{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:16:06Z","timestamp":1760242566901,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2017,11,17]],"date-time":"2017-11-17T00:00:00Z","timestamp":1510876800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"2017 Special Project of Cultivation and Development of  Innovation Base","award":["Z171100002217024"],"award-info":[{"award-number":["Z171100002217024"]}]},{"name":"Research Funds of Beijing Social Science","award":["16GLC069"],"award-info":[{"award-number":["16GLC069"]}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["2016MS70","2017XS098","2017XS099"],"award-info":[{"award-number":["2016MS70","2017XS098","2017XS099"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Traditional stochastic dominance rules are so strict and qualitative conditions that generally a stochastic dominance relation between two alternatives does not exist. To solve the problem, we firstly supplement the definitions of almost stochastic dominance (ASD). Then, we propose a new definition of stochastic dominance degree (SDD) that is based on the idea of ASD. The new definition takes both the objective mean and stakeholders\u2019 subjective preference into account, and can measure both standard and almost stochastic dominance degree. The new definition contains four kinds of SDD corresponding to different stakeholders (rational investors, risk averters, risk seekers, and prospect investors). The operator in the definition can also be changed to fit in with different circumstances. On the basis of the new SDD definition, we present a method to solve stochastic multiple criteria decision-making problem. The numerical experiment shows that the new method could produce a more accurate result according to the utility situations of stakeholders. Moreover, even when it is difficult to elicit the group utility distribution of stakeholders, or when the group utility distribution is ambiguous, the method can still rank alternatives.<\/jats:p>","DOI":"10.3390\/e19110606","type":"journal-article","created":{"date-parts":[[2017,11,17]],"date-time":"2017-11-17T12:00:20Z","timestamp":1510920020000},"page":"606","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A New Stochastic Dominance Degree Based on Almost Stochastic Dominance and Its Application in Decision Making"],"prefix":"10.3390","volume":"19","author":[{"given":"Yunna","family":"Wu","sequence":"first","affiliation":[{"name":"School of Economics and Management, North China Electric Power University, Beijing 102206, China"},{"name":"Beijing Key Laboratory of New Energy and Low-Carbon Development, North China Electric Power University, Changping, Beijing 102206, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9994-7493","authenticated-orcid":false,"given":"Xiaokun","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Economics and Management, North China Electric Power University, Beijing 102206, China"},{"name":"Beijing Key Laboratory of New Energy and Low-Carbon Development, North China Electric Power University, Changping, Beijing 102206, China"}]},{"given":"Hu","family":"Xu","sequence":"additional","affiliation":[{"name":"China Development Bank, Guizhou Branch, Guizhou 550000, China"}]},{"given":"Chuanbo","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Economics and Management, North China Electric Power University, Beijing 102206, China"},{"name":"Beijing Key Laboratory of New Energy and Low-Carbon Development, North China Electric Power University, Changping, Beijing 102206, China"}]},{"given":"Ruhang","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Economics and Management, North China Electric Power University, Beijing 102206, China"},{"name":"Beijing Key Laboratory of New Energy and Low-Carbon Development, North China Electric Power University, Changping, Beijing 102206, China"}]}],"member":"1968","published-online":{"date-parts":[[2017,11,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.knosys.2014.04.024","article-title":"Probability weighted means as surrogates for stochastic dominance in decision making","volume":"66","author":"Yager","year":"2014","journal-title":"Knowl. Based Syst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/j.ejor.2013.09.013","article-title":"Decision making with imprecise probabilities and utilities by means of statistical preference and stochastic dominance","volume":"234","author":"Montes","year":"2014","journal-title":"Eur. J. Oper. Res."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Wu, Y., Xu, C., and Xu, H. (2016). Optimal site selection of tidal power plants using a novel method: A case in China. Energies, 9.","DOI":"10.3390\/en9100832"},{"key":"ref_4","first-page":"1","article-title":"An almost stochastic dominance based method for stochastic multiple attributes decision making","volume":"11","author":"Wu","year":"2017","journal-title":"Intell. Decis.Technol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1016\/j.omega.2007.03.001","article-title":"Probabilistic dominance criteria for comparing uncertain alternatives: A tutorial","volume":"37","author":"Graves","year":"2009","journal-title":"Omega"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Hodder, J.E., Jackwerth, J.C., and Kolokolova, O. (2014). Improved Portfolio Choice Using Second Order Stochastic Dominance, Social Science Electronic Publishing.","DOI":"10.1093\/rof\/rfu025"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/s10479-009-0549-9","article-title":"Stochastic dominance of portfolio insurance strategies","volume":"185","author":"Zagst","year":"2011","journal-title":"Ann. Oper. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"868","DOI":"10.1016\/j.csda.2012.07.030","article-title":"Stochastic dominance with imprecise information","volume":"71","author":"Montes","year":"2014","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1142\/S0219024912500173","article-title":"Metrization of stochastic dominance rules","volume":"15","author":"Stoyanov","year":"2012","journal-title":"Int. J. Theor. Appl. Finance"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/j.knosys.2016.09.023","article-title":"Uncertain multi-attributes decision making method based on interval number with probability distribution weighted operators and stochastic dominance degree","volume":"113","author":"Wu","year":"2016","journal-title":"Knowl. Based Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1074","DOI":"10.1287\/mnsc.48.8.1074.169","article-title":"Preferred by \u201call\u201d and preferred by \u201cmost\u201d decision makers: Almost stochastic dominance","volume":"48","author":"Leshno","year":"2002","journal-title":"Manag. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1016\/j.ejor.2003.06.008","article-title":"Preference and veto thresholds in multicriteria analysis based on stochastic dominance","volume":"158","author":"Nowak","year":"2004","journal-title":"Eur. J. Oper. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2907","DOI":"10.1080\/03610926.2012.689066","article-title":"Supplier evaluation and selection based on stochastic dominance: A quality-based approach","volume":"43","author":"Shu","year":"2014","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Post, T. (2015). Critical Values for Almost Stochastic Dominance Based on Relative Risk Aversion, Social Science Electronic Publishing.","DOI":"10.2139\/ssrn.2666209"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1287\/mnsc.2015.2325","article-title":"Portfolio analysis using stochastic dominance, relative entropy, and empirical likelihood","volume":"63","author":"Post","year":"2017","journal-title":"Manag. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1111\/j.1467-937X.2005.00350.x","article-title":"Consistent testing for stochastic dominance under general sampling schemes","volume":"72","author":"Linton","year":"2005","journal-title":"Rev. Econ. Stud."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"544","DOI":"10.1016\/j.cie.2009.12.001","article-title":"A method based on stochastic dominance degrees for stochastic multiple criteria decision making","volume":"58","author":"Zhang","year":"2010","journal-title":"Comput. Ind. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4139","DOI":"10.1016\/j.ins.2011.05.013","article-title":"A method for stochastic multiple criteria decision making based on dominance degrees","volume":"181","author":"Liu","year":"2011","journal-title":"Inf. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/j.knosys.2014.07.005","article-title":"Stochastic multiple criteria decision making with aspiration level based on prospect stochastic dominance","volume":"70","author":"Tan","year":"2014","journal-title":"Knowl. Based Syst."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Levy, H. (2006). Stochastic dominance: Investment decision making under uncertainty. Studies in Risk and Uncertainty, Springer.","DOI":"10.1007\/0-387-29311-6"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1250","DOI":"10.1287\/mnsc.1120.1616","article-title":"Revisiting almost second-degree stochastic dominance","volume":"59","author":"Tzeng","year":"2013","journal-title":"Manag. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1023\/A:1007730226688","article-title":"Stochastic dominance and prospect dominance with subjective weighting functions","volume":"16","author":"Levy","year":"1998","journal-title":"J. Risk Uncertain."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1626","DOI":"10.1016\/j.ejor.2005.10.003","article-title":"Aspiration level approach in stochastic mcdm problems","volume":"177","author":"Nowak","year":"2007","journal-title":"Eur. J. Oper. Res."},{"key":"ref_24","unstructured":"Blavatskyy, P.R. (2012). Probabilistic Choice and Stochastic Dominance, Springer."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Huang, R.J., Tsetlin, I., Tzeng, L.Y., and Winkler, R.L. (2015). Generalized Almost Stochastic Dominance, Institute for Operations Research and the Management Sciences (INFORMS).","DOI":"10.1287\/opre.2014.1340"},{"key":"ref_26","unstructured":"Hwang, C.-L., and Lin, M.-J. (2012). Group Decision Making under Multiple Criteria: Methods and Applications, Springer Science & Business Media."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1016\/j.econlet.2013.08.020","article-title":"A note on almost stochastic dominance","volume":"121","author":"Guo","year":"2013","journal-title":"Econ. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1016\/j.ins.2014.02.077","article-title":"A stochastic ahp decision making methodology for imprecise preferences","volume":"270","author":"Jalao","year":"2014","journal-title":"Inf. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/BF01078868","article-title":"Stochastic dominance in multicriterion analysis under risk","volume":"39","author":"Martel","year":"1995","journal-title":"Theory Decis."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/11\/606\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:49:56Z","timestamp":1760208596000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/11\/606"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,11,17]]},"references-count":29,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2017,11]]}},"alternative-id":["e19110606"],"URL":"https:\/\/doi.org\/10.3390\/e19110606","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2017,11,17]]}}}