{"id":"https://openalex.org/W2039045183","doi":"https://doi.org/10.1007/s00500-008-0309-9","title":"The modes of convergence in the approximation of fuzzy random optimization problems","display_name":"The modes of convergence in the approximation of fuzzy random optimization problems","publication_year":2008,"publication_date":"2008-04-22","ids":{"openalex":"https://openalex.org/W2039045183","doi":"https://doi.org/10.1007/s00500-008-0309-9","mag":"2039045183"},"language":"en","primary_location":{"id":"doi:10.1007/s00500-008-0309-9","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s00500-008-0309-9","pdf_url":null,"source":{"id":"https://openalex.org/S65753830","display_name":"Soft Computing","issn_l":"1432-7643","issn":["1432-7643","1433-7479"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Soft Computing","raw_type":"journal-article"},"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":null,"display_name":"Yan-Kui Liu","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":"Yan-Kui Liu","raw_affiliation_strings":["College of Mathematics & Computer Science, Hebei University, 071002, Baoding, Hebei, China","Hebei University, College of Mathematics & Computer Science, 071002, Baoding, Hebei, China#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Mathematics & Computer Science, Hebei University, 071002, Baoding, Hebei, China","institution_ids":["https://openalex.org/I43337087"]},{"raw_affiliation_string":"Hebei University, College of Mathematics & Computer Science, 071002, Baoding, Hebei, China#TAB#","institution_ids":["https://openalex.org/I43337087"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100415129","display_name":"Zhiqiang Liu","orcid":"https://orcid.org/0000-0003-3937-9559"},"institutions":[{"id":"https://openalex.org/I168719708","display_name":"City University of Hong Kong","ror":"https://ror.org/03q8dnn23","country_code":"HK","type":"education","lineage":["https://openalex.org/I168719708"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Zhi-Qiang Liu","raw_affiliation_strings":["School of Creative Media, City University of Hong Kong, Hong Kong, China","City University of Hong Kong, School of Creative Media, 071002, Hong Kong, Hebei, China#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Creative Media, City University of Hong Kong, Hong Kong, China","institution_ids":["https://openalex.org/I168719708"]},{"raw_affiliation_string":"City University of Hong Kong, School of Creative Media, 071002, Hong Kong, Hebei, China#TAB#","institution_ids":["https://openalex.org/I168719708"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5008929688","display_name":"Jinwu Gao","orcid":"https://orcid.org/0000-0003-4806-927X"},"institutions":[{"id":"https://openalex.org/I78988378","display_name":"Renmin University of China","ror":"https://ror.org/041pakw92","country_code":"CN","type":"education","lineage":["https://openalex.org/I78988378"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jinwu Gao","raw_affiliation_strings":["School of Information, Renmin University of China, 100872, Beijing, China","Renmin University of China, School of Information, 100872, Beijing, Hebei, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Information, Renmin University of China, 100872, Beijing, China","institution_ids":["https://openalex.org/I78988378"]},{"raw_affiliation_string":"Renmin University of China, School of Information, 100872, Beijing, Hebei, China","institution_ids":["https://openalex.org/I78988378"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":null,"fwci":5.2631,"has_fulltext":false,"cited_by_count":27,"citation_normalized_percentile":{"value":0.95548575,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"13","issue":"2","first_page":"117","last_page":"125"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12135","display_name":"Fuzzy Systems and Optimization","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9976000189781189,"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/T10545","display_name":"Optimization and Variational Analysis","score":0.9828000068664551,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/compact-convergence","display_name":"Compact convergence","score":0.7546443939208984},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.7259318232536316},{"id":"https://openalex.org/keywords/normal-convergence","display_name":"Normal convergence","score":0.7255268692970276},{"id":"https://openalex.org/keywords/modes-of-convergence","display_name":"Modes of convergence (annotated index)","score":0.7199708223342896},{"id":"https://openalex.org/keywords/convergence-tests","display_name":"Convergence tests","score":0.6928207874298096},{"id":"https://openalex.org/keywords/convergence-of-random-variables","display_name":"Convergence of random variables","score":0.6568317413330078},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6473753452301025},{"id":"https://openalex.org/keywords/proofs-of-convergence-of-random-variables","display_name":"Proofs of convergence of random variables","score":0.6287068724632263},{"id":"https://openalex.org/keywords/dominated-convergence-theorem","display_name":"Dominated convergence theorem","score":0.6100181937217712},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.49767354130744934},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4884034991264343},{"id":"https://openalex.org/keywords/weak-convergence","display_name":"Weak convergence","score":0.4819783568382263},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4037643074989319},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.3444617688655853},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2535908818244934},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.18354544043540955},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.1567395031452179},{"id":"https://openalex.org/keywords/algebra-of-random-variables","display_name":"Algebra of random variables","score":0.15659889578819275},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.10262680053710938},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.0518779456615448}],"concepts":[{"id":"https://openalex.org/C5871375","wikidata":"https://www.wikidata.org/wiki/Q1780715","display_name":"Compact convergence","level":4,"score":0.7546443939208984},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.7259318232536316},{"id":"https://openalex.org/C14928880","wikidata":"https://www.wikidata.org/wiki/Q1512559","display_name":"Normal convergence","level":4,"score":0.7255268692970276},{"id":"https://openalex.org/C48216909","wikidata":"https://www.wikidata.org/wiki/Q6889156","display_name":"Modes of convergence (annotated index)","level":5,"score":0.7199708223342896},{"id":"https://openalex.org/C538505008","wikidata":"https://www.wikidata.org/wiki/Q3979448","display_name":"Convergence tests","level":4,"score":0.6928207874298096},{"id":"https://openalex.org/C95763700","wikidata":"https://www.wikidata.org/wiki/Q578985","display_name":"Convergence of random variables","level":3,"score":0.6568317413330078},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6473753452301025},{"id":"https://openalex.org/C206190082","wikidata":"https://www.wikidata.org/wiki/Q7250031","display_name":"Proofs of convergence of random variables","level":5,"score":0.6287068724632263},{"id":"https://openalex.org/C38487692","wikidata":"https://www.wikidata.org/wiki/Q1067156","display_name":"Dominated convergence theorem","level":5,"score":0.6100181937217712},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.49767354130744934},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4884034991264343},{"id":"https://openalex.org/C57945734","wikidata":"https://www.wikidata.org/wiki/Q7977941","display_name":"Weak convergence","level":3,"score":0.4819783568382263},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4037643074989319},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.3444617688655853},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2535908818244934},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.18354544043540955},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.1567395031452179},{"id":"https://openalex.org/C89231458","wikidata":"https://www.wikidata.org/wiki/Q4723976","display_name":"Algebra of random variables","level":4,"score":0.15659889578819275},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.10262680053710938},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0518779456615448},{"id":"https://openalex.org/C174265945","wikidata":"https://www.wikidata.org/wiki/Q1142889","display_name":"Isolated point","level":4,"score":0.0},{"id":"https://openalex.org/C47776270","wikidata":"https://www.wikidata.org/wiki/Q1455249","display_name":"Topological vector space","level":3,"score":0.0},{"id":"https://openalex.org/C76178495","wikidata":"https://www.wikidata.org/wiki/Q4808784","display_name":"Asset (computer security)","level":2,"score":0.0},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.0},{"id":"https://openalex.org/C81332173","wikidata":"https://www.wikidata.org/wiki/Q179899","display_name":"Topological space","level":2,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s00500-008-0309-9","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s00500-008-0309-9","pdf_url":null,"source":{"id":"https://openalex.org/S65753830","display_name":"Soft Computing","issn_l":"1432-7643","issn":["1432-7643","1433-7479"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Soft Computing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":35,"referenced_works":["https://openalex.org/W25274176","https://openalex.org/W1483175286","https://openalex.org/W1530855127","https://openalex.org/W1558926007","https://openalex.org/W1791236723","https://openalex.org/W1977914365","https://openalex.org/W1979163195","https://openalex.org/W1980346738","https://openalex.org/W1983729076","https://openalex.org/W1987811848","https://openalex.org/W1997113692","https://openalex.org/W2005434213","https://openalex.org/W2006605559","https://openalex.org/W2009721924","https://openalex.org/W2012008892","https://openalex.org/W2012647743","https://openalex.org/W2035014483","https://openalex.org/W2049091291","https://openalex.org/W2049230000","https://openalex.org/W2054020565","https://openalex.org/W2056974582","https://openalex.org/W2073545186","https://openalex.org/W2088799860","https://openalex.org/W2111022135","https://openalex.org/W2124613385","https://openalex.org/W2125306618","https://openalex.org/W2125520958","https://openalex.org/W2128530362","https://openalex.org/W2912189486","https://openalex.org/W3143971390","https://openalex.org/W4205215823","https://openalex.org/W4249936906","https://openalex.org/W4254329280","https://openalex.org/W4254535626","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W4253827225","https://openalex.org/W2198540137","https://openalex.org/W2779146629","https://openalex.org/W2022177602","https://openalex.org/W363847523","https://openalex.org/W2376926889","https://openalex.org/W3095868147","https://openalex.org/W1530855127","https://openalex.org/W2391852390","https://openalex.org/W191482792"],"abstract_inverted_index":null,"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":4},{"year":2012,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
