{"id":"https://openalex.org/W4411028501","doi":"https://doi.org/10.3390/sym17060868","title":"Aczel\u2013Alsina Shapley Choquet Integral Operators for Multi-Criteria Decision Making in Complex Intuitionistic Fuzzy Environments","display_name":"Aczel\u2013Alsina Shapley Choquet Integral Operators for Multi-Criteria Decision Making in Complex Intuitionistic Fuzzy Environments","publication_year":2025,"publication_date":"2025-06-03","ids":{"openalex":"https://openalex.org/W4411028501","doi":"https://doi.org/10.3390/sym17060868"},"language":"en","primary_location":{"id":"doi:10.3390/sym17060868","doi":"https://doi.org/10.3390/sym17060868","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17060868","pdf_url":null,"source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","norway-1","poland-70","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://doi.org/10.3390/sym17060868","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5024929276","display_name":"Ikhtesham Ullah","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I3131573596","display_name":"Abbottabad University of Science and Technology","ror":"https://ror.org/0549hdw45","country_code":"PK","type":"education","lineage":["https://openalex.org/I3131573596"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Ikhtesham Ullah","raw_affiliation_strings":["Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22500, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22500, Pakistan","institution_ids":["https://openalex.org/I3131573596"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101785778","display_name":"Muhammad Sajjad Ali Khan","orcid":"https://orcid.org/0000-0003-3683-6582","observed_orcids":["https://orcid.org/0000-0003-3683-6582"]},"institutions":[{"id":"https://openalex.org/I16076960","display_name":"COMSATS University Islamabad","ror":"https://ror.org/00nqqvk19","country_code":"PK","type":"education","lineage":["https://openalex.org/I16076960"]},{"id":"https://openalex.org/I4387156092","display_name":"Khushal Khan Khattak University Karak","ror":"https://ror.org/006knb923","country_code":"PK","type":"education","lineage":["https://openalex.org/I4387156092"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Muhammad Sajjad Ali Khan","raw_affiliation_strings":["Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Abbottabad 22060, Pakistan","Department of Mathematics, Khushal Khan Khattak University, Karak 27200, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Abbottabad 22060, Pakistan","institution_ids":["https://openalex.org/I16076960"]},{"raw_affiliation_string":"Department of Mathematics, Khushal Khan Khattak University, Karak 27200, Pakistan","institution_ids":["https://openalex.org/I4387156092"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5045227516","display_name":"Kamran Kamran","orcid":"https://orcid.org/0000-0001-8217-3308","observed_orcids":["https://orcid.org/0000-0001-8217-3308"]},"institutions":[{"id":"https://openalex.org/I206573129","display_name":"Islamia College University","ror":"https://ror.org/02p2c1595","country_code":"PK","type":"education","lineage":["https://openalex.org/I206573129"]}],"countries":["PK"],"is_corresponding":true,"raw_author_name":"Kamran","raw_affiliation_strings":["Department of Mathematics, Islamia College Peshawar, Peshawer 25000, Pakistan"],"raw_orcid":"https://orcid.org/0000-0001-8217-3308","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Islamia College Peshawar, Peshawer 25000, Pakistan","institution_ids":["https://openalex.org/I206573129"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5063184842","display_name":"Fawad Hussain","orcid":"https://orcid.org/0000-0002-3050-8564","observed_orcids":["https://orcid.org/0000-0002-3050-8564"]},"institutions":[{"id":"https://openalex.org/I3131573596","display_name":"Abbottabad University of Science and Technology","ror":"https://ror.org/0549hdw45","country_code":"PK","type":"education","lineage":["https://openalex.org/I3131573596"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Fawad Hussain","raw_affiliation_strings":["Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22500, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22500, Pakistan","institution_ids":["https://openalex.org/I3131573596"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5026770895","display_name":"Madad Khan","orcid":"https://orcid.org/0000-0003-0044-961X","observed_orcids":["https://orcid.org/0000-0003-0044-961X"]},"institutions":[{"id":"https://openalex.org/I16076960","display_name":"COMSATS University Islamabad","ror":"https://ror.org/00nqqvk19","country_code":"PK","type":"education","lineage":["https://openalex.org/I16076960"]},{"id":"https://openalex.org/I4387156092","display_name":"Khushal Khan Khattak University Karak","ror":"https://ror.org/006knb923","country_code":"PK","type":"education","lineage":["https://openalex.org/I4387156092"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Madad Khan","raw_affiliation_strings":["Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Abbottabad 22060, Pakistan","Department of Mathematics, Khushal Khan Khattak University, Karak 27200, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Abbottabad 22060, Pakistan","institution_ids":["https://openalex.org/I16076960"]},{"raw_affiliation_string":"Department of Mathematics, Khushal Khan Khattak University, Karak 27200, Pakistan","institution_ids":["https://openalex.org/I4387156092"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5007601852","display_name":"Ioan\u2010Lucian Popa","orcid":"https://orcid.org/0000-0001-8042-1806","observed_orcids":["https://orcid.org/0000-0001-8042-1806"]},"institutions":[{"id":"https://openalex.org/I3123573182","display_name":"Transilvania University of Bra\u0219ov","ror":"https://ror.org/01cg9ws23","country_code":"RO","type":"education","lineage":["https://openalex.org/I3123573182"]},{"id":"https://openalex.org/I4210140223","display_name":"1 Decembrie 1918 University","ror":"https://ror.org/04577k168","country_code":"RO","type":"education","lineage":["https://openalex.org/I4210140223"]}],"countries":["RO"],"is_corresponding":true,"raw_author_name":"Ioan-Lucian Popa","raw_affiliation_strings":["Department of Computing, Mathematics and Electronics, \u201c1 Decembrie 1918\u201d University of Alba Iulia, 510009 Alba Iulia, Romania","Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania"],"raw_orcid":"https://orcid.org/0000-0001-8042-1806","affiliations":[{"raw_affiliation_string":"Department of Computing, Mathematics and Electronics, \u201c1 Decembrie 1918\u201d University of Alba Iulia, 510009 Alba Iulia, Romania","institution_ids":["https://openalex.org/I4210140223"]},{"raw_affiliation_string":"Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania","institution_ids":["https://openalex.org/I3123573182"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5076083494","display_name":"Hela Elmannai","orcid":"https://orcid.org/0000-0003-2571-1848","observed_orcids":["https://orcid.org/0000-0003-2571-1848"]},"institutions":[{"id":"https://openalex.org/I106778892","display_name":"Princess Nourah bint Abdulrahman University","ror":"https://ror.org/05b0cyh02","country_code":"SA","type":"education","lineage":["https://openalex.org/I106778892"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Hela Elmannai","raw_affiliation_strings":["Department of Information Technology, College of Computer and Information Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0003-2571-1848","affiliations":[{"raw_affiliation_string":"Department of Information Technology, College of Computer and Information Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia","institution_ids":["https://openalex.org/I106778892"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":7,"corresponding_author_ids":["https://openalex.org/A5007601852","https://openalex.org/A5045227516"],"corresponding_institution_ids":["https://openalex.org/I206573129","https://openalex.org/I3123573182","https://openalex.org/I4210140223"],"apc_list":{"value":2400,"currency":"CHF","value_usd":2896},"apc_paid":{"value":2400,"currency":"CHF","value_usd":2896},"fwci":1.58,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.82875917,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"17","issue":"6","first_page":"868","last_page":"868"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10050","display_name":"Multi-Criteria Decision Making","score":0.9965999722480774,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10050","display_name":"Multi-Criteria Decision Making","score":0.9965999722480774,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11671","display_name":"Fuzzy and Soft Set Theory","score":0.002199999988079071,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T12709","display_name":"Optimization and Mathematical Programming","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/complex-intuitionistic-fuzzy-sets","display_name":"complex intuitionistic fuzzy sets","score":0.9610000252723694},{"id":"https://openalex.org/keywords/aggregation-operators","display_name":"aggregation operators","score":0.8389999866485596},{"id":"https://openalex.org/keywords/multi-criteria-decision-making","display_name":"multi-criteria decision making","score":0.9639999866485596},{"id":"https://openalex.org/keywords/weighted-average-operator","display_name":"weighted averaging operator","score":0.6000000238418579},{"id":"https://openalex.org/keywords/weighted-geometric-operator","display_name":"weighted geometric operator","score":0.984000027179718}],"concepts":[{"id":"https://openalex.org/C112799922","wikidata":"https://www.wikidata.org/wiki/Q5104911","display_name":"Choquet integral","level":3,"score":0.8646254539489746},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6422208547592163},{"id":"https://openalex.org/C100516043","wikidata":"https://www.wikidata.org/wiki/Q4455138","display_name":"Fuzzy measure theory","level":5,"score":0.5055019855499268},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.4954777956008911},{"id":"https://openalex.org/C199022921","wikidata":"https://www.wikidata.org/wiki/Q240046","display_name":"Shapley value","level":3,"score":0.4842545688152313},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3592725992202759},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.3331815004348755},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.26543471217155457},{"id":"https://openalex.org/C42011625","wikidata":"https://www.wikidata.org/wiki/Q1055058","display_name":"Fuzzy set","level":3,"score":0.23258733749389648},{"id":"https://openalex.org/C177142836","wikidata":"https://www.wikidata.org/wiki/Q44455","display_name":"Game theory","level":2,"score":0.20421263575553894},{"id":"https://openalex.org/C1883856","wikidata":"https://www.wikidata.org/wiki/Q3407463","display_name":"Fuzzy number","level":4,"score":0.19184601306915283},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.13581976294517517}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3390/sym17060868","doi":"https://doi.org/10.3390/sym17060868","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17060868","pdf_url":null,"source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","norway-1","poland-70","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.3390/sym17060868","doi":"https://doi.org/10.3390/sym17060868","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17060868","pdf_url":null,"source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","cwts-core","norway-1","poland-70","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320322484","display_name":"Princess Nourah Bint Abdulrahman University","ror":"https://ror.org/05b0cyh02"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":51,"referenced_works":["https://openalex.org/W145436905","https://openalex.org/W1422679120","https://openalex.org/W1427155388","https://openalex.org/W1481142035","https://openalex.org/W1525212704","https://openalex.org/W1829951598","https://openalex.org/W1918839972","https://openalex.org/W1980564456","https://openalex.org/W1981107087","https://openalex.org/W1998031161","https://openalex.org/W2003115337","https://openalex.org/W2008818676","https://openalex.org/W2011231726","https://openalex.org/W2013932252","https://openalex.org/W2031241748","https://openalex.org/W2033810643","https://openalex.org/W2079457935","https://openalex.org/W2089953456","https://openalex.org/W2091617649","https://openalex.org/W2092580384","https://openalex.org/W2097403045","https://openalex.org/W2105651716","https://openalex.org/W2131233583","https://openalex.org/W2270529708","https://openalex.org/W2289047752","https://openalex.org/W2345848912","https://openalex.org/W2466751077","https://openalex.org/W2514754342","https://openalex.org/W2578456536","https://openalex.org/W2750671045","https://openalex.org/W2781213756","https://openalex.org/W2871390967","https://openalex.org/W2883333523","https://openalex.org/W2883366251","https://openalex.org/W2917013821","https://openalex.org/W2995299237","https://openalex.org/W3118617226","https://openalex.org/W3154506182","https://openalex.org/W3207182758","https://openalex.org/W3217806805","https://openalex.org/W4211007335","https://openalex.org/W4221106348","https://openalex.org/W4234672050","https://openalex.org/W4296299137","https://openalex.org/W4312205568","https://openalex.org/W4316664435","https://openalex.org/W4386380361","https://openalex.org/W4388997289","https://openalex.org/W4399358035","https://openalex.org/W4405282742","https://openalex.org/W4406245633"],"related_works":["https://openalex.org/W2871390967","https://openalex.org/W2979734128","https://openalex.org/W2995542036","https://openalex.org/W2883333523","https://openalex.org/W2946111325","https://openalex.org/W3012607315","https://openalex.org/W7204185478","https://openalex.org/W3166255030","https://openalex.org/W3190875766","https://openalex.org/W3150208416"],"abstract_inverted_index":{"Complex":[0,94,107],"Intuitionistic":[1,95,108],"Fuzzy":[2,96,109],"Sets":[3],"(CIFSs)":[4],"are":[5,133,138],"an":[6],"advanced":[7],"form":[8],"of":[9,125,167],"intuitionistic":[10],"fuzzy":[11],"sets":[12],"that":[13,174],"utilize":[14],"complex":[15,142],"numbers":[16],"to":[17,140,185],"effectively":[18],"manage":[19],"uncertainty":[20],"and":[21,51,58,66,105,131,143,161,181],"hesitation":[22],"in":[23,146],"multi-criteria":[24],"decision":[25],"making":[26],"(MCDM).":[27],"This":[28],"paper":[29],"introduces":[30],"the":[31,49,62,82,86,93,106,154,165,168,175],"Shapley":[32,89,101,114],"Choquet":[33,90,102,115],"integral":[34],"(SCI),":[35],"which":[36,68],"is":[37],"a":[38],"powerful":[39],"tool":[40],"for":[41],"integrating":[42],"information":[43,145],"from":[44],"various":[45],"sources":[46],"while":[47],"considering":[48],"importance":[50],"interactions":[52],"among":[53],"criteria.":[54],"To":[55],"address":[56],"ambiguity":[57],"inconsistency,":[59],"we":[60],"apply":[61],"Aczel\u2013Alsina":[63,87,97,110],"(AA)":[64],"t-norm":[65],"t-conorm,":[67],"offer":[69],"greater":[70],"flexibility":[71],"than":[72],"traditional":[73],"norms.":[74],"We":[75],"propose":[76],"two":[77],"novel":[78],"aggregation":[79],"operators":[80,137],"within":[81],"CIFS":[83],"framework":[84],"using":[85],"Generalized":[88,100,113],"Integral":[91,103,116],"(AAGSCI):":[92],"Weighted":[98,111],"Average":[99],"(CIFAAWAGSCI)":[104],"Geometric":[112],"(CIFAAWGGSCI),":[117],"along":[118],"with":[119],"their":[120],"special":[121],"cases.":[122],"The":[123],"properties":[124],"these":[126],"operators,":[127],"including":[128],"idempotency,":[129],"boundedness,":[130],"monotonicity,":[132],"thoroughly":[134],"investigated.":[135],"These":[136],"designed":[139],"evaluate":[141],"asymmetric":[144],"real-life":[147],"problems.":[148],"A":[149],"case":[150],"study":[151],"on":[152,159],"selecting":[153],"optimal":[155],"bridge":[156],"design":[157],"based":[158],"structural":[160],"aesthetic":[162],"criteria":[163],"demonstrates":[164],"applicability":[166],"proposed":[169,176],"method.":[170],"Our":[171],"results":[172],"indicate":[173],"method":[177],"yields":[178],"more":[179],"consistent":[180],"reliable":[182],"outcomes":[183],"compared":[184],"existing":[186],"approaches.":[187]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":1}],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2025-10-10T00:00:00"}