{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T05:34:59Z","timestamp":1769751299724,"version":"3.49.0"},"reference-count":35,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T00:00:00Z","timestamp":1557705600000},"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 study the portfolio selection problem considering transaction costs under multiple periods. For non-professional investors, it is a critical factor to choose an appropriate model among multiple portfolio selection models in investment. Based on the credibility measure, we formulate a multi-period polynomial portfolio selection model to gather the risk indicators involving variance, semi-variance, entropy, and semi-entropy, helping investors bet on assets. According to the polynomial goal programming (PGP) approach, investors can conquer the fields by combining apposite indicators to build appropriate models. Subsequently, an adjusted genetic algorithm on the foundation of the penalty function is designed to obtain the optimal solution of this multi-period model. The results indicate that the PGP method is suitable for investors to choose the model and assigns the proper models to investors with different risk preferences.<\/jats:p>","DOI":"10.3390\/e21050491","type":"journal-article","created":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T11:00:57Z","timestamp":1557745257000},"page":"491","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Performance of Different Risk Indicators in a Multi-Period Polynomial Portfolio Selection Problem Based on the Credibility Measure"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7835-8879","authenticated-orcid":false,"given":"Jian","family":"Zhou","sequence":"first","affiliation":[{"name":"School of Management, Shanghai University, Shanghai 200444, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jie","family":"Shen","sequence":"additional","affiliation":[{"name":"School of Management, Shanghai University, Shanghai 200444, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ziheng","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Economics, Shanghai University of Finance and Economics, Shanghai 200433, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yujie","family":"Gu","sequence":"additional","affiliation":[{"name":"School of Management, Shanghai University, Shanghai 200444, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mingxuan","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Economics and Management, Tongji University, Shanghai 200082, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,13]]},"reference":[{"key":"ref_1","first-page":"77","article-title":"Portfolio selection","volume":"7","author":"Markowitz","year":"1952","journal-title":"J. Financ."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/1350486042000254015","article-title":"Mean-semivariance efficient frontier: A downside risk model for portfolio selection","volume":"12","author":"Ballestero","year":"2005","journal-title":"Appl. Math. Financ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.eswa.2017.02.033","article-title":"Mean-semivariance portfolio optimization with multiobjective evolutionary algorithms and technical analysis rules","volume":"79","author":"Macedo","year":"2017","journal-title":"Expert Syst. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Beraldi, P., Violi, A., Ferrara, M., Ciancio, C., and Pansera, B.A. (2019). Dealing with complex transaction costs in portfolio management. Ann. Oper. Res., 1\u201316.","DOI":"10.1007\/s10479-019-03210-5"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/S0377-2217(98)00033-2","article-title":"Portfolio selection based on upper and lower exponential possibility distributions","volume":"114","author":"Tanaka","year":"1999","journal-title":"Eur. J. Oper. Res."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/j.amc.2003.10.019","article-title":"On admissible efficient portfolio selection problem","volume":"159","author":"Zhang","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1109\/TFUZZ.2002.800692","article-title":"Expected value of fuzzy variable and fuzzy expected value models","volume":"10","author":"Liu","year":"2002","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cam.2007.06.009","article-title":"Mean-semivariance models for fuzzy portfolio selection","volume":"217","author":"Huang","year":"2008","journal-title":"J. Comput. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Fang, S.C., Rajasekera, J.R., and Tsao, H.S.J. (1997). Entropy Optimization and Mathematical Programming, Springer.","DOI":"10.1007\/978-1-4615-6131-6"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"145","DOI":"10.2298\/YJOR120912031R","article-title":"Derivation of some new distributions in statistical mechanics using maximum entropy approach","volume":"1","author":"Ray","year":"2014","journal-title":"Yugoslav J. Oper. Res."},{"key":"ref_12","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_13","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/j.cam.2008.09.010","article-title":"Portfolio selection based on fuzzy cross-entropy","volume":"228","author":"Qin","year":"2009","journal-title":"J. Comput. Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1007\/s12597-017-0311-z","article-title":"Multi objective mean-variance-skewness model with Burg\u2019s entropy and fuzzy return for portfolio optimization","volume":"55","author":"Ray","year":"2018","journal-title":"Opsearch"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1627","DOI":"10.1109\/TFUZZ.2016.2543753","article-title":"Mean-semi-entropy models of fuzzy portfolio selection","volume":"24","author":"Zhou","year":"2016","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3598","DOI":"10.1080\/00207721.2015.1102359","article-title":"Multi-criteria decision-making method based on a cross-entropy with interval neutrosophic sets","volume":"47","author":"Tian","year":"2016","journal-title":"Int. J. Syst. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1016\/j.amc.2007.04.036","article-title":"Multi-period semi-variance portfolio selection: Model and numerical solution","volume":"194","author":"Yan","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1007\/s12190-008-0086-8","article-title":"A class of multi-period semi-variance portfolio selection with a four-factor futures price model","volume":"29","author":"Yan","year":"2009","journal-title":"J. Appl. Math. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/s40305-018-0191-9","article-title":"Time-consistent portfolio policy for asset-liability mean-variance model with state-dependent risk aversion","volume":"6","author":"Peng","year":"2018","journal-title":"J. Oper. Res. Soc. China"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1026","DOI":"10.1016\/j.ejor.2016.04.055","article-title":"Fuzzy multi-period portfolio selection with different investment horizons","volume":"254","author":"Guo","year":"2016","journal-title":"Eur. J. Oper. Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1007\/s00291-013-0335-6","article-title":"Credibilitic mean-variance model for multi-period portfolio selection problem with risk control","volume":"36","author":"Zhang","year":"2014","journal-title":"OR Spectr."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1302","DOI":"10.1016\/j.physa.2017.11.058","article-title":"Credibilistic multi-period portfolio optimization based on scenario tree","volume":"492","author":"Mohebbi","year":"2018","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/j.eswa.2016.11.014","article-title":"Modelling and constructing membership function for uncertain portfolio parameters: A credibilistic framework","volume":"71","author":"Jalota","year":"2017","journal-title":"Expert Syst. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/S0165-0114(99)80004-9","article-title":"Fuzzy sets as a basis for a theory of possibility","volume":"100","author":"Zadeh","year":"1999","journal-title":"Fuzzy Sets Syst."},{"key":"ref_25","first-page":"149","article-title":"A theory of approximate reasoning","volume":"9","author":"Zadeh","year":"1979","journal-title":"Mach. Intell."},{"key":"ref_26","unstructured":"Liu, B. (2007). Uncertainty Theory, Springer."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1016\/j.insmatheco.2011.05.008","article-title":"Portfolio adjusting optimization with added assets and transaction costs based on credibility measures","volume":"49","author":"Zhang","year":"2011","journal-title":"Insurance Math. Econ."},{"key":"ref_28","first-page":"1","article-title":"Portfolio selection based on distance between fuzzy variables","volume":"2014","author":"Qian","year":"2014","journal-title":"Math. Probl. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1109\/TFUZZ.2007.894975","article-title":"Entropy of credibility distributions for fuzzy variables","volume":"16","author":"Li","year":"2008","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1007\/s12652-017-0478-4","article-title":"Time consistent fuzzy multi-period rolling portfolio optimization with adaptive risk aversion factor","volume":"8","author":"Zhou","year":"2017","journal-title":"J. Ambient Intell. Humaniz. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/j.eswa.2017.10.056","article-title":"A polynomial goal programming model for portfolio optimization based on entropy and higher moments","volume":"94","author":"Aksarayli","year":"2018","journal-title":"Expert Syst. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/BF02408382","article-title":"Portfolio selection with skewness: A multiple-objective approach","volume":"1","author":"Lai","year":"1991","journal-title":"Rev. Quant. Financ. Account."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1287\/opre.1080.0566","article-title":"Portfolio selection with robust estimation","volume":"57","author":"DeMiguel","year":"2009","journal-title":"Oper. Res."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1016\/S0045-7825(99)00389-8","article-title":"An efficient constraint handling method for genetic algorithms","volume":"186","author":"Deb","year":"2000","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.fss.2011.05.013","article-title":"A multi-objective genetic algorithm for cardinality constrained fuzzy portfolio selection","volume":"188","author":"Segura","year":"2012","journal-title":"Fuzzy Sets Syst."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/5\/491\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:51:32Z","timestamp":1760187092000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/5\/491"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,13]]},"references-count":35,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["e21050491"],"URL":"https:\/\/doi.org\/10.3390\/e21050491","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5,13]]}}}