{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T20:41:54Z","timestamp":1757623314290,"version":"3.44.0"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2025,7,9]],"date-time":"2025-07-09T00:00:00Z","timestamp":1752019200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,7,9]],"date-time":"2025-07-09T00:00:00Z","timestamp":1752019200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Slovenian Agency for Research and Innovation","award":["grant P1-0222"],"award-info":[{"award-number":["grant P1-0222"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,10]]},"DOI":"10.1007\/s40314-025-03317-5","type":"journal-article","created":{"date-parts":[[2025,7,10]],"date-time":"2025-07-10T09:24:10Z","timestamp":1752139450000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Extremal solvability of interval multi-linear systems in max-plus"],"prefix":"10.1007","volume":"44","author":[{"given":"Sedighe","family":"Khaleghzade","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4472-3609","authenticated-orcid":false,"given":"Mostafa","family":"Zangiabadi","sequence":"additional","affiliation":[]},{"given":"Aljo\u0161a","family":"Peperko","sequence":"additional","affiliation":[]},{"given":"Tina","family":"\u0160filigoj","sequence":"additional","affiliation":[]},{"given":"Masoud","family":"Hajarian","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,7,9]]},"reference":[{"key":"3317_CR1","first-page":"1","volume":"3","author":"HR Afshin","year":"2016","unstructured":"Afshin HR, Shojaeifard AR (2016) Max-Plus algebra on tensors and its properties. Wavelet and Linear Algebra 3:1\u201311","journal-title":"Wavelet and Linear Algebra"},{"key":"3317_CR2","volume-title":"Synchronization and Linearity","author":"F Baccelli","year":"1992","unstructured":"Baccelli F, Cohen G, Olsder GJ, Quadrat GP (1992) Synchronization and Linearity. Wiley Series in Probability and Mathematical Statistics, Wiley, New York"},{"key":"3317_CR3","doi-asserted-by":"publisher","first-page":"697","DOI":"10.1016\/j.camwa.2019.07.024","volume":"79","author":"H Bozorgmanesh","year":"2020","unstructured":"Bozorgmanesh H, Hajarian M, Chronopoulos AT (2020) Interval tensors and their application in solving multi-linear systems of equations. Comput Math Appl 79:697\u2013715","journal-title":"Comput Math Appl"},{"key":"3317_CR4","doi-asserted-by":"crossref","unstructured":"Butkovi\u010d P (2010) Max-linear systems theory and algorithms. Springer Science and Business Media","DOI":"10.1007\/978-1-84996-299-5"},{"key":"3317_CR5","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1016\/S0024-3795(01)00405-0","volume":"340","author":"K Cechl\u00e1rov\u00e1","year":"2002","unstructured":"Cechl\u00e1rov\u00e1 K, Cuninghame-Green RA (2002) Interval systems of max-separable linear equations. Linear Algebra Appl 340:215\u2013224","journal-title":"Linear Algebra Appl"},{"key":"3317_CR6","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1109\/TAC.1985.1103925","volume":"30","author":"G Cohen","year":"1985","unstructured":"Cohen G, Dubois D, Quadrat JP, Viot M (1985) A linear-system theoretic view of discrete-event processes and its use for performance evaluation in manufacturing. IEEE Trans Autom Control 30:210\u2013220","journal-title":"IEEE Trans Autom Control"},{"key":"3317_CR7","unstructured":"Cuninghame-Green RA (2012) Minimax algebra. Vol. 166, Springer Science and Business Media"},{"key":"3317_CR8","doi-asserted-by":"publisher","first-page":"689","DOI":"10.1007\/s10915-015-0156-7","volume":"68","author":"W Ding","year":"2016","unstructured":"Ding W, Wei Y (2016) Solving multi-linear systems with $$M$$-tensors. J Sci Comput 68:689\u2013715","journal-title":"J Sci Comput"},{"key":"3317_CR9","doi-asserted-by":"publisher","first-page":"1277","DOI":"10.1007\/s11464-017-0669-5","volume":"12","author":"S Friedland","year":"2017","unstructured":"Friedland S, Qi L, Wei Y, Yang Q (2017) Tensor and hypergraph. Frontiers of Mathematics in China 12:1277\u20131277","journal-title":"Frontiers of Mathematics in China"},{"key":"3317_CR10","doi-asserted-by":"crossref","unstructured":"Gaubert S, Plus M (1997) Methods and applications of (max,+) linear algebra. In: Annual Symposium on Theoretical Aspects of Computer Science. Springer, Berlin, Heidelberg","DOI":"10.1007\/BFb0023465"},{"key":"3317_CR11","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1016\/j.trb.2006.02.003","volume":"41","author":"RMP Goverde","year":"2007","unstructured":"Goverde RMP (2007) Railway timetable stability analysis using max-plus system theory. Transportation Research Part B: Methodological 41:179\u2013201","journal-title":"Transportation Research Part B: Methodological"},{"key":"3317_CR12","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1007\/s10626-014-0205-7","volume":"26","author":"M Kersbergen","year":"2016","unstructured":"Kersbergen M, Rudan J, van den Boom T, De Schutter B (2016) Towards railway traffic management using switching max-plus-linear systems. Discrete Event Dynamic Systems 26:183\u2013223","journal-title":"Discrete Event Dynamic Systems"},{"key":"3317_CR13","first-page":"704","volume":"58","author":"S Khaleghzade","year":"2022","unstructured":"Khaleghzade S, Zangiabadi M, Peperko A, Hajarian M (2022) Interval multi-linear systems for tensors in the max-plus algebra and their application in solving the job shop problem. Kybernetika 58:704\u2013728","journal-title":"Kybernetika"},{"key":"3317_CR14","doi-asserted-by":"publisher","first-page":"649","DOI":"10.1007\/s11464-014-0377-3","volume":"10","author":"X Li","year":"2015","unstructured":"Li X, Ng MK (2015) Solving sparse non-negative tensor equations: algorithms and applications. Frontiers of Mathematics in China 10:649\u2013680","journal-title":"Frontiers of Mathematics in China"},{"key":"3317_CR15","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1016\/j.laa.2005.02.011","volume":"403","author":"H My\u0161kova","year":"2005","unstructured":"My\u0161kova H (2005) Interval systems of max-separable linear equations. Linear Algebra Appl 403:263\u2013272","journal-title":"Linear Algebra Appl"},{"key":"3317_CR16","doi-asserted-by":"publisher","first-page":"1992","DOI":"10.1016\/j.laa.2012.04.051","volume":"437","author":"H My\u0161kova","year":"2012","unstructured":"My\u0161kova H (2012) Interval max-plus systems of linear equations. Linear Algebra Appl 437:1992\u20132000","journal-title":"Linear Algebra Appl"},{"key":"3317_CR17","unstructured":"Olsder GJ (1989) Applications of the theory of stochastic discrete-event systems to array processors and scheduling in public transportation. In: Proceedings of the 28th IEEE Conference on Decision and Control"},{"key":"3317_CR18","doi-asserted-by":"crossref","unstructured":"Qi L, Luo Z (2017) Tensor analysis: spectral theory and special tensors. Vol. 151, SIAM","DOI":"10.1137\/1.9781611974751"},{"key":"3317_CR19","doi-asserted-by":"publisher","first-page":"148","DOI":"10.1016\/j.laa.2017.10.001","volume":"537","author":"C Wang","year":"2018","unstructured":"Wang C, Tao Y (2018) Interval strong solutions of interval systems of max-plus linear equations. Linear Algebra Appl 537:148\u2013159","journal-title":"Linear Algebra Appl"},{"key":"3317_CR20","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2019.112569","volume":"368","author":"X Wang","year":"2020","unstructured":"Wang X, Che M, Wei Y (2020) Neural network approach for solving nonsingular multi-linear tensor systems. J Comput Appl Math 368:112569","journal-title":"J Comput Appl Math"},{"key":"3317_CR21","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1016\/j.neucom.2019.03.025","volume":"351","author":"X Wang","year":"2019","unstructured":"Wang X, Che M, Wei Y (2019) Neural networks based approach solving multi-linear systems with $$M$$-tensors. Neurocomputing 351:33\u201342","journal-title":"Neurocomputing"},{"key":"3317_CR22","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1016\/j.neucom.2021.11.108","volume":"472","author":"X Wang","year":"2022","unstructured":"Wang X, Mo C, Qiao S, Wei Y (2022) Predefined-time convergent neural networks for solving the time-varying nonsingular multi-linear tensor equations. Neurocomputing 472:68\u201384","journal-title":"Neurocomputing"},{"key":"3317_CR23","doi-asserted-by":"publisher","first-page":"16079","DOI":"10.1109\/TASE.2025.3575246","volume":"22","author":"X Wang","year":"2025","unstructured":"Wang X, Stanimirovi\u0107 PS, Wei Y (2025) Dynamic approaches for finding least squares solution of time-varying multi-linear systems. IEEE Trans Autom Sci Eng 22:16079\u201316090","journal-title":"IEEE Trans Autom Sci Eng"},{"key":"3317_CR24","unstructured":"Wei Y, Ding W (2016) Theory and Computation of Tensors: Multi-Dimensional Arrays. Academic Press"},{"key":"3317_CR25","doi-asserted-by":"publisher","first-page":"88","DOI":"10.1016\/j.sysconle.2016.07.005","volume":"96","author":"H Zhang","year":"2016","unstructured":"Zhang H, Tao Y, Zhang Z (2016) Strong solvability of interval max-plus systems and applications to optimal control. Syst Control Lett 96:88\u201394","journal-title":"Syst Control Lett"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03317-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03317-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03317-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,9]],"date-time":"2025-09-09T05:04:34Z","timestamp":1757394274000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03317-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,9]]},"references-count":25,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["3317"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03317-5","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2025,7,9]]},"assertion":[{"value":"18 February 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 June 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 June 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 July 2025","order":4,"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 Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"349"}}