{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T06:16:08Z","timestamp":1778220968617,"version":"3.51.4"},"reference-count":42,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2019,3,20]],"date-time":"2019-03-20T00:00:00Z","timestamp":1553040000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Thailand Research Fund (TRF) and the King Mongkut's University of Technology Thonburi (KMUTT)","award":["Grant No. RSA6080047"],"award-info":[{"award-number":["Grant No. RSA6080047"]}]},{"name":"The Rajamangala University of Technology Thanyaburi (RMUTTT)","award":["Grant No.NSF62D0604"],"award-info":[{"award-number":["Grant No.NSF62D0604"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, a generalized picture fuzzy soft set is proposed, which is an extension of the picture fuzzy soft sets. We investigate the basic properties of picture fuzzy soft sets and define an F-subset, M-subset, extended union, extended intersection, restricted union, restricted intersection and also prove the De Morgan\u2019s laws for picture fuzzy soft information. We investigate upper and lower substitution for both picture fuzzy sets and generalized picture fuzzy soft sets. Meanwhile, the related proofs are given in detail. Finally, we propose an algorithm to deal with generalized picture fuzzy soft information. To show the supremacy and effectiveness of the proposed technique, we illustrate a descriptive example using generalized picture fuzzy soft information. Results indicate that the proposed technique is more generalized and effective over all the existing structures of fuzzy soft sets.<\/jats:p>","DOI":"10.3390\/sym11030415","type":"journal-article","created":{"date-parts":[[2019,3,21]],"date-time":"2019-03-21T04:11:56Z","timestamp":1553141516000},"page":"415","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":83,"title":["Generalized Picture Fuzzy Soft Sets and Their Application in Decision Support Systems"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7983-706X","authenticated-orcid":false,"given":"Muhammad Jabir","family":"Khan","sequence":"first","affiliation":[{"name":"KMUTT Fixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut\u2019s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5463-4581","authenticated-orcid":false,"given":"Poom","family":"Kumam","sequence":"additional","affiliation":[{"name":"KMUTT Fixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut\u2019s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand"},{"name":"KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut\u2019s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8616-8829","authenticated-orcid":false,"given":"Shahzaib","family":"Ashraf","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul wali Khan University, Mardan 23200, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8773-4821","authenticated-orcid":false,"given":"Wiyada","family":"Kumam","sequence":"additional","affiliation":[{"name":"Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT), Thanyaburi, Pathumthani 12110, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,3,20]]},"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. Contr."},{"key":"ref_2","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_3","doi-asserted-by":"crossref","first-page":"610","DOI":"10.1109\/21.229476","article-title":"Vague sets","volume":"23","author":"Gau","year":"1993","journal-title":"IEEE Trans. Syst. Man Cybernet."},{"key":"ref_4","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_5","doi-asserted-by":"crossref","unstructured":"(1999). Molodtsov, Soft set theory-first results. Comput. Math. Appl., 37, 19\u201331.","DOI":"10.1016\/S0898-1221(99)00056-5"},{"key":"ref_6","first-page":"589","article-title":"Fuzzy soft sets","volume":"9","author":"Maji","year":"2001","journal-title":"J. Fuzzy Math."},{"key":"ref_7","first-page":"677","article-title":"Intuitionistic fuzzy soft sets","volume":"9","author":"Maji","year":"2001","journal-title":"J. Fuzzy Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1016\/j.camwa.2009.04.019","article-title":"Combination of interval-valued fuzzy set and soft set","volume":"58","author":"Yang","year":"2009","journal-title":"Comput. Math. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1425","DOI":"10.1016\/j.camwa.2009.12.006","article-title":"Generalised fuzzy soft sets","volume":"59","author":"Majumdar","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1016\/j.camwa.2009.10.015","article-title":"Vague soft sets and their properties","volume":"59","author":"Xu","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3329","DOI":"10.1016\/j.asoc.2011.01.003","article-title":"A note on soft sets, rough soft sets and fuzzy soft sets","volume":"11","author":"Ali","year":"2011","journal-title":"Appl. Soft Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"5844","DOI":"10.1016\/j.apm.2012.01.036","article-title":"The trapezoidal fuzzy soft set and its application in MCDM","volume":"36","author":"Xiao","year":"2012","journal-title":"Appl. Math. Model."},{"key":"ref_13","first-page":"57","article-title":"Neutrosophic soft set","volume":"5","author":"Maji","year":"2013","journal-title":"Ann. Fuzzy Math. Inform."},{"key":"ref_14","first-page":"130","article-title":"Intuitionistic neutrosophic soft set","volume":"8","author":"Broumi","year":"2013","journal-title":"J. Inf. Comput. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"4915","DOI":"10.1016\/j.apm.2012.10.015","article-title":"The multi-fuzzy soft set and its application in decision making","volume":"37","author":"Yang","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_16","first-page":"643785","article-title":"Hesitant fuzzy soft set and its applications in multicriteria decision making","volume":"2014","author":"Wang","year":"2014","journal-title":"J. Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3552","DOI":"10.1016\/j.asoc.2013.03.015","article-title":"Generalized intuitionistic fuzzy soft sets with applications in decision-making","volume":"13","author":"Agarwal","year":"2013","journal-title":"Appl. Soft Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1109\/TFUZZ.2018.2860967","article-title":"Another view on generalized intuitionistic fuzzy soft sets and related multi attribute decision making methods","volume":"27","author":"Feng","year":"2018","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3308","DOI":"10.1016\/j.camwa.2010.03.015","article-title":"Soft matrix theory and its decision making","volume":"59","author":"Cagman","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1016\/j.asoc.2014.08.042","article-title":"Soft discernibility matrix and its applications in decision making","volume":"24","author":"Feng","year":"2014","journal-title":"Appl. Soft Comput."},{"key":"ref_21","first-page":"409","article-title":"Picture fuzzy sets","volume":"30","author":"Cuong","year":"2014","journal-title":"J. Comput. Sci. Cybern."},{"key":"ref_22","first-page":"2857","article-title":"Correlation coefficients for picture fuzzy sets","volume":"27","author":"Singh","year":"2014","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.eswa.2014.07.026","article-title":"DPFCM: A novel distributed picture fuzzy clustering method on picture fuzzy sets","volume":"2","author":"Son","year":"2015","journal-title":"Expert Syst. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Thong, P.H., and Son, L.H. (2015, January 9\u201311). A new approach to multi-variables fuzzy forecasting using picture fuzzy clustering and picture fuzzy rules interpolation method. Proceedings of the 6th International Conference on Knowledge and Systems Engineering, Hanoi, Vietnam.","DOI":"10.1007\/978-3-319-11680-8_54"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1016\/j.asoc.2016.05.009","article-title":"Generalized picture distance measure and applications to picture fuzzy clustering","volume":"46","author":"Son","year":"2016","journal-title":"Appl. Soft Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10700-016-9249-5","article-title":"Measuring analogousness in picture fuzzy sets: From picture distance measures to picture association measures","volume":"16","author":"Son","year":"2017","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_27","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":"Son","year":"2016","journal-title":"Appl. Intell."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.engappai.2016.08.009","article-title":"Picture fuzzy clustering for complex data","volume":"56","author":"Thong","year":"2016","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.knosys.2016.06.023","article-title":"A novel automatic picture fuzzy clustering method based on particle swarm optimization and picture composite cardinality","volume":"109","author":"Thong","year":"2016","journal-title":"Knowl. Based Syst."},{"key":"ref_30","first-page":"713","article-title":"Picture fuzzy aggregation operator and their application to multiple attribute decision making","volume":"33","author":"Wei","year":"2017","journal-title":"J. Int. Fuzzy Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"491","DOI":"10.3846\/16111699.2016.1197147","article-title":"Picture fuzzy cross-entropy for multiple attribute decision making problems","volume":"17","author":"Wei","year":"2016","journal-title":"J. Bus. Econ. Manag."},{"key":"ref_32","first-page":"1711","article-title":"Adjustable soft discernibility matrix based on picture fuzzy soft sets and its application in decision making","volume":"29","author":"Yang","year":"2015","journal-title":"J. Int. Fuzzy Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"5275","DOI":"10.1007\/s13369-017-2625-9","article-title":"Some picture fuzzy aggregation operators and their applications to multi criteria decision-making","volume":"42","author":"Garg","year":"2017","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1615\/Int.J.UncertaintyQuantification.2017020096","article-title":"Algorithm for picture fuzzy multiple attribute decision making based on new distance measure","volume":"7","author":"Peng","year":"2017","journal-title":"Int. J. Uncertain. Quant."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Liu, Z., Qin, K., and Pei, Z. (2017). A Method for Fuzzy Soft Sets in Decision-Making Based on an Ideal Solution. Symmetry, 9.","DOI":"10.3390\/sym9100246"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Ashraf, S., Mahmood, T., Abdullah, S., and Khan, Q. (2018). Different approaches to multi-criteria group decision making problems for picture fuzzy environment. Bull. Braz. Math. Soc. New Ser., 1\u201325.","DOI":"10.1007\/s00574-018-0103-y"},{"key":"ref_37","first-page":"59","article-title":"Novel concept of cubic picture fuzzy sets","volume":"24","author":"Ashraf","year":"2018","journal-title":"J. New Theory"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Zeng, S., Asharf, S., Arif, M., and Abdullah, S. (2019). Application of Exponential Jensen Picture Fuzzy Divergence Measure in Multi-Criteria Group Decision Making. Mathematics, 7.","DOI":"10.3390\/math7020191"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1","DOI":"10.20454\/ijas.2019.1506","article-title":"Solution of multi-criteria group decision making problem based on picture linguistic informations","volume":"8","author":"Muhammad","year":"2019","journal-title":"Int. J. Algebra Stat."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.asoc.2018.10.021","article-title":"Picture fuzzy Dombi aggregation operator: Application to MADM process","volume":"74","author":"Jana","year":"2019","journal-title":"Appl. Soft Comput. J."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Chen, J., and Ye, J. (2017). Some single-valued neutrosophic Dombi weighted aggregation operators for multiple attribute decision-making. Symmetry, 9.","DOI":"10.3390\/sym9060082"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1057\/s41274-017-0190-y","article-title":"Some intuitionistic fuzzy Dombi bonferroni mean operators and their application to multi-attribute group decision making","volume":"69","author":"Liu","year":"2018","journal-title":"J. Oper. Res. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/3\/415\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:39:23Z","timestamp":1760186363000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/3\/415"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,20]]},"references-count":42,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,3]]}},"alternative-id":["sym11030415"],"URL":"https:\/\/doi.org\/10.3390\/sym11030415","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,3,20]]}}}