{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,4]],"date-time":"2026-05-04T04:13:35Z","timestamp":1777868015608,"version":"3.51.4"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T00:00:00Z","timestamp":1760659200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T00:00:00Z","timestamp":1760659200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["71971163"],"award-info":[{"award-number":["71971163"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Annual Project of Shaanxi Provincial Social Science Foundation, China","award":["2023D014"],"award-info":[{"award-number":["2023D014"]}]},{"name":"Scientific Research Project of Shaanxi Provincial Department of Education, China","award":["23JK0220"],"award-info":[{"award-number":["23JK0220"]}]},{"name":"Ministry of Education of the People\u2019s Republic of China Humanities and Social Sciences Youth Foundation","award":["23YJCZH116"],"award-info":[{"award-number":["23YJCZH116"]}]},{"DOI":"10.13039\/501100012166","name":"National Key R&D Program of China","doi-asserted-by":"crossref","award":["2022YFA1004000"],"award-info":[{"award-number":["2022YFA1004000"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100012166","name":"National Key R&D Program of China","doi-asserted-by":"crossref","award":["2022YFA1004001"],"award-info":[{"award-number":["2022YFA1004001"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["OR Spectrum"],"published-print":{"date-parts":[[2026,3]]},"DOI":"10.1007\/s00291-025-00836-9","type":"journal-article","created":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T17:13:17Z","timestamp":1760721197000},"page":"75-115","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A new meta-frontier data envelopment analysis framework in two-stage structure"],"prefix":"10.1007","volume":"48","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8068-7716","authenticated-orcid":false,"given":"Ruiyue","family":"Lin","sequence":"first","affiliation":[]},{"given":"Zhiping","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Ertao","family":"Xu","sequence":"additional","affiliation":[]},{"given":"Zongxin","family":"Li","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,10,17]]},"reference":[{"issue":"4","key":"836_CR1","doi-asserted-by":"publisher","first-page":"1077","DOI":"10.1007\/s00291-021-00643-y","volume":"43","author":"Q An","year":"2021","unstructured":"An Q, Wang P, Yang H et al (2021) Fixed cost allocation in two-stage system using DEA from a noncooperative view. OR Spectr 43(4):1077\u20131102. https:\/\/doi.org\/10.1007\/s00291-021-00643-y","journal-title":"OR Spectr"},{"issue":"4","key":"836_CR2","doi-asserted-by":"publisher","first-page":"1177","DOI":"10.1007\/s00291-022-00682-z","volume":"44","author":"M Azadi","year":"2022","unstructured":"Azadi M, Karimi B, Ho W et al (2022) Assessing green performance of power plants by multiple hybrid returns to scale technologies. OR Spectr 44(4):1177\u20131211. https:\/\/doi.org\/10.1007\/s00291-022-00682-z","journal-title":"OR Spectr"},{"key":"836_CR3","doi-asserted-by":"publisher","first-page":"1613","DOI":"10.1287\/mnsc.30.9.1078","volume":"32","author":"RD Banker","year":"1984","unstructured":"Banker RD, Charnes A, Cooper WW (1984) Some models for estimating technical and scale inefficiencies in DEA. Manage Sci 32:1613\u20131627. https:\/\/doi.org\/10.1287\/mnsc.30.9.1078","journal-title":"Manage Sci"},{"issue":"1","key":"836_CR4","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1016\/0304-405X(75)90025-2","volume":"2","author":"VS Bawa","year":"1975","unstructured":"Bawa VS (1975) Optimal rules for ordering uncertain prospects. J Financ Econ 2(1):95\u2013121. https:\/\/doi.org\/10.1016\/0304-405X(75)90025-2","journal-title":"J Financ Econ"},{"issue":"2","key":"836_CR5","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1006\/jeth.1996.0096","volume":"70","author":"RG Chambers","year":"1996","unstructured":"Chambers RG, Chung Y, F\u00e4re R (1996) Benefit and distance functions. J Econ Theory 70(2):407\u2013419. https:\/\/doi.org\/10.1006\/jeth.1996.0096","journal-title":"J Econ Theory"},{"issue":"3\u20134","key":"836_CR6","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1002\/nav.3800090303","volume":"9","author":"A Charnes","year":"1962","unstructured":"Charnes A, Cooper WW (1962) Programming with linear fractional functionals. Nav Res Logist 9(3\u20134):181\u2013186. https:\/\/doi.org\/10.1002\/nav.3800090303","journal-title":"Nav Res Logist"},{"issue":"6","key":"836_CR7","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1016\/0377-2217(78)90138-8","volume":"2","author":"A Charnes","year":"1978","unstructured":"Charnes A, Cooper WW, Rhodes E (1978) Measuring the efficiency of decision making units. Eur J Oper Res 2(6):429\u2013444. https:\/\/doi.org\/10.1016\/0377-2217(78)90138-8","journal-title":"Eur J Oper Res"},{"issue":"1","key":"836_CR8","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1016\/j.ejor.2019.06.053","volume":"280","author":"L Chen","year":"2020","unstructured":"Chen L, Huang Y, Li MJ et al (2020) Meta-frontier analysis using cross-efficiency method for performance evaluation. Eur J Oper Res 280(1):219\u2013229. https:\/\/doi.org\/10.1016\/j.ejor.2019.06.053","journal-title":"Eur J Oper Res"},{"key":"836_CR9","doi-asserted-by":"publisher","DOI":"10.1007\/s00291-024-00753-3","author":"L Chen","year":"2024","unstructured":"Chen L, Lan YX, Wang S (2024) Convex or non-convex? A new meta-frontier data envelopment analysis framework considering technology compatibility. OR Spectr. https:\/\/doi.org\/10.1007\/s00291-024-00753-3","journal-title":"OR Spectr"},{"issue":"3","key":"836_CR10","doi-asserted-by":"publisher","first-page":"1170","DOI":"10.1016\/j.ejor.2008.05.011","volume":"196","author":"Y Chen","year":"2009","unstructured":"Chen Y, Cook WD, Li N et al (2009) Additive efficiency decomposition in two-stage DEA. Eur J Oper Res 196(3):1170\u20131176. https:\/\/doi.org\/10.1016\/j.ejor.2008.05.011","journal-title":"Eur J Oper Res"},{"issue":"3","key":"836_CR11","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1016\/j.ejor.2012.11.021","volume":"226","author":"Y Chen","year":"2013","unstructured":"Chen Y, Cook WD, Kao C et al (2013) Network DEA pitfalls: divisional efficiency and frontier projection under general network structures. Eur J Oper Res 226(3):507\u2013515. https:\/\/doi.org\/10.1016\/j.ejor.2012.11.021","journal-title":"Eur J Oper Res"},{"key":"836_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.resconrec.2020.105255","volume":"168","author":"Y Chen","year":"2021","unstructured":"Chen Y, Wang M, Feng C et al (2021) Total factor energy efficiency in Chinese manufacturing industry under industry and regional heterogeneities. Resour Conserv Recycl 168:105255. https:\/\/doi.org\/10.1016\/j.resconrec.2020.105255","journal-title":"Resour Conserv Recycl"},{"issue":"5","key":"836_CR13","doi-asserted-by":"publisher","first-page":"1392","DOI":"10.1016\/j.eneco.2012.06.003","volume":"34","author":"CR Chiu","year":"2012","unstructured":"Chiu CR, Liou JL, Wu PI et al (2012) Decomposition of the environmental inefficiency of the meta-frontier with undesirable output. Energy Econ 34(5):1392\u20131399. https:\/\/doi.org\/10.1016\/j.eneco.2012.06.003","journal-title":"Energy Econ"},{"key":"836_CR14","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1023\/A:1007701304281","volume":"11","author":"WW Cooper","year":"1999","unstructured":"Cooper WW, Park KS, Pastor JT (1999) Ram: a range adjusted measure of inefficiency for use with additive models, and relations to other models and measures in dea. J Product 11:5\u201342. https:\/\/doi.org\/10.1023\/A:1007701304281","journal-title":"J Product"},{"key":"836_CR15","doi-asserted-by":"publisher","unstructured":"F\u00e4re R, Grosskopf S (2000) Network DEA Socio-Econ Plan Sci 34(1):35\u201349. https:\/\/doi.org\/10.1016\/S0038-0121(99)00012-9","DOI":"10.1016\/S0038-0121(99)00012-9"},{"key":"836_CR16","doi-asserted-by":"publisher","first-page":"62","DOI":"10.1016\/j.omega.2015.07.007","volume":"61","author":"DU Galagedera","year":"2016","unstructured":"Galagedera DU, Watson J, Premachandra I et al (2016) Modeling leakage in two-stage DEA models: an application to us mutual fund families. Omega 61:62\u201377. https:\/\/doi.org\/10.1016\/j.omega.2015.07.007","journal-title":"Omega"},{"issue":"4","key":"836_CR17","doi-asserted-by":"publisher","first-page":"1213","DOI":"10.1007\/s00291-022-00683-y","volume":"44","author":"A Hatami-Marbini","year":"2022","unstructured":"Hatami-Marbini A, Arabmaldar A, Asu JO (2022) Robust productivity growth and efficiency measurement with undesirable outputs: evidence from the oil industry. OR Spectr 44(4):1213\u20131254. https:\/\/doi.org\/10.1007\/s00291-022-00683-y","journal-title":"OR Spectr"},{"issue":"1","key":"836_CR18","doi-asserted-by":"publisher","first-page":"418","DOI":"10.1016\/j.ejor.2006.11.041","volume":"185","author":"C Kao","year":"2008","unstructured":"Kao C, Hwang SN (2008) Efficiency decomposition in two-stage data envelopment analysis: an application to non-life insurance companies in taiwan. Eur J Oper Res 185(1):418\u2013429. https:\/\/doi.org\/10.1016\/j.ejor.2006.11.041","journal-title":"Eur J Oper Res"},{"issue":"1","key":"836_CR19","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1007\/s00291-021-00653-w","volume":"44","author":"M Khoveyni","year":"2022","unstructured":"Khoveyni M, Eslami R (2022) Merging two-stage series network structures: a DEA-based approach. OR Spectr 44(1):273\u2013302. https:\/\/doi.org\/10.1007\/s00291-021-00653-w","journal-title":"OR Spectr"},{"issue":"3","key":"836_CR20","doi-asserted-by":"publisher","first-page":"1043","DOI":"10.1016\/j.ejor.2021.01.005","volume":"293","author":"R Lin","year":"2021","unstructured":"Lin R, Liu Q (2021) Multiplier dynamic data envelopment analysis based on directional distance function: an application to mutual funds. Eur J Oper Res 293(3):1043\u20131057. https:\/\/doi.org\/10.1016\/j.ejor.2021.01.005","journal-title":"Eur J Oper Res"},{"issue":"2","key":"836_CR21","doi-asserted-by":"publisher","first-page":"735","DOI":"10.1016\/j.ejor.2023.10.034","volume":"314","author":"R Lin","year":"2024","unstructured":"Lin R, Peng Y (2024) A new cross-efficiency meta-frontier analysis method with good ability to identify technology gaps. Eur J Oper Res 314(2):735\u2013746. https:\/\/doi.org\/10.1016\/j.ejor.2023.10.034","journal-title":"Eur J Oper Res"},{"issue":"3","key":"836_CR22","doi-asserted-by":"publisher","first-page":"821","DOI":"10.1007\/s00291-017-0475-1","volume":"39","author":"R Lin","year":"2017","unstructured":"Lin R, Chen Z, Hu Q et al (2017) Dynamic network DEA approach with diversification to multi-period performance evaluation of funds. OR Spectr 39(3):821\u2013860. https:\/\/doi.org\/10.1007\/s00291-017-0475-1","journal-title":"OR Spectr"},{"key":"836_CR23","volume-title":"Riskmetrics technical document","author":"JP Morgan","year":"1996","unstructured":"Morgan JP (1996) Riskmetrics technical document. RiskMetrics, New York"},{"key":"836_CR24","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1007\/s00181-007-0119-4","volume":"34","author":"CJ O\u2019Donnell","year":"2008","unstructured":"O\u2019Donnell CJ, Rao DSP, Battese GE (2008) Meta frontier frameworks for the study of firm-level efficiencies and technology ratios. Empir Econ 34:231\u2013255. https:\/\/doi.org\/10.1007\/s00181-007-0119-4","journal-title":"Empir Econ"},{"issue":"12","key":"836_CR25","doi-asserted-by":"publisher","first-page":"3302","DOI":"10.1016\/j.jbankfin.2012.07.018","volume":"36","author":"IM Premachandra","year":"2012","unstructured":"Premachandra IM, Zhu J, Watson J, Galagedera DUA (2012) Best-performing us mutual fund families from 1993 to 2008: evidence from a novel two-stage DEA model for efficiency decomposition. J Bank Finance 36(12):3302\u20133317. https:\/\/doi.org\/10.1016\/j.jbankfin.2012.07.018","journal-title":"J Bank Finance"},{"key":"836_CR26","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/j.seps.2022.101315","volume":"2","author":"RT Rockafellar","year":"2000","unstructured":"Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2:21\u201342. https:\/\/doi.org\/10.1016\/j.seps.2022.101315","journal-title":"J Risk"},{"key":"836_CR27","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1016\/j.cie.2017.05.028","volume":"113","author":"J Sun","year":"2017","unstructured":"Sun J, Wang C, Ji X et al (2017) Performance evaluation of heterogeneous bank supply chain systems from the perspective of measurement and decomposition. Comput Ind Eng 113:891\u2013903. https:\/\/doi.org\/10.1016\/j.cie.2017.05.028","journal-title":"Comput Ind Eng"},{"key":"836_CR28","first-page":"63","volume":"43","author":"JL Treynor","year":"1965","unstructured":"Treynor JL (1965) How to rate management of investment funds. Harv Bus Rev 43:63\u201375","journal-title":"Harv Bus Rev"},{"issue":"3","key":"836_CR29","doi-asserted-by":"publisher","first-page":"1013","DOI":"10.1016\/j.ejor.2018.02.048","volume":"269","author":"B Walheer","year":"2018","unstructured":"Walheer B (2018) Aggregation of meta-frontier technology gap ratios: the case of European sectors in 1995\u20132015. Eur J Oper Res 269(3):1013\u20131026. https:\/\/doi.org\/10.1016\/j.ejor.2018.02.048","journal-title":"Eur J Oper Res"},{"key":"836_CR30","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1016\/j.eneco.2016.04.005","volume":"56","author":"Q Wang","year":"2016","unstructured":"Wang Q, Su B, Zhou P et al (2016) Measuring total-factor co2 emission performance and technology gaps using a non-radial directional distance function: a modified approach. Energy Econ 56:475\u2013482. https:\/\/doi.org\/10.1016\/j.eneco.2016.04.005","journal-title":"Energy Econ"},{"issue":"5","key":"836_CR31","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1002\/nav.21815","volume":"65","author":"Q Wang","year":"2018","unstructured":"Wang Q, Hang Y, Hu JL et al (2018) An alternative meta-frontier framework for measuring the heterogeneity of technology. Nav Res Logist 65(5):427\u2013445. https:\/\/doi.org\/10.1002\/nav.21815","journal-title":"Nav Res Logist"},{"key":"836_CR32","doi-asserted-by":"publisher","DOI":"10.1016\/j.cie.2022.108462","volume":"171","author":"Q Wang","year":"2022","unstructured":"Wang Q, Li L, Sun J (2022) Environmental efficiency of port and regional system: a two-stage network efficiency model. Comput Ind Eng 171:108462. https:\/\/doi.org\/10.1016\/j.cie.2022.108462","journal-title":"Comput Ind Eng"},{"issue":"4","key":"836_CR33","doi-asserted-by":"publisher","first-page":"909","DOI":"10.1016\/j.omega.2008.07.001","volume":"37","author":"J Wu","year":"2009","unstructured":"Wu J, Liang L, Chen Y (2009) DEA game cross-efficiency approach to olympic rankings. Omega 37(4):909\u2013918. https:\/\/doi.org\/10.1016\/j.omega.2008.07.001","journal-title":"Omega"},{"issue":"1","key":"836_CR34","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1007\/s10479-019-03436-3","volume":"287","author":"MM Yu","year":"2020","unstructured":"Yu MM, Chen LH (2020) A meta-frontier network data envelopment analysis approach for the measurement of technological bias with network production structure. Ann Oper Res 287(1):495\u2013514. https:\/\/doi.org\/10.1007\/s10479-019-03436-3","journal-title":"Ann Oper Res"},{"key":"836_CR35","doi-asserted-by":"publisher","DOI":"10.1016\/j.omega.2023.102888","volume":"119","author":"MM Yu","year":"2023","unstructured":"Yu MM, Rakshit I (2023) Assessing the dynamic efficiency and technology gap of airports under different ownerships: a union dynamic NDEA approach. Omega 119:102888. https:\/\/doi.org\/10.1016\/j.omega.2023.102888","journal-title":"Omega"},{"issue":"1","key":"836_CR36","doi-asserted-by":"publisher","first-page":"254","DOI":"10.1016\/j.ejor.2021.10.054","volume":"301","author":"MM Yu","year":"2022","unstructured":"Yu MM, See KF, Hsiao B (2022) Integrating group frontier and meta-frontier directional distance functions to evaluate the efficiency of production units. Eur J Oper Res 301(1):254\u2013276. https:\/\/doi.org\/10.1016\/j.ejor.2021.10.054","journal-title":"Eur J Oper Res"}],"container-title":["OR Spectrum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00291-025-00836-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00291-025-00836-9","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00291-025-00836-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T11:20:12Z","timestamp":1777548012000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00291-025-00836-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,17]]},"references-count":36,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["836"],"URL":"https:\/\/doi.org\/10.1007\/s00291-025-00836-9","relation":{},"ISSN":["0171-6468","1436-6304"],"issn-type":[{"value":"0171-6468","type":"print"},{"value":"1436-6304","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,17]]},"assertion":[{"value":"22 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 September 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 October 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}