{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T18:29:52Z","timestamp":1769192992002,"version":"3.49.0"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2018,5,31]],"date-time":"2018-05-31T00:00:00Z","timestamp":1527724800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Glob Optim"],"published-print":{"date-parts":[[2019,7]]},"DOI":"10.1007\/s10898-018-0668-4","type":"journal-article","created":{"date-parts":[[2018,5,31]],"date-time":"2018-05-31T04:07:12Z","timestamp":1527739632000},"page":"443-465","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Multi-parametric global optimization approach for tri-level mixed-integer linear optimization problems"],"prefix":"10.1007","volume":"74","author":[{"given":"Styliani","family":"Avraamidou","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6220-818X","authenticated-orcid":false,"given":"Efstratios N.","family":"Pistikopoulos","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,5,31]]},"reference":[{"issue":"1","key":"668_CR1","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1016\/j.cor.2013.06.009","volume":"41","author":"N Alguacil","year":"2014","unstructured":"Alguacil, N., Delgadillo, A., Arroyo, J.: A trilevel programming approach for electric grid defense planning. Comput. Oper. Res. 41(1), 282\u2013290 (2014)","journal-title":"Comput. Oper. Res."},{"issue":"11","key":"668_CR2","doi-asserted-by":"publisher","first-page":"1021","DOI":"10.1057\/jors.1988.172","volume":"39","author":"G Anandalingam","year":"1988","unstructured":"Anandalingam, G.: A mathematical programming model of decentralized multi-level systems. J. Oper. Res. Soc. 39(11), 1021\u20131033 (1988)","journal-title":"J. Oper. Res. Soc."},{"key":"668_CR3","unstructured":"Avraamidou, S., Diangelakis, N.A., Pistikopoulos, E.N.: Mixed integer bilevel optimization through multi-parametric programming. In: Foundations of Computer Aided Process Operations \/ Chemical Process Control 2017 (2017) http:\/\/folk.ntnu.no\/skoge\/prost\/proceedings\/focapo-cpc-2017\/FOCAPO-CPC%202017%20Contributed%20Papers\/73_FOCAPO_Contributed.pdf"},{"issue":"5","key":"668_CR4","doi-asserted-by":"publisher","first-page":"711","DOI":"10.1109\/TSMC.1984.6313291","volume":"14","author":"J Bard","year":"1984","unstructured":"Bard, J.: An investigation of the linear three level programming problem. IEEE Trans. Syst. Man Cybern. 14(5), 711\u2013717 (1984)","journal-title":"IEEE Trans. Syst. Man Cybern."},{"issue":"1","key":"668_CR5","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1007\/BF02098170","volume":"34","author":"C Blair","year":"1992","unstructured":"Blair, C.: The computational complexity of multi-level linear programs. Ann. Oper. Res. 34(1), 13\u201319 (1992)","journal-title":"Ann. Oper. Res."},{"issue":"6","key":"668_CR6","doi-asserted-by":"publisher","first-page":"530","DOI":"10.1287\/inte.1060.0252","volume":"36","author":"G Brown","year":"2006","unstructured":"Brown, G., Carlyle, M., Salmern, J., Wood, K.: Defending critical infrastructure. Interfaces 36(6), 530\u2013544 (2006)","journal-title":"Interfaces"},{"issue":"6","key":"668_CR7","doi-asserted-by":"publisher","first-page":"3069","DOI":"10.1109\/TPWRS.2014.2313841","volume":"29","author":"B Chen","year":"2014","unstructured":"Chen, B., Wang, J., Wang, L., He, Y., Wang, Z.: Robust optimization for transmission expansion planning: minimax cost vs. minimax regret. IEEE Trans. Power Syst. 29(6), 3069\u20133077 (2014)","journal-title":"IEEE Trans. Power Syst."},{"issue":"2","key":"668_CR8","doi-asserted-by":"publisher","first-page":"469","DOI":"10.1016\/j.ejor.2004.01.047","volume":"166","author":"S Dempe","year":"2005","unstructured":"Dempe, S., Kalashnikov, V., Rios-Mercado, R.Z.: Discrete bilevel programming: application to a natural gas cash-out problem. Eur. J. Oper. Res. 166(2), 469\u2013488 (2005)","journal-title":"Eur. J. Oper. Res."},{"issue":"4","key":"668_CR9","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1080\/02331934.2012.696641","volume":"63","author":"S Dempe","year":"2014","unstructured":"Dempe, S., Mordukhovich, B., Zemkoho, A.: Necessary optimality conditions in pessimistic bilevel programming. Optimization 63(4), 505\u2013533 (2014)","journal-title":"Optimization"},{"issue":"4","key":"668_CR10","doi-asserted-by":"publisher","first-page":"609","DOI":"10.1007\/s10898-006-9100-6","volume":"38","author":"NP Faisca","year":"2007","unstructured":"Faisca, N.P., Dua, V., Rustem, B., Saraiva, P.M., Pistikopoulos, E.N.: Parametric global optimisation for bilevel programming. J. Glob. Optim. 38(4), 609\u2013623 (2007)","journal-title":"J. Glob. Optim."},{"key":"668_CR11","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1007\/s10287-007-0062-z","volume":"6","author":"NP Faisca","year":"2009","unstructured":"Faisca, N.P., Saraiva, P.M., Rustem, B., Pistikopoulos, E.N.: A multi-parametric programming approach for multilevel hierarchical and decentralised optimisation problems. CMS 6, 377\u2013397 (2009)","journal-title":"CMS"},{"key":"668_CR12","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780195100563.001.0001","volume-title":"Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications","author":"C Floudas","year":"1995","unstructured":"Floudas, C.: Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications. Oxford University Press, Oxford (1995)"},{"issue":"7","key":"668_CR13","doi-asserted-by":"publisher","first-page":"406","DOI":"10.1287\/mnsc.18.7.406","volume":"18","author":"T Gal","year":"1972","unstructured":"Gal, T., Nedoma, J.: Multiparametric linear programming. Manag. Sci. 18(7), 406\u2013422 (1972)","journal-title":"Manag. Sci."},{"key":"668_CR14","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1016\/j.ins.2016.08.022","volume":"370\u2013371","author":"J Han","year":"2016","unstructured":"Han, J., Zhang, G., Hu, Y., Lu, J.: A solution to bi\/tri-level programming problems using particle swarm optimization. Inf. Sci. 370\u2013371, 519\u2013537 (2016)","journal-title":"Inf. Sci."},{"issue":"5","key":"668_CR15","doi-asserted-by":"publisher","first-page":"1194","DOI":"10.1137\/0913069","volume":"13","author":"P Hansen","year":"1992","unstructured":"Hansen, P., Jaumard, B., Savard, G.: New branch-and-bound rules for linear bilevel programming. SIAM J. Sci. Stat. Comput. 13(5), 1194\u20131217 (1992)","journal-title":"SIAM J. Sci. Stat. Comput."},{"issue":"10","key":"668_CR16","doi-asserted-by":"publisher","first-page":"1808","DOI":"10.1016\/j.automatica.2007.03.008","volume":"43","author":"C Jones","year":"2007","unstructured":"Jones, C., Kerrigan, E., Maciejowski, J.: Lexicographic perturbation for multiparametric linear programming with applications to control. Automatica 43(10), 1808\u20131816 (2007)","journal-title":"Automatica"},{"issue":"3","key":"668_CR17","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1016\/0165-0114(95)00086-0","volume":"77","author":"YJ Lai","year":"1996","unstructured":"Lai, Y.J.: Hierarchical optimization: a satisfactory solution. Fuzzy Sets Syst. 77(3), 321\u2013335 (1996)","journal-title":"Fuzzy Sets Syst."},{"issue":"5","key":"668_CR18","doi-asserted-by":"publisher","first-page":"911","DOI":"10.1287\/opre.38.5.911","volume":"38","author":"JT Moore","year":"1990","unstructured":"Moore, J.T., Bard, J.F.: The mixed integer linear bilevel programming problem. Oper. Res. 38(5), 911\u2013921 (1990)","journal-title":"Oper. Res."},{"issue":"4","key":"668_CR19","doi-asserted-by":"publisher","first-page":"2013","DOI":"10.1109\/TPWRS.2014.2349031","volume":"30","author":"A Moreira","year":"2015","unstructured":"Moreira, A., Street, A., Arroyo, J.: An adjustable robust optimization approach for contingency-constrained transmission expansion planning. IEEE Trans. Power Syst. 30(4), 2013\u20132022 (2015)","journal-title":"IEEE Trans. Power Syst."},{"key":"668_CR20","doi-asserted-by":"publisher","DOI":"10.1002\/aic.15717","author":"C Ning","year":"2017","unstructured":"Ning, C., You, F.: Data-driven adaptive nested robust optimization: General modeling framework and efficient computational algorithm for decision making under uncertainty. AIChE J (2017). https:\/\/doi.org\/10.1002\/aic.15717","journal-title":"AIChE J"},{"key":"668_CR21","doi-asserted-by":"publisher","first-page":"587","DOI":"10.1007\/s10898-016-0463-z","volume":"69","author":"R Oberdieck","year":"2016","unstructured":"Oberdieck, R., Diangelakis, N., Avraamidou, S., Pistikopoulos, E.: On unbounded and binary parameters in multi-parametric programming: applications to mixed-integer bilevel optimization and duality theory. J. Glob. Optim. 69, 587\u2013606 (2016a)","journal-title":"J. Glob. Optim."},{"issue":"2\u20133","key":"668_CR22","doi-asserted-by":"publisher","first-page":"527","DOI":"10.1007\/s10898-014-0143-9","volume":"59","author":"R Oberdieck","year":"2014","unstructured":"Oberdieck, R., Wittmann-Hohlbein, M., Pistikopoulos, E.: A branch and bound method for the solution of multiparametric mixed integer linear programming problems. J. Glob. Optim. 59(2\u20133), 527\u2013543 (2014)","journal-title":"J. Glob. Optim."},{"key":"668_CR23","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1016\/j.cherd.2016.09.034","volume":"116","author":"R Oberdieck","year":"2016","unstructured":"Oberdieck, R., Diangelakis, N., Nascu, I., Papathanasiou, M., Sun, M., Avraamidou, S., Pistikopoulos, E.: On multi-parametric programming and its applications in process systems engineering. Chem. Eng. Res. Des. 116, 61\u201382 (2016b)","journal-title":"Chem. Eng. Res. Des."},{"issue":"33","key":"668_CR24","doi-asserted-by":"publisher","first-page":"8979","DOI":"10.1021\/acs.iecr.6b01913","volume":"55","author":"R Oberdieck","year":"2016","unstructured":"Oberdieck, R., Diangelakis, N., Papathanasiou, M., Nascu, I., Pistikopoulos, E.: Pop\u2014parametric optimization toolbox. Ind. Eng. Chem. Res. 55(33), 8979\u20138991 (2016c)","journal-title":"Ind. Eng. Chem. Res."},{"key":"668_CR25","doi-asserted-by":"crossref","unstructured":"Olaru, S., Dumur, D.: On the continuity and complexity of control laws based on multiparametric linear programs, pp. 5465\u20135470 (2006)","DOI":"10.1109\/CDC.2006.377330"},{"issue":"2","key":"668_CR26","doi-asserted-by":"publisher","first-page":"1151","DOI":"10.1016\/j.ejor.2005.08.024","volume":"176","author":"S Pramanik","year":"2007","unstructured":"Pramanik, S., Roy, T.: Fuzzy goal programming approach to multilevel programming problems. Eur. J. Oper. Res. 176(2), 1151\u20131166 (2007)","journal-title":"Eur. J. Oper. Res."},{"issue":"1","key":"668_CR27","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/s10898-008-9291-0","volume":"44","author":"GK Saharidis","year":"2009","unstructured":"Saharidis, G.K., Ierapetritou, M.G.: Resolution method for mixed integer bi-level linear problems based on decomposition technique. J. Global Optim. 44(1), 29\u201351 (2009)","journal-title":"J. Global Optim."},{"issue":"6","key":"668_CR28","doi-asserted-by":"publisher","first-page":"2957","DOI":"10.1016\/j.eswa.2013.10.027","volume":"41","author":"M Sakawa","year":"2014","unstructured":"Sakawa, M., Matsui, T.: Interactive fuzzy stochastic multi-level 0\u20131 programming using tabu search and probability maximization. Expert Syst. Appl. 41(6), 2957\u20132963 (2014)","journal-title":"Expert Syst. Appl."},{"issue":"2","key":"668_CR29","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1016\/S0898-1221(98)00118-7","volume":"36","author":"M Sakawa","year":"1998","unstructured":"Sakawa, M., Nishizaki, I., Uemura, Y.: Interactive fuzzy programming for multilevel linear programming problems. Comput. Math. Appl. 36(2), 71\u201386 (1998)","journal-title":"Comput. Math. Appl."},{"issue":"3","key":"668_CR30","doi-asserted-by":"publisher","first-page":"580","DOI":"10.1016\/S0377-2217(98)00019-8","volume":"114","author":"M Sakawa","year":"1999","unstructured":"Sakawa, M., Nishizaki, I., Hitaka, M.: Interactive fuzzy programming for multi-level 0\u20131 programming problems through genetic algorithms. Eur. J. Oper. Res. 114(3), 580\u2013588 (1999)","journal-title":"Eur. J. Oper. Res."},{"issue":"1","key":"668_CR31","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1016\/0305-0548(95)00007-9","volume":"23","author":"HS Shih","year":"1996","unstructured":"Shih, H.S., Lai, Y.J., Lee, E.: Fuzzy approach for multi-level programming problems. Comput. Oper. Res. 23(1), 73\u201391 (1996)","journal-title":"Comput. Oper. Res."},{"issue":"5","key":"668_CR32","doi-asserted-by":"publisher","first-page":"594","DOI":"10.1057\/palgrave.jors.2601112","volume":"52","author":"S Sinha","year":"2001","unstructured":"Sinha, S.: A comment on Anandalingam (1988). A mathematical programming model of decentralized multi-level systems. J. Oper. Res. Soc. 39: 1021\u20131033. J. Oper. Res. Soc. 52(5), 594\u2013596 (2001)","journal-title":"J. Oper. Res. Soc."},{"issue":"9","key":"668_CR33","doi-asserted-by":"publisher","first-page":"1259","DOI":"10.1016\/S0305-0548(02)00061-8","volume":"30","author":"S Sinha","year":"2003","unstructured":"Sinha, S.: Fuzzy mathematical programming applied to multi-level programming problems. Comput. Oper. Res. 30(9), 1259\u20131268 (2003)","journal-title":"Comput. Oper. Res."},{"key":"668_CR34","doi-asserted-by":"publisher","first-page":"253","DOI":"10.3182\/20050703-6-CZ-1902.00903","volume":"16","author":"J Spjtvold","year":"2005","unstructured":"Spjtvold, J., Tndel, P., Johansen, T.: A method for obtaining continuous solutions to multiparametric linear programs. IFAC Proc. Vol. 16, 253\u2013258 (2005)","journal-title":"IFAC Proc. Vol."},{"issue":"1","key":"668_CR35","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1109\/TPWRS.2013.2278700","volume":"29","author":"A Street","year":"2014","unstructured":"Street, A., Moreira, A., Arroyo, J.: Energy and reserve scheduling under a joint generation and transmission security criterion: an adjustable robust optimization approach. IEEE Trans. Power Syst. 29(1), 3\u201314 (2014)","journal-title":"IEEE Trans. Power Syst."},{"issue":"4","key":"668_CR36","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1016\/0305-0548(86)90023-7","volume":"13","author":"UP Wen","year":"1986","unstructured":"Wen, U.P., Bialas, W.: The hybrid algorithm for solving the three-level linear programming problem. Comput. Oper. Res. 13(4), 367\u2013377 (1986)","journal-title":"Comput. Oper. Res."},{"issue":"1","key":"668_CR37","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1023\/A:1022610103712","volume":"93","author":"D White","year":"1997","unstructured":"White, D.: Penalty function approach to linear trilevel programming. J. Optim. Theory Appl. 93(1), 183\u2013197 (1997)","journal-title":"J. Optim. Theory Appl."},{"key":"668_CR38","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1016\/j.cor.2013.07.016","volume":"41","author":"P Xu","year":"2014","unstructured":"Xu, P., Wang, L.Z.: An exact algorithm for the bilevel mixed integer linear programming problem under three simplifying assumptions. Comput. Oper. Res. 41, 309\u2013318 (2014)","journal-title":"Comput. Oper. Res."},{"issue":"2","key":"668_CR39","doi-asserted-by":"publisher","first-page":"470","DOI":"10.1016\/j.cie.2013.07.012","volume":"66","author":"X Xu","year":"2013","unstructured":"Xu, X., Meng, Z., Shen, R.: A tri-level programming model based on conditional value-at-risk for three-stage supply chain management. Comput. Ind. Eng. 66(2), 470\u2013475 (2013)","journal-title":"Comput. Ind. Eng."},{"issue":"4","key":"668_CR40","doi-asserted-by":"publisher","first-page":"712","DOI":"10.1109\/TSMCC.2007.897487","volume":"37","author":"Y Yao","year":"2007","unstructured":"Yao, Y., Edmunds, T., Papageorgiou, D., Alvarez, R.: Trilevel optimization in power network defense. IEEE Trans. Syst. Man Cybern. Part C Appl. Rev. 37(4), 712\u2013718 (2007)","journal-title":"IEEE Trans. Syst. Man Cybern. Part C Appl. Rev."}],"container-title":["Journal of Global Optimization"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-018-0668-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10898-018-0668-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-018-0668-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T19:35:56Z","timestamp":1661369756000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10898-018-0668-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,31]]},"references-count":40,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,7]]}},"alternative-id":["668"],"URL":"https:\/\/doi.org\/10.1007\/s10898-018-0668-4","relation":{},"ISSN":["0925-5001","1573-2916"],"issn-type":[{"value":"0925-5001","type":"print"},{"value":"1573-2916","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,5,31]]},"assertion":[{"value":"2 July 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 May 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 May 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}