{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T12:44:48Z","timestamp":1768826688247,"version":"3.49.0"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2019,12,9]],"date-time":"2019-12-09T00:00:00Z","timestamp":1575849600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,12,9]],"date-time":"2019-12-09T00:00:00Z","timestamp":1575849600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100007397","name":"Univerzita Karlova v Praze","doi-asserted-by":"publisher","award":["SVV\u20132017\u2013260452"],"award-info":[{"award-number":["SVV\u20132017\u2013260452"]}],"id":[{"id":"10.13039\/100007397","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007397","name":"Univerzita Karlova v Praze","doi-asserted-by":"publisher","award":["SVV\u20132017\u2013260452"],"award-info":[{"award-number":["SVV\u20132017\u2013260452"]}],"id":[{"id":"10.13039\/100007397","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001824","name":"Grantov\u00e1 Agentura Cesk\u00e9 Republiky","doi-asserted-by":"publisher","award":["P403-18-04735S"],"award-info":[{"award-number":["P403-18-04735S"]}],"id":[{"id":"10.13039\/501100001824","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Grantov\u00e1 Agentura Cesk\u00e9 Republiky","award":["P403-18-04735S"],"award-info":[{"award-number":["P403-18-04735S"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2020,2]]},"DOI":"10.1007\/s10957-019-01610-y","type":"journal-article","created":{"date-parts":[[2019,12,9]],"date-time":"2019-12-09T07:02:35Z","timestamp":1575874955000},"page":"565-580","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Duality Gap in Interval Linear Programming"],"prefix":"10.1007","volume":"184","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7955-4692","authenticated-orcid":false,"given":"Jana","family":"Novotn\u00e1","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7340-8491","authenticated-orcid":false,"given":"Milan","family":"Hlad\u00edk","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8524-4036","authenticated-orcid":false,"given":"Tom\u00e1\u0161","family":"Masa\u0159\u00edk","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,12,9]]},"reference":[{"key":"1610_CR1","first-page":"85","volume-title":"Linear Programming-New Frontiers in Theory and Applications, Chap.\u00a02","author":"M Hlad\u00edk","year":"2012","unstructured":"Hlad\u00edk, M.: Interval linear programming: a survey. In: Mann, Z.A. (ed.) Linear Programming-New Frontiers in Theory and Applications, Chap.\u00a02, pp. 85\u2013120. Nova Science Publishers, New York (2012)"},{"key":"1610_CR2","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1007\/0-387-32698-7_3","volume-title":"Linear Optimization Problems with Inexact Data, Chap.\u00a03","author":"J Rohn","year":"2006","unstructured":"Rohn, J.: Interval linear programming. In: Fiedler, M., Nedoma, J., Ram\u00edk, J., Rohn, J., Zimmermann, K. (eds.) Linear Optimization Problems with Inexact Data, Chap.\u00a03, pp. 79\u2013100. Springer, New York (2006)"},{"key":"1610_CR3","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1007\/0-387-32698-7_2","volume-title":"Linear Optimization Problems with Inexact Data, Chap.\u00a02","author":"J Rohn","year":"2006","unstructured":"Rohn, J.: Solvability of systems of interval linear equations and inequalities. In: Fiedler, M., Nedoma, J., Ram\u00edk, J., Rohn, J., Zimmermann, K. (eds.) Linear Optimization Problems with Inexact Data, Chap.\u00a02, pp. 35\u201377. Springer, New York (2006)"},{"key":"1610_CR4","unstructured":"Novotn\u00e1, J., Hlad\u00edk, M., Masa\u0159\u00edk, T.: In: Zadnik Stirn, L., et al. (ed.) Proceedings of the 14th International Symposium on Operational Research SOR\u201917, Bled, Slovenia, September 27\u201329, 2017, pp. 501\u2013506. Slovenian Society Informatika, Ljubljana, Slovenia (2017)"},{"issue":"3","key":"1610_CR5","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1111\/j.1475-3995.2009.00737.x","volume":"17","author":"V Gabrel","year":"2010","unstructured":"Gabrel, V., Murat, C., Remli, N.: Linear programming with interval right hand sides. Int. Trans. Oper. Res. 17(3), 397\u2013408 (2010). https:\/\/doi.org\/10.1111\/j.1475-3995.2009.00737.x","journal-title":"Int. Trans. Oper. Res."},{"key":"1610_CR6","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1007\/978-3-319-67308-0_37","volume-title":"Optimization and Decision Science: Methodologies and Applications, Springer Proceedings in Mathematics and Statistics","author":"R Cerulli","year":"2017","unstructured":"Cerulli, R., D\u2019Ambrosio, C., Gentili, M.: Best and worst values of the optimal cost of the interval transportation problem. In: Sforza, A., Sterle, C. (eds.) Optimization and Decision Science: Methodologies and Applications, Springer Proceedings in Mathematics and Statistics, vol. 217, pp. 367\u2013374. Springer, Cham (2017)"},{"key":"1610_CR7","volume-title":"Introduction to Interval Computations. Computer Science and Applied Mathematics","author":"G Alefeld","year":"1983","unstructured":"Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Computer Science and Applied Mathematics. Academic Press, New York (1983)"},{"key":"1610_CR8","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717716","volume-title":"Introduction to Interval Analysis","author":"RE Moore","year":"2009","unstructured":"Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009). https:\/\/doi.org\/10.1137\/1.9780898717716"},{"key":"1610_CR9","volume-title":"Linear Optimization Problems with Inexact Data","author":"M Fiedler","year":"2006","unstructured":"Fiedler, M., Nedoma, J., Ram\u00edk, J., Rohn, J., Zimmermann, K.: Linear Optimization Problems with Inexact Data. Springer, New York (2006)"},{"key":"1610_CR10","doi-asserted-by":"publisher","DOI":"10.1007\/s10100-018-0580-5","author":"E Garajov\u00e1","year":"2019","unstructured":"Garajov\u00e1, E., Hlad\u00edk, M., Rada, M.: Interval linear programming under transformations: optimal solutions and optimal value range. Cent. Eur. J. Oper. Res. (2019). https:\/\/doi.org\/10.1007\/s10100-018-0580-5. (in press)","journal-title":"Cent. Eur. J. Oper. Res."},{"issue":"2","key":"1610_CR11","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1080\/03081087.2016.1180339","volume":"65","author":"M Hlad\u00edk","year":"2017","unstructured":"Hlad\u00edk, M.: Transformations of interval linear systems of equations and inequalities. Linear Multilinear Algebra 65(2), 211\u2013223 (2017). https:\/\/doi.org\/10.1080\/03081087.2016.1180339","journal-title":"Linear Multilinear Algebra"},{"key":"1610_CR12","doi-asserted-by":"crossref","unstructured":"Rohn, J.: Duality in interval linear programming. In: Nickel, K. (ed.) Interval Mathematics, Proc. Int. Symp., Freiburg, 1980, pp. 521\u2013529. Academic Press, New York (1980)","DOI":"10.1016\/B978-0-12-518850-0.50044-5"},{"issue":"8","key":"1610_CR13","doi-asserted-by":"publisher","first-page":"1288","DOI":"10.1057\/jors.2009.81","volume":"61","author":"V Gabrel","year":"2010","unstructured":"Gabrel, V., Murat, C.: Robustness and duality in linear programming. J. Oper. Res. Soc. 61(8), 1288\u20131296 (2010). https:\/\/doi.org\/10.1057\/jors.2009.81","journal-title":"J. Oper. Res. Soc."},{"issue":"2","key":"1610_CR14","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1016\/j.orl.2004.04.011","volume":"33","author":"P Serafini","year":"2005","unstructured":"Serafini, P.: Linear programming with variable matrix entries. Oper. Res. Lett. 33(2), 165\u2013170 (2005). https:\/\/doi.org\/10.1016\/j.orl.2004.04.011","journal-title":"Oper. Res. Lett."},{"key":"1610_CR15","unstructured":"Beeck, H.: Linear programming with inexact data. Technick\u00fd report TUM-ISU-7830, Technical University of Munich, Munich (1978)"},{"key":"1610_CR16","doi-asserted-by":"publisher","first-page":"591","DOI":"10.1016\/j.cam.2015.07.034","volume":"292","author":"A Mostafaee","year":"2016","unstructured":"Mostafaee, A., Hlad\u00edk, M., \u010cern\u00fd, M.: Inverse linear programming with interval coefficients. J. Comput. Appl. Math. 292, 591\u2013608 (2016). https:\/\/doi.org\/10.1016\/j.cam.2015.07.034","journal-title":"J. Comput. Appl. Math."},{"issue":"2","key":"1610_CR17","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1057\/palgrave.jors.2600891","volume":"51","author":"JW Chinneck","year":"2000","unstructured":"Chinneck, J.W., Ramadan, K.: Linear programming with interval coefficients. J. Oper. Res. Soc. 51(2), 209\u2013220 (2000). https:\/\/doi.org\/10.1057\/palgrave.jors.2600891","journal-title":"J. Oper. Res. Soc."},{"issue":"3","key":"1610_CR18","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1007\/s10700-009-9060-7","volume":"8","author":"M Hlad\u00edk","year":"2009","unstructured":"Hlad\u00edk, M.: Optimal value range in interval linear programming. Fuzzy Optim. Decis. Mak. 8(3), 283\u2013294 (2009). https:\/\/doi.org\/10.1007\/s10700-009-9060-7","journal-title":"Fuzzy Optim. Decis. Mak."},{"issue":"3","key":"1610_CR19","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1023\/A:1009987227018","volume":"3","author":"J Rohn","year":"1997","unstructured":"Rohn, J.: Complexity of some linear problems with interval data. Reliab. Comput. 3(3), 315\u2013323 (1997)","journal-title":"Reliab. Comput."},{"issue":"1","key":"1610_CR20","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1007\/s11590-012-0589-y","volume":"8","author":"M Hlad\u00edk","year":"2014","unstructured":"Hlad\u00edk, M.: How to determine basis stability in interval linear programming. Optim. Lett. 8(1), 375\u2013389 (2014). https:\/\/doi.org\/10.1007\/s11590-012-0589-y","journal-title":"Optim. Lett."},{"issue":"Suppl. 3","key":"1610_CR21","doi-asserted-by":"publisher","first-page":"677","DOI":"10.1002\/zamm.200108115114","volume":"81","author":"J Kon\u00ed\u010dkov\u00e1","year":"2001","unstructured":"Kon\u00ed\u010dkov\u00e1, J.: Sufficient condition of basis stability of an interval linear programming problem. ZAMM Z. Angew. Math. Mech. 81(Suppl. 3), 677\u2013678 (2001). https:\/\/doi.org\/10.1002\/zamm.200108115114","journal-title":"ZAMM Z. Angew. Math. Mech."},{"key":"1610_CR22","volume-title":"Mathematical Programming","author":"S Vajda","year":"1961","unstructured":"Vajda, S.: Mathematical Programming. Addison-Wesley, Reading (1961)"},{"key":"1610_CR23","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1080\/02331938108842705","volume":"12","author":"W Gerlach","year":"1981","unstructured":"Gerlach, W.: Zur L\u00f6sung linearer Ungleichungssysteme bei St\u00f6rung der rechten Seite und der Koeffizientenmatrix. Math. Operationsforsch. Stat. Ser. Optimization 12, 41\u201343 (1981)","journal-title":"Math. Operationsforsch. Stat. Ser. Optimization"},{"key":"1610_CR24","unstructured":"Machost, B.: Numerische Behandlung des Simplexverfahrens mit intervallanalytischen Methoden. Tech. Rep.\u00a030, Berichte der Gesellschaft f\u00fcr Mathematik und Datenverarbeitung, Bonn (1970)"},{"issue":"1\u20132","key":"1610_CR25","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1080\/03081089508818341","volume":"38","author":"J Rohn","year":"1994","unstructured":"Rohn, J., Kreslov\u00e1, J.: Linear interval inequalities. Linear Multilinear Algebra 38(1\u20132), 79\u201382 (1994). https:\/\/doi.org\/10.1080\/03081089508818341","journal-title":"Linear Multilinear Algebra"},{"issue":"1","key":"1610_CR26","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1007\/BF02243426","volume":"26","author":"J Rohn","year":"1981","unstructured":"Rohn, J.: Strong solvability of interval linear programming problems. Computing 26(1), 79\u201382 (1981). https:\/\/doi.org\/10.1007\/BF02243426","journal-title":"Computing"},{"key":"1610_CR27","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1007\/978-3-319-67308-0_40","volume-title":"Springer Proceedings in Mathematics & Statistics","author":"Elif Garajov\u00e1","year":"2017","unstructured":"Garajov\u00e1, E., Hlad\u00edk, M., Rada, M.: On the properties of interval linear programs with a fixed coefficient matrix. In: Springer Proceedings in Mathematics and Statistics, pp. 393\u2013401. Springer (2017). https:\/\/doi.org\/10.1007\/978-3-319-67308-0_40"},{"issue":"11","key":"1610_CR28","doi-asserted-by":"publisher","first-page":"4156","DOI":"10.1016\/j.laa.2013.02.012","volume":"438","author":"M Hlad\u00edk","year":"2013","unstructured":"Hlad\u00edk, M.: Weak and strong solvability of interval linear systems of equations and inequalities. Linear Algebra Appl. 438(11), 4156\u20134165 (2013). https:\/\/doi.org\/10.1016\/j.laa.2013.02.012","journal-title":"Linear Algebra Appl."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-019-01610-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10957-019-01610-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-019-01610-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,12,8]],"date-time":"2020-12-08T00:38:53Z","timestamp":1607387933000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10957-019-01610-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,9]]},"references-count":28,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,2]]}},"alternative-id":["1610"],"URL":"https:\/\/doi.org\/10.1007\/s10957-019-01610-y","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,12,9]]},"assertion":[{"value":"7 December 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 November 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 December 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}