{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T15:02:21Z","timestamp":1775228541119,"version":"3.50.1"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,11,9]],"date-time":"2024-11-09T00:00:00Z","timestamp":1731110400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,11,9]],"date-time":"2024-11-09T00:00:00Z","timestamp":1731110400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["DP21010022"],"award-info":[{"award-number":["DP21010022"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,2]]},"DOI":"10.1007\/s40314-024-02992-0","type":"journal-article","created":{"date-parts":[[2024,11,9]],"date-time":"2024-11-09T03:21:24Z","timestamp":1731122484000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Efficient methods for verifying monotonicity of 2-additive fuzzy measures"],"prefix":"10.1007","volume":"44","author":[{"given":"Gleb","family":"Beliakov","sequence":"first","affiliation":[]},{"given":"Chaquen","family":"Beliakov Amaya","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,11,9]]},"reference":[{"key":"2992_CR1","doi-asserted-by":"crossref","unstructured":"Anderson DT, Keller JM, Havens TC (2010) Learning fuzzy-valued fuzzy measures for the fuzzy-valued Sugeno fuzzy integral. In Computational Intelligence for Knowledge-Based Systems Design: 13th International Conference on Information Processing and Management of Uncertainty, IPMU 2010, Dortmund, Germany, June 28-July 2, 2010. Proceedings 13, pages 502\u2013511. Springer","DOI":"10.1007\/978-3-642-14049-5_52"},{"issue":"4","key":"2992_CR2","doi-asserted-by":"publisher","first-page":"469","DOI":"10.1145\/235815.235821","volume":"22","author":"CB Barber","year":"1996","unstructured":"Barber CB, Dobkin DP, Huhdanpaa H (1996) The quickhull algorithm for convex hulls. ACM Trans Math Softw 22(4):469\u2013483","journal-title":"ACM Trans Math Softw"},{"issue":"1","key":"2992_CR3","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1016\/j.fss.2008.07.004","volume":"160","author":"G Beliakov","year":"2009","unstructured":"Beliakov G (2009) Construction of aggregation functions from data using linear programming. Fuzzy Sets Syst 160(1):65\u201375","journal-title":"Fuzzy Sets Syst"},{"key":"2992_CR4","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1016\/j.ejor.2021.11.004","volume":"301","author":"G Beliakov","year":"2021","unstructured":"Beliakov G (2021) Knapsack problems with dependencies through non-additive measures and Choquet integral. Eur J Oper Res 301:277\u2013286","journal-title":"Eur J Oper Res"},{"key":"2992_CR5","doi-asserted-by":"publisher","first-page":"469","DOI":"10.1016\/j.ins.2020.08.113","volume":"546","author":"G Beliakov","year":"2021","unstructured":"Beliakov G (2021) On the derivatives of set functions in matrix representation. Inf Sci 546:469\u2013480","journal-title":"Inf Sci"},{"key":"2992_CR6","unstructured":"Beliakov G (2021) fmtools package, version 4, https:\/\/personal-sites.deakin.edu.au\/~gleb\/fmtools.html"},{"key":"2992_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.knosys.2019.105134","volume":"189","author":"G Beliakov","year":"2020","unstructured":"Beliakov G, Divakov D (2020) On representation of fuzzy measures for learning Choquet and Sugeno integrals. Knowl-Based Syst 189:1\u20135","journal-title":"Knowl-Based Syst"},{"key":"2992_CR8","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1016\/j.ins.2019.11.024","volume":"514","author":"G Beliakov","year":"2020","unstructured":"Beliakov G, Gagolewski M, James S (2020) Robust fitting for the Sugeno integral with respect to general fuzzy measures. Inf Sci 514:449\u2013461","journal-title":"Inf Sci"},{"key":"2992_CR9","volume-title":"Discrete fuzzy measures: computational aspects","author":"G Beliakov","year":"2019","unstructured":"Beliakov G, James S, Wu J-Z (2019) Discrete fuzzy measures: computational aspects. Springer, Berlin"},{"key":"2992_CR10","doi-asserted-by":"publisher","first-page":"100","DOI":"10.1016\/j.ins.2019.04.042","volume":"494","author":"G Beliakov","year":"2019","unstructured":"Beliakov G, Wu J-Z (2019) Learning fuzzy measures from data: simplifications and optimisation strategies. Inf Sci 494:100\u2013113","journal-title":"Inf Sci"},{"key":"2992_CR11","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1016\/j.ins.2021.07.017","volume":"577","author":"G Beliakov","year":"2021","unstructured":"Beliakov G, Wu J-Z (2021) Random generation of capacities and its application in comprehensive decision aiding. Inf Sci 577:424\u2013435","journal-title":"Inf Sci"},{"key":"2992_CR12","doi-asserted-by":"crossref","unstructured":"Beliakov G, Wu J-Z, Ding W (2024) Representation, optimization and generation of fuzzy measures. Inform Fus pp 102295","DOI":"10.1016\/j.inffus.2024.102295"},{"key":"2992_CR13","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511804441","volume-title":"Convex optimization","author":"S Boyd","year":"2004","unstructured":"Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge"},{"issue":"2","key":"2992_CR14","doi-asserted-by":"publisher","first-page":"524","DOI":"10.1007\/s10957-012-0115-5","volume":"156","author":"A Causa","year":"2013","unstructured":"Causa A, Raciti F (2013) A purely geometric approach to the problem of computing the projection of a point on a simplex. J Optim Theory Appl 156(2):524\u2013528","journal-title":"J Optim Theory Appl"},{"key":"2992_CR15","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-031-01558-8","volume-title":"Computational aspects of cooperative game theory","author":"G Chalkiadakis","year":"2012","unstructured":"Chalkiadakis G, Elkind E, Wooldridge M (2012) Computational aspects of cooperative game theory. Morgan and Claypool Publishers, San Rafael"},{"issue":"3","key":"2992_CR16","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1016\/0165-4896(89)90056-5","volume":"17","author":"A Chateauneuf","year":"1989","unstructured":"Chateauneuf A, Jaffray J-Y (1989) Some characterizations of lower probabilities and other monotone capacities through the use of M\u00f6bius inversion. Math Soc Sci 17(3):263\u2013283","journal-title":"Math Soc Sci"},{"key":"2992_CR17","doi-asserted-by":"publisher","first-page":"131","DOI":"10.5802\/aif.53","volume":"5","author":"G Choquet","year":"1954","unstructured":"Choquet G (1954) Theory of capacities. Annales de l\u2019institut Fourier 5:131\u2013295","journal-title":"Annales de l\u2019institut Fourier"},{"key":"2992_CR18","doi-asserted-by":"publisher","first-page":"50","DOI":"10.1016\/j.ins.2019.05.079","volume":"501","author":"EF Combarro","year":"2019","unstructured":"Combarro EF, de Saracho JH, D\u00edaz I (2019) Minimals plus: an improved algorithm for the random generation of linear extensions of partially ordered sets. Inf Sci 501:50\u201367","journal-title":"Inf Sci"},{"issue":"10","key":"2992_CR19","doi-asserted-by":"publisher","first-page":"3046","DOI":"10.1016\/j.cor.2005.02.034","volume":"33","author":"EF Combarro","year":"2006","unstructured":"Combarro EF, Miranda P (2006) Identification of fuzzy measures from sample data with genetic algorithms. Comput Oper Res 33(10):3046\u20133066","journal-title":"Comput Oper Res"},{"key":"2992_CR20","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1007\/978-3-319-03155-2_6","volume-title":"Non-additive measures","author":"K Fujimoto","year":"2014","unstructured":"Fujimoto K (2014) Cooperative game as non-additive measure. In: Torra V, Narukawa Y, Sugeno M (eds) Non-additive measures. Springer, Berlin, New York, pp 131\u2013171"},{"key":"2992_CR21","doi-asserted-by":"publisher","unstructured":"Fukuda K (2020) Polyhedral computation ETH Zurich. https:\/\/doi.org\/10.3929\/ethz-b-000426218","DOI":"10.3929\/ethz-b-000426218"},{"key":"2992_CR22","doi-asserted-by":"crossref","unstructured":"Grabisch M (1995) A new algorithm for identifying fuzzy measures and its application to pattern recognition. In Fuzzy Systems, 1995. International Joint Conference of the Fourth IEEE International Conference on Fuzzy Systems and The Second International Fuzzy Engineering Symposium., Proceedings of 1995 IEEE Int, pages 145\u2013150. IEEE","DOI":"10.1109\/FUZZY.1995.409673"},{"issue":"6","key":"2992_CR23","doi-asserted-by":"publisher","first-page":"567","DOI":"10.1016\/0167-8655(96)00020-7","volume":"17","author":"M Grabisch","year":"1996","unstructured":"Grabisch M (1996) The representation of importance and interaction of features by fuzzy measures. Pattern Recogn Lett 17(6):567\u2013575","journal-title":"Pattern Recogn Lett"},{"issue":"2","key":"2992_CR24","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1016\/S0165-0114(97)00168-1","volume":"92","author":"M Grabisch","year":"1997","unstructured":"Grabisch M (1997) k-order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst 92(2):167\u2013189","journal-title":"Fuzzy Sets Syst"},{"key":"2992_CR25","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-30690-2","volume-title":"Set functions. Games and capacities in decision making","author":"M Grabisch","year":"2016","unstructured":"Grabisch M (2016) Set functions. Games and capacities in decision making. Springer, Berlin"},{"issue":"2","key":"2992_CR26","doi-asserted-by":"publisher","first-page":"766","DOI":"10.1016\/j.ejor.2007.02.025","volume":"186","author":"M Grabisch","year":"2008","unstructured":"Grabisch M, Kojadinovic I, Meyer P (2008) A review of methods for capacity identification in Choquet integral based multi-attribute utility theory: Applications of the Kappalab R package. Eur J Oper Res 186(2):766\u2013785","journal-title":"Eur J Oper Res"},{"key":"2992_CR27","doi-asserted-by":"publisher","unstructured":"Grabisch M, Labreuche C, Sun P (2023) An approximation algorithm for random generation of capacities. Order. https:\/\/doi.org\/10.1007\/s11083-023-09630-0","DOI":"10.1007\/s11083-023-09630-0"},{"key":"2992_CR28","doi-asserted-by":"crossref","unstructured":"H\u00fcllermeier E, Tehrani AF (2013) Efficient learning of classifiers based on the 2-additive Choquet integral. In: A MCN (ed) Computational intelligence in intelligent data analysis. Springer, Berlin, pp 17\u201329","DOI":"10.1007\/978-3-642-32378-2_2"},{"key":"2992_CR29","doi-asserted-by":"crossref","unstructured":"Lust T, Rolland A (2014) 2-additive Choquet optimal solutions in multiobjective optimization problems. In: Laurent A et al. (Eds) IPMU 2014, Part I, CCIS 442, pages 256\u2013265","DOI":"10.1007\/978-3-319-08795-5_27"},{"issue":"5","key":"2992_CR30","doi-asserted-by":"publisher","first-page":"730","DOI":"10.1134\/S0965542516050146","volume":"56","author":"VN Malozemov","year":"2016","unstructured":"Malozemov VN, Tamasyan GS (2016) Two fast algorithms for projecting a point onto the canonical simplex. Comput Math Math Phys 56(5):730\u2013743","journal-title":"Comput Math Math Phys"},{"key":"2992_CR31","doi-asserted-by":"publisher","first-page":"308","DOI":"10.1016\/j.ejor.2019.10.036","volume":"283","author":"B Mayag","year":"2019","unstructured":"Mayag B, Bouyssou D (2019) Necessary and possible interaction between criteria in a 2-additive Choquet integral model. Eur J Oper Res 283:308\u2013320","journal-title":"Eur J Oper Res"},{"issue":"3","key":"2992_CR32","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1007\/s11238-010-9198-3","volume":"71","author":"B Mayag","year":"2011","unstructured":"Mayag B, Grabisch M, Labreuche C (2011) A representation of preferences by the Choquet integral with respect to a 2-additive capacity. Theor Decis 71(3):297\u2013324","journal-title":"Theor Decis"},{"issue":"2","key":"2992_CR33","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1016\/S0165-0114(98)00429-1","volume":"107","author":"L Mikenina","year":"1999","unstructured":"Mikenina L, Zimmermann H-J (1999) Improved feature selection and classification by the 2-additive fuzzy measure. Fuzzy Sets Syst 107(2):197\u2013218","journal-title":"Fuzzy Sets Syst"},{"key":"2992_CR34","doi-asserted-by":"publisher","first-page":"2864","DOI":"10.1109\/TFUZZ.2019.2945243","volume":"28","author":"P Miranda","year":"2020","unstructured":"Miranda P, Garc\u00eda-Segador P (2020) Combinatorial structure of the polytope of 2-additive measures. IEEE Trans Fuzzy Syst 28:2864\u20132874","journal-title":"IEEE Trans Fuzzy Syst"},{"key":"2992_CR35","doi-asserted-by":"publisher","DOI":"10.1002\/9780470317013","volume-title":"Spatial tessellations: concepts and applications of voronoi diagrams. Wiley Series in Probability and Statistics","author":"A Okabe","year":"2000","unstructured":"Okabe A, Boots B, Sugihara K, Chiu S (2000) Spatial tessellations: concepts and applications of voronoi diagrams. Wiley Series in Probability and Statistics. Wiley, New York"},{"key":"2992_CR36","volume-title":"Game theory","author":"G Owen","year":"1982","unstructured":"Owen G (1982) Game theory. Academic Press, London"},{"key":"2992_CR37","doi-asserted-by":"publisher","first-page":"945","DOI":"10.1016\/j.ejor.2019.10.005","volume":"282","author":"GD Pelegrina","year":"2020","unstructured":"Pelegrina GD, Duarte LT, Grabisch M, Romano JMT (2020) The multilinear model in multicriteria decision making: the case of 2-additive capacities and contributions to parameter identification. Eur J Oper Res 282:945\u2013956","journal-title":"Eur J Oper Res"},{"key":"2992_CR38","doi-asserted-by":"publisher","DOI":"10.1017\/9781009257794","volume-title":"Polytopes and graphs. Cambridge studies in advanced mathematics","author":"G Pineda Villavicencio","year":"2024","unstructured":"Pineda Villavicencio G (2024) Polytopes and graphs. Cambridge studies in advanced mathematics. Cambridge University Press, Cambridge"},{"issue":"2","key":"2992_CR39","doi-asserted-by":"publisher","first-page":"288","DOI":"10.1134\/S1990478916020137","volume":"10","author":"GS Tamasyan","year":"2016","unstructured":"Tamasyan GS, Prosolupov EV, Angelov TA (2016) Comparative study of two fast algorithms for projecting a point to the standard simplex. J Appl Ind Math 10(2):288\u2013301","journal-title":"J Appl Ind Math"},{"key":"2992_CR40","doi-asserted-by":"publisher","first-page":"128","DOI":"10.1007\/BF01580381","volume":"11","author":"P Wolfe","year":"1976","unstructured":"Wolfe P (1976) Finding the nearest point in a polytope. Math Progr 11:128\u2013149","journal-title":"Math Progr"},{"key":"2992_CR41","doi-asserted-by":"publisher","first-page":"1732","DOI":"10.1002\/int.22119","volume":"34","author":"J-Z Wu","year":"2019","unstructured":"Wu J-Z, Beliakov G (2019) Nonadditive robust ordinal regression with nonadditivity index and multiple goal linear programming. Int J Intell Syst 34:1732\u20131752","journal-title":"Int J Intell Syst"},{"key":"2992_CR42","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2023.120080","volume":"659","author":"J-Z Wu","year":"2024","unstructured":"Wu J-Z, Beliakov G, James S, Gagolewski M (2024) Random generation of linearly constrained fuzzy measures and domain coverage performance evaluation. Inf Sci 659:120080","journal-title":"Inf Sci"},{"issue":"6","key":"2992_CR43","doi-asserted-by":"publisher","first-page":"2094","DOI":"10.1109\/TFUZZ.2015.2403851","volume":"23","author":"J-Z Wu","year":"2015","unstructured":"Wu J-Z, Yang S, Zhang Q, Ding S (2015) 2-additive capacity identification methods from multicriteria correlation preference information. IEEE Trans Fuzzy Syst 23(6):2094\u20132106","journal-title":"IEEE Trans Fuzzy Syst"},{"key":"2992_CR44","doi-asserted-by":"publisher","first-page":"185543","DOI":"10.1109\/ACCESS.2019.2960539","volume":"7","author":"R-J Xi","year":"2019","unstructured":"Xi R-J, Wu Z-J, Beliakov G (2019) Matrix representation of capacity-based multicriteria decision analysis. IEEE Access 7:185543\u2013185553","journal-title":"IEEE Access"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-024-02992-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-024-02992-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-024-02992-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,4]],"date-time":"2025-03-04T05:16:12Z","timestamp":1741065372000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-024-02992-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,9]]},"references-count":44,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,2]]}},"alternative-id":["2992"],"URL":"https:\/\/doi.org\/10.1007\/s40314-024-02992-0","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,9]]},"assertion":[{"value":"1 April 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 August 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 October 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 November 2024","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 there is no Conflict of interest in this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest."}}],"article-number":"25"}}