{"id":"https://openalex.org/W2746400625","doi":"https://doi.org/10.3390/sym9080158","title":"Intuitionistic-Fuzzy Goals in Zero-Sum Multi Criteria Matrix Games","display_name":"Intuitionistic-Fuzzy Goals in Zero-Sum Multi Criteria Matrix Games","publication_year":2017,"publication_date":"2017-08-15","ids":{"openalex":"https://openalex.org/W2746400625","doi":"https://doi.org/10.3390/sym9080158","mag":"2746400625"},"language":"en","primary_location":{"id":"doi:10.3390/sym9080158","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym9080158","pdf_url":"https://www.mdpi.com/2073-8994/9/8/158/pdf?version=1502804393","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/2073-8994/9/8/158/pdf?version=1502804393","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5068963911","display_name":"Zia Bashir","orcid":"https://orcid.org/0000-0002-6051-8413"},"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":"Zia Bashir","raw_affiliation_strings":["Department of Mathematics, Quaid-i-Azam University, Islamabad-45320, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Quaid-i-Azam University, Islamabad-45320, Pakistan","institution_ids":["https://openalex.org/I12469534"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5069226693","display_name":"Jaros\u0142aw W\u0105tr\u00f3bski","orcid":"https://orcid.org/0000-0002-4415-9414"},"institutions":[{"id":"https://openalex.org/I155313962","display_name":"West Pomeranian University of Technology in Szczecin","ror":"https://ror.org/0596m7f19","country_code":"PL","type":"education","lineage":["https://openalex.org/I155313962"]}],"countries":["PL"],"is_corresponding":true,"raw_author_name":"Jaros\u0142aw W\u0105tr\u00f3bski","raw_affiliation_strings":["Department of Web Systems Analysis and Data Processing in the Faculty of Computer Science and Information Technology, West Pomeranian University of Technology, Szczecin 71-210, Poland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Web Systems Analysis and Data Processing in the Faculty of Computer Science and Information Technology, West Pomeranian University of Technology, Szczecin 71-210, Poland","institution_ids":["https://openalex.org/I155313962"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5045904242","display_name":"Tabasam Rashid","orcid":"https://orcid.org/0000-0002-8691-1088"},"institutions":[{"id":"https://openalex.org/I87482320","display_name":"University of Management and Technology","ror":"https://ror.org/0095xcq10","country_code":"PK","type":"education","lineage":["https://openalex.org/I87482320"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Tabasam Rashid","raw_affiliation_strings":["Department of Mathematics, University of Management and Technology, Lahore-54770, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Management and Technology, Lahore-54770, Pakistan","institution_ids":["https://openalex.org/I87482320"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5030757656","display_name":"Wojciech Sa\u0142abun","orcid":"https://orcid.org/0000-0001-7076-2519"},"institutions":[{"id":"https://openalex.org/I155313962","display_name":"West Pomeranian University of Technology in Szczecin","ror":"https://ror.org/0596m7f19","country_code":"PL","type":"education","lineage":["https://openalex.org/I155313962"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Wojciech Sa\u0142abun","raw_affiliation_strings":["Department of Artificial Intelligence method and Applied Mathematics in the Faculty of Computer Science and Information Technology, West Pomeranian University of Technology, Szczecin 71-210, Poland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Artificial Intelligence method and Applied Mathematics in the Faculty of Computer Science and Information Technology, West Pomeranian University of Technology, Szczecin 71-210, Poland","institution_ids":["https://openalex.org/I155313962"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5055782027","display_name":"Jawad Ali","orcid":"https://orcid.org/0000-0002-3372-0034"},"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":"Jawad Ali","raw_affiliation_strings":["Department of Mathematics, Quaid-i-Azam University, Islamabad-45320, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Quaid-i-Azam University, Islamabad-45320, Pakistan","institution_ids":["https://openalex.org/I12469534"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5069226693"],"corresponding_institution_ids":["https://openalex.org/I155313962"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":2.8104,"has_fulltext":true,"cited_by_count":26,"citation_normalized_percentile":{"value":0.908172,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":98},"biblio":{"volume":"9","issue":"8","first_page":"158","last_page":"158"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10050","display_name":"Multi-Criteria Decision Making","score":0.9994000196456909,"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":0.9994000196456909,"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":0.9987000226974487,"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":0.9871000051498413,"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/vagueness","display_name":"Vagueness","score":0.6504241228103638},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5326675176620483},{"id":"https://openalex.org/keywords/zero","display_name":"Zero (linguistics)","score":0.5275784730911255},{"id":"https://openalex.org/keywords/fuzzy-set-operations","display_name":"Fuzzy set operations","score":0.5061299204826355},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.5015482902526855},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.49191713333129883},{"id":"https://openalex.org/keywords/zero-sum-game","display_name":"Zero-sum game","score":0.49184346199035645},{"id":"https://openalex.org/keywords/fuzzy-number","display_name":"Fuzzy number","score":0.48499295115470886},{"id":"https://openalex.org/keywords/fuzzy-mathematics","display_name":"Fuzzy mathematics","score":0.46435287594795227},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.43345725536346436},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.42356282472610474},{"id":"https://openalex.org/keywords/game-theory","display_name":"Game theory","score":0.4181404113769531},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.41390275955200195},{"id":"https://openalex.org/keywords/fuzzy-set","display_name":"Fuzzy set","score":0.4077439606189728},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.21759477257728577},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.2085229754447937}],"concepts":[{"id":"https://openalex.org/C2776825360","wikidata":"https://www.wikidata.org/wiki/Q1411921","display_name":"Vagueness","level":3,"score":0.6504241228103638},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5326675176620483},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.5275784730911255},{"id":"https://openalex.org/C148671577","wikidata":"https://www.wikidata.org/wiki/Q5511133","display_name":"Fuzzy set operations","level":4,"score":0.5061299204826355},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.5015482902526855},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.49191713333129883},{"id":"https://openalex.org/C136356330","wikidata":"https://www.wikidata.org/wiki/Q156612","display_name":"Zero-sum game","level":3,"score":0.49184346199035645},{"id":"https://openalex.org/C1883856","wikidata":"https://www.wikidata.org/wiki/Q3407463","display_name":"Fuzzy number","level":4,"score":0.48499295115470886},{"id":"https://openalex.org/C166586474","wikidata":"https://www.wikidata.org/wiki/Q5511129","display_name":"Fuzzy mathematics","level":5,"score":0.46435287594795227},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.43345725536346436},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.42356282472610474},{"id":"https://openalex.org/C177142836","wikidata":"https://www.wikidata.org/wiki/Q44455","display_name":"Game theory","level":2,"score":0.4181404113769531},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.41390275955200195},{"id":"https://openalex.org/C42011625","wikidata":"https://www.wikidata.org/wiki/Q1055058","display_name":"Fuzzy set","level":3,"score":0.4077439606189728},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.21759477257728577},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.2085229754447937},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym9080158","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym9080158","pdf_url":"https://www.mdpi.com/2073-8994/9/8/158/pdf?version=1502804393","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:ebdc17fca4ed4c76b2c30eb04f50ac3a","is_oa":true,"landing_page_url":"https://doaj.org/article/ebdc17fca4ed4c76b2c30eb04f50ac3a","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 9, Iss 8, p 158 (2017)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/9/8/158/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym9080158","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; Volume 9; Issue 8; Pages: 158","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym9080158","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym9080158","pdf_url":"https://www.mdpi.com/2073-8994/9/8/158/pdf?version=1502804393","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":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320328132","display_name":"Zachodniopomorski Uniwersytet Technologiczny w Szczecinie","ror":"https://ror.org/0596m7f19"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2746400625.pdf","grobid_xml":"https://content.openalex.org/works/W2746400625.grobid-xml"},"referenced_works_count":49,"referenced_works":["https://openalex.org/W148430901","https://openalex.org/W1528458083","https://openalex.org/W1536339261","https://openalex.org/W1605217022","https://openalex.org/W1965096608","https://openalex.org/W1968437988","https://openalex.org/W1980564456","https://openalex.org/W1982770836","https://openalex.org/W1984329312","https://openalex.org/W1984810795","https://openalex.org/W1991138088","https://openalex.org/W2002512455","https://openalex.org/W2011536216","https://openalex.org/W2014193286","https://openalex.org/W2023290810","https://openalex.org/W2026774912","https://openalex.org/W2037209710","https://openalex.org/W2058693307","https://openalex.org/W2066089804","https://openalex.org/W2067833990","https://openalex.org/W2069109824","https://openalex.org/W2070428630","https://openalex.org/W2070988335","https://openalex.org/W2075358004","https://openalex.org/W2080974154","https://openalex.org/W2087656541","https://openalex.org/W2096127377","https://openalex.org/W2099867994","https://openalex.org/W2102759796","https://openalex.org/W2107527466","https://openalex.org/W2117367938","https://openalex.org/W2148261435","https://openalex.org/W2165933233","https://openalex.org/W2176210190","https://openalex.org/W2320794415","https://openalex.org/W2507537394","https://openalex.org/W2508454452","https://openalex.org/W2512729965","https://openalex.org/W2605330569","https://openalex.org/W2738618086","https://openalex.org/W2739633147","https://openalex.org/W2912565176","https://openalex.org/W2963326510","https://openalex.org/W3125430941","https://openalex.org/W4206742934","https://openalex.org/W4211007335","https://openalex.org/W4239496270","https://openalex.org/W4292158025","https://openalex.org/W4292295629"],"related_works":["https://openalex.org/W4206081075","https://openalex.org/W2253053392","https://openalex.org/W2121560876","https://openalex.org/W1470403688","https://openalex.org/W4242275735","https://openalex.org/W2366385051","https://openalex.org/W2211233","https://openalex.org/W2037428035","https://openalex.org/W2614887008","https://openalex.org/W2048452567"],"abstract_inverted_index":{"The":[0,14],"classical":[1],"matrix":[2,37],"theory":[3,17],"is":[4,46,63,69,109],"deficient":[5],"to":[6,22,48,71,111],"express":[7],"the":[8,11,83,95,113],"vagueness":[9],"of":[10,97],"real":[12],"life.":[13],"fuzzy":[15,40,60,91,102],"set":[16],"has":[18,28],"been":[19,30],"successfully":[20],"applied":[21],"bridge":[23],"this":[24,44],"gap.":[25],"Much":[26],"work":[27,81],"already":[29],"done":[31],"on":[32,86],"a":[33,52,87],"two-person":[34,54],"zero":[35,55],"sum":[36,56],"game":[38,57,89],"with":[39,58,90,100],"goals.":[41,61],"In":[42],"continuation,":[43],"paper":[45],"dedicated":[47],"define":[49],"and":[50],"study":[51,85],"multi-criteria":[53,88],"intuitionistic":[59,101],"It":[62],"shown":[64],"that":[65],"solving":[66,72],"such":[67],"games":[68],"equivalent":[70],"two":[73],"crisp":[74],"multi":[75],"object":[76],"linear":[77,98],"programming":[78,99],"problems.":[79],"Our":[80],"generalizes":[82],"previous":[84],"goals":[92],"by":[93],"adopting":[94],"approach":[96],"sets.":[103],"Finally,":[104],"an":[105],"illustrative":[106],"numerical":[107],"example":[108],"provided":[110],"elaborate":[112],"proposed":[114],"approach.":[115]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":3}],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2017-08-31T00:00:00"}
