{"id":"https://openalex.org/W2914895754","doi":"https://doi.org/10.3390/sym11020248","title":"The Pentagonal Fuzzy Number:Its Different Representations, Properties, Ranking, Defuzzification and Application in Game Problems","display_name":"The Pentagonal Fuzzy Number:Its Different Representations, Properties, Ranking, Defuzzification and Application in Game Problems","publication_year":2019,"publication_date":"2019-02-16","ids":{"openalex":"https://openalex.org/W2914895754","doi":"https://doi.org/10.3390/sym11020248","mag":"2914895754"},"language":"en","primary_location":{"id":"doi:10.3390/sym11020248","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11020248","pdf_url":"https://www.mdpi.com/2073-8994/11/2/248/pdf?version=1550574740","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/2073-8994/11/2/248/pdf?version=1550574740","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5062935023","display_name":"Avishek Chakraborty","orcid":"https://orcid.org/0000-0002-8684-3716"},"institutions":[{"id":"https://openalex.org/I98365261","display_name":"Indian Institute of Engineering Science and Technology, Shibpur","ror":"https://ror.org/02ytfzr55","country_code":"IN","type":"education","lineage":["https://openalex.org/I98365261"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Avishek Chakraborty","raw_affiliation_strings":["Department of Basic Science, Narula Institute of Technology, Agarpara, Kolkata-700109, India","Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, India"],"affiliations":[{"raw_affiliation_string":"Department of Basic Science, Narula Institute of Technology, Agarpara, Kolkata-700109, India","institution_ids":[]},{"raw_affiliation_string":"Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, India","institution_ids":["https://openalex.org/I98365261"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5041388199","display_name":"Sankar Prasad Mondal","orcid":"https://orcid.org/0000-0003-4690-2598"},"institutions":[{"id":"https://openalex.org/I99601430","display_name":"Maulana Abul Kalam Azad University of Technology, West Bengal","ror":"https://ror.org/030tcae29","country_code":"IN","type":"education","lineage":["https://openalex.org/I99601430"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Sankar Prasad Mondal","raw_affiliation_strings":["Department of Natural Science, Maulana Abul Kalam Azad University of Technology, West Bengal, Haringhata, Nadia, West Bengal 741249, India"],"affiliations":[{"raw_affiliation_string":"Department of Natural Science, Maulana Abul Kalam Azad University of Technology, West Bengal, Haringhata, Nadia, West Bengal 741249, India","institution_ids":["https://openalex.org/I99601430"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5067540712","display_name":"Shariful Alam","orcid":"https://orcid.org/0000-0001-8263-117X"},"institutions":[{"id":"https://openalex.org/I98365261","display_name":"Indian Institute of Engineering Science and Technology, Shibpur","ror":"https://ror.org/02ytfzr55","country_code":"IN","type":"education","lineage":["https://openalex.org/I98365261"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Shariful Alam","raw_affiliation_strings":["Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, India"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, India","institution_ids":["https://openalex.org/I98365261"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5028762211","display_name":"Ali Ahmadian","orcid":"https://orcid.org/0000-0002-0106-7050"},"institutions":[{"id":"https://openalex.org/I130343225","display_name":"Universiti Putra Malaysia","ror":"https://ror.org/02e91jd64","country_code":"MY","type":"education","lineage":["https://openalex.org/I130343225"]}],"countries":["MY"],"is_corresponding":true,"raw_author_name":"Ali Ahmadian","raw_affiliation_strings":["Laboratory of Computational Sciences and Mathematical Physics, Institute for Mathematical Research, Universiti Putra Malaysia, Serdang, Selangor 43400 UPM, Malaysia"],"affiliations":[{"raw_affiliation_string":"Laboratory of Computational Sciences and Mathematical Physics, Institute for Mathematical Research, Universiti Putra Malaysia, Serdang, Selangor 43400 UPM, Malaysia","institution_ids":["https://openalex.org/I130343225"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5054484335","display_name":"Norazak Senu","orcid":"https://orcid.org/0000-0001-8614-8281"},"institutions":[{"id":"https://openalex.org/I130343225","display_name":"Universiti Putra Malaysia","ror":"https://ror.org/02e91jd64","country_code":"MY","type":"education","lineage":["https://openalex.org/I130343225"]}],"countries":["MY"],"is_corresponding":false,"raw_author_name":"Norazak Senu","raw_affiliation_strings":["Laboratory of Computational Sciences and Mathematical Physics, Institute for Mathematical Research, Universiti Putra Malaysia, Serdang, Selangor 43400 UPM, Malaysia"],"affiliations":[{"raw_affiliation_string":"Laboratory of Computational Sciences and Mathematical Physics, Institute for Mathematical Research, Universiti Putra Malaysia, Serdang, Selangor 43400 UPM, Malaysia","institution_ids":["https://openalex.org/I130343225"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5013897423","display_name":"Debashis De","orcid":"https://orcid.org/0000-0002-9688-9806"},"institutions":[{"id":"https://openalex.org/I99601430","display_name":"Maulana Abul Kalam Azad University of Technology, West Bengal","ror":"https://ror.org/030tcae29","country_code":"IN","type":"education","lineage":["https://openalex.org/I99601430"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Debashis De","raw_affiliation_strings":["Department of Computer Science and Engineering, Maulana Abul Kalam Azad University of Technology, West Bengal, Haringhata, Nadia, West Bengal 741249, India"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science and Engineering, Maulana Abul Kalam Azad University of Technology, West Bengal, Haringhata, Nadia, West Bengal 741249, India","institution_ids":["https://openalex.org/I99601430"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5085181870","display_name":"Soheil Salahshour","orcid":"https://orcid.org/0000-0003-1390-3551"},"institutions":[{"id":"https://openalex.org/I4210103000","display_name":"Islamic Azad University Mobarakeh","ror":"https://ror.org/01a3xr262","country_code":"IR","type":"education","lineage":["https://openalex.org/I110525433","https://openalex.org/I4210103000"]}],"countries":["IR"],"is_corresponding":false,"raw_author_name":"Soheil Salahshour","raw_affiliation_strings":["Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Iran"],"affiliations":[{"raw_affiliation_string":"Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Iran","institution_ids":["https://openalex.org/I4210103000"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":7,"corresponding_author_ids":["https://openalex.org/A5028762211"],"corresponding_institution_ids":["https://openalex.org/I130343225"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":9.8642,"has_fulltext":false,"cited_by_count":93,"citation_normalized_percentile":{"value":0.98617669,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":96,"max":100},"biblio":{"volume":"11","issue":"2","first_page":"248","last_page":"248"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12135","display_name":"Fuzzy Systems and Optimization","score":0.9998000264167786,"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"}},"topics":[{"id":"https://openalex.org/T12135","display_name":"Fuzzy Systems and Optimization","score":0.9998000264167786,"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"}},{"id":"https://openalex.org/T10050","display_name":"Multi-Criteria Decision Making","score":0.9973000288009644,"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/T10820","display_name":"Fuzzy Logic and Control Systems","score":0.9904000163078308,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/defuzzification","display_name":"Defuzzification","score":0.7455663084983826},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6601513624191284},{"id":"https://openalex.org/keywords/ranking","display_name":"Ranking (information retrieval)","score":0.6318163871765137},{"id":"https://openalex.org/keywords/fuzzy-number","display_name":"Fuzzy number","score":0.5966236591339111},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.5857003927230835},{"id":"https://openalex.org/keywords/interval","display_name":"Interval (graph theory)","score":0.47950297594070435},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.455166220664978},{"id":"https://openalex.org/keywords/fuzzy-set","display_name":"Fuzzy set","score":0.41821861267089844},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3071986734867096},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.30305740237236023},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.11849448084831238}],"concepts":[{"id":"https://openalex.org/C170260401","wikidata":"https://www.wikidata.org/wiki/Q5251735","display_name":"Defuzzification","level":5,"score":0.7455663084983826},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6601513624191284},{"id":"https://openalex.org/C189430467","wikidata":"https://www.wikidata.org/wiki/Q7293293","display_name":"Ranking (information retrieval)","level":2,"score":0.6318163871765137},{"id":"https://openalex.org/C1883856","wikidata":"https://www.wikidata.org/wiki/Q3407463","display_name":"Fuzzy number","level":4,"score":0.5966236591339111},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.5857003927230835},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.47950297594070435},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.455166220664978},{"id":"https://openalex.org/C42011625","wikidata":"https://www.wikidata.org/wiki/Q1055058","display_name":"Fuzzy set","level":3,"score":0.41821861267089844},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3071986734867096},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.30305740237236023},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.11849448084831238}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym11020248","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11020248","pdf_url":"https://www.mdpi.com/2073-8994/11/2/248/pdf?version=1550574740","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:f5dc5697ff1b4bf9b2faa2b69b678caa","is_oa":true,"landing_page_url":"https://doaj.org/article/f5dc5697ff1b4bf9b2faa2b69b678caa","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"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 2, p 248 (2019)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/11/2/248/","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/sym11020248","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/sym11020248","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11020248","pdf_url":"https://www.mdpi.com/2073-8994/11/2/248/pdf?version=1550574740","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":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2914895754.pdf","grobid_xml":"https://content.openalex.org/works/W2914895754.grobid-xml"},"referenced_works_count":45,"referenced_works":["https://openalex.org/W1515119244","https://openalex.org/W1606783562","https://openalex.org/W1965387747","https://openalex.org/W1966731082","https://openalex.org/W1969566617","https://openalex.org/W1974946744","https://openalex.org/W1983429281","https://openalex.org/W1986230455","https://openalex.org/W1987249808","https://openalex.org/W1993565652","https://openalex.org/W2019482291","https://openalex.org/W2021804535","https://openalex.org/W2025383595","https://openalex.org/W2026832772","https://openalex.org/W2036861478","https://openalex.org/W2043001097","https://openalex.org/W2049441685","https://openalex.org/W2051798785","https://openalex.org/W2052842391","https://openalex.org/W2057570629","https://openalex.org/W2061861081","https://openalex.org/W2062014224","https://openalex.org/W2063567490","https://openalex.org/W2069069851","https://openalex.org/W2084122390","https://openalex.org/W2092092154","https://openalex.org/W2092958951","https://openalex.org/W2131060089","https://openalex.org/W2153339908","https://openalex.org/W2185574174","https://openalex.org/W2233793374","https://openalex.org/W2402686429","https://openalex.org/W2403358908","https://openalex.org/W2520364703","https://openalex.org/W2610832977","https://openalex.org/W2766765374","https://openalex.org/W2768188132","https://openalex.org/W2783060941","https://openalex.org/W2792688412","https://openalex.org/W2794220118","https://openalex.org/W4211007335","https://openalex.org/W6632248184","https://openalex.org/W6679602806","https://openalex.org/W6686654129","https://openalex.org/W6726737809"],"related_works":["https://openalex.org/W2773005380","https://openalex.org/W2381957916","https://openalex.org/W4388451300","https://openalex.org/W2060361571","https://openalex.org/W1879432897","https://openalex.org/W2256857057","https://openalex.org/W3167814444","https://openalex.org/W3176863924","https://openalex.org/W2023513928","https://openalex.org/W4361984403"],"abstract_inverted_index":{"In":[0],"this":[1,89],"paper,":[2],"different":[3],"measures":[4],"of":[5,22,33,48,56,62,88,122],"interval-valued":[6,24],"pentagonal":[7,50],"fuzzy":[8,25,51],"numbers":[9,26],"(IVPFN)":[10],"associated":[11],"with":[12],"assorted":[13],"membership":[14,40],"functions":[15,41],"(MF)":[16],"were":[17,83,94],"explored,":[18],"considering":[19],"significant":[20],"exposure":[21],"multifarious":[23],"in":[27,96,108,127],"neoteric":[28],"studies.":[29],"Also,":[30],"the":[31,44,49,54,85,102,120,123,128],"idea":[32],"MF":[34,66],"is":[35],"generalized":[36],"somewhat":[37],"to":[38,118],"nonlinear":[39,65],"for":[42,59,112],"viewing":[43],"symmetries":[45],"and":[46,64,78,117],"asymmetries":[47],"structures.":[52],"Accordingly,":[53],"construction":[55],"level":[57],"sets,":[58],"each":[60],"case":[61],"linear":[63],"was":[67,73],"also":[68,84],"carried":[69],"out.":[70],"Besides,":[71],"defuzzification":[72],"undertaken":[74],"using":[75],"three":[76],"methods":[77],"a":[79,97,109],"ranking":[80],"method,":[81],"which":[82],"main":[86],"features":[87],"framework.":[90],"The":[91],"developed":[92],"intellects":[93],"implemented":[95],"game":[98],"problem":[99],"by":[100],"taking":[101],"parameters":[103,124],"as":[104],"PFNs,":[105],"ultimately":[106],"resulting":[107],"new":[110],"direction":[111],"modeling":[113],"real":[114],"world":[115],"problems":[116],"comprehend":[119],"uncertainty":[121],"more":[125],"precisely":[126],"evaluation":[129],"process.":[130]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":12},{"year":2024,"cited_by_count":13},{"year":2023,"cited_by_count":25},{"year":2022,"cited_by_count":15},{"year":2021,"cited_by_count":11},{"year":2020,"cited_by_count":12},{"year":2019,"cited_by_count":3}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}