{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,16]],"date-time":"2026-09-16T00:02:59Z","timestamp":1789516979850,"version":"build-2803163510"},"reference-count":32,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2017,2,21]],"date-time":"2017-02-21T00:00:00Z","timestamp":1487635200000},"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>This paper aims to develop a risk-free protection index model for portfolio selection based on the uncertain theory. First, the returns of risk assets are assumed as uncertain variables and subject to reputable experts\u2019 evaluations. Second, under this assumption, combining with the risk-free interest rate we define a risk-free protection index (RFPI), which can measure the protection degree when the loss of risk assets happens. Third, note that the proportion entropy serves as a complementary means to reduce the risk by the preset diversification requirement. We put forward a risk-free protection index model with an entropy constraint under an uncertainty framework by applying the RFPI, Huang\u2019s risk index model (RIM), and mean-variance-entropy model (MVEM). Furthermore, to solve our portfolio model, an algorithm is given to estimate the uncertain expected return and standard deviation of different risk assets by applying the Delphi method. Finally, an example is provided to show that the risk-free protection index model performs better than the traditional MVEM and RIM.<\/jats:p>","DOI":"10.3390\/e19020080","type":"journal-article","created":{"date-parts":[[2017,2,22]],"date-time":"2017-02-22T11:40:52Z","timestamp":1487763652000},"page":"80","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A Risk-Free Protection Index Model for Portfolio Selection with Entropy Constraint under an Uncertainty Framework"],"prefix":"10.3390","volume":"19","author":[{"given":"Jianwei","family":"Gao","sequence":"first","affiliation":[{"name":"School of Economics and Management, North China Electric Power University, Beijing 102206, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huicheng","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Economics and Management, North China Electric Power University, Beijing 102206, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,2,21]]},"reference":[{"key":"ref_1","first-page":"77","article-title":"Portfolio selection","volume":"7","author":"Markowitz","year":"1952","journal-title":"J. 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