{"id":"https://openalex.org/W4293761295","doi":"https://doi.org/10.1002/int.23003","title":"Extension of GRA method for multiattribute group decision making problem under linguistic Pythagorean fuzzy setting with incomplete weight information","display_name":"Extension of GRA method for multiattribute group decision making problem under linguistic Pythagorean fuzzy setting with incomplete weight information","publication_year":2022,"publication_date":"2022-08-29","ids":{"openalex":"https://openalex.org/W4293761295","doi":"https://doi.org/10.1002/int.23003"},"language":"en","primary_location":{"id":"doi:10.1002/int.23003","doi":"https://doi.org/10.1002/int.23003","is_oa":false,"landing_page_url":"https://doi.org/10.1002/int.23003","pdf_url":null,"source":{"id":"https://openalex.org/S57950554","display_name":"International Journal of Intelligent Systems","issn_l":"0884-8173","issn":["0884-8173","1098-111X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","ccf-c","cwts-core","dongbi-c","jufo-1","norway-1","poland-100","russia-white-list-1"],"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Intelligent Systems","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","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/I12469534","display_name":"Quaid-i-Azam University","ror":"https://ror.org/04s9hft57","country_code":"PK","type":"education","lineage":["https://openalex.org/I12469534"]},{"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, Khushal Khan Khattak University, Karak, KPK, Pakistan","Department of Mathematics Quaid\u2010i\u2010Azam University Islamabad Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Khushal Khan Khattak University, Karak, KPK, Pakistan","institution_ids":["https://openalex.org/I4387156092"]},{"raw_affiliation_string":"Department of Mathematics Quaid\u2010i\u2010Azam University Islamabad Pakistan","institution_ids":["https://openalex.org/I12469534"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5050424009","display_name":"Chiranjibe Jana","orcid":"https://orcid.org/0000-0002-4541-5336","observed_orcids":["https://orcid.org/0000-0002-4541-5336","https://orcid.org/0000-0002-0252-9712"]},"institutions":[{"id":"https://openalex.org/I41549804","display_name":"Vidyasagar University","ror":"https://ror.org/027jsza11","country_code":"IN","type":"education","lineage":["https://openalex.org/I41549804"]}],"countries":["IN"],"is_corresponding":true,"raw_author_name":"Chiranjibe Jana","raw_affiliation_strings":["Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, India"],"raw_orcid":"https://orcid.org/0000-0002-4541-5336","affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, India","institution_ids":["https://openalex.org/I41549804"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101197113","display_name":"Muhammad Tahir Khan","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I12469534","display_name":"Quaid-i-Azam University","ror":"https://ror.org/04s9hft57","country_code":"PK","type":"education","lineage":["https://openalex.org/I12469534"]},{"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 Tahir Khan","raw_affiliation_strings":["Department of Mathematics Khushal Khan Khattak University Karak KPK Pakistan","Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics Khushal Khan Khattak University Karak KPK Pakistan","institution_ids":["https://openalex.org/I4387156092"]},{"raw_affiliation_string":"Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan","institution_ids":["https://openalex.org/I12469534"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5051931629","display_name":"Waqas Mahmood","orcid":"https://orcid.org/0000-0003-1171-8782","observed_orcids":["https://orcid.org/0000-0003-1171-8782"]},"institutions":[{"id":"https://openalex.org/I12469534","display_name":"Quaid-i-Azam University","ror":"https://ror.org/04s9hft57","country_code":"PK","type":"education","lineage":["https://openalex.org/I12469534"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Waqas Mahmood","raw_affiliation_strings":["Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan","institution_ids":["https://openalex.org/I12469534"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5039205000","display_name":"Madhumangal Pal","orcid":"https://orcid.org/0000-0002-6709-836X","observed_orcids":["https://orcid.org/0000-0002-6709-836X","https://orcid.org/0000-0002-7131-7131"]},"institutions":[{"id":"https://openalex.org/I41549804","display_name":"Vidyasagar University","ror":"https://ror.org/027jsza11","country_code":"IN","type":"education","lineage":["https://openalex.org/I41549804"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Madhumangal Pal","raw_affiliation_strings":["Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, India","institution_ids":["https://openalex.org/I41549804"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5082956031","display_name":"Wali Khan Mashwani","orcid":"https://orcid.org/0000-0002-5081-741X","observed_orcids":["https://orcid.org/0000-0002-5081-741X"]},"institutions":[{"id":"https://openalex.org/I178777954","display_name":"Kohat University of Science and Technology","ror":"https://ror.org/057d2v504","country_code":"PK","type":"education","lineage":["https://openalex.org/I178777954"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Wali Khan Mashwani","raw_affiliation_strings":["Institute of Numerical Sciences Kohat University of Science & Technology Kohat Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Numerical Sciences Kohat University of Science & Technology Kohat Pakistan","institution_ids":["https://openalex.org/I178777954"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5050424009"],"corresponding_institution_ids":["https://openalex.org/I41549804"],"apc_list":{"value":4000,"currency":"USD","value_usd":4000},"apc_paid":null,"fwci":4.8976,"has_fulltext":false,"cited_by_count":39,"citation_normalized_percentile":{"value":0.95653626,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":93,"max":99},"biblio":{"volume":"37","issue":"11","first_page":"9726","last_page":"9749"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10050","display_name":"Multi-Criteria Decision Making","score":0.9980000257492065,"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":0.9980000257492065,"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":0.00039999998989515007,"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/T12368","display_name":"Grey System Theory Applications","score":0.00019999999494757503,"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/grey-relational-analysis","display_name":"grey relational analysis","score":0.9279999732971191},{"id":"https://openalex.org/keywords/multi-attribute-group-decision-making","display_name":"multi-attribute group decision making","score":0.9890000224113464},{"id":"https://openalex.org/keywords/incomplete-attribute-weights","display_name":"incomplete attribute weights","score":0.9440000057220459},{"id":"https://openalex.org/keywords/distance-metrics","display_name":"distance measures","score":0.9279999732971191},{"id":"https://openalex.org/keywords/entropy-measures","display_name":"entropy measures","score":0.9980000257492065},{"id":"https://openalex.org/keywords/positive-ideal-solution","display_name":"positive ideal solution","score":0.8349999785423279},{"id":"https://openalex.org/keywords/negative-ideal-solution","display_name":"negative ideal solution","score":0.9929999709129333}],"concepts":[{"id":"https://openalex.org/C2776825360","wikidata":"https://www.wikidata.org/wiki/Q1411921","display_name":"Vagueness","level":3,"score":0.8577326536178589},{"id":"https://openalex.org/C64734493","wikidata":"https://www.wikidata.org/wiki/Q5608296","display_name":"Grey relational analysis","level":2,"score":0.768085241317749},{"id":"https://openalex.org/C100461760","wikidata":"https://www.wikidata.org/wiki/Q11518","display_name":"Pythagorean theorem","level":2,"score":0.7260221242904663},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.6708119511604309},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.5622029900550842},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5575171113014221},{"id":"https://openalex.org/C20701700","wikidata":"https://www.wikidata.org/wiki/Q388791","display_name":"Group decision-making","level":2,"score":0.5446594953536987},{"id":"https://openalex.org/C25343380","wikidata":"https://www.wikidata.org/wiki/Q277521","display_name":"Relation (database)","level":2,"score":0.5308367013931274},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.5224472880363464},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.47067150473594666},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.44329822063446045},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.42289164662361145},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.41097259521484375},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.345703661441803},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.16145873069763184},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1002/int.23003","doi":"https://doi.org/10.1002/int.23003","is_oa":false,"landing_page_url":"https://doi.org/10.1002/int.23003","pdf_url":null,"source":{"id":"https://openalex.org/S57950554","display_name":"International Journal of Intelligent Systems","issn_l":"0884-8173","issn":["0884-8173","1098-111X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","ccf-c","cwts-core","dongbi-c","jufo-1","norway-1","poland-100","russia-white-list-1"],"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Intelligent Systems","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":42,"referenced_works":["https://openalex.org/W594625170","https://openalex.org/W1480444690","https://openalex.org/W1549868337","https://openalex.org/W1577818368","https://openalex.org/W1660289406","https://openalex.org/W1668569279","https://openalex.org/W1829951598","https://openalex.org/W1980564456","https://openalex.org/W1981107087","https://openalex.org/W1984935922","https://openalex.org/W1986849065","https://openalex.org/W1990041962","https://openalex.org/W1995654381","https://openalex.org/W2013932252","https://openalex.org/W2015274034","https://openalex.org/W2041240839","https://openalex.org/W2044626008","https://openalex.org/W2054712770","https://openalex.org/W2076395105","https://openalex.org/W2079457935","https://openalex.org/W2084242363","https://openalex.org/W2087991154","https://openalex.org/W2092238922","https://openalex.org/W2157041604","https://openalex.org/W2406581770","https://openalex.org/W2461507070","https://openalex.org/W2512019570","https://openalex.org/W2791638390","https://openalex.org/W2801931701","https://openalex.org/W2891745665","https://openalex.org/W2906568202","https://openalex.org/W2947345116","https://openalex.org/W2982447614","https://openalex.org/W3118296807","https://openalex.org/W3135383876","https://openalex.org/W3179338307","https://openalex.org/W3183326526","https://openalex.org/W3195892533","https://openalex.org/W4211007335","https://openalex.org/W4240278385","https://openalex.org/W4245152641","https://openalex.org/W4252640433"],"related_works":["https://openalex.org/W2978578535","https://openalex.org/W2801931701","https://openalex.org/W2891745665","https://openalex.org/W2791638390","https://openalex.org/W4313218077","https://openalex.org/W3203598377","https://openalex.org/W3089275498","https://openalex.org/W1995654381","https://openalex.org/W2948368673","https://openalex.org/W2899115505"],"abstract_inverted_index":{"Linguistic":[0],"Pythagorean":[1],"fuzzy":[2],"numbers":[3],"(LPFNs)":[4],"are":[5,28,97,111],"better":[6],"tools":[7],"for":[8,72,88],"dealing":[9],"with":[10,23,92,151],"imprecision":[11],"and":[12,30,69,86,107,129,144,146],"vagueness.":[13],"This":[14],"article":[15],"develops":[16],"a":[17,50,65,75],"new":[18,51,66],"multiattribute":[19],"group":[20],"decision-making":[21,76],"approach":[22,77,150],"LPFNs.":[24,73],"The":[25,99],"attribute":[26,35],"values":[27],"LPFNs,":[29],"the":[31,34,40,43,54,82,121,127,130,142,148],"information":[32,96],"about":[33],"weight":[36,95],"is":[37,61,78,117,137],"incomplete.":[38],"Extended":[39],"notion":[41],"of":[42,53,101,123],"traditional":[44],"grey":[45,102,124],"relational":[46,115],"analysis":[47],"(GRA)":[48],"method,":[49],"extension":[52],"GRA":[55,84],"method":[56,85],"based":[57,80],"on":[58,81],"LPFN":[59],"details":[60],"introduced.":[62],"We":[63],"develop":[64],"distance":[67],"measure":[68,71],"entropy":[70],"Moreover,":[74],"proposed":[79],"traditional.":[83],"steps":[87],"solving":[89],"LPFMAGDM":[90],"problems":[91],"incompletely":[93],"known":[94],"given.":[98],"degree":[100,116,122],"relation":[103,125],"between":[104],"positive":[105],"(PIS)":[106],"negative-ideal":[108],"solutions":[109],"(NIS)":[110],"determined.":[112],"A":[113],"relative":[114],"considered":[118],"by":[119],"calculating":[120],"to":[126,140],"LPF-PIS":[128],"LPF-NIS,":[131],"respectively.":[132],"Finally,":[133],"an":[134],"illustrative":[135],"example":[136],"also":[138],"given":[139],"show":[141],"application":[143],"effectiveness":[145],"compare":[147],"developed":[149],"existing":[152],"methods.":[153]},"counts_by_year":[{"year":2026,"cited_by_count":7},{"year":2025,"cited_by_count":10},{"year":2024,"cited_by_count":11},{"year":2023,"cited_by_count":9},{"year":2022,"cited_by_count":2}],"updated_date":"2026-10-30T12:39:25.321706","created_date":"2022-08-31T00:00:00"}