{"id":"https://openalex.org/W2992846308","doi":"https://doi.org/10.3390/e21121205","title":"An Entropy-Based Cross-Efficiency under Variable Returns to Scale","display_name":"An Entropy-Based Cross-Efficiency under Variable Returns to Scale","publication_year":2019,"publication_date":"2019-12-07","ids":{"openalex":"https://openalex.org/W2992846308","doi":"https://doi.org/10.3390/e21121205","mag":"2992846308"},"language":"en","primary_location":{"id":"doi:10.3390/e21121205","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e21121205","pdf_url":"https://www.mdpi.com/1099-4300/21/12/1205/pdf?version=1576737382","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"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":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/21/12/1205/pdf?version=1576737382","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5027903032","display_name":"Chun-Hsiung Su","orcid":null},"institutions":[{"id":"https://openalex.org/I42474999","display_name":"Vanung University","ror":"https://ror.org/04r3dny05","country_code":"TW","type":"education","lineage":["https://openalex.org/I42474999"]}],"countries":["TW"],"is_corresponding":false,"raw_author_name":"Chun-Hsiung Su","raw_affiliation_strings":["Department of Tourism and Leisure Management, Vanung University, Zhongli, Taoyuan 320, Taiwan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Tourism and Leisure Management, Vanung University, Zhongli, Taoyuan 320, Taiwan","institution_ids":["https://openalex.org/I42474999"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5089598856","display_name":"Tim Lu","orcid":null},"institutions":[{"id":"https://openalex.org/I42474999","display_name":"Vanung University","ror":"https://ror.org/04r3dny05","country_code":"TW","type":"education","lineage":["https://openalex.org/I42474999"]}],"countries":["TW"],"is_corresponding":true,"raw_author_name":"Tim Lu","raw_affiliation_strings":["Department of Marketing and Logistics, Vanung University, Zhongli, Taoyuan 320, Taiwan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Marketing and Logistics, Vanung University, Zhongli, Taoyuan 320, Taiwan","institution_ids":["https://openalex.org/I42474999"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5089598856"],"corresponding_institution_ids":["https://openalex.org/I42474999"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.3954,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.68660258,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":95},"biblio":{"volume":"21","issue":"12","first_page":"1205","last_page":"1205"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10357","display_name":"Efficiency Analysis Using DEA","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/T10357","display_name":"Efficiency Analysis Using DEA","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/T11954","display_name":"Energy Efficiency and Management","score":0.9646000266075134,"subfield":{"id":"https://openalex.org/subfields/2105","display_name":"Renewable Energy, Sustainability and the Environment"},"field":{"id":"https://openalex.org/fields/21","display_name":"Energy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10050","display_name":"Multi-Criteria Decision Making","score":0.958899974822998,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/data-envelopment-analysis","display_name":"Data envelopment analysis","score":0.7369543313980103},{"id":"https://openalex.org/keywords/returns-to-scale","display_name":"Returns to scale","score":0.6453366875648499},{"id":"https://openalex.org/keywords/ranking","display_name":"Ranking (information retrieval)","score":0.5845993757247925},{"id":"https://openalex.org/keywords/variable","display_name":"Variable (mathematics)","score":0.5660534501075745},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.5449508428573608},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.504379153251648},{"id":"https://openalex.org/keywords/cross-entropy","display_name":"Cross entropy","score":0.480476051568985},{"id":"https://openalex.org/keywords/scale","display_name":"Scale (ratio)","score":0.4700768291950226},{"id":"https://openalex.org/keywords/measure","display_name":"Measure (data warehouse)","score":0.4262453317642212},{"id":"https://openalex.org/keywords/index","display_name":"Index (typography)","score":0.4250311255455017},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4009827673435211},{"id":"https://openalex.org/keywords/econometrics","display_name":"Econometrics","score":0.35044851899147034},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3326371908187866},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.2843194007873535},{"id":"https://openalex.org/keywords/production","display_name":"Production (economics)","score":0.2609281539916992},{"id":"https://openalex.org/keywords/data-mining","display_name":"Data mining","score":0.2332337498664856},{"id":"https://openalex.org/keywords/economics","display_name":"Economics","score":0.11348477005958557},{"id":"https://openalex.org/keywords/principle-of-maximum-entropy","display_name":"Principle of maximum entropy","score":0.10700404644012451},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.1026863157749176}],"concepts":[{"id":"https://openalex.org/C22088475","wikidata":"https://www.wikidata.org/wiki/Q647974","display_name":"Data envelopment analysis","level":2,"score":0.7369543313980103},{"id":"https://openalex.org/C165083353","wikidata":"https://www.wikidata.org/wiki/Q282942","display_name":"Returns to scale","level":3,"score":0.6453366875648499},{"id":"https://openalex.org/C189430467","wikidata":"https://www.wikidata.org/wiki/Q7293293","display_name":"Ranking (information retrieval)","level":2,"score":0.5845993757247925},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.5660534501075745},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.5449508428573608},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.504379153251648},{"id":"https://openalex.org/C167981619","wikidata":"https://www.wikidata.org/wiki/Q1685498","display_name":"Cross entropy","level":3,"score":0.480476051568985},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.4700768291950226},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.4262453317642212},{"id":"https://openalex.org/C2777382242","wikidata":"https://www.wikidata.org/wiki/Q6017816","display_name":"Index (typography)","level":2,"score":0.4250311255455017},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4009827673435211},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.35044851899147034},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3326371908187866},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.2843194007873535},{"id":"https://openalex.org/C2778348673","wikidata":"https://www.wikidata.org/wiki/Q739302","display_name":"Production (economics)","level":2,"score":0.2609281539916992},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.2332337498664856},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.11348477005958557},{"id":"https://openalex.org/C9679016","wikidata":"https://www.wikidata.org/wiki/Q1417473","display_name":"Principle of maximum entropy","level":2,"score":0.10700404644012451},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.1026863157749176},{"id":"https://openalex.org/C136764020","wikidata":"https://www.wikidata.org/wiki/Q466","display_name":"World Wide Web","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","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},{"id":"https://openalex.org/C139719470","wikidata":"https://www.wikidata.org/wiki/Q39680","display_name":"Macroeconomics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/e21121205","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e21121205","pdf_url":"https://www.mdpi.com/1099-4300/21/12/1205/pdf?version=1576737382","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"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":"Entropy","raw_type":"journal-article"},{"id":"pmh:oai:mdpi.com:/1099-4300/21/12/1205/","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/e21121205","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":"Entropy","raw_type":"Text"},{"id":"pmh:oai:pubmedcentral.nih.gov:7514551","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/7514551","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"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":"Entropy (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e21121205","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e21121205","pdf_url":"https://www.mdpi.com/1099-4300/21/12/1205/pdf?version=1576737382","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"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":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.6399999856948853,"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities"}],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2992846308.pdf","grobid_xml":"https://content.openalex.org/works/W2992846308.grobid-xml"},"referenced_works_count":28,"referenced_works":["https://openalex.org/W1964018898","https://openalex.org/W1977985128","https://openalex.org/W1979635627","https://openalex.org/W1993768205","https://openalex.org/W1995875735","https://openalex.org/W2008088727","https://openalex.org/W2012651708","https://openalex.org/W2016007736","https://openalex.org/W2021951270","https://openalex.org/W2024537555","https://openalex.org/W2035934853","https://openalex.org/W2052762591","https://openalex.org/W2061653342","https://openalex.org/W2062897808","https://openalex.org/W2075855625","https://openalex.org/W2076452041","https://openalex.org/W2129763896","https://openalex.org/W2141532588","https://openalex.org/W2149327851","https://openalex.org/W2156577211","https://openalex.org/W2215780646","https://openalex.org/W2529501758","https://openalex.org/W2561019521","https://openalex.org/W2755314791","https://openalex.org/W2789071027","https://openalex.org/W2939800659","https://openalex.org/W2942973311","https://openalex.org/W2978399793"],"related_works":["https://openalex.org/W2285000485","https://openalex.org/W2989675511","https://openalex.org/W3123982235","https://openalex.org/W2269864442","https://openalex.org/W2491316436","https://openalex.org/W166369153","https://openalex.org/W2101201995","https://openalex.org/W2994140020","https://openalex.org/W651998079","https://openalex.org/W2187519334"],"abstract_inverted_index":{"Cross-efficiency":[0],"evaluation":[1,22,59,106],"is":[2,24,69,148,155],"an":[3,156,169],"effective":[4,157],"methodology":[5],"for":[6,27,171],"discriminating":[7],"among":[8],"a":[9,74,104,145],"set":[10],"of":[11,52,111,116,139,153],"decision-making":[12],"units":[13],"(DMUs)":[14],"through":[15],"both":[16],"self-":[17],"and":[18,91,124],"peer-evaluation":[19],"methods.":[20],"This":[21],"technique":[23],"usually":[25],"used":[26,182],"data":[28],"envelopment":[29],"analysis":[30],"(DEA)":[31],"models":[32],"with":[33,108,141,174],"constant":[34],"returns":[35,54],"to":[36,39,55,71,159,167,183],"scale":[37,56],"due":[38],"the":[40,58,66,96,109,121,129,151,161,185],"fact":[41],"that":[42],"negative":[43,62,83,135],"efficiencies":[44,84,88],"never":[45],"happen":[46],"in":[47,94,188],"this":[48,81,100,163,189],"case.":[49],"For":[50,137],"cases":[51],"variable":[53],"(VRSs),":[57],"may":[60,85],"generate":[61],"cross-efficiencies.":[63],"However,":[64],"when":[65],"production":[67],"technology":[68,110],"known":[70],"be":[72,78],"VRS,":[73],"VRS":[75],"model":[76],"must":[77],"used.":[79],"In":[80,99],"case,":[82],"occur.":[86],"Negative":[87],"are":[89],"unreasonable":[90],"cause":[92],"difficulties":[93],"calculating":[95],"final":[97],"cross-efficiency.":[98],"paper,":[101],"we":[102],"propose":[103],"cross-efficiency":[105,114,143],"method,":[107],"VRS.":[112],"The":[113],"intervals":[115],"DMUs":[117,140,173],"were":[118],"derived":[119],"from":[120],"associated":[122],"aggressive":[123],"benevolent":[125],"formulations.":[126],"More":[127],"importantly,":[128],"proposed":[130,187],"approach":[131,186],"does":[132],"not":[133],"produce":[134],"efficiencies.":[136],"comparison":[138],"their":[142],"intervals,":[144],"numerical":[146],"index":[147,170],"required.":[149],"Since":[150],"concept":[152,164],"entropy":[154],"tool":[158],"measure":[160],"uncertainty,":[162],"was":[165,181],"employed":[166],"build":[168],"ranking":[172],"cross":[175],"efficiency":[176],"intervals.":[177],"A":[178],"real-case":[179],"example":[180],"illustrate":[184],"paper.":[190]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}