{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T08:36:54Z","timestamp":1768984614515,"version":"3.49.0"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,7,16]],"date-time":"2022-07-16T00:00:00Z","timestamp":1657929600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,7,16]],"date-time":"2022-07-16T00:00:00Z","timestamp":1657929600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Ann Oper Res"],"published-print":{"date-parts":[[2022,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Applications of data envelopment analysis (DEA) often include inputs and outputs represented as percentages, ratios and averages, collectively referred to as ratio measures. It is known that conventional DEA models cannot correctly incorporate such measures. To address this gap, the authors have previously developed new variable and constant returns-to-scale models and computational procedures suitable for the treatment of ratio measures. The focus of this new paper is on the scale characteristics of the variable returns-to-scale production frontiers with ratio inputs and outputs. This includes the notions of the most productive scale size (MPSS), scale and overall efficiency as measures of divergence from MPSS. Additional development concerns alternative notions of returns to scale arising in models with ratio measures. To keep the exposition as general as possible and suitable in different contexts, we allow all scale characteristics to be evaluated with respect to any selected subsets of volume and ratio inputs and outputs, while keeping the remaining measures constant. Overall, this new paper aims at expanding the range of techniques available in applications with ratio measures.<\/jats:p>","DOI":"10.1007\/s10479-022-04862-6","type":"journal-article","created":{"date-parts":[[2022,7,16]],"date-time":"2022-07-16T11:02:54Z","timestamp":1657969374000},"page":"383-423","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Scale characteristics of variable returns-to-scale production technologies with ratio inputs and outputs"],"prefix":"10.1007","volume":"318","author":[{"given":"Ole Bent","family":"Olesen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Niels Christian","family":"Petersen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7555-6589","authenticated-orcid":false,"given":"Victor V.","family":"Podinovski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,7,16]]},"reference":[{"issue":"3","key":"4862_CR1","doi-asserted-by":"publisher","first-page":"568","DOI":"10.2307\/2525845","volume":"13","author":"SN Afriat","year":"1972","unstructured":"Afriat, S. N. (1972). Efficiency estimation of production functions. International Economic Review, 13(3), 568\u2013598.","journal-title":"International Economic Review"},{"issue":"1","key":"4862_CR2","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1016\/0377-2217(84)90006-7","volume":"17","author":"RD Banker","year":"1984","unstructured":"Banker, R. D. (1984). Estimating most productive scale size using data envelopment analysis. European Journal of Operational Research, 17(1), 35\u201344.","journal-title":"European Journal of Operational Research"},{"issue":"9","key":"4862_CR3","doi-asserted-by":"publisher","first-page":"1078","DOI":"10.1287\/mnsc.30.9.1078","volume":"30","author":"RD Banker","year":"1984","unstructured":"Banker, R. D., Charnes, A., & Cooper, W. W. (1984). Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science, 30(9), 1078\u20131092.","journal-title":"Management Science"},{"issue":"4","key":"4862_CR4","doi-asserted-by":"publisher","first-page":"513","DOI":"10.1287\/opre.34.4.513","volume":"34","author":"RD Banker","year":"1986","unstructured":"Banker, R. D., & Morey, R. C. (1986). Efficiency analysis for exogenously fixed inputs and outputs. Operations Research, 34(4), 513\u2013521.","journal-title":"Operations Research"},{"issue":"1","key":"4862_CR5","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1016\/0377-2217(92)90178-C","volume":"62","author":"RD Banker","year":"1992","unstructured":"Banker, R. D., & Thrall, R. M. (1992). Estimation of returns to scale using data envelopment analysis. European Journal of Operational Research, 62(1), 74\u201384.","journal-title":"European Journal of Operational Research"},{"issue":"1","key":"4862_CR6","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1287\/opre.50.1.3.17780","volume":"50","author":"RE Bixby","year":"2002","unstructured":"Bixby, R. E. (2002). Solving real-world linear programs: A decade or more of progress. Operations Research, 50(1), 3\u201315.","journal-title":"Operations Research"},{"key":"4862_CR7","doi-asserted-by":"crossref","unstructured":"Chambers, R. G., & F\u00e4re, R. (2008). A \u201ccalculus\u201d for data envelopment analysis. Journal of Productivity Analysis, 30(3), 169\u2013175.","DOI":"10.1007\/s11123-008-0104-8"},{"issue":"6","key":"4862_CR8","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, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6), 429\u2013444.","journal-title":"European Journal of Operational Research"},{"issue":"2","key":"4862_CR9","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1007\/s11123-010-0190-2","volume":"35","author":"WW Cooper","year":"2011","unstructured":"Cooper, W. W., Pastor, J. T., Borras, F., Aparicio, J., & Pastor, D. (2011). BAM: A bounded adjusted measure of efficiency for use with bounded additive models. Journal of Productivity Analysis, 35(2), 85\u201394.","journal-title":"Journal of Productivity Analysis"},{"key":"4862_CR10","doi-asserted-by":"crossref","unstructured":"Cooper, W. W., Seiford, L. M., & Tone, K. (2007). Data envelopment analysis: A comprehensive text with models, applications, references and DEA-solver software (2nd edn.). Springer.","DOI":"10.1007\/978-0-387-45283-8"},{"key":"4862_CR11","unstructured":"Deprins, D., Simar, L., & Tulkens, H. (1984). Measuring labor-efficiency in post offices. In M. Marchand, P. Pestieau, & H. Tulkens (Eds.), The performance of public enterprises: Concepts and measurements (pp. 243\u2013267). North-Holland."},{"issue":"2","key":"4862_CR12","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1016\/S0377-2217(00)00149-1","volume":"132","author":"RG Dyson","year":"2001","unstructured":"Dyson, R. G., Allen, R., Camanho, A. S., Podinovski, V. V., Sarrico, C. S., & Shale, E. A. (2001). Pitfalls and protocols in DEA. European Journal of Operational Research, 132(2), 245\u2013259.","journal-title":"European Journal of Operational Research"},{"issue":"2","key":"4862_CR13","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1007\/s10589-011-9420-4","volume":"52","author":"JM Elble","year":"2012","unstructured":"Elble, J. M., & Sahinidis, N. V. (2012). Scaling linear optimization problems prior to application of the simplex method. Computational Optimization and Applications, 52(2), 345\u2013371.","journal-title":"Computational Optimization and Applications"},{"issue":"1","key":"4862_CR14","doi-asserted-by":"publisher","first-page":"486","DOI":"10.1016\/j.apm.2007.11.018","volume":"33","author":"A Emrouznejad","year":"2009","unstructured":"Emrouznejad, A., & Amin, G. R. (2009). DEA models for ratio data: Convexity consideration. Applied Mathematical Modelling, 33(1), 486\u2013498.","journal-title":"Applied Mathematical Modelling"},{"key":"4862_CR15","doi-asserted-by":"crossref","unstructured":"F\u00e4re, R., Grosskopf, S., & Logan, J. (1983a). The relative efficiency of Illinois electric utilities. Resources and Energy, 5(4), 349\u2013367.","DOI":"10.1016\/0165-0572(83)90033-6"},{"key":"4862_CR16","doi-asserted-by":"crossref","unstructured":"F\u00e4re, R., Grosskopf, S., & Lovell, C. A. K. (1983b). The structure of technical efficiency. Scandinavian Journal of Economics, 85(2), 181\u2013190.","DOI":"10.2307\/3439477"},{"key":"4862_CR17","doi-asserted-by":"crossref","unstructured":"F\u00e4re, R., Grosskopf, S., & Lovell, C. A. K. (1985). The measurement of efficiency of production. Kluwer Academic Publishers.","DOI":"10.1007\/978-94-015-7721-2"},{"issue":"1","key":"4862_CR18","doi-asserted-by":"publisher","first-page":"150","DOI":"10.1016\/0022-0531(78)90060-1","volume":"19","author":"R F\u00e4re","year":"1978","unstructured":"F\u00e4re, R., & Lovell, C. A. K. (1978). Measuring the technical efficiency of production. Journal of Economic Theory, 19(1), 150\u2013162.","journal-title":"Journal of Economic Theory"},{"issue":"10","key":"4862_CR19","doi-asserted-by":"publisher","first-page":"1023","DOI":"10.1057\/palgrave.jors.2601741","volume":"55","author":"FR F\u00f8rsund","year":"2004","unstructured":"F\u00f8rsund, F. R., & Hjalmarsson, L. (2004). Calculating scale elasticity in DEA models. Journal of the Operational Research Society, 55(10), 1023\u20131038.","journal-title":"Journal of the Operational Research Society"},{"key":"4862_CR20","doi-asserted-by":"crossref","unstructured":"Frisch, R. (1965). Theory of production. D. Reidel Publishing Company.","DOI":"10.1007\/978-94-017-6161-1"},{"issue":"4","key":"4862_CR21","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1007\/BF01073549","volume":"4","author":"B Golany","year":"1993","unstructured":"Golany, B., & Roll, Y. (1993). Some extensions of techniques to handle non-discretionary factors in data envelopment analysis. Journal of Productivity Analysis, 4(4), 419\u2013432.","journal-title":"Journal of Productivity Analysis"},{"issue":"2","key":"4862_CR22","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1016\/j.ejor.2004.05.008","volume":"168","author":"P Hadjicostas","year":"2006","unstructured":"Hadjicostas, P., & Soteriou, A. C. (2006). One-sided elasticities and technical efficiency in multi-output production: A theoretical framework. European Journal of Operational Research, 168(2), 425\u2013449.","journal-title":"European Journal of Operational Research"},{"issue":"1","key":"4862_CR23","doi-asserted-by":"publisher","first-page":"230","DOI":"10.1016\/j.ejor.2019.07.005","volume":"280","author":"A Mostafaee","year":"2020","unstructured":"Mostafaee, A., & Soleimani-damaneh, M. (2020). Global sub-increasing and global sub-decreasing returns to scale in free disposal hull technologies: Definition, characterization and calculation. European Journal of Operational Research, 280(1), 230\u2013241.","journal-title":"European Journal of Operational Research"},{"issue":"2","key":"4862_CR24","doi-asserted-by":"publisher","first-page":"446","DOI":"10.1016\/j.ejor.2015.03.013","volume":"245","author":"OB Olesen","year":"2015","unstructured":"Olesen, O. B., Petersen, N. C., & Podinovski, V. V. (2015). Efficiency analysis with ratio measures. European Journal of Operational Research, 245(2), 446\u2013462.","journal-title":"European Journal of Operational Research"},{"issue":"2","key":"4862_CR25","doi-asserted-by":"publisher","first-page":"640","DOI":"10.1016\/j.ejor.2017.02.021","volume":"261","author":"OB Olesen","year":"2017","unstructured":"Olesen, O. B., Petersen, N. C., & Podinovski, V. V. (2017). Efficiency measures and computational approaches for data envelopment analysis models with ratio inputs and outputs. European Journal of Operational Research, 261(2), 640\u2013655.","journal-title":"European Journal of Operational Research"},{"issue":"3","key":"4862_CR26","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1007\/s11123-022-00631-6","volume":"57","author":"OB Olesen","year":"2022","unstructured":"Olesen, O. B., Petersen, N. C., & Podinovski, V. V. (2022). The structure of production technologies with ratio inputs and outputs. Journal of Productivity Analysis, 57(3), 255\u2013267. https:\/\/doi.org\/10.1007\/s11123-022-00631-6.","journal-title":"Journal of Productivity Analysis"},{"key":"4862_CR27","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1016\/j.omega.2015.02.009","volume":"56","author":"JT Pastor","year":"2015","unstructured":"Pastor, J. T., Aparicio, J., Alcaraz, J., Vidal, F., & Pastor, D. (2015). An enhanced BAM for unbounded or partially bounded CRS additive models. Omega, 56, 16\u201324.","journal-title":"Omega"},{"issue":"3","key":"4862_CR28","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1007\/s11123-012-0324-9","volume":"40","author":"JT Pastor","year":"2013","unstructured":"Pastor, J. T., Aparicio, J., Monge, J. F., & Pastor, D. (2013). Modeling CRS bounded additive DEA models and characterizing their Pareto-efficient points. Journal of Productivity Analysis, 40(3), 285\u2013292.","journal-title":"Journal of Productivity Analysis"},{"key":"4862_CR29","doi-asserted-by":"crossref","unstructured":"Podinovski, V. V. (2004a). Efficiency and global scale characteristics on the \u201cno free lunch\u201d assumption only. Journal of Productivity Analysis, 22(3), 227\u2013257.","DOI":"10.1007\/s11123-004-7575-z"},{"key":"4862_CR30","doi-asserted-by":"crossref","unstructured":"Podinovski, V. V. (2004b). Local and global returns to scale in performance measurement. Journal of the Operational Research Society, 55(2), 170\u2013178.","DOI":"10.1057\/palgrave.jors.2601680"},{"issue":"2","key":"4862_CR31","doi-asserted-by":"publisher","first-page":"552","DOI":"10.1016\/j.ejor.2003.09.008","volume":"161","author":"VV Podinovski","year":"2005","unstructured":"Podinovski, V. V. (2005). Selective convexity in DEA models. European Journal of Operational Research, 161(2), 552\u2013563.","journal-title":"European Journal of Operational Research"},{"issue":"3","key":"4862_CR32","doi-asserted-by":"publisher","first-page":"970","DOI":"10.1016\/j.ejor.2016.09.029","volume":"258","author":"VV Podinovski","year":"2017","unstructured":"Podinovski, V. V. (2017). Returns to scale in convex production technologies. European Journal of Operational Research, 258(3), 970\u2013982.","journal-title":"European Journal of Operational Research"},{"issue":"1","key":"4862_CR33","doi-asserted-by":"publisher","first-page":"236","DOI":"10.1287\/opre.2015.1457","volume":"64","author":"VV Podinovski","year":"2016","unstructured":"Podinovski, V. V., Chambers, R. G., Atici, K. B., & Deineko, I. D. (2016). Marginal values and returns to scale for nonparametric production frontiers. Operations Research, 64(1), 236\u2013250.","journal-title":"Operations Research"},{"issue":"3","key":"4862_CR34","doi-asserted-by":"publisher","first-page":"553","DOI":"10.1016\/0377-2217(94)00346-7","volume":"90","author":"J Ruggiero","year":"1996","unstructured":"Ruggiero, J. (1996). On the measurement of technical efficiency in the public sector. European Journal of Operational Research, 90(3), 553\u2013565.","journal-title":"European Journal of Operational Research"},{"key":"4862_CR35","doi-asserted-by":"crossref","unstructured":"Sahoo, B. K., & Tone, K. (2015). Scale elasticity in non-parametric DEA approach. In J. Zhu (Ed.), Data envelopment analysis: A handbook of models and methods (pp. 269\u2013290). Springer.","DOI":"10.1007\/978-1-4899-7553-9_9"},{"key":"4862_CR36","unstructured":"Shephard, R. W. (1974). Indirect production functions. Mathematical Systems in Economics. Anton Hain."},{"issue":"3","key":"4862_CR37","doi-asserted-by":"publisher","first-page":"592","DOI":"10.1016\/j.ejor.2013.01.012","volume":"228","author":"V Zelenyuk","year":"2013","unstructured":"Zelenyuk, V. (2013). A scale elasticity measure for directional distance function and its dual: Theory and DEA estimation. European Journal of Operational Research, 228(3), 592\u2013600.","journal-title":"European Journal of Operational Research"}],"container-title":["Annals of Operations Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-022-04862-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10479-022-04862-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-022-04862-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,4]],"date-time":"2022-11-04T19:59:52Z","timestamp":1667591992000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10479-022-04862-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,16]]},"references-count":37,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,11]]}},"alternative-id":["4862"],"URL":"https:\/\/doi.org\/10.1007\/s10479-022-04862-6","relation":{},"ISSN":["0254-5330","1572-9338"],"issn-type":[{"value":"0254-5330","type":"print"},{"value":"1572-9338","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,16]]},"assertion":[{"value":"24 June 2022","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 July 2022","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no conflicting interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}}]}}