{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T14:50:48Z","timestamp":1775141448577,"version":"3.50.1"},"reference-count":28,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2015,5,14]],"date-time":"2015-05-14T00:00:00Z","timestamp":1431561600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we define the portfolio return as fuzzy average yield and risk as hybrid-entropy and variance to deal with the portfolio selection problem with both random uncertainty and fuzzy uncertainty, and propose a mean-variance hybrid-entropy model (MVHEM). A multi-objective genetic algorithm named Non-dominated Sorting Genetic Algorithm II (NSGA-II) is introduced to solve the model. We make empirical comparisons by using the data from the Shanghai and Shenzhen stock exchanges in China. The results show that the MVHEM generally performs better than the traditional portfolio selection models.<\/jats:p>","DOI":"10.3390\/e17053319","type":"journal-article","created":{"date-parts":[[2015,5,15]],"date-time":"2015-05-15T07:57:02Z","timestamp":1431676622000},"page":"3319-3331","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["A Mean-Variance Hybrid-Entropy Model for Portfolio Selection with Fuzzy Returns"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4270-2903","authenticated-orcid":false,"given":"Rongxi","family":"Zhou","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yu","family":"Zhan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ru","family":"Cai","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guanqun","family":"Tong","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,5,14]]},"reference":[{"key":"ref_1","first-page":"77","article-title":"Portfolio selection","volume":"7","author":"Markowitz","year":"1952","journal-title":"J. Financ."},{"key":"ref_2","first-page":"219","article-title":"Fuzzy portfolio selection and its applications to decision making","volume":"13","author":"Watada","year":"1997","journal-title":"Tatra Mt. Math. Publ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/0165-0114(96)84605-7","article-title":"A fuzzy control model for dynamic portfolio management","volume":"78","author":"Ostermask","year":"1998","journal-title":"Fuzzy Sets Syst."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/j.ejor.2009.05.003","article-title":"Mean-variance-skewness model for portfolio selection with fuzzy returns","volume":"202","author":"Li","year":"2010","journal-title":"Eur. J. Oper. Res."},{"key":"ref_5","first-page":"253","article-title":"Financial Decision Making Using Computational Intelligence","volume":"70","author":"Doumpos","year":"2012","journal-title":"Springer Optim. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/j.fss.2013.09.002","article-title":"A new fuzzy programming approach for multi-period portfolio optimization with return demand and risk control","volume":"246","author":"Zhang","year":"2014","journal-title":"Fuzzy Sets Syst."},{"key":"ref_7","unstructured":"Markowitz, H. (1959). Portfolio Selection: Efficient Diversification of Investments, Wiley."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1287\/mnsc.37.5.519","article-title":"Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market","volume":"37","author":"Konno","year":"1991","journal-title":"Manag. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1287\/mnsc.44.5.673","article-title":"A minimax portfolio selection rule with linear programming solution","volume":"44","author":"Young","year":"1998","journal-title":"Manag. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"957","DOI":"10.1287\/mnsc.46.7.957.12039","article-title":"Portfolio optimization under a minimax rule","volume":"46","author":"Cai","year":"2000","journal-title":"Manag. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/0304-405X(75)90025-2","article-title":"Optimal rules for ordering uncertain prospects","volume":"2","author":"Bawa","year":"1975","journal-title":"J. Financ. Econ."},{"key":"ref_12","first-page":"181","article-title":"Multi-objective mean-variance-skewness model for portfolio optimization","volume":"9","author":"Jana","year":"2007","journal-title":"Appl. Math. Optim."},{"key":"ref_13","first-page":"98","article-title":"Study on the Model Selection in Portfolio Management\u2014Models and empirical analysis based on Chinese stock market","volume":"30","author":"Yu","year":"2013","journal-title":"J. Quant. Tech. Econ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4909","DOI":"10.3390\/e15114909","article-title":"Applications of Entropy in Finance: A Review","volume":"15","author":"Zhou","year":"2013","journal-title":"Entropy"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1080\/00036847200000017","article-title":"Entropy, market risk, and the selection of efficient portfolios","volume":"4","author":"Philippatos","year":"1972","journal-title":"Appl. Econ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1080\/10835547.2001.12089636","article-title":"MPT and the downside risk framework: A comment on two recent studies","volume":"7","author":"Cheng","year":"2001","journal-title":"J. Real Estate Portf. Manag."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1108\/15265940510574754","article-title":"Theory of portfolio and risk based on incremental entropy","volume":"6","author":"Ou","year":"2005","journal-title":"J. Risk Financ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1080\/07474930801960394","article-title":"Optimal portfolio diversification using the maximum entropy principle","volume":"27","author":"Bera","year":"2008","journal-title":"Economet. Rev."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1096","DOI":"10.1109\/TFUZZ.2008.924200","article-title":"Mean-entropy models for fuzzy portfolio selection","volume":"16","author":"Huang","year":"2008","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"117","DOI":"10.3390\/e13010117","article-title":"Mean-Variance-Skewness-Entropy Measures: A Multi-Objective Approach for Portfolio Selection","volume":"13","author":"Usta","year":"2011","journal-title":"Entropy"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/s10479-009-0550-3","article-title":"Portfolio selection using \u03bb mean and hybrid entropy","volume":"185","author":"Xu","year":"2011","journal-title":"Ann. Oper. Res."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/j.amc.2014.04.006","article-title":"Diversified portfolios with different entropy measures","volume":"241","author":"Yu","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_23","first-page":"853","article-title":"Portfolio selection model with the measures of information entropy-incremental entropy-skewness","volume":"5","author":"Zhou","year":"2013","journal-title":"Adv. Inf. Sci. Serv. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/j.ejor.2012.04.023","article-title":"A possibilistic mean-semivariance-entropy model for multi-period portfolio selection with transaction costs","volume":"222","author":"Zhang","year":"2012","journal-title":"Eur. J. Oper. Res."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s10700-006-0016-x","article-title":"A survey of credibility theory","volume":"5","author":"Liu","year":"2006","journal-title":"Fuzzy Optim. Decision Making."},{"key":"ref_26","first-page":"127","article-title":"Network Models in Economics and Finance","volume":"100","author":"Kalyagin","year":"2014","journal-title":"Springer Optim. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/S0019-9958(72)90199-4","article-title":"A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory","volume":"20","author":"Termini","year":"1972","journal-title":"Inf. Control."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1109\/4235.996017","article-title":"A fast and elitist multi-objective genetic algorithm; NSGA-II","volume":"6","author":"Deb","year":"2002","journal-title":"IEEE Trans. on Evolut. Comput."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/5\/3319\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:46:18Z","timestamp":1760215578000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/5\/3319"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,5,14]]},"references-count":28,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2015,5]]}},"alternative-id":["e17053319"],"URL":"https:\/\/doi.org\/10.3390\/e17053319","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,5,14]]}}}