{"id":"https://openalex.org/W2927009511","doi":"https://doi.org/10.3390/sym11030440","title":"A New Ranking Methodology for Pythagorean Trapezoidal Uncertain Linguistic Fuzzy Sets Based on Einstein Operations","display_name":"A New Ranking Methodology for Pythagorean Trapezoidal Uncertain Linguistic Fuzzy Sets Based on Einstein Operations","publication_year":2019,"publication_date":"2019-03-25","ids":{"openalex":"https://openalex.org/W2927009511","doi":"https://doi.org/10.3390/sym11030440","mag":"2927009511"},"language":"en","primary_location":{"id":"doi:10.3390/sym11030440","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11030440","pdf_url":"https://www.mdpi.com/2073-8994/11/3/440/pdf?version=1553522532","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,"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"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/11/3/440/pdf?version=1553522532","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101587424","display_name":"Arshad Ahmad Khan","orcid":"https://orcid.org/0000-0002-7581-1735"},"institutions":[{"id":"https://openalex.org/I89652312","display_name":"Northwest A&F University","ror":"https://ror.org/0051rme32","country_code":"CN","type":"education","lineage":["https://openalex.org/I89652312"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Arshad Ahmad Khan","raw_affiliation_strings":["College of Economic and Management, Northwest A & F University Yangling, Shaanxi 712100, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Economic and Management, Northwest A & F University Yangling, Shaanxi 712100, China","institution_ids":["https://openalex.org/I89652312"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5077817609","display_name":"Saleem Abdullah","orcid":"https://orcid.org/0000-0002-7474-5115"},"institutions":[{"id":"https://openalex.org/I181063083","display_name":"Abdul Wali Khan University Mardan","ror":"https://ror.org/03b9y4e65","country_code":"PK","type":"education","lineage":["https://openalex.org/I181063083"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Saleem Abdullah","raw_affiliation_strings":["Department of Mathematics, Abdul WAli Khan University, Mardan 23200, Khyber Pakhtunkhwa, Pakistan"],"raw_orcid":"https://orcid.org/0000-0002-7474-5115","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Abdul WAli Khan University, Mardan 23200, Khyber Pakhtunkhwa, Pakistan","institution_ids":["https://openalex.org/I181063083"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101908596","display_name":"Muhammad Shakeel","orcid":"https://orcid.org/0000-0001-5825-9697"},"institutions":[{"id":"https://openalex.org/I263208278","display_name":"Hazara University","ror":"https://ror.org/018y22094","country_code":"PK","type":"education","lineage":["https://openalex.org/I263208278"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Muhammad Shakeel","raw_affiliation_strings":["Department of Mathematics and Statistics, Hazara Univeristy, Mansehra 21120, Khyber Pakhtunkhwa, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, Hazara Univeristy, Mansehra 21120, Khyber Pakhtunkhwa, Pakistan","institution_ids":["https://openalex.org/I263208278"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5037217269","display_name":"Faisal Khan","orcid":"https://orcid.org/0000-0002-5031-6121"},"institutions":[{"id":"https://openalex.org/I188760350","display_name":"Ollscoil na Gaillimhe \u2013 University of Galway","ror":"https://ror.org/03bea9k73","country_code":"IE","type":"education","lineage":["https://openalex.org/I188760350"]}],"countries":["IE"],"is_corresponding":false,"raw_author_name":"Faisal Khan","raw_affiliation_strings":["College of Engineering and Informatics, National University Ireland, Galway H91 CF50, Ireland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Engineering and Informatics, National University Ireland, Galway H91 CF50, Ireland","institution_ids":["https://openalex.org/I188760350"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5074344560","display_name":"Noor Ul Amin","orcid":null},"institutions":[{"id":"https://openalex.org/I263208278","display_name":"Hazara University","ror":"https://ror.org/018y22094","country_code":"PK","type":"education","lineage":["https://openalex.org/I263208278"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Noor ul Amin","raw_affiliation_strings":["Department of Information Technology, Hazara University, Mansehra 21120, Khyber Pakhtunkhwa, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Information Technology, Hazara University, Mansehra 21120, Khyber Pakhtunkhwa, Pakistan","institution_ids":["https://openalex.org/I263208278"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5014293912","display_name":"Jianchao Luo","orcid":null},"institutions":[{"id":"https://openalex.org/I89652312","display_name":"Northwest A&F University","ror":"https://ror.org/0051rme32","country_code":"CN","type":"education","lineage":["https://openalex.org/I89652312"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Jianchao Luo","raw_affiliation_strings":["College of Economic and Management, Northwest A & F University Yangling, Shaanxi 712100, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Economic and Management, Northwest A & F University Yangling, Shaanxi 712100, China","institution_ids":["https://openalex.org/I89652312"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5014293912"],"corresponding_institution_ids":["https://openalex.org/I89652312"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.8519,"has_fulltext":true,"cited_by_count":11,"citation_normalized_percentile":{"value":0.86119451,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"11","issue":"3","first_page":"440","last_page":"440"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10050","display_name":"Multi-Criteria Decision Making","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9620000123977661,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12135","display_name":"Fuzzy Systems and Optimization","score":0.9539999961853027,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/pythagorean-theorem","display_name":"Pythagorean theorem","score":0.9504580497741699},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.7809722423553467},{"id":"https://openalex.org/keywords/einstein","display_name":"Einstein","score":0.6583585143089294},{"id":"https://openalex.org/keywords/ranking","display_name":"Ranking (information retrieval)","score":0.6390341520309448},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.6116660833358765},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.49772241711616516},{"id":"https://openalex.org/keywords/group-decision-making","display_name":"Group decision-making","score":0.4779072701931},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4692656397819519},{"id":"https://openalex.org/keywords/process","display_name":"Process (computing)","score":0.4444957673549652},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.36497312784194946},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.2996392846107483},{"id":"https://openalex.org/keywords/law","display_name":"Law","score":0.058881282806396484},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.05358967185020447}],"concepts":[{"id":"https://openalex.org/C100461760","wikidata":"https://www.wikidata.org/wiki/Q11518","display_name":"Pythagorean theorem","level":2,"score":0.9504580497741699},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.7809722423553467},{"id":"https://openalex.org/C146846114","wikidata":"https://www.wikidata.org/wiki/Q901448","display_name":"Einstein","level":2,"score":0.6583585143089294},{"id":"https://openalex.org/C189430467","wikidata":"https://www.wikidata.org/wiki/Q7293293","display_name":"Ranking (information retrieval)","level":2,"score":0.6390341520309448},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.6116660833358765},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.49772241711616516},{"id":"https://openalex.org/C20701700","wikidata":"https://www.wikidata.org/wiki/Q388791","display_name":"Group decision-making","level":2,"score":0.4779072701931},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4692656397819519},{"id":"https://openalex.org/C98045186","wikidata":"https://www.wikidata.org/wiki/Q205663","display_name":"Process (computing)","level":2,"score":0.4444957673549652},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.36497312784194946},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.2996392846107483},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.058881282806396484},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.05358967185020447},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym11030440","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11030440","pdf_url":"https://www.mdpi.com/2073-8994/11/3/440/pdf?version=1553522532","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,"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"},{"id":"pmh:oai:doaj.org/article:dd4f54fa25b9469b884a0932f8ac7edf","is_oa":true,"landing_page_url":"https://doaj.org/article/dd4f54fa25b9469b884a0932f8ac7edf","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 11, Iss 3, p 440 (2019)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/11/3/440/","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/sym11030440","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym11030440","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11030440","pdf_url":"https://www.mdpi.com/2073-8994/11/3/440/pdf?version=1553522532","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,"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":[{"id":"https://metadata.un.org/sdg/16","score":0.800000011920929,"display_name":"Peace, Justice and strong institutions"}],"awards":[{"id":"https://openalex.org/G2109416509","display_name":"\u519c\u4e1a\u4fe1\u7528\u62c5\u4fdd\u5236\u5ea6\u6709\u6548\u6027\u8bc4\u4ef7\u3001\u98ce\u9669\u63a7\u5236\u4e0e\u4f53\u7cfb\u6784\u5efa\u7814\u7a76","funder_award_id":"71873100","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6755661989","display_name":"\u519c\u6751\u571f\u5730\u627f\u5305\u7ecf\u8425\u6743\u62b5\u62bc\u878d\u8d44\u8bd5\u70b9\u6548\u679c\u8bc4\u4ef7\u3001\u8fd0\u4f5c\u6a21\u5f0f\u4e0e\u652f\u6301\u653f\u7b56\u7814\u7a76","funder_award_id":"71573210","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2927009511.pdf","grobid_xml":"https://content.openalex.org/works/W2927009511.grobid-xml"},"referenced_works_count":35,"referenced_works":["https://openalex.org/W1489487994","https://openalex.org/W1584867433","https://openalex.org/W1829951598","https://openalex.org/W1916037859","https://openalex.org/W1980564456","https://openalex.org/W2008103229","https://openalex.org/W2013932252","https://openalex.org/W2022326739","https://openalex.org/W2041027988","https://openalex.org/W2078423530","https://openalex.org/W2079457935","https://openalex.org/W2104077967","https://openalex.org/W2345848912","https://openalex.org/W2393506224","https://openalex.org/W2396040005","https://openalex.org/W2497207950","https://openalex.org/W2559660532","https://openalex.org/W2595446892","https://openalex.org/W2759308881","https://openalex.org/W2766645648","https://openalex.org/W2791638390","https://openalex.org/W2795272090","https://openalex.org/W2796572160","https://openalex.org/W2800763202","https://openalex.org/W2802356646","https://openalex.org/W2888420659","https://openalex.org/W2888601022","https://openalex.org/W2889385013","https://openalex.org/W2908721168","https://openalex.org/W2943466888","https://openalex.org/W2963701940","https://openalex.org/W4211007335","https://openalex.org/W6640025919","https://openalex.org/W6735131503","https://openalex.org/W6750423631"],"related_works":["https://openalex.org/W3035601835","https://openalex.org/W2997325129","https://openalex.org/W2747160131","https://openalex.org/W2930412558","https://openalex.org/W3204527492","https://openalex.org/W2889230580","https://openalex.org/W2884025808","https://openalex.org/W2624109864","https://openalex.org/W2906568202","https://openalex.org/W2151097448"],"abstract_inverted_index":{"In":[0],"this":[1],"article,":[2],"we":[3,82],"proposed":[4,74,94,119,138],"new":[5],"Pythagorean":[6,14,25,38,99],"trapezoidal":[7,15,26,39,100],"uncertain":[8,16,27,40,101],"linguistic":[9,17,28,41,102],"fuzzy":[10,18,29,42,103],"aggregation":[11,61,79,95],"information\u2014namely,":[12],"the":[13,24,37,49,59,66,72,93,98,113,118,128,137,154],"Einstein":[19,30,43,50],"weighted":[20,32,45],"averaging":[21,33,46],"(PTULFEWA)":[22],"operator,":[23,35],"ordered":[31],"(PTULFEOWA)":[34],"and":[36,63,115,142],"hybrid":[44],"(PTULFEHWA)":[47],"operator\u2014using":[48],"operational":[51],"laws.":[52],"We":[53],"studied":[54],"some":[55],"important":[56],"properties":[57],"of":[58,117,127],"suggested":[60],"operators":[62,96],"showed":[64],"that":[65,136,146],"PTULFEHWA":[67],"is":[68,140],"more":[69,144],"general":[70],"than":[71],"other":[73],"operators,":[75],"which":[76],"simplifies":[77],"these":[78],"operators.":[80],"Furthermore,":[81],"presented":[83,129],"a":[84,122],"multiple":[85],"attribute":[86],"group":[87],"decision":[88],"making":[89],"(MADM)":[90],"process":[91],"for":[92,150],"under":[97],"(PTULF)":[104],"environment.":[105],"A":[106],"numerical":[107],"example":[108],"was":[109,125],"constructed":[110],"to":[111,134],"determine":[112],"effectiveness":[114],"practicality":[116],"approach.":[120],"Lastly,":[121],"comparative":[123],"analysis":[124],"performed":[126],"approach":[130],"with":[131],"existing":[132],"approaches":[133],"show":[135],"method":[139],"consistent":[141],"provides":[143],"information":[145],"may":[147],"be":[148],"useful":[149],"complex":[151],"problems":[152],"in":[153],"decision-making":[155],"process.":[156]},"counts_by_year":[{"year":2023,"cited_by_count":2},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":1}],"updated_date":"2026-06-06T09:05:17.133730","created_date":"2025-10-10T00:00:00"}