{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T22:19:41Z","timestamp":1773440381659,"version":"3.50.1"},"reference-count":51,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,9,7]],"date-time":"2021-09-07T00:00:00Z","timestamp":1630972800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>As an extension of intuitionistic fuzzy sets, the theory of picture fuzzy sets not only deals with the degrees of rejection and acceptance but also considers the degree of refusal during a decision-making process; therefore, by incorporating this competency of picture fuzzy sets, the goal of this study is to propose a novel hybrid model called picture fuzzy soft expert sets by combining picture fuzzy sets with soft expert sets for dealing with uncertainties in different real-world group decision-making problems. The proposed hybrid model is a more generalized form of intuitionistic fuzzy soft expert sets. Some novel desirable properties of the proposed model, namely, subset, equality, complement, union and intersection, are investigated together with their corresponding examples. Two well-known operations AND and OR are also studied for the developed model. Further, a decision-making method supporting by an algorithmic format under the proposed approach is presented. Moreover, an illustrative application is provided for its better demonstration, which is subjected to the selection of a suitable company of virtual reality devices. Finally, a comparison of the initiated method is explored with some existing models, including intuitionistic fuzzy soft expert sets.<\/jats:p>","DOI":"10.3390\/e23091176","type":"journal-article","created":{"date-parts":[[2021,9,7]],"date-time":"2021-09-07T10:24:25Z","timestamp":1631010265000},"page":"1176","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["A New Group Decision-Making Technique under Picture Fuzzy Soft Expert Information"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7855-508X","authenticated-orcid":false,"given":"Fairouz","family":"Tchier","sequence":"first","affiliation":[{"name":"Department of Mathematics, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5316-3063","authenticated-orcid":false,"given":"Ghous","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Division of Science and Technology, University of Education, Lahore 54770, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6794-7942","authenticated-orcid":false,"given":"Muhammad","family":"Gulzar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University Faisalabad, Faisalabad 38000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8522-1942","authenticated-orcid":false,"given":"Dragan","family":"Pamu\u010dar","sequence":"additional","affiliation":[{"name":"Department of Logistics, Military Academy, University of Defence in Belgrade, 11000 Belgrade, Serbia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3877-8059","authenticated-orcid":false,"given":"Ganesh","family":"Ghorai","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,7]]},"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":"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_3","doi-asserted-by":"crossref","unstructured":"Bo\u017eani\u0107, D., Mili\u0107, A., Te\u0161i\u0107, D., Salabun, W., and Pamu\u010dar, D. (2021). D numbers\u2013FUCOM\u2013fuzzy RAFSI model for selecting the group of construction machines for enabling mobility. Facta Univ. Ser. Mech. Eng., 1\u201326.","DOI":"10.22190\/FUME210318047B"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Chen, C.H. (2020). A novel multi-criteria decision-making model for building material supplier selection based on entropy-AHP weighted TOPSIS. Entropy, 22.","DOI":"10.3390\/e22020259"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Dymova, L., Kaczmarek, K., Sevastjanov, P., and Kulawik, J. (2021). A fuzzy multiple criteria decision making approach with a complete user friendly computer implementation. Entropy, 23.","DOI":"10.3390\/e23020203"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"114508","DOI":"10.1016\/j.eswa.2020.114508","article-title":"Multi-criteria decision analysis towards robust service quality measurement","volume":"170","author":"Pamucar","year":"2021","journal-title":"Expert Syst. Appl."},{"key":"ref_7","unstructured":"Cuong, B.C. (2013). Picture Fuzzy Sets-First Results. Part 1 and Part 2, Seminar Neuro-Fuzzy Systems with Applications, Institute of Mathematics. Technical Report."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"591","DOI":"10.3233\/IFS-141338","article-title":"Correlation coefficients for picture fuzzy sets","volume":"28","author":"Singh","year":"2015","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_9","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_10","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 multicriteria decision-making","volume":"42","author":"Garg","year":"2017","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"555","DOI":"10.15388\/Informatica.2018.181","article-title":"TODIM method for picture fuzzy multiple attribute decision making","volume":"29","author":"Wei","year":"2018","journal-title":"Informatica"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1007\/s00574-018-0103-y","article-title":"Different approaches to multi-criteria group decision making problems for picture fuzzy environment","volume":"50","author":"Ashraf","year":"2019","journal-title":"Bull. Brazil. Math. Soc. New Ser."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"104","DOI":"10.31181\/dmame2104104s","article-title":"Career selection of students using hybridized distance measure based on picture fuzzy set and rough set theory","volume":"4","author":"Sahu","year":"2021","journal-title":"Decis. Mak. Appl. Manag. Eng."},{"key":"ref_14","first-page":"77","article-title":"Some similarity measures for picture fuzzy sets and their applications","volume":"15","author":"Wei","year":"2018","journal-title":"Iran. J. Fuzzy Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"10115","DOI":"10.3233\/JIFS-202839","article-title":"CPT-TODIM method for picture fuzzy multiple attribute group decision making and its application to food enterprise quality credit evaluation","volume":"40","author":"Jiang","year":"2021","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1329","DOI":"10.1007\/s00521-020-05034-y","article-title":"An optimization study based on Dijkstra algorithm for a network with trapezoidal picture fuzzy numbers","volume":"33","author":"Akram","year":"2021","journal-title":"Neural Comput. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"51236","DOI":"10.1109\/ACCESS.2019.2910844","article-title":"New operations on interval-valued picture fuzzy set, interval-valued picture fuzzy soft set and their applications","volume":"7","author":"Khalil","year":"2019","journal-title":"IEEE Access"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2103","DOI":"10.3233\/JIFS-190812","article-title":"An adjustable weighted soft discernibility matrix based on generalized picture fuzzy soft set and its applications in decision making","volume":"38","author":"Khan","year":"2020","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"101873","DOI":"10.1016\/j.scs.2019.101873","article-title":"MULTIMOORA based MCDM model for site selection of car sharing station under picture fuzzy environment","volume":"53","author":"Lin","year":"2020","journal-title":"Sustain. Cities Soc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"682","DOI":"10.24818\/EA\/2019\/52\/682","article-title":"Picture fuzzy weighted distance measures and their application to investment selection","volume":"21","author":"Liu","year":"2019","journal-title":"Amfiteatru Econ."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Liu, D., Luo, Y., and Liu, Z. (2020). The linguistic picture fuzzy set and its application in multi-criteria decision-making: An illustration to the TOPSIS and TODIM methods based on entropy weight. Symmetry, 12.","DOI":"10.3390\/sym12071170"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"114644","DOI":"10.1016\/j.eswa.2021.114644","article-title":"Picture fuzzy extension of the CODAS method for multi-criteria vehicle shredding facility location","volume":"175","author":"Karagoz","year":"2021","journal-title":"Expert Syst. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Zhao, R., Luo, M., and Li, S. (2021). A dynamic distance measure of picture fuzzy sets and its application. Symmetry, 13.","DOI":"10.3390\/sym13030436"},{"key":"ref_24","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_25","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/S0898-1221(99)00056-5","article-title":"Soft set theory-First results","volume":"37","author":"Molodtsov","year":"1999","journal-title":"Comput. Math. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1547","DOI":"10.1016\/j.camwa.2008.11.009","article-title":"On some new operations in soft set theory","volume":"57","author":"Ali","year":"2009","journal-title":"Comput. Math. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1077","DOI":"10.1016\/S0898-1221(02)00216-X","article-title":"An application of soft sets in a decision-making problem","volume":"44","author":"Maji","year":"2002","journal-title":"Comput. Math. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2381","DOI":"10.1007\/s10462-018-9617-3","article-title":"A novel type of soft rough covering and its application to multicriteria group decision making","volume":"52","author":"Zhan","year":"2019","journal-title":"Art. Intell. Rev."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.ins.2019.04.054","article-title":"Covering-based general multigranulation intuitionistic fuzzy rough sets and corresponding applications to multi-attribute group decision-making","volume":"494","author":"Zhang","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1711","DOI":"10.3233\/IFS-151648","article-title":"Adjustable soft discernibility matrix based on picture fuzzy soft sets and its applications in decision making","volume":"29","author":"Yang","year":"2015","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.eswa.2018.07.060","article-title":"Group decision-making methods based on hesitant N-soft sets","volume":"115","author":"Akram","year":"2019","journal-title":"Expert Syst. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/j.inffus.2018.05.005","article-title":"Necessary and possible hesitant fuzzy sets: A novel model for group decision making","volume":"46","author":"Alcantud","year":"2019","journal-title":"Inf. Fusion"},{"key":"ref_33","first-page":"757868","article-title":"Soft expert sets","volume":"2011","author":"Alkhazaleh","year":"2011","journal-title":"Adv. Deci. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1349","DOI":"10.4236\/am.2014.59127","article-title":"Fuzzy soft expert set and its application","volume":"5","author":"Alkhazaleh","year":"2014","journal-title":"Appl. Math."},{"key":"ref_35","first-page":"1","article-title":"Multi criteria decision making problem with soft expert set","volume":"78","author":"Arockiarani","year":"2013","journal-title":"Int. J. Comput. Appl."},{"key":"ref_36","first-page":"208","article-title":"Possibility fuzzy soft expert set","volume":"12","author":"Bashir","year":"2012","journal-title":"Open J. Appl. Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1585","DOI":"10.3233\/JIFS-151652","article-title":"Cubic soft expert sets and their application in decision making","volume":"31","author":"Qayyum","year":"2016","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_38","first-page":"89","article-title":"Intuitionistic fuzzy soft expert sets and its application in decision making","volume":"1","author":"Broumi","year":"2015","journal-title":"J. New Theory"},{"key":"ref_39","first-page":"175","article-title":"Bipolar fuzzy soft expert set and its application in decision making","volume":"10","author":"Hassan","year":"2017","journal-title":"Int. J. Appl. Deci. Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"10381","DOI":"10.1007\/s13369-020-04733-x","article-title":"Decision-making method based on fuzzy N-soft expert sets","volume":"45","author":"Ali","year":"2020","journal-title":"Arab. J. Sci. Eng."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"12051","DOI":"10.1007\/s00521-021-05850-w","article-title":"Novel MCGDM analysis under m-polar fuzzy soft expert sets","volume":"33","author":"Akram","year":"2021","journal-title":"Neural Comput. Appl."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"5874216","DOI":"10.1155\/2021\/5874216","article-title":"Ranking effectiveness of COVID-19 tests Using fuzzy bipolar soft expert sets","volume":"2021","author":"Ali","year":"2021","journal-title":"Math. Prob. Eng."},{"key":"ref_43","unstructured":"Cuong, B.C., Son, L.H., Phong, P.H., Ngan, R.T., and Thao, N.X. (2015). Some Operators on Interval-Valued Picture Fuzzy Sets and a Picture Clustering Algorithm on Picture Fuzzy Sets, Springer."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"18857","DOI":"10.1007\/s00500-020-05116-y","article-title":"Multi-valued picture fuzzy soft sets and their applications in group decision-making problems","volume":"24","author":"Jan","year":"2020","journal-title":"Soft Comput."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Mihelj, M., Novak, D., and Begu\u0161, S. (2014). Virtual Reality Technology and Applications, Springer Science+Business Media.","DOI":"10.1007\/978-94-007-6910-6"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Akram, M., Ali, G., and Alcantud, J.C.R. (2021). Parameter reduction analysis under interval-valued m-polar fuzzy soft information. Art. Intell. Rev., 1\u201342.","DOI":"10.1007\/s10462-021-10027-x"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"105","DOI":"10.31181\/rme2001020105t","article-title":"Einstein geometric aggregation operators using a novel complex interval-valued Pythagorean fuzzy setting with application in green supplier chain management","volume":"2","author":"Ali","year":"2021","journal-title":"Rep. Mech. Eng."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"135","DOI":"10.31181\/rme200102135a","article-title":"Resolving a location selection problem by means of an integrated AHP-RAFSI approach","volume":"2","author":"Alosta","year":"2021","journal-title":"Rep. Mech. Eng."},{"key":"ref_49","first-page":"419","article-title":"Prioritizing the weights of the evaluation criteria under fuzziness: The fuzzy full consistency method\u2013FUCOM-F","volume":"18","author":"Pamucar","year":"2020","journal-title":"Facta Univ. Ser. Mech. Eng."},{"key":"ref_50","first-page":"409","article-title":"Picture fuzzy sets","volume":"30","author":"Cuong","year":"2014","journal-title":"J. Comput. Sci."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1111\/j.1460-2466.1992.tb00811.x","article-title":"Virtual reality technology: A tutorial","volume":"42","author":"Biocca","year":"1992","journal-title":"J. Commun."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/9\/1176\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:57:59Z","timestamp":1760165879000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/9\/1176"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,7]]},"references-count":51,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["e23091176"],"URL":"https:\/\/doi.org\/10.3390\/e23091176","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9,7]]}}}