generalized Pythagorean fuzzy geometric aggregation operators using Einstein t-norm and t-conorm for multicriteria decision-making process","volume":"32","author":"Garg","year":"2017","journal-title":"Int. J. Intell. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2556","DOI":"10.1002\/int.22160","article-title":"On generalized similarity measures for Pythagorean fuzzy sets and their applications to multiple attribute decision-making","volume":"34","author":"Verma","year":"2019","journal-title":"Int. J. Intell. Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"7419","DOI":"10.1109\/ACCESS.2018.2890097","article-title":"Algorithms for interval-valued pythagorean fuzzy sets in emergency decision making based on multiparametric similarity measures and WDBA","volume":"7","author":"Peng","year":"2019","journal-title":"IEEE Access"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"955","DOI":"10.2991\/ijcis.d.190820.001","article-title":"A Novel Distance Measure for Pythagorean Fuzzy Sets and its Applications to the Technique for Order Preference by Similarity to Ideal Solutions","volume":"12","author":"Zhou","year":"2019","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Khan, A.A., Ashraf, S., Abdullah, S., Qiyas, M., Luo, J., and Khan, S.U. (2019). Pythagorean fuzzy Dombi aggregation operators and their application in decision support system. Symmetry, 11.","DOI":"10.3390\/sym11030383"},{"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":"2017","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1002\/int.21927","article-title":"Some q-rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision making","volume":"33","author":"Liu","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1002\/int.21933","article-title":"Some q-rung orthopai fuzzy Bonferroni mean operators and their application to multi-attribute group decision making","volume":"33","author":"Liu","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2255","DOI":"10.1002\/int.22028","article-title":"Exponential operation and aggregation operator for q-rung orthopair fuzzy set and their decision-making method with a new score function","volume":"33","author":"Peng","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1426","DOI":"10.1002\/int.21985","article-title":"Some q-rung orthopair fuzzy Heronian mean operators in multiple attribute decision making","volume":"33","author":"Wei","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"834","DOI":"10.1109\/TFUZZ.2018.2826452","article-title":"Multiple-attribute decision making based on Archimedean Bonferroni operators of q-rung orthopair fuzzy numbers","volume":"27","author":"Liu","year":"2018","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1900","DOI":"10.1002\/int.22001","article-title":"Multiple attribute decision-making method for dealing with heterogeneous relationship among attributes and unknown attribute weight information under q-rung orthopair fuzzy environment","volume":"33","author":"Liu","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"802","DOI":"10.1002\/int.21968","article-title":"Minkowski-type distance measures for generalized orthopair fuzzy sets","volume":"33","author":"Du","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2139","DOI":"10.1002\/int.22007","article-title":"Another view on q-rung orthopair fuzzy sets","volume":"33","author":"Ali","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Liu, P., Chen, S.M., and Wang, P. (2018). Multiple-attribute group decision-making based on q-rung orthopair fuzzy power maclaurin symmetric mean operators. IEEE Trans. Syst. Man Cybern. Syst., 1\u201316.","DOI":"10.1109\/TSMC.2018.2852948"},{"key":"ref_37","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_38","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1109\/91.995119","article-title":"Complex fuzzy sets","volume":"10","author":"Ramot","year":"2002","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1016\/0165-0114(89)90122-X","article-title":"Fuzzy Complex Numbers","volume":"33","author":"Buckley","year":"1989","journal-title":"Fuzzy Sets Syst."},{"key":"ref_40","unstructured":"Nguyen, H.T., Kandel, A., and Kreinovich, V. (2000, January 7\u201310). Complex Fuzzy Sets: Towards New Foundations. Proceedings of the Ninth IEEE International Conference on Fuzzy Systems. FUZZ-IEEE 2000 (Cat. No. 00CH37063), San Antonio, TX, USA."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1227","DOI":"10.1016\/j.ijar.2009.05.010","article-title":"Operation properties and \u03b4-equalities of complex fuzzy sets","volume":"50","author":"Zhang","year":"2009","journal-title":"Int. J. Approx. Reason."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1063\/1.4757515","article-title":"Complex intuitionistic fuzzy sets","volume":"1482","author":"Alkouri","year":"2012","journal-title":"AIP Conf. Proc."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Kumar, T., and Bajaj, R.K. (2014). On complex intuitionistic fuzzy soft sets with distance measures and entropies. J. Math.","DOI":"10.1155\/2014\/972198"},{"key":"ref_44","first-page":"1898","article-title":"The algebraic structures of complex intuitionistic fuzzy soft sets associated with groups and subgroups","volume":"26","author":"Quek","year":"2019","journal-title":"Sci. Iran."},{"key":"ref_45","first-page":"149","article-title":"Complex multi-fuzzy soft expert set and its application","volume":"14","author":"Hassan","year":"2019","journal-title":"Int. J. Math. Comput. Sci."},{"key":"ref_46","first-page":"305","article-title":"A note on fuzzy soft topological spaces","volume":"3","author":"Roy","year":"2012","journal-title":"Ann. Fuzzy Math. Inform."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"65002","DOI":"10.1109\/ACCESS.2018.2877921","article-title":"Complex multi-fuzzy soft set: Its entropy and similarity measure","volume":"6","author":"Hassan","year":"2018","journal-title":"IEEE Access"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1002\/int.21933","article-title":"A Method to Multi-Attribute Group Decision-Making Problem with Complex q-Rung Orthopair Linguistic Information Based on Heronian Mean Operators","volume":"33","author":"Liu","year":"2018","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"2679","DOI":"10.1007\/s13369-018-3413-x","article-title":"Some generalized complex intuitionistic fuzzy aggregation operators and their application to multicriteria decision-making process","volume":"44","author":"Garg","year":"2019","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"496","DOI":"10.1007\/s10489-018-1290-3","article-title":"A robust correlation coefficient measure of complex intuitionistic fuzzy sets and their applications in decision-making","volume":"49","author":"Garg","year":"2019","journal-title":"Appl. Intell."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"e12325","DOI":"10.1111\/exsy.12325","article-title":"Complex intuitionistic fuzzy power aggregation operators and their applications in multicriteria decision-making","volume":"35","author":"Rani","year":"2018","journal-title":"Expert Syst."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Ullah, K., Mahmood, T., Ali, Z., and Jan, N. (2019). On some distance measures of complex Pythagorean fuzzy sets and their applications in pattern recognition. Complex. Intell. Syst., 1\u201313.","DOI":"10.1007\/s40747-019-0103-6"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Akram, M., and Naz, S. (2019). A novel decision-making approach under complex Pythagorean fuzzy environment. Math. Comput. Appl., 24.","DOI":"10.3390\/mca24030073"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1179","DOI":"10.1109\/TFUZZ.2006.890678","article-title":"Intuitionistic fuzzy aggregation operators","volume":"15","author":"Xu","year":"2007","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"546","DOI":"10.1007\/s10588-017-9242-8","article-title":"Confidence levels based Pythagorean fuzzy aggregation operators and its application to decision-making process","volume":"23","author":"Garg","year":"2017","journal-title":"Comput. Math. Organ. Theory"}],"container-title":["Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2078-2489\/11\/1\/5\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:44:15Z","timestamp":1760190255000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2078-2489\/11\/1\/5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,20]]},"references-count":55,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["info11010005"],"URL":"https:\/\/doi.org\/10.3390\/info11010005","relation":{},"ISSN":["2078-2489"],"issn-type":[{"value":"2078-2489","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,12,20]]}}}