{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,22]],"date-time":"2026-07-22T10:43:10Z","timestamp":1784716990125,"version":"3.55.0"},"reference-count":19,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T00:00:00Z","timestamp":1767916800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T00:00:00Z","timestamp":1767916800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key R&D Program of China","doi-asserted-by":"crossref","award":["2023YFA1011302"],"award-info":[{"award-number":["2023YFA1011302"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12171063"],"award-info":[{"award-number":["12171063"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the Project for Creative Research Groups at Institutions of Higher Education in Chongqing","award":["CXQT20014"],"award-info":[{"award-number":["CXQT20014"]}]},{"DOI":"10.13039\/501100005230","name":"Natural Science Foundation of Chongqing","doi-asserted-by":"crossref","award":["CSTB2025NSCQ-GPX0994"],"award-info":[{"award-number":["CSTB2025NSCQ-GPX0994"]}],"id":[{"id":"10.13039\/501100005230","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Science and Technology Research Program of Chongqing Municipal Education Commission","award":["KJQN202500510"],"award-info":[{"award-number":["KJQN202500510"]}]},{"name":"Science and Technology Research Program of Chongqing Municipal Education Commission","award":["KJZD-K202300509"],"award-info":[{"award-number":["KJZD-K202300509"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2026,2]]},"DOI":"10.1007\/s10957-025-02907-x","type":"journal-article","created":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T06:51:19Z","timestamp":1767941479000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Pareto Optimality of Critical State Vectors in Vector Sets with State Attributes"],"prefix":"10.1007","volume":"208","author":[{"given":"Yuanmei","family":"Xia","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hongsheng","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-6709-7011","authenticated-orcid":false,"given":"Kequan","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,1,9]]},"reference":[{"key":"2907_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-0685-4","volume-title":"Reliability Evaluation of Engineering Systems: Concepts and Techniques","author":"R Billinton","year":"1992","unstructured":"Billinton, R., Allan, R.N.: Reliability Evaluation of Engineering Systems: Concepts and Techniques, 2nd edn. Springer, New York (1992)","edition":"2"},{"key":"2907_CR2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-1860-4","volume-title":"Reliability Evaluation of Power Systems","author":"R Billinton","year":"1996","unstructured":"Billinton, R., Allan, R.N.: Reliability Evaluation of Power Systems, 2nd edn. Plenum Press, New York (1996)","edition":"2"},{"issue":"3","key":"2907_CR3","doi-asserted-by":"publisher","first-page":"1238","DOI":"10.1109\/59.32623","volume":"4","author":"R Billinton","year":"1989","unstructured":"Billinton, R., Kumar, S., Chowdhury, N., Chu, K., Debnath, K., Goel, L., Khan, E., Kos, P., Nourbakhsh, G., Oteng-Adjei, J.: A reliability test system for educational purposes-basic data. IEEE Trans. Power Syst. 4(3), 1238\u20131244 (1989)","journal-title":"IEEE Trans. Power Syst."},{"issue":"3","key":"2907_CR4","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1016\/S0378-7796(98)00064-9","volume":"47","author":"R Billinton","year":"1998","unstructured":"Billinton, R., Zhang, W.: State extension in adequacy evaluation of composite power systems-concept and algorithm. Electr. Power Syst. Res. 47(3), 189\u2013195 (1998)","journal-title":"Electr. Power Syst. Res."},{"issue":"3","key":"2907_CR5","doi-asserted-by":"publisher","first-page":"467","DOI":"10.1287\/moor.8.3.467","volume":"8","author":"EU Choo","year":"1983","unstructured":"Choo, E.U., Atkins, D.R.: Proper efficiency in nonconvex multicriteria programming. Math. Oper. Res. 8(3), 467\u2013470 (1983)","journal-title":"Math. Oper. Res."},{"key":"2907_CR6","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511809088","volume-title":"Introduction to Lattices and Order","author":"BA Davey","year":"2002","unstructured":"Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, New York (2002)","edition":"2"},{"key":"2907_CR7","volume-title":"Multicriteria Optimization","author":"M Ehrgott","year":"2005","unstructured":"Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)"},{"issue":"3","key":"2907_CR8","doi-asserted-by":"publisher","first-page":"618","DOI":"10.1016\/0022-247X(68)90201-1","volume":"22","author":"AM Geoffrion","year":"1968","unstructured":"Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22(3), 618\u2013630 (1968)","journal-title":"J. Math. Anal. Appl."},{"key":"2907_CR9","volume-title":"Efficient Theory of Multiobjective Programming","author":"YD Hu","year":"1994","unstructured":"Hu, Y.D.: Efficient Theory of Multiobjective Programming. Shanghai Science and Technology Press, Shanghai (1994)"},{"key":"2907_CR10","unstructured":"Koopmans, T.C.: Analysis of production as an efficient combination of activities. In: Koopmans, T.C., editor, Activity Analysis of Production and Allocation. Cowles Commission Monograph\u2013Volume 13, pp. 33\u201397. John Wiley and Sons, New York (1951)"},{"key":"2907_CR11","doi-asserted-by":"crossref","unstructured":"Kuhn, H.W., Tucker, A.W.: Nonlinear Programming. In: Neyman, J., (ed.), Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability\u2013Volume 2, pp. 481\u2013492. University of California Press, Berkeley (1951)","DOI":"10.1525\/9780520411586-036"},{"key":"2907_CR12","volume-title":"Risk Assessment of Power Systems: Models, Methods, and Applications","author":"WY Li","year":"2005","unstructured":"Li, W.Y.: Risk Assessment of Power Systems: Models, Methods, and Applications. Wiley-IEEE Press, Hoboken (2005)"},{"issue":"6","key":"2907_CR13","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1016\/S0893-9659(99)00087-7","volume":"12","author":"J-C Liu","year":"1999","unstructured":"Liu, J.-C.: $$\\varvec {\\epsilon }$$-properly efficient solutions to nondifferentiable multiobjective programming problems. Appl. Math. Lett. 12(6), 109\u2013113 (1999)","journal-title":"Appl. Math. Lett."},{"issue":"1","key":"2907_CR14","doi-asserted-by":"publisher","first-page":"1178","DOI":"10.1109\/TPWRS.2023.3258319","volume":"39","author":"ZY Liu","year":"2024","unstructured":"Liu, Z.Y., Tang, P.T., Hou, K., Zhu, L.W., Zhao, J.B., Jia, H.J., Pei, W.: A Lagrange-multiplier-based reliability assessment for power systems considering topology and injection uncertainties. IEEE Trans. Power Syst. 39(1), 1178\u20131189 (2024)","journal-title":"IEEE Trans. Power Syst."},{"issue":"2","key":"2907_CR15","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1007\/BF00936165","volume":"43","author":"P Loridan","year":"1984","unstructured":"Loridan, P.: $$\\varvec {\\epsilon }$$-solutions in vector minimization problems. J. Optim. Theory Appl. 43(2), 265\u2013276 (1984)","journal-title":"J. Optim. Theory Appl."},{"key":"2907_CR16","doi-asserted-by":"crossref","unstructured":"Luc, D.T.: Pareto optimality. In: Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L., (ed.), Pareto Optimality, Game Theory and Equilibria. Springer Optimization and Its Applications\u2013Volume 17, pp. 481\u2013515. Springer, New York (2008)","DOI":"10.1007\/978-0-387-77247-9_18"},{"key":"2907_CR17","volume-title":"Manuale di Economia Politica","author":"V Pareto","year":"1906","unstructured":"Pareto, V.: Manuale di Economia Politica. Societ\u00e0 Editrice Libraria, Milano (1906)"},{"issue":"2","key":"2907_CR18","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1360\/SSM-2023-0047","volume":"54","author":"YM Xia","year":"2024","unstructured":"Xia, Y.M., Zhao, K.Q., Gao, Y.: Generalized Pascoletti-Serafini scalarization for multi-objective optimization. Sci. Sin. Math. 54(2), 231\u2013244 (2024)","journal-title":"Sci. Sin. Math."},{"issue":"2","key":"2907_CR19","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1360\/SCM-2019-0004","volume":"51","author":"KQ Zhao","year":"2021","unstructured":"Zhao, K.Q., Yang, X.M., Xia, Y.M.: Complete scalarization of multi-objective optimization problems. Sci. Sin. Math. 51(2), 411\u2013424 (2021)","journal-title":"Sci. Sin. Math."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02907-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-025-02907-x","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02907-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T04:53:05Z","timestamp":1776747185000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-025-02907-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,9]]},"references-count":19,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,2]]}},"alternative-id":["2907"],"URL":"https:\/\/doi.org\/10.1007\/s10957-025-02907-x","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,9]]},"assertion":[{"value":"19 December 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 December 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 January 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"This study does not have any conflicts to disclose.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"77"}}