{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,15]],"date-time":"2026-09-15T23:17:15Z","timestamp":1789514235477,"version":"build-2803163510"},"reference-count":27,"publisher":"SAGE Publications","issue":"1","license":[{"start":{"date-parts":[[2019,6,27]],"date-time":"2019-06-27T00:00:00Z","timestamp":1561593600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"published-print":{"date-parts":[[2019,7,9]]},"abstract":"<jats:p>Pythagorean fuzzy sets (PFSs) and interval-valued Pythagorean fuzzy sets (IVPFSs) play a vital role in decision-making processes. In this paper based on PFS and IVPFS we introduce the concept of Cubic Pythagorean fuzzy set in which membership degree is an IVPFS and non-membership degree is a PFS. We define some basic operation of Cubic Pythagorean fuzzy numbers (CPFNs). We define score and accuracy functions to compare CPFNs. We also define distance between CPFNs. Based on the defined operations we develop Cubic Pythagorean fuzzy weighted averaging (CPFWA) operator and Cubic Pythagorean fuzzy weighted geometric (CPFWG) operator. We discuss some properties of the developed operators such as idempotancy, boundedness and monotonicity. Moreover, we give a multi-attribute decision making, to show the validity and effectiveness of the developed approach. Finally, we compare our approach with the existing methods.<\/jats:p>","DOI":"10.3233\/jifs-18382","type":"journal-article","created":{"date-parts":[[2019,6,28]],"date-time":"2019-06-28T13:09:54Z","timestamp":1561727394000},"page":"1529-1544","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":39,"title":["Cubic Pythagorean fuzzy sets and their application to multi-attribute decision making with unknown weight information"],"prefix":"10.1177","volume":"37","author":[{"given":"Syed Zaheer","family":"Abbas","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics Beijing Institute of Technology Beijing, P.R. China"},{"name":"Department of Mathematics and Statistics Hazara University, Mansehra, KPK, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muhammad Sajjad","family":"Ali Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics Hazara University, Mansehra, KPK, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Saleem","family":"Abdullah","sequence":"additional","affiliation":[{"name":"Department of Mathematics Abdul Wali Khan University, Mardan, KPK, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huafei","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics Beijing Institute of Technology Beijing, P.R. China"},{"name":"Beijing key Laboratory on MCAACI, Beijing Institute of Technology, Beijing, P.R. China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fawad","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department of Mathematics Abbottabad University of Science and Technology, Abbottabad, KPK, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"179","published-online":{"date-parts":[[2019,6,27]]},"reference":[{"key":"e_1_3_1_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-0114(86)80034-3"},{"key":"e_1_3_1_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(89)90215-7"},{"key":"e_1_3_1_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2008.10.012"},{"key":"e_1_3_1_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejor.2009.05.009"},{"issue":"1","key":"e_1_3_1_6_2","first-page":"83","article-title":"Annals of Fuzzy Mathematics and Informatics","volume":"4","author":"Jun Y.B.","year":"2012","unstructured":"Y.B.Jun, C.S.Kim, Ki and O.Yang, Annals of Fuzzy Mathematics and Informatics, Cubic sets 4(1) (2012), 83\u201398.","journal-title":"Cubic sets"},{"key":"e_1_3_1_7_2","first-page":"1","article-title":"A novel extension of cubic sets and its applications in BCK\/BCI-algebras","author":"Jun Y.B.","year":"2018","unstructured":"Y.B.Jun, A novel extension of cubic sets and its applications in BCK\/BCI-algebras, Annals of Fuzzy Mathematics and Informatics Volume x, No. x, (Month 2018), pp. 1-xx.","journal-title":"Annals of Fuzzy Mathematics and Informatics"},{"key":"e_1_3_1_8_2","doi-asserted-by":"publisher","DOI":"10.3233\/JIFS-17811"},{"key":"e_1_3_1_9_2","doi-asserted-by":"crossref","unstructured":"M.S.A.Khan S.Abdullah A.Ali and K.Rahman Pythagorean Hesitant Fuzzy Information Aggregation and Their Application to Multi-Attribute Group Decision-Making Problems J Intell Syst (2018) 1-18. https:\/\/doi.org\/10.1515\/jisys-2017-0231.","DOI":"10.1515\/jisys-2017-0231"},{"key":"e_1_3_1_10_2","doi-asserted-by":"publisher","DOI":"10.3233\/JIFS-171164"},{"key":"e_1_3_1_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-2217(98)00040-X"},{"key":"e_1_3_1_12_2","doi-asserted-by":"crossref","unstructured":"W.Liang X.Zhang and M.Liu The Maximizing Deviation Method Based on Interval-Valued Pythagorean Fuzzy Weighted Aggregating Operator for Multiple Criteria Group Decision Analysis Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume (2015) Article ID 746572 pp. 15.","DOI":"10.1155\/2015\/746572"},{"key":"e_1_3_1_13_2","doi-asserted-by":"publisher","DOI":"10.1002\/int.21738"},{"key":"e_1_3_1_14_2","first-page":"1","article-title":"Fundamental Properties of Interval-Valued Pythagorean Fuzzy Aggregation Operators","volume":"00","author":"Peng X.","year":"2015","unstructured":"X.Peng and Y.Yang, Fundamental Properties of Interval-Valued Pythagorean Fuzzy Aggregation Operators, International Journal of Intelligent Systems 00 (2015), 1\u201344.","journal-title":"International Journal of Intelligent Systems"},{"key":"e_1_3_1_15_2","first-page":"1","article-title":"Pythagorean Fuzzy Choquet Integral Based MABAC Method for Multiple Attribute Group Decision Making","volume":"00","author":"Peng X.","year":"2016","unstructured":"X.Peng and Y.Yang, Pythagorean Fuzzy Choquet Integral Based MABAC Method for Multiple Attribute Group Decision Making, International Journal of Intelligent Systems 00 (2016), 1\u201332.","journal-title":"International Journal of Intelligent Systems"},{"key":"e_1_3_1_16_2","first-page":"122","article-title":"Some properties of Pythagorean fuzzy hybrid averaging aggregation operator","volume":"7","author":"Rahman K.","year":"2017","unstructured":"K.Rahman, M.S.A.Khan, S.Abdullah and F.Husain, Some properties of Pythagorean fuzzy hybrid averaging aggregation operator, Journal of Applied Environmental and Biological Sciences 7 (2017), 122\u2013133.","journal-title":"Journal of Applied Environmental and Biological Sciences"},{"key":"e_1_3_1_17_2","doi-asserted-by":"publisher","DOI":"10.71330\/thenucleus.2017.107"},{"key":"e_1_3_1_18_2","article-title":"Pythagorean fuzzy hybrid geometric aggregation operator and their applications to multiple attribute decision making","volume":"14","author":"Rahman K.","unstructured":"K.Rahman, S.Abdullah, M.S.A.Khan and M.Shakeel, Pythagorean fuzzy hybrid geometric aggregation operator and their applications to multiple attribute decision making, International Journal of Computer Science and Information Security (IJCSIS) 14(6) (2016).","journal-title":"International Journal of Computer Science and Information Security (IJCSIS)"},{"key":"e_1_3_1_19_2","first-page":"67","article-title":"Pythagorean fuzzy ordered weighted geometric aggregation operator and their application to multiple attribute group decision making","volume":"7","author":"Rahman K.","year":"2017","unstructured":"K.Rahman, S.Abdullah, F.Hussain, M.S.A.Khan and M.Shakeel, Pythagorean fuzzy ordered weighted geometric aggregation operator and their application to multiple attribute group decision making, Journal of Applied Environmental and Biological Sciences 7 (2017), 67\u201383.","journal-title":"Journal of Applied Environmental and Biological Sciences"},{"key":"e_1_3_1_20_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.knosys.2008.03.038"},{"key":"e_1_3_1_21_2","first-page":"24","article-title":"Using the method of maximizing deviations to make decision for multi-indices","volume":"7","author":"Wang Y.M.","year":"1998","unstructured":"Y.M.Wang, Using the method of maximizing deviations to make decision for multi-indices, System Engineering and Electronics 7 (1998), 24\u201326.","journal-title":"System Engineering and Electronics"},{"key":"e_1_3_1_22_2","doi-asserted-by":"publisher","DOI":"10.1002\/int.21584"},{"key":"e_1_3_1_23_2","first-page":"57","article-title":"Pythagorean fuzzy subsets","volume":"24","author":"Yager R.R.","year":"2013","unstructured":"R.R.Yager, Pythagorean fuzzy subsets. In: Proc Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, Canada; 24-28 June, (2013) pp. 57\u201361.","journal-title":"In: Proc Joint IFSA World Congress and NAFIPS Annual Meeting"},{"key":"e_1_3_1_24_2","doi-asserted-by":"publisher","DOI":"10.1109\/TFUZZ.2013.2278989"},{"key":"e_1_3_1_25_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cie.2009.10.005"},{"key":"e_1_3_1_26_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0019-9958(65)90241-X"},{"key":"e_1_3_1_27_2","doi-asserted-by":"publisher","DOI":"10.1002\/int.21676"},{"key":"e_1_3_1_28_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2015.10.012"}],"container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/JIFS-18382","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.3233\/JIFS-18382","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/JIFS-18382","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T09:38:43Z","timestamp":1777455523000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/JIFS-18382"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,27]]},"references-count":27,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,7,9]]}},"alternative-id":["10.3233\/JIFS-18382"],"URL":"https:\/\/doi.org\/10.3233\/jifs-18382","relation":{},"ISSN":["1064-1246","1875-8967"],"issn-type":[{"value":"1064-1246","type":"print"},{"value":"1875-8967","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,6,27]]}}}