{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:12:29Z","timestamp":1776834749575,"version":"3.51.2"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,7,28]],"date-time":"2020-07-28T00:00:00Z","timestamp":1595894400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,7,28]],"date-time":"2020-07-28T00:00:00Z","timestamp":1595894400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100004608","name":"Natural Science Foundation of Jiangsu Province","doi-asserted-by":"publisher","award":["BK20190723"],"award-info":[{"award-number":["BK20190723"]}],"id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"publisher"}]},{"name":"NUPTSF","award":["NY219137"],"award-info":[{"award-number":["NY219137"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61673011"],"award-info":[{"award-number":["61673011"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004608","name":"Natural Science Foundation of Jiangsu Province","doi-asserted-by":"publisher","award":["BK20170916"],"award-info":[{"award-number":["BK20170916"]}],"id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Soft Comput"],"published-print":{"date-parts":[[2021,1]]},"DOI":"10.1007\/s00500-020-05206-x","type":"journal-article","created":{"date-parts":[[2020,7,28]],"date-time":"2020-07-28T20:02:48Z","timestamp":1595966568000},"page":"1099-1112","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Saddle point equilibrium model for uncertain discrete systems"],"prefix":"10.1007","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7880-8742","authenticated-orcid":false,"given":"Yun","family":"Sun","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongyan","family":"Yan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuanguo","family":"Zhu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,7,28]]},"reference":[{"key":"5206_CR1","doi-asserted-by":"crossref","unstructured":"Atashpaz-Gargari E, Lucas C (2007) Imperialist competitive algorithm: an algorithm for optimization inspired by imperialistic competition. In: IEEE Congress on Evolutionary Computation, Singapore","DOI":"10.1109\/CEC.2007.4425083"},{"key":"5206_CR2","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1007\/BFb0040234","volume-title":"Approximation of differential games of pursuit-evasion by discrete-time games. Differential games-developments in modelling and computation","author":"M Bardi","year":"1991","unstructured":"Bardi M, Soravia P (1991) Approximation of differential games of pursuit-evasion by discrete-time games. Differential games-developments in modelling and computation. Springer, Berlin, pp 131\u2013143"},{"issue":"2","key":"5206_CR3","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1016\/0022-0396(84)90040-8","volume":"53","author":"EN Barron","year":"1984","unstructured":"Barron EN, Evans LC, Jensen R (1984) Viscosity solutions of Isaacs\u2019 equations and differential games with Lipschitz controls. J Differ Equ 53(2):213\u2013233","journal-title":"J Differ Equ"},{"issue":"4","key":"5206_CR4","doi-asserted-by":"publisher","first-page":"539","DOI":"10.1007\/BF00935757","volume":"33","author":"T Basar","year":"1981","unstructured":"Basar T (1981) On the saddle-point solution of a class of stochastic differential games. J Optim Theory Appl 33(4):539\u2013556","journal-title":"J Optim Theory Appl"},{"key":"5206_CR5","first-page":"933","volume":"50","author":"RL Bishop","year":"1960","unstructured":"Bishop RL (1960) Duopoly: collusion or warfare. Am Econ Rev 50:933\u2013961","journal-title":"Am Econ Rev"},{"issue":"2","key":"5206_CR6","first-page":"32","volume":"8","author":"X Chen","year":"2011","unstructured":"Chen X (2011) American option pricing formula for uncertain financial market. Int J Oper Res 8(2):32\u201337","journal-title":"Int J Oper Res"},{"issue":"6","key":"5206_CR7","doi-asserted-by":"publisher","first-page":"911","DOI":"10.2307\/1909667","volume":"39","author":"S Clemhout","year":"1971","unstructured":"Clemhout S, Leitmann G, Wan HY Jr (1971) A differential game model of duopoly. Econometrica 39(6):911\u2013938","journal-title":"Econometrica"},{"key":"5206_CR8","doi-asserted-by":"publisher","first-page":"105546","DOI":"10.1016\/j.asoc.2019.105546","volume":"82","author":"O Darwish","year":"2019","unstructured":"Darwish O, Al-Fuqaha A, Brahim GB, Jenhani I, Vasilakos A (2019) Statistical hierarchical analysis with deep neural network-based framework for covert timing channels detection. Appl Soft Comput 82:105546","journal-title":"Appl Soft Comput"},{"key":"5206_CR9","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-019-04042-y","author":"A Dey","year":"2019","unstructured":"Dey A, Miyani G, Sil A (2019) Application of artificial neural network (ANN) for estimating reliable service life of reinforced concrete (RC) structure bookkeeping factors responsible for deterioration mechanism. Soft Comput. https:\/\/doi.org\/10.1007\/s00500-019-04042-y","journal-title":"Soft Comput"},{"issue":"5","key":"5206_CR10","doi-asserted-by":"publisher","first-page":"301","DOI":"10.2514\/3.27758","volume":"10","author":"J Deyst","year":"1973","unstructured":"Deyst J, Price CF (1973) Optimal stochastic guidance laws for tactical missiles. J Spacecr Rock 10(5):301\u2013308","journal-title":"J Spacecr Rock"},{"key":"5206_CR11","volume-title":"Competition among the few","author":"WJ Fellner","year":"1949","unstructured":"Fellner WJ (1949) Competition among the few. Kelley, New York"},{"key":"5206_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-018-03689-3","author":"Y Feng","year":"2018","unstructured":"Feng Y, Dai L, Gao J, Cheng G (2018) Uncertain pursuit-evasion game. Soft Comput. https:\/\/doi.org\/10.1007\/s00500-018-03689-3","journal-title":"Soft Comput"},{"issue":"2","key":"5206_CR13","doi-asserted-by":"publisher","first-page":"496","DOI":"10.1137\/S036301290444280X","volume":"45","author":"S Hamad\u00e8ne","year":"2006","unstructured":"Hamad\u00e8ne S (2006) Mixed zero-sum stochastic differential game and American game options. SIAM J Control Optim 45(2):496\u2013518","journal-title":"SIAM J Control Optim"},{"key":"5206_CR14","volume-title":"Differential games","author":"R Isaacs","year":"1975","unstructured":"Isaacs R (1975) Differential games, 2nd edn. Kruger Publishing Company, Huntington","edition":"2"},{"issue":"8","key":"5206_CR15","first-page":"3229","volume":"15","author":"Y Kang","year":"2012","unstructured":"Kang Y, Zhu Y (2012) Bang-bang optimal control for multi-stage uncertain systems. Inf An Int Interdiscipl J 15(8):3229\u20133237","journal-title":"Inf An Int Interdiscipl J"},{"issue":"3","key":"5206_CR16","doi-asserted-by":"publisher","first-page":"902","DOI":"10.1109\/TAC.2017.2719158","volume":"63","author":"H Kebriaei","year":"2017","unstructured":"Kebriaei H, Iannelli L (2017) Discrete-time robust hierarchical linear\u2013quadratic dynamic games. IEEE Trans Autom Control 63(3):902\u2013909","journal-title":"IEEE Trans Autom Control"},{"key":"5206_CR17","doi-asserted-by":"crossref","unstructured":"Kifer Y (2000) Game options. Finance Stochast 4(4):443\u2013463","DOI":"10.1007\/PL00013527"},{"key":"5206_CR18","doi-asserted-by":"crossref","unstructured":"Liu B (2013) Polyrectangular theorem and independence of uncertain vectors. J Uncertain Anal Appl 1. Article 9","DOI":"10.1186\/2195-5468-1-9"},{"key":"5206_CR19","volume-title":"Uncertainty theory","author":"B Liu","year":"2007","unstructured":"Liu B (2007) Uncertainty theory, 2nd edn. Springer, Berlin","edition":"2"},{"key":"5206_CR20","unstructured":"Liu B (2008) Fuzzy process, hybrid process and uncertain process. J Uncertain Syst 2(1):3\u201316"},{"issue":"1","key":"5206_CR21","first-page":"3","volume":"3","author":"B Liu","year":"2009","unstructured":"Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3\u201310","journal-title":"J Uncertain Syst"},{"key":"5206_CR22","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-89484-1","volume-title":"Theory and practice of uncertain programming","author":"B Liu","year":"2009","unstructured":"Liu B (2009) Theory and practice of uncertain programming, 2nd edn. Springer, Berlin","edition":"2"},{"key":"5206_CR23","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-13959-8","volume-title":"Uncertainty theory: a branch of mathematics for modeling human uncertainty","author":"B Liu","year":"2010","unstructured":"Liu B (2010) Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer, Berlin"},{"issue":"1","key":"5206_CR24","first-page":"3","volume":"6","author":"B Liu","year":"2012","unstructured":"Liu B (2012) Why is there a need for uncertainty theory? J Uncertain Syst 6(1):3\u201310","journal-title":"J Uncertain Syst"},{"key":"5206_CR25","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-44354-5","volume-title":"Uncertainty theory","author":"B Liu","year":"2015","unstructured":"Liu B (2015) Uncertainty theory, 4th edn. Springer, Berlin","edition":"4"},{"issue":"4","key":"5206_CR26","doi-asserted-by":"publisher","first-page":"e12410","DOI":"10.1111\/exsy.12410","volume":"36","author":"H Naderpour","year":"2019","unstructured":"Naderpour H, Nagai K, Haji M, Mirrashid M (2019) Adaptive neuro-fuzzy inference modelling and sensitivity analysis for capacity estimation of fiber reinforced polymer-strengthened circular reinforced concrete columns. Expert Syst 36(4):e12410","journal-title":"Expert Syst"},{"issue":"2","key":"5206_CR27","first-page":"18","volume":"8","author":"J Peng","year":"2011","unstructured":"Peng J, Yao K (2011) A new option pricing model for stocks in uncertainty markets. Int J Oper Res 8(2):18\u201326","journal-title":"Int J Oper Res"},{"issue":"Suppl. 1","key":"5206_CR28","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1142\/S0218488513400060","volume":"21","author":"L Sheng","year":"2013","unstructured":"Sheng L, Zhu Y (2013) Optimistic value model of uncertain optimal control. Int J Uncertain Fuzziness Knowl-Based Syst 21(Suppl. 1):75\u201387","journal-title":"Int J Uncertain Fuzziness Knowl-Based Syst"},{"issue":"6","key":"5206_CR29","doi-asserted-by":"publisher","first-page":"4082","DOI":"10.1137\/140953642","volume":"52","author":"J Sun","year":"2014","unstructured":"Sun J, Yong J (2014) Linear quadratic stochastic differential games: open-loop and closed-loop saddle points. SIAM J Control Optim 52(6):4082\u20134121","journal-title":"SIAM J Control Optim"},{"key":"5206_CR30","doi-asserted-by":"publisher","first-page":"605","DOI":"10.1007\/s10845-014-1003-7","volume":"28","author":"Y Sun","year":"2017","unstructured":"Sun Y, Zhu Y (2017) Bang\u2013bang property for an uncertain saddle point problem. J Intell Manuf 28:605\u2013613","journal-title":"J Intell Manuf"},{"key":"5206_CR31","doi-asserted-by":"publisher","first-page":"2063","DOI":"10.1002\/mma.4733","volume":"41","author":"Y Sun","year":"2018","unstructured":"Sun Y, Ding C, Zhu Y (2018) Saddle point equilibrium under uncertain environment. Math Methods Appl Sci 41:2063\u20132073","journal-title":"Math Methods Appl Sci"},{"issue":"1","key":"5206_CR32","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1137\/0307011","volume":"7","author":"P Varaiya","year":"1969","unstructured":"Varaiya P, Lin J (1969) Existence of saddle points in differential games. SIAM J Control 7(1):141\u2013157","journal-title":"SIAM J Control"},{"issue":"3","key":"5206_CR33","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1007\/BF00930575","volume":"4","author":"G Wang","year":"1969","unstructured":"Wang G, Leitmann G (1969) Necessary and sufficient conditions for multistage two-person zero-sum games. J Optim Theory Appl 4(3):145\u2013155","journal-title":"J Optim Theory Appl"},{"key":"5206_CR34","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1016\/j.orl.2013.02.008","volume":"41","author":"J Wang","year":"2010","unstructured":"Wang J, Wang P (2010) Counterterror measures and economic growth: a differential game. Oper Res Lett 41:285\u2013289","journal-title":"Oper Res Lett"},{"issue":"4","key":"5206_CR35","doi-asserted-by":"publisher","first-page":"957","DOI":"10.1109\/TNNLS.2016.2638863","volume":"29","author":"Q Wei","year":"2017","unstructured":"Wei Q, Liu D, Lin Q, Song R (2017) Adaptive dynamic programming for discrete-time zero-sum games. IEEE Trans Neural Netw Learn Syst 29(4):957\u2013969","journal-title":"IEEE Trans Neural Netw Learn Syst"},{"key":"5206_CR36","doi-asserted-by":"publisher","first-page":"527","DOI":"10.1007\/s10845-014-0996-2","volume":"28","author":"H Yan","year":"2017","unstructured":"Yan H, Zhu Y (2017) Bang-bang control model with optimistic value criterion for uncertain switched systems. J Intell Manuf 28:527\u2013534","journal-title":"J Intell Manuf"},{"key":"5206_CR37","doi-asserted-by":"crossref","first-page":"267","DOI":"10.3934\/jimo.2016016","volume":"13","author":"H Yan","year":"2017","unstructured":"Yan H, Sun Y, Zhu Y (2017) A linear\u2013quadratic control problem of uncertain discrete-time switched systems. J Ind Manag Optim 13:267\u2013282","journal-title":"J Ind Manag Optim"},{"issue":"4","key":"5206_CR38","doi-asserted-by":"publisher","first-page":"819","DOI":"10.1109\/TFUZZ.2015.2486809","volume":"24","author":"X Yang","year":"2015","unstructured":"Yang X, Gao J (2015) Linear\u2013quadratic uncertain differential game with application to resource extraction problem. IEEE Trans Fuzzy Syst 24(4):819\u2013826","journal-title":"IEEE Trans Fuzzy Syst"},{"key":"5206_CR39","doi-asserted-by":"crossref","unstructured":"Yang X, Gao J (2013) Uncertain differential games with application to capitalism. J Uncertain Anal Appl 1. Article 17","DOI":"10.1186\/2195-5468-1-17"},{"issue":"7","key":"5206_CR40","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1080\/01969722.2010.511552","volume":"41","author":"Y Zhu","year":"2010","unstructured":"Zhu Y (2010) Uncertain optimal control with application to a portfolio selection model. Cybern Syst 41(7):535\u2013547","journal-title":"Cybern Syst"},{"issue":"4","key":"5206_CR41","first-page":"278","volume":"6","author":"Y Zhu","year":"2012","unstructured":"Zhu Y (2012) Functions of uncertain variables and uncertain programming. J Uncertain Syst 6(4):278\u2013288","journal-title":"J Uncertain Syst"}],"container-title":["Soft Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-020-05206-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00500-020-05206-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-020-05206-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,4]],"date-time":"2023-10-04T23:48:24Z","timestamp":1696463304000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00500-020-05206-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,28]]},"references-count":41,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["5206"],"URL":"https:\/\/doi.org\/10.1007\/s00500-020-05206-x","relation":{},"ISSN":["1432-7643","1433-7479"],"issn-type":[{"value":"1432-7643","type":"print"},{"value":"1433-7479","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,28]]},"assertion":[{"value":"28 July 2020","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with ethical standards"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"This article does not contain any studies with human participants or animals performed by any of the authors.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}}]}}