{"id":"https://openalex.org/W3133878563","doi":"https://doi.org/10.3390/sym13030436","title":"A Dynamic Distance Measure of Picture Fuzzy Sets and Its Application","display_name":"A Dynamic Distance Measure of Picture Fuzzy Sets and Its Application","publication_year":2021,"publication_date":"2021-03-08","ids":{"openalex":"https://openalex.org/W3133878563","doi":"https://doi.org/10.3390/sym13030436","mag":"3133878563"},"language":"en","primary_location":{"id":"doi:10.3390/sym13030436","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13030436","pdf_url":"https://www.mdpi.com/2073-8994/13/3/436/pdf?version=1615955010","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/2073-8994/13/3/436/pdf?version=1615955010","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5103044194","display_name":"Ruirui Zhao","orcid":"https://orcid.org/0000-0002-5337-5813"},"institutions":[{"id":"https://openalex.org/I88830068","display_name":"Shaanxi Normal University","ror":"https://ror.org/0170z8493","country_code":"CN","type":"education","lineage":["https://openalex.org/I88830068"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Ruirui Zhao","raw_affiliation_strings":["College of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an 710062, China","College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China"],"raw_orcid":"https://orcid.org/0000-0002-5337-5813","affiliations":[{"raw_affiliation_string":"College of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an 710062, China","institution_ids":["https://openalex.org/I88830068"]},{"raw_affiliation_string":"College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China","institution_ids":["https://openalex.org/I88830068"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5006235641","display_name":"Minxia Luo","orcid":"https://orcid.org/0000-0001-9400-3533"},"institutions":[{"id":"https://openalex.org/I55538621","display_name":"China Jiliang University","ror":"https://ror.org/05v1y0t93","country_code":"CN","type":"education","lineage":["https://openalex.org/I55538621"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Minxia Luo","raw_affiliation_strings":["Department of Information and Computing Science, China Jiliang University, Hangzhou 310018, China"],"raw_orcid":"https://orcid.org/0000-0001-9400-3533","affiliations":[{"raw_affiliation_string":"Department of Information and Computing Science, China Jiliang University, Hangzhou 310018, China","institution_ids":["https://openalex.org/I55538621"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5102968426","display_name":"Sheng-Gang Li","orcid":"https://orcid.org/0000-0003-2998-726X"},"institutions":[{"id":"https://openalex.org/I88830068","display_name":"Shaanxi Normal University","ror":"https://ror.org/0170z8493","country_code":"CN","type":"education","lineage":["https://openalex.org/I88830068"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Shenggang Li","raw_affiliation_strings":["College of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an 710062, China","College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an 710062, China","institution_ids":["https://openalex.org/I88830068"]},{"raw_affiliation_string":"College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China","institution_ids":["https://openalex.org/I88830068"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5006235641"],"corresponding_institution_ids":["https://openalex.org/I55538621"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":3.0478,"has_fulltext":false,"cited_by_count":33,"citation_normalized_percentile":{"value":0.91072508,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"13","issue":"3","first_page":"436","last_page":"436"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10050","display_name":"Multi-Criteria Decision Making","score":0.9991999864578247,"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":0.9991999864578247,"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":0.9506999850273132,"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"}},{"id":"https://openalex.org/T12135","display_name":"Fuzzy Systems and Optimization","score":0.9501000046730042,"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/counterintuitive","display_name":"Counterintuitive","score":0.8448941707611084},{"id":"https://openalex.org/keywords/measure","display_name":"Measure (data warehouse)","score":0.6928070187568665},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.6193599700927734},{"id":"https://openalex.org/keywords/fuzzy-measure-theory","display_name":"Fuzzy measure theory","score":0.5617966651916504},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.5583525896072388},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5453556776046753},{"id":"https://openalex.org/keywords/fuzzy-number","display_name":"Fuzzy number","score":0.5079925656318665},{"id":"https://openalex.org/keywords/point","display_name":"Point (geometry)","score":0.4776824116706848},{"id":"https://openalex.org/keywords/fuzzy-set","display_name":"Fuzzy set","score":0.47584033012390137},{"id":"https://openalex.org/keywords/defuzzification","display_name":"Defuzzification","score":0.4725865125656128},{"id":"https://openalex.org/keywords/type-2-fuzzy-sets-and-systems","display_name":"Type-2 fuzzy sets and systems","score":0.44864752888679504},{"id":"https://openalex.org/keywords/fuzzy-classification","display_name":"Fuzzy classification","score":0.4444434940814972},{"id":"https://openalex.org/keywords/fuzzy-set-operations","display_name":"Fuzzy set operations","score":0.43233394622802734},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.42677509784698486},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3571932315826416},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.320533812046051},{"id":"https://openalex.org/keywords/data-mining","display_name":"Data mining","score":0.29298868775367737}],"concepts":[{"id":"https://openalex.org/C101097943","wikidata":"https://www.wikidata.org/wiki/Q5176983","display_name":"Counterintuitive","level":2,"score":0.8448941707611084},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.6928070187568665},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.6193599700927734},{"id":"https://openalex.org/C100516043","wikidata":"https://www.wikidata.org/wiki/Q4455138","display_name":"Fuzzy measure theory","level":5,"score":0.5617966651916504},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.5583525896072388},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5453556776046753},{"id":"https://openalex.org/C1883856","wikidata":"https://www.wikidata.org/wiki/Q3407463","display_name":"Fuzzy number","level":4,"score":0.5079925656318665},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.4776824116706848},{"id":"https://openalex.org/C42011625","wikidata":"https://www.wikidata.org/wiki/Q1055058","display_name":"Fuzzy set","level":3,"score":0.47584033012390137},{"id":"https://openalex.org/C170260401","wikidata":"https://www.wikidata.org/wiki/Q5251735","display_name":"Defuzzification","level":5,"score":0.4725865125656128},{"id":"https://openalex.org/C17350324","wikidata":"https://www.wikidata.org/wiki/Q7860605","display_name":"Type-2 fuzzy sets and systems","level":5,"score":0.44864752888679504},{"id":"https://openalex.org/C127385683","wikidata":"https://www.wikidata.org/wiki/Q1475696","display_name":"Fuzzy classification","level":4,"score":0.4444434940814972},{"id":"https://openalex.org/C148671577","wikidata":"https://www.wikidata.org/wiki/Q5511133","display_name":"Fuzzy set operations","level":4,"score":0.43233394622802734},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.42677509784698486},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3571932315826416},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.320533812046051},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.29298868775367737},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","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},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym13030436","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13030436","pdf_url":"https://www.mdpi.com/2073-8994/13/3/436/pdf?version=1615955010","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:08a01d2a4a284840a318405769e9b455","is_oa":true,"landing_page_url":"https://doaj.org/article/08a01d2a4a284840a318405769e9b455","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 13, Iss 3, p 436 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/13/3/436/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym13030436","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 13; Issue 3; Pages: 436","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym13030436","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13030436","pdf_url":"https://www.mdpi.com/2073-8994/13/3/436/pdf?version=1615955010","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","display_name":"Peace, Justice and strong institutions","score":0.8299999833106995}],"awards":[{"id":"https://openalex.org/G1804066553","display_name":null,"funder_award_id":"11771263","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G273029900","display_name":null,"funder_award_id":"GK201503013","funder_id":"https://openalex.org/F4320335787","funder_display_name":"Fundamental Research Funds for the Central Universities"},{"id":"https://openalex.org/G3499628877","display_name":null,"funder_award_id":"2018TS059","funder_id":"https://openalex.org/F4320335787","funder_display_name":"Fundamental Research Funds for the Central Universities"},{"id":"https://openalex.org/G4665617217","display_name":null,"funder_award_id":"61773019","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"},{"id":"https://openalex.org/F4320335787","display_name":"Fundamental Research Funds for the Central Universities","ror":null}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3133878563.pdf"},"referenced_works_count":34,"referenced_works":["https://openalex.org/W1505735190","https://openalex.org/W1966808269","https://openalex.org/W1979507251","https://openalex.org/W1980564456","https://openalex.org/W1982675899","https://openalex.org/W2006873874","https://openalex.org/W2055108933","https://openalex.org/W2067648457","https://openalex.org/W2081282960","https://openalex.org/W2084063803","https://openalex.org/W2084101582","https://openalex.org/W2087501680","https://openalex.org/W2144086263","https://openalex.org/W2155776784","https://openalex.org/W2165024836","https://openalex.org/W2208115510","https://openalex.org/W2385257316","https://openalex.org/W2463044584","https://openalex.org/W2620114837","https://openalex.org/W2669156789","https://openalex.org/W2734543528","https://openalex.org/W2767160721","https://openalex.org/W2775587901","https://openalex.org/W2803372772","https://openalex.org/W2809020038","https://openalex.org/W2885009725","https://openalex.org/W2896634720","https://openalex.org/W2912565176","https://openalex.org/W2955299959","https://openalex.org/W2969157164","https://openalex.org/W3000706679","https://openalex.org/W3128670286","https://openalex.org/W4211007335","https://openalex.org/W6683094535"],"related_works":["https://openalex.org/W2184441312","https://openalex.org/W2674573885","https://openalex.org/W1485268022","https://openalex.org/W4361984403","https://openalex.org/W1975005191","https://openalex.org/W1487720066","https://openalex.org/W1542808721","https://openalex.org/W2331106272","https://openalex.org/W2148720919","https://openalex.org/W2960454922"],"abstract_inverted_index":{"Picture":[0],"fuzzy":[1,9,35,43,74,96,102],"sets,":[2,10],"which":[3],"are":[4,49],"the":[5,30,60,69,116],"extension":[6],"of":[7,41,72,94],"intuitionistic":[8],"can":[11],"deal":[12],"with":[13],"inconsistent":[14],"information":[15],"better":[16],"in":[17,85],"practical":[18],"applications.":[19],"A":[20],"distance":[21,39,62,92,118],"measure":[22,93,119],"is":[23,58,120],"an":[24],"important":[25],"mathematical":[26],"tool":[27],"to":[28,78],"calculate":[29],"difference":[31],"degree":[32,71],"between":[33],"picture":[34,42,73,95,101],"sets.":[36,75],"Although":[37],"some":[38,50],"measures":[40,63],"sets":[44,97],"have":[45],"been":[46],"constructed,":[47],"there":[48],"unreasonable":[51,81],"and":[52,82,109,122],"counterintuitive":[53,83],"cases.":[54],"The":[55],"main":[56],"reason":[57],"that":[59,115],"existing":[61],"do":[64],"not":[65],"or":[66],"seldom":[67],"consider":[68],"refusal":[70],"In":[76],"order":[77],"solve":[79],"these":[80],"cases,":[84],"this":[86],"paper,":[87],"we":[88,113],"propose":[89],"a":[90,100,106],"dynamic":[91],"based":[98],"on":[99],"point":[103],"operator.":[104],"Through":[105],"numerical":[107],"comparison":[108],"multi-criteria":[110],"decision-making":[111],"problems,":[112],"show":[114],"proposed":[117],"reasonable":[121],"effective.":[123]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":13},{"year":2024,"cited_by_count":8},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2}],"updated_date":"2026-06-26T08:34:08.712188","created_date":"2025-10-10T00:00:00"}