{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T14:50:26Z","timestamp":1777301426563,"version":"3.51.4"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,1,1]],"date-time":"2023-01-01T00:00:00Z","timestamp":1672531200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,1,1]],"date-time":"2023-01-01T00:00:00Z","timestamp":1672531200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Universidad de Cadiz"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Fuzzy Optim Decis Making"],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper deals with an inverse data envelopment analysis (DEA) based on the non-radial slacks-based model in the presence of uncertainty employing both integer and continuous interval data. To this matter, suitable technology and formulation for the DEA are proposed using arithmetic and partial orders for interval numbers. The inverse DEA is discussed from the following question: if the output of <jats:inline-formula><jats:alternatives><jats:tex-math>$$DMU_o$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>D<\/mml:mi>\n                    <mml:mi>M<\/mml:mi>\n                    <mml:msub>\n                      <mml:mi>U<\/mml:mi>\n                      <mml:mi>o<\/mml:mi>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> increases from <jats:inline-formula><jats:alternatives><jats:tex-math>$$Y_o$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>Y<\/mml:mi>\n                    <mml:mi>o<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> to <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\beta _o$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>\u03b2<\/mml:mi>\n                    <mml:mi>o<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, such the new <jats:italic>DMU<\/jats:italic> is given by <jats:inline-formula><jats:alternatives><jats:tex-math>$$(\\alpha _o^*,\\beta )$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msubsup>\n                      <mml:mi>\u03b1<\/mml:mi>\n                      <mml:mi>o<\/mml:mi>\n                      <mml:mo>\u2217<\/mml:mo>\n                    <\/mml:msubsup>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>\u03b2<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> belongs to the technology, and its inefficiency score is not less than t-percent, how much should the inputs of the DMU increase? A new model of inverse DEA is offered to respond to the previous question, whose interval Pareto solutions are characterized using the Pareto solution of a related multiple-objective nonlinear programming (MONLP). Necessary and sufficient conditions for input estimation are proposed when output is increased. A functional example is presented on data to illustrate the new model and methodology, with continuous and integer interval variables.<\/jats:p>","DOI":"10.1007\/s10700-022-09403-1","type":"journal-article","created":{"date-parts":[[2023,1,1]],"date-time":"2023-01-01T08:41:25Z","timestamp":1672562485000},"page":"587-609","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Using slacks-based model to solve inverse DEA with integer intervals for input estimation"],"prefix":"10.1007","volume":"22","author":[{"given":"Atefeh","family":"Younesi","sequence":"first","affiliation":[]},{"given":"Farhad Hosseinzadeh","family":"Lotfi","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0290-6477","authenticated-orcid":false,"given":"Manuel","family":"Arana-Jim\u00e9nez","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,1,1]]},"reference":[{"key":"9403_CR1","doi-asserted-by":"crossref","unstructured":"Apt, K. R., & Zoeteweij, P. A. (2004). Comparative study of arithmetic constraints on integer intervals. In International workshop on constraint solving and constraint logic programming, (vol. 3010, pp. 1\u201324).","DOI":"10.1007\/978-3-540-24662-6_1"},{"key":"9403_CR2","doi-asserted-by":"crossref","unstructured":"Arana-Jim\u00e9nez, M. (Ed.). (2010). Optimiality conditions in vector optimization. Bussum: Bentham Science Publishers Ltd.","DOI":"10.2174\/97816080511061100101"},{"key":"9403_CR3","doi-asserted-by":"publisher","first-page":"6556","DOI":"10.1002\/mma.4474","volume":"40","author":"M Arana-Jim\u00e9nez","year":"2017","unstructured":"Arana-Jim\u00e9nez, M., & Antczak, T. (2017). The minimal criterion for the equivalence between local and global optimal solutions in nondifferentiable optimization problem. Mathematical Methods in the Applied Sciences, 40, 6556\u20136564.","journal-title":"Mathematical Methods in the Applied Sciences"},{"issue":"3","key":"9403_CR4","doi-asserted-by":"publisher","first-page":"709","DOI":"10.1051\/ro\/2016058","volume":"51","author":"M Arana-Jim\u00e9nez","year":"2017","unstructured":"Arana-Jim\u00e9nez, M., & Salles Neto, L. L. (2017). Sufficient condition for partial efficiency in a bicriteria nonlinear cutting stock problem. AIRO-Operations Research, 51(3), 709\u2013717.","journal-title":"AIRO-Operations Research"},{"key":"9403_CR5","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1016\/j.fss.2020.12.011","volume":"422","author":"M Arana-Jim\u00e9nez","year":"2021","unstructured":"Arana-Jim\u00e9nez, M., Snchez-Gil, M. C., Younesi, A., & Lozano, S. (2021). integer interval DEA; an axiomatic derivation of the technology and an additive, slacks-based model. Fuzzy Sets and Systems, 422, 83\u2013105.","journal-title":"Fuzzy Sets and Systems"},{"issue":"9","key":"9403_CR6","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 efficiencies in data envelopment analysis. Management Science, 30(9), 1078\u20131092.","journal-title":"Management Science"},{"issue":"6","key":"9403_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, W. W., & Rhodes, E. (1978). Measuring the efficiencies of DMUs. European Journal of Operational Research, 2(6), 429\u2013444.","journal-title":"European Journal of Operational Research"},{"key":"9403_CR8","doi-asserted-by":"publisher","first-page":"105362","DOI":"10.1016\/j.ssci.2021.105362","volume":"142","author":"L Chen","year":"2021","unstructured":"Chen, L., Gao, Y., Li, M. J., Wang, Y. M., & Liao, L. H. (2021). A new inverse data envelopment analysis approach to achieve China\u2019s road transportation safety objectives. Safety Science, 142, 105362.","journal-title":"Safety Science"},{"key":"9403_CR9","doi-asserted-by":"publisher","first-page":"115344","DOI":"10.1016\/j.eswa.2021.115344","volume":"183","author":"L Chen","year":"2021","unstructured":"Chen, L., & Wang, Y. M. (2021). Limitation and optimization of inputs and outputs in the inverse data envelopment analysis under variable returns to scale. Expert Systems with Applications, 183, 115344.","journal-title":"Expert Systems with Applications"},{"key":"9403_CR10","doi-asserted-by":"publisher","first-page":"1605","DOI":"10.1051\/ro\/2020029","volume":"55","author":"S Ghobadi","year":"2021","unstructured":"Ghobadi, S. (2021). Merging decision-making units with interval data. RAIRO Operations Research, 55, 1605\u20131631.","journal-title":"RAIRO Operations Research"},{"issue":"5\u20136","key":"9403_CR11","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1016\/j.mcm.2005.08.005","volume":"43","author":"A Hadi-Vencheh","year":"2006","unstructured":"Hadi-Vencheh, A., & Foroughi, A. A. (2006). A generalized DEA model for inputs\/outputs estimation. Mathematical and Computer Modelling, 43(5\u20136), 447\u2013457.","journal-title":"Mathematical and Computer Modelling"},{"key":"9403_CR12","doi-asserted-by":"publisher","DOI":"10.1080\/02331934.2014.974599","author":"A Hadi-Vencheh","year":"2014","unstructured":"Hadi-Vencheh, A., Hatami-Marbini, A. G., Beigi, Z., & Gholami, K. (2014). An inverse optimization model for imprecise data envelopment analysis. Optimization: A Journal of Mathematical Programming and Operations Research. https:\/\/doi.org\/10.1080\/02331934.2014.974599.","journal-title":"Optimization: A Journal of Mathematical Programming and Operations Research"},{"key":"9403_CR13","doi-asserted-by":"publisher","DOI":"10.1111\/exsy.12302","author":"A Hadi-Vencheh","year":"2018","unstructured":"Hadi-Vencheh, A., Wanke, P., Jamshidi, A., & Chen, Z. (2018). Sustainability of Chinese airlines: A modified slack-based measure model for $$CO_2$$ emissions. Expert Systems. https:\/\/doi.org\/10.1111\/exsy.12302.","journal-title":"Expert Systems"},{"key":"9403_CR14","doi-asserted-by":"publisher","DOI":"10.1155\/571896","author":"GR Jahanshahloo","year":"2014","unstructured":"Jahanshahloo, G. R., Hosseinzadeh Lotfi, F., Rostamy-Malkhalifeh, M., & Ghobadi, S. (2014). Using enhanced Russel model to solve inverse data envelopment analysis problems. The Scientific World Journal. https:\/\/doi.org\/10.1155\/571896.","journal-title":"The Scientific World Journal"},{"issue":"2","key":"9403_CR15","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1016\/j.amc.2003.08.001","volume":"156","author":"GR Jahanshahloo","year":"2004","unstructured":"Jahanshahloo, G. R., Hosseinzadeh Lotfi, F., Shoja, N., Tohidi, G., & Razavyan, S. (2004). Input estimation and identification of extra inputs in inverse DEA models. Applied Mathematics and Computation, 156(2), 427\u2013437.","journal-title":"Applied Mathematics and Computation"},{"key":"9403_CR16","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1016\/j.ejor.2014.07.002","volume":"240","author":"GR Jahanshahloo","year":"2015","unstructured":"Jahanshahloo, G. R., Soleimani-damaneh, M., & Ghobadi, S. (2015). Inverse DEA under inter-temporal dependence using multiple-objective programming. European Journal of Operational Research, 240, 447\u2013456.","journal-title":"European Journal of Operational Research"},{"key":"9403_CR17","doi-asserted-by":"publisher","first-page":"6078","DOI":"10.1080\/00207543.2019.1657245","volume":"58","author":"HA Le Thi","year":"2020","unstructured":"Le Thi, H. A. (2020). DC programming and DCA for supply chain and production management: State-of-the-art models and methods. International Journal of Production Research, 58, 6078\u20136114.","journal-title":"International Journal of Production Research"},{"issue":"10","key":"9403_CR18","doi-asserted-by":"publisher","first-page":"3004","DOI":"10.1016\/j.cor.2005.02.031","volume":"33","author":"S Lozano","year":"2006","unstructured":"Lozano, S., & Villa, G. (2006). Data envelopment analysis of integer-valued inputs and outputs. Computers and Operations Research, 33(10), 3004\u20133014.","journal-title":"Computers and Operations Research"},{"issue":"6","key":"9403_CR19","first-page":"187","volume":"6","author":"KG Shinto","year":"2019","unstructured":"Shinto, K. G., & Sushama, C. M. (2019). An inverse DEA model for input\/output estimation with integer restriction. Journal of Mathematics and Statistical science, 6(6), 187\u2013194.","journal-title":"Journal of Mathematics and Statistical science"},{"key":"9403_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.fss.2018.04.009","volume":"362","author":"L Stefanini","year":"2019","unstructured":"Stefanini, L., & Arana-Jim\u00e9nez, M. (2019). Karush\u2013Kuhn\u2013Tucker conditions for interval and fuzzy optimization in several variables under total and directional generalized differentiability. Fuzzy Sets and Systems, 362, 1\u201334.","journal-title":"Fuzzy Sets and Systems"},{"key":"9403_CR21","doi-asserted-by":"publisher","first-page":"184","DOI":"10.3390\/e23020184","volume":"23","author":"Y Tan","year":"2021","unstructured":"Tan, Y., Jamshidi, A., Hadi-Vencheh, A., & Wanke, P. (2021). Hotel performance in the UK: The role of information entropy in a novel Slack-based Data Envelopment Analysis. Entropy, 23, 184. https:\/\/doi.org\/10.3390\/e23020184.","journal-title":"Entropy"},{"key":"9403_CR22","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/S0377-2217(99)00007-7","volume":"121","author":"QL Wei","year":"2000","unstructured":"Wei, Q. L., Zhang, J., & Zhang, X. (2000). Inverse DEA model for inputs\/outputs estimate. European Journal of Operational Research, 121, 151\u2013163.","journal-title":"European Journal of Operational Research"},{"key":"9403_CR23","doi-asserted-by":"publisher","DOI":"10.1108\/JM2-05-2014-0036515-535","author":"A Yousefi","year":"2016","unstructured":"Yousefi, A., & Hadi-Vencheh, A. (2016). Selecting Six Sigma projects: MCDM or DEA? Journal of Modelling in Management. https:\/\/doi.org\/10.1108\/JM2-05-2014-0036515-535.","journal-title":"Journal of Modelling in Management"},{"key":"9403_CR24","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1111\/j.1475-3995.1999.tb00166.x","volume":"6","author":"XS Zhang","year":"1999","unstructured":"Zhang, X. S., & Cui, J. C. (1999). A project evaluation system in the state economic information system of china an operations research pravtice in public sector. International Transactions in Operational Research, 6, 441\u2013452.","journal-title":"International Transactions in Operational Research"},{"key":"9403_CR25","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.cie.2020.106368","volume":"142","author":"GJ Zhang","year":"2020","unstructured":"Zhang, G. J., & Cui, J. C. (2020). A general inverse DEA model for non-radial DEA. Computers and Industrial Engineering, 142, 1\u20137.","journal-title":"Computers and Industrial Engineering"}],"container-title":["Fuzzy Optimization and Decision Making"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10700-022-09403-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10700-022-09403-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10700-022-09403-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,28]],"date-time":"2023-09-28T01:10:09Z","timestamp":1695863409000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10700-022-09403-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,1]]},"references-count":25,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["9403"],"URL":"https:\/\/doi.org\/10.1007\/s10700-022-09403-1","relation":{},"ISSN":["1568-4539","1573-2908"],"issn-type":[{"value":"1568-4539","type":"print"},{"value":"1573-2908","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,1]]},"assertion":[{"value":"22 November 2022","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 January 2023","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}