{"id":"https://openalex.org/W2804747855","doi":"https://doi.org/10.3390/sym10060199","title":"Some Normal Intuitionistic Fuzzy Heronian Mean Operators Using Hamacher Operation and Their Application","display_name":"Some Normal Intuitionistic Fuzzy Heronian Mean Operators Using Hamacher Operation and Their Application","publication_year":2018,"publication_date":"2018-06-01","ids":{"openalex":"https://openalex.org/W2804747855","doi":"https://doi.org/10.3390/sym10060199","mag":"2804747855"},"language":"en","primary_location":{"id":"doi:10.3390/sym10060199","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10060199","pdf_url":"https://www.mdpi.com/2073-8994/10/6/199/pdf?version=1527994481","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/2073-8994/10/6/199/pdf?version=1527994481","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100612323","display_name":"Guofang Zhang","orcid":"https://orcid.org/0000-0003-3484-5864"},"institutions":[{"id":"https://openalex.org/I43337087","display_name":"Hebei University","ror":"https://ror.org/01p884a79","country_code":"CN","type":"education","lineage":["https://openalex.org/I43337087"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Guofang Zhang","raw_affiliation_strings":["Key Lab. of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding 071002, China"],"affiliations":[{"raw_affiliation_string":"Key Lab. of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding 071002, China","institution_ids":["https://openalex.org/I43337087"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100328657","display_name":"Zhiming Zhang","orcid":"https://orcid.org/0000-0002-1462-4944"},"institutions":[{"id":"https://openalex.org/I43337087","display_name":"Hebei University","ror":"https://ror.org/01p884a79","country_code":"CN","type":"education","lineage":["https://openalex.org/I43337087"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhiming Zhang","raw_affiliation_strings":["Key Lab. of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding 071002, China"],"affiliations":[{"raw_affiliation_string":"Key Lab. of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding 071002, China","institution_ids":["https://openalex.org/I43337087"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5077465737","display_name":"Hang Kong","orcid":null},"institutions":[{"id":"https://openalex.org/I43337087","display_name":"Hebei University","ror":"https://ror.org/01p884a79","country_code":"CN","type":"education","lineage":["https://openalex.org/I43337087"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Hang Kong","raw_affiliation_strings":["Key Lab. of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding 071002, China"],"affiliations":[{"raw_affiliation_string":"Key Lab. of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding 071002, China","institution_ids":["https://openalex.org/I43337087"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5100612323"],"corresponding_institution_ids":["https://openalex.org/I43337087"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":2.8609,"has_fulltext":true,"cited_by_count":21,"citation_normalized_percentile":{"value":0.91053224,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"10","issue":"6","first_page":"199","last_page":"199"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10050","display_name":"Multi-Criteria Decision Making","score":0.9998000264167786,"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":0.9998000264167786,"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":0.9678999781608582,"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/T14175","display_name":"Intuitionistic Fuzzy Systems Applications","score":0.9327999949455261,"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/mathematics","display_name":"Mathematics","score":0.6723165512084961},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.5327451229095459},{"id":"https://openalex.org/keywords/group-decision-making","display_name":"Group decision-making","score":0.43893855810165405},{"id":"https://openalex.org/keywords/einstein","display_name":"Einstein","score":0.4268348217010498},{"id":"https://openalex.org/keywords/distribution","display_name":"Distribution (mathematics)","score":0.41139721870422363},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.37250766158103943},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3245794475078583},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.07867497205734253}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6723165512084961},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.5327451229095459},{"id":"https://openalex.org/C20701700","wikidata":"https://www.wikidata.org/wiki/Q388791","display_name":"Group decision-making","level":2,"score":0.43893855810165405},{"id":"https://openalex.org/C146846114","wikidata":"https://www.wikidata.org/wiki/Q901448","display_name":"Einstein","level":2,"score":0.4268348217010498},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.41139721870422363},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.37250766158103943},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3245794475078583},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.07867497205734253},{"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/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"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/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","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/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym10060199","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10060199","pdf_url":"https://www.mdpi.com/2073-8994/10/6/199/pdf?version=1527994481","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:ea41cd6aeb6b43878523f73aff4aa125","is_oa":true,"landing_page_url":"https://doaj.org/article/ea41cd6aeb6b43878523f73aff4aa125","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 10, Iss 6, p 199 (2018)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/10/6/199/","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/sym10060199","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/sym10060199","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10060199","pdf_url":"https://www.mdpi.com/2073-8994/10/6/199/pdf?version=1527994481","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":[{"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16","score":0.800000011920929}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2804747855.pdf","grobid_xml":"https://content.openalex.org/works/W2804747855.grobid-xml"},"referenced_works_count":66,"referenced_works":["https://openalex.org/W1937491830","https://openalex.org/W1945942604","https://openalex.org/W1965777053","https://openalex.org/W1969352744","https://openalex.org/W1971875039","https://openalex.org/W1976915331","https://openalex.org/W1980564456","https://openalex.org/W1981107087","https://openalex.org/W1985678092","https://openalex.org/W2002636607","https://openalex.org/W2007538282","https://openalex.org/W2008103229","https://openalex.org/W2013908206","https://openalex.org/W2015282411","https://openalex.org/W2016698635","https://openalex.org/W2032765402","https://openalex.org/W2036378460","https://openalex.org/W2038073747","https://openalex.org/W2048394994","https://openalex.org/W2064581049","https://openalex.org/W2072394128","https://openalex.org/W2076296214","https://openalex.org/W2151159878","https://openalex.org/W2166034382","https://openalex.org/W2233779637","https://openalex.org/W2271577254","https://openalex.org/W2301121348","https://openalex.org/W2323551416","https://openalex.org/W2329317779","https://openalex.org/W2344622620","https://openalex.org/W2380537178","https://openalex.org/W2382719933","https://openalex.org/W2383338108","https://openalex.org/W2403315466","https://openalex.org/W2519497782","https://openalex.org/W2530680027","https://openalex.org/W2560319565","https://openalex.org/W2577928796","https://openalex.org/W2584316876","https://openalex.org/W2592818683","https://openalex.org/W2606031165","https://openalex.org/W2606422429","https://openalex.org/W2609661822","https://openalex.org/W2613185130","https://openalex.org/W2664214589","https://openalex.org/W2740484758","https://openalex.org/W2766645648","https://openalex.org/W2766716840","https://openalex.org/W2767097575","https://openalex.org/W2767853298","https://openalex.org/W2780754879","https://openalex.org/W2782797181","https://openalex.org/W2792640078","https://openalex.org/W2793980993","https://openalex.org/W2805289078","https://openalex.org/W2806820182","https://openalex.org/W2963701940","https://openalex.org/W4211007335","https://openalex.org/W4230289555","https://openalex.org/W4292078529","https://openalex.org/W6697734666","https://openalex.org/W6727012351","https://openalex.org/W6728217526","https://openalex.org/W6749967550","https://openalex.org/W6814828556","https://openalex.org/W7060990614"],"related_works":["https://openalex.org/W2151097448","https://openalex.org/W2796809906","https://openalex.org/W2348660983","https://openalex.org/W1976254693","https://openalex.org/W2384801745","https://openalex.org/W2369673506","https://openalex.org/W2079612625","https://openalex.org/W1928552208","https://openalex.org/W2040240542","https://openalex.org/W1551306197"],"abstract_inverted_index":{"Hamacher":[0,84,89,144,148,153,159,215],"operation":[1,10,17,216],"is":[2,77,100,191,203,304],"a":[3,13,70,177,195],"generalization":[4],"of":[5,15,29,83,94,183,271,297,312],"the":[6,19,63,81,112,119,168,172,198,208,235,241,250,264,269,272,298,308,313],"algebraic":[7],"and":[8,11,35,69,124,137,158,175,201,221,225,230,255,278,282,294,310],"Einstein":[9],"expresses":[12],"family":[14],"binary":[16],"in":[18,53,66,127,316],"unit":[20],"interval":[21],"[0,1].":[22],"Heronian":[23,103,145,150,155,162,242],"mean":[24,104,146,151,156,163,243],"can":[25,48,267,288],"deal":[26,246],"with":[27,247],"correlations":[28],"different":[30,273],"criteria":[31,274],"or":[32,252,275],"input":[33,122,253,276],"arguments":[34,277],"does":[36,257],"not":[37,105,258],"bring":[38,259],"out":[39],"repeated":[40,261],"calculation.":[41,262],"The":[42,181],"normal":[43,50,209],"intuitionistic":[44],"fuzzy":[45],"numbers":[46],"(NIFNs)":[47],"depict":[49],"distribution":[51,210],"information":[52],"practical":[54],"decision":[55,196,292,299],"making.":[56],"A":[57],"decision-making":[58,74],"problem":[59,174],"was":[60],"researched":[61],"under":[62,197,234],"NIFN":[64,131,199,236],"environment":[65,200],"this":[67,184,317],"study,":[68],"new":[71,178,185],"multi-criteria":[72],"group":[73],"(MCGDM)":[75],"approach":[76,186,314],"herein":[78],"introduced":[79],"on":[80],"basis":[82],"operation.":[85],"Firstly,":[86],"according":[87],"to":[88,129,171,217,226,245],"operation,":[90],"some":[91,135,280],"operational":[92],"laws":[93],"NIFNs":[95],"are":[96,187],"presented.":[97],"Secondly,":[98],"it":[99,190,202,213,239,256],"noted":[101],"that":[102],"only":[106],"takes":[107],"into":[108],"account":[109],"mutuality":[110],"between":[111,121,249],"attribute":[113],"values":[114],"once,":[115],"but":[116],"also":[117],"considers":[118],"correlation":[120],"argument":[123],"itself.":[125],"Therefore,":[126,263],"order":[128],"aggregate":[130],"information,":[132],"we":[133,166],"developed":[134,176],"operators":[136,142,170],"studied":[138],"their":[139],"properties.":[140],"These":[141],"include":[143],"(NIFHHM),":[147],"weighted":[149,160],"(NIFHWHM),":[152],"geometric":[154,161],"(NIFHGHM),":[157],"(NIFHWGHM).":[164],"Furthermore,":[165],"applied":[167],"proposed":[169,265],"MCGDM":[173,179,223,284],"approach.":[180],"characteristics":[182],"that:":[188],"(1)":[189],"suitable":[192],"for":[193,206,291,306],"making":[194,293],"more":[204,228,231],"reasonable":[205,281],"aggregating":[207],"data;":[211],"(2)":[212],"utilizes":[214],"provide":[218],"an":[219,302],"effective":[220],"powerful":[222],"algorithm":[224],"make":[227],"reliable":[229,283],"flexible":[232],"decisions":[233],"circumstance;":[237],"(3)":[238],"uses":[240],"operator":[244],"interrelations":[248],"attributes":[251],"arguments,":[254],"about":[260],"method":[266],"describe":[268],"interaction":[270],"offer":[279],"aggregation":[285],"operators,":[286],"which":[287],"open":[289],"avenues":[290],"broaden":[295],"perspectives":[296],"experts.":[300],"Lastly,":[301],"application":[303],"given":[305],"showing":[307],"effectiveness":[309],"feasibility":[311],"presented":[315],"paper.":[318]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":2}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}
