{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:18:46Z","timestamp":1760239126409,"version":"build-2065373602"},"reference-count":91,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2020,10,13]],"date-time":"2020-10-13T00:00:00Z","timestamp":1602547200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The subtraction of fuzzy numbers (FNs) is not an inverse operator to FNs addition. The family of all oriented FNs (OFNs) may be considered as symmetrical closure of all the FNs family in that the subtraction is an inverse operation to addition. An imprecise present value is modelled by a trapezoidal oriented FN (TrOFN). Then, the expected discount factor (EDF) is a TrOFFN too. This factor may be applied as a premise for invest-making. Proposed decision strategies are dependent on a comparison of an oriented fuzzy profit index and the specific profitability threshold. This way we get an investment recommendation described as a fuzzy subset on the fixed rating scale. Risk premium measure is a special case of profit index. Further in the paper, the Sharpe\u2019s ratio, the Jensen\u2019s ratio, the Treynor\u2019s ratio, the Sortino\u2019s ratio, Roy\u2019s criterion and the Modiglianis\u2019 coefficient are generalised for the case when an EDF is given as a TrOFN. In this way, we get many different imprecise recommendations. For this reason, an imprecise recommendation management module is described. Obtained results show that the proposed theory can be used as a theoretical background for financial robo-advisers. All theoretical considerations are illustrated by means of a simple empirical case study.<\/jats:p>","DOI":"10.3390\/sym12101672","type":"journal-article","created":{"date-parts":[[2020,10,17]],"date-time":"2020-10-17T07:23:22Z","timestamp":1602919402000},"page":"1672","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["On Application Oriented Fuzzy Numbers for Imprecise Investment Recommendations"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6011-7855","authenticated-orcid":false,"given":"Anna","family":"\u0141yczkowska-Han\u0107kowiak","sequence":"first","affiliation":[{"name":"Institute of Economy and Finance, WSB University in Poznan, ul. Powsta\u0144c\u00f3w Wielkopolskich 5, 61-895 Pozna\u0144, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,13]]},"reference":[{"key":"ref_1","unstructured":"Klopotek, M., Wierzcho\u0144, S.T., and Michalewicz, M. (2002, January 3\u20136). Fuzzy numbers with algebraic operations: Algorithmic approach. Proceedings of the Intelligent Information Systems 2002, Sopot, Poland."},{"key":"ref_2","first-page":"51","article-title":"On fuzzy number calculus","volume":"16","year":"2006","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Piasecki, K. (2018). Revision of the Kosi\u0144ski\u2019s Theory of Ordered Fuzzy Numbers. Axioms, 7.","DOI":"10.3390\/axioms7010016"},{"key":"ref_4","first-page":"493","article-title":"The Directed Inference for the Kosinski\u2019s Fuzzy Number Model","volume":"Volume 427","author":"Abraham","year":"2015","journal-title":"Proceedings of the Second International Afro-European Conference for Industrial Advancement"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/978-3-319-19324-3_23","article-title":"The Directed Compatibility between Ordered Fuzzy Numbers\u2014A Base Tool for a Direction Sensitive Fuzzy Information Processing","volume":"119","author":"Prokopowicz","year":"2015","journal-title":"Artif. Intell. Soft Comput."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Piasecki, K. (2019). Relation \u201cgreater than or equal to\u201d between ordered fuzzy numbers. Appl. Syst. Innov., 2.","DOI":"10.3390\/asi2030026"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Piasecki, K., and \u0141yczkowska-Han\u0107kowiak, A. (2020). Representation of Japanese Candlesticks by Oriented Fuzzy Numbers. Econometrics, 8.","DOI":"10.3390\/econometrics8010001"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1983","DOI":"10.1093\/rfs\/hhz014","article-title":"The promises and pitfalls of Robo-Advising","volume":"32","author":"Prabhala","year":"2019","journal-title":"Rev. Financ. Stud."},{"key":"ref_9","first-page":"77","article-title":"Portfolio Selection","volume":"7","author":"Markowitz","year":"1952","journal-title":"J. Financ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1915","DOI":"10.1093\/rfs\/hhm075","article-title":"Optimal versus naive diversification: How inefficient is the 1\/N portfolio strategy","volume":"22","author":"Garlappi","year":"2009","journal-title":"Rev. Financ. Stud."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1002\/asmb.2066","article-title":"Concentrated portfolio selection models based on historical data","volume":"31","author":"Chen","year":"2015","journal-title":"Applied Stoch. Models Bus. Ind."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1016\/j.ejor.2015.09.005","article-title":"Dynamic mean-risk portfolio selection with multiple risk measures in continuous time","volume":"249","author":"Gao","year":"2016","journal-title":"Eur. J. Oper. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1443","DOI":"10.1016\/S0378-4266(02)00271-6","article-title":"Conditional value-at-risk for general loss distributions","volume":"26","author":"Rockafellar","year":"2002","journal-title":"J. Bank. Financ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2283","DOI":"10.1002\/int.22029","article-title":"Modelling stock selection using ordered weighted averaging operator","volume":"33","author":"Hajjami","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1111\/1467-9965.00100","article-title":"Optimal dynamic portfolio selection: Multiperiod mean-variance formulation","volume":"10","author":"Li","year":"2000","journal-title":"Math. Financ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1016\/j.ejor.2018.01.009","article-title":"Time consistent multi-period robust risk measures and portfolio selection models with regime-switching","volume":"268","author":"Liu","year":"2018","journal-title":"Eur. J. Oper. Res."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Roman, R.-C., Precup, R.-E., and Petriu, E.M. (2020). Hybrid Data-Driven Fuzzy Active Disturbance Rejection Control for Tower Crane Systems. Eur. J. Control.","DOI":"10.1016\/j.ejcon.2020.08.001"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Zhu, Z., Pan, Y., Zhou, Q., and Lu, C. (2020). Event-Triggered Adaptive Fuzzy Control for Stochastic Nonlinear Systems with Unmeasured States and Unknown Backlash-Like Hysteresis. IEEE Trans. Fuzzy Syst.","DOI":"10.1109\/TFUZZ.2020.2973950"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1023\/A:1020907229361","article-title":"On fuzzy portfolio selection problems","volume":"1","author":"Wang","year":"2002","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/S0020-0255(02)00157-3","article-title":"Capital budgeting techniques using discounted fuzzy versus probabilistic cash flows","volume":"142","author":"Kahraman","year":"2002","journal-title":"Inf. Sci."},{"key":"ref_21","unstructured":"Peng, J., Mok, H.M.K., and Tse, W.M. (2005, January 18\u201321). Credibility programming approach to fuzzy portfolio selection problems. Proceedings of the International Conference on Machine Learning and Cybernetics, Guangzhou, China."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1016\/j.ejor.2006.04.010","article-title":"Two new models for portfolio selection with stochastic returns taking fuzzy information","volume":"180","author":"Huang","year":"2007","journal-title":"Eur. J. Oper. Res."},{"key":"ref_23","first-page":"383","article-title":"Portfolio selection with fuzzy return","volume":"18","author":"Huang","year":"2007","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_24","first-page":"76","article-title":"Optimal portfolio selection models with uncertain returns","volume":"3","author":"Yan","year":"2009","journal-title":"Mod. Appl. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Zhang, Y., Li, X., Wong, H.S., and Tan, L. (2009, January 20\u201324). Fuzzy multi-objective portfolio selection model with transaction costs. Proceedings of the IEEE International Conference on Fuzzy Systems, Jeju Island, Korea.","DOI":"10.1109\/FUZZY.2009.5277135"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Fang, Y., Lai, K.K., and Wang, S. (2008). Fuzzy Portfolio Optimization. Theory and Methods, Springer. Lecture Notes in Economics and Mathematical Systems, 609.","DOI":"10.1007\/978-3-540-77926-1"},{"key":"ref_27","first-page":"557","article-title":"Fuzzy Portfolio Optimization Model with Fuzzy Numbers","volume":"Volume 100","author":"Li","year":"2011","journal-title":"Nonlinear Mathematics for Uncertainty and Its Applications, Advances in Intelligent and Soft Computing"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"628295","DOI":"10.1155\/2012\/628295","article-title":"Fuzzy Investment Portfolio Selection Models Based on Interval Analysis Approach","volume":"2012","author":"Guo","year":"2012","journal-title":"Math. Probl. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/s10700-012-9126-9","article-title":"Optimizing fuzzy portfolio selection problems by parametric quadratic programming","volume":"11","author":"Wu","year":"2012","journal-title":"Fuzzy Optim. Decis. Making"},{"key":"ref_30","first-page":"5935","article-title":"Portfolio selection model with interval values based on fuzzy probability distribution functions","volume":"8","author":"Lin","year":"2012","journal-title":"Int. J. Innov. Comput. Inf. Control"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"704","DOI":"10.1016\/j.insmatheco.2013.09.005","article-title":"Fuzzy portfolio optimization model under real constraints","volume":"53","author":"Liu","year":"2013","journal-title":"Insur. Math. Econ."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1016\/j.econmod.2013.07.023","article-title":"Multi-period portfolio optimization under possibility measures","volume":"35","author":"Zhang","year":"2013","journal-title":"Econ. Model."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1016\/j.ins.2012.07.005","article-title":"Multi-objective portfolio selection model with fuzzy random returns and a compromise approach-based genetic algorithm","volume":"220","author":"Li","year":"2013","journal-title":"Inf. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Gupta, P., Mehlawat, M.K., Inuiguchi, M., and Chandra, S. (2014). Fuzzy Portfolio Optimization. Advances in Hybrid Multi-Criteria Methodologies, Springer. Studies in Fuzziness and Soft Computing 316.","DOI":"10.1007\/978-3-642-54652-5"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.ins.2016.01.042","article-title":"Credibilistic mean-entropy models for multi-period portfolio selection with multi-choice aspiration levels","volume":"345","author":"Mehlawat","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_36","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_37","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.asoc.2015.11.005","article-title":"Evolutionary multi-objective optimization algorithms for fuzzy portfolio selection","volume":"39","author":"Saborido","year":"2016","journal-title":"Appl. Soft Comput."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/j.knosys.2017.10.019","article-title":"International asset allocation optimization with fuzzy return","volume":"139","author":"Liu","year":"2018","journal-title":"Knowl. Based Syst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.knosys.2017.12.020","article-title":"Portfolio selection and risk investment under the hesitant fuzzy environment","volume":"144","author":"Zhou","year":"2018","journal-title":"Knowl. Based Syst."},{"key":"ref_40","first-page":"373","article-title":"Score-hesitation trade-off and portfolio selection under intuitionistic fuzzy environment","volume":"268","author":"Zhou","year":"2019","journal-title":"Int. J. Intell. Syst."},{"key":"ref_41","first-page":"37","article-title":"Basis of financial arithmetic from the viewpoint of the utility theory","volume":"22","author":"Piasecki","year":"2012","journal-title":"Oper. Res. Decis."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Hao, F.F., and Liu, Y.K. (2008, January 18\u201320). Portfolio Selection Problem in Fuzzy Random Decision Systems. Proceedings of the 3rd International Conference on Innovative Computing Information and Control, Dalian, China.","DOI":"10.1109\/ICICIC.2008.423"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Hasuike, T., Katagiri, H., and Ishii, H. (2007, January 23\u201326). Portfolio selection problems with random fuzzy variable returns. Proceedings of the 2007 IEEE International Fuzzy Systems Conference, London, UK.","DOI":"10.1109\/FUZZY.2007.4295402"},{"key":"ref_44","first-page":"106","article-title":"Assessing the probabilistic fuzzy Net Present Value for a capital, Investment choice using fuzzy arithmetic","volume":"22","author":"Tsao","year":"2005","journal-title":"J. Chin. Inst. Ind. Eng."},{"key":"ref_45","first-page":"65","article-title":"Effectiveness of securities with fuzzy probabilistic return","volume":"21","author":"Piasecki","year":"2011","journal-title":"Oper. Res. Decis."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"179","DOI":"10.2478\/slgr-2014-0024","article-title":"On Imprecise Investment Recommendations","volume":"37","author":"Piasecki","year":"2014","journal-title":"Stud. Logic Gramm. Rhetor."},{"key":"ref_47","unstructured":"Choudhry, T., and Mizerka, J. (2018). Two-Asset Portfolio with Triangular Fuzzy Present Values\u2014An Alternative Approach. Contemporary Trends in Accounting, Finance and Financial Institutions, Springer. Springer Proceedings in Business and Economics."},{"key":"ref_48","first-page":"183","article-title":"Multi-asset portfolio with trapezoidal fuzzy present values","volume":"64","author":"Piasecki","year":"2018","journal-title":"Stat. Rev."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Tarczy\u0144ski, W., and Nermend, K. (2019). Investment Strategies Determined by Present Value Given as Trapezoidal Fuzzy Numbers. Effective Investments on Capital Markets. Springer Proceedings in Business and Economics, Springer.","DOI":"10.1007\/978-3-030-21274-2"},{"key":"ref_50","unstructured":"Prazak, P. (2017, January 13\u201315). Multiple asset portfolio with present value given as a discrete fuzzy number. Proceedings of the 35th International Conference Mathematical Methods in Economics MME 2017, Hradec Kr\u00e1lov\u00e9, Czech Republic."},{"key":"ref_51","unstructured":"Ward, T.L. (1985, January 11\u201313). Discounted fuzzy cash flow analysis. Proceedings of the 1985 Fall Industrial Engineering Conference, Stuttgart, Germany."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/0165-0114(87)90128-X","article-title":"The fuzzy mathematics of finance","volume":"21","author":"Buckley","year":"1987","journal-title":"Fuzzy Sets Syst."},{"key":"ref_53","unstructured":"Greenhut, J.G., Norman, G., and Temponi, C.T. (1995, January 17\u201320). Towards a fuzzy theory of oligopolistic competition. Proceedings of the IEEE ISUMA-NAFIPS, College Park, MD, USA."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1016\/j.ins.2004.05.007","article-title":"Fuzzy financial profitability analyses of demand side management alternatives from participant perspective","volume":"169","author":"Sheen","year":"2005","journal-title":"Inf. Sci."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/0956-5221(89)90021-3","article-title":"Fuzzy numbers and Net Present Value","volume":"5","author":"Gutierrez","year":"1989","journal-title":"Scand. J. Mgmt."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/S0165-0114(98)00088-8","article-title":"Fuzzy capital budgeting","volume":"111","author":"Kuchta","year":"2000","journal-title":"Fuzzy Sets Syst."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1051\/ejess:2001115","article-title":"Discounted cash-flows analysis. An interactive fuzzy arithmetic approach","volume":"15","author":"Lesage","year":"2001","journal-title":"Eur. J. Econ. Soc. Syst."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/0165-0114(81)90032-4","article-title":"Concepts of probabilistic sets","volume":"5","author":"Hiroto","year":"1981","journal-title":"Fuzzy Sets Syst."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"564","DOI":"10.1016\/j.ins.2016.07.044","article-title":"The fuzzy SAW and fuzzy TOPSIS procedures based on ordered fuzzy numbers","volume":"369","author":"Roszkowska","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1020","DOI":"10.1016\/j.asoc.2016.09.027","article-title":"Fuzzy TOPSIS method with ordered fuzzy numbers for flow control in a manufacturing system","volume":"52","author":"Rudnik","year":"2017","journal-title":"Appl. Soft Comput."},{"key":"ref_61","first-page":"1569860","article-title":"On application of ordered fuzzy numbers in ranking linguistically evaluated negotiation offers","volume":"2018","author":"Piasecki","year":"2018","journal-title":"Adv. Fuzzy Syst."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/j.eswa.2018.09.023","article-title":"A doubly extended TOPSIS method for group decision making based on ordered fuzzy numbers","volume":"116","author":"Kacprzak","year":"2018","journal-title":"Expert Syst. Appl."},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Piasecki, K., Roszkowska, E., and \u0141yczkowska-Han\u0107kowiak, A. (2019). Simple Additive Weighting Method Equipped with Fuzzy Ranking of Evaluated Alternatives. Symmetry, 11.","DOI":"10.3390\/sym11040482"},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Piasecki, K., Roszkowska, E., and \u0141yczkowska-Han\u0107kowiak, A. (2019). Impact of the Orientation of the Ordered Fuzzy Assessment on the Simple Additive Weighted Method. Symmetry, 11.","DOI":"10.3390\/sym11091104"},{"key":"ref_65","first-page":"50","article-title":"Ordered fuzzy numbers approach to an investment project evaluation","volume":"4","year":"2013","journal-title":"Manag. Prod. Eng. Rev."},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"\u0141yczkowska-Han\u0107kowiak, A. (2017). Behavioural present value determined by ordered fuzzy number. SSRN Electron. J., 1.","DOI":"10.2139\/ssrn.2988243"},{"key":"ref_67","unstructured":"Piasecki, K. (2017, January 13\u201315). Expected return rate determined as oriented fuzzy number. Proceedings of the 35th International Conference Mathematical Methods in Economics Conference, Hradec Kr\u00e1lov\u00e9, Czech Republic."},{"key":"ref_68","unstructured":"\u0141yczkowska-Han\u0107kowiak, A., and Piasecki, K. (2018, January 12\u201314). Two-assets portfolio with trapezoidal oriented fuzzy present values. Proceedings of the 36th International Conference Mathematical Methods in Economics Conference, Jind\u0159ich\u016fv Hradec, Czech Republic."},{"key":"ref_69","first-page":"41","article-title":"Present value of portfolio of assets with present values determined by trapezoidal ordered fuzzy numbers","volume":"28","author":"Piasecki","year":"2018","journal-title":"Oper. Res. Decis."},{"key":"ref_70","doi-asserted-by":"crossref","unstructured":"\u0141yczkowska-Han\u0107kowiak, A. (2019). Sharpe\u2019s Ratio for Oriented Fuzzy Discount Factor. Mathematics, 7.","DOI":"10.3390\/math7030272"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"89","DOI":"10.2478\/slgr-2014-0019","article-title":"Optimizing Firm Inventory Costs as a Fuzzy Problem","volume":"37","author":"Kacprzak","year":"2014","journal-title":"Stud. Logic Gramm. Rhetor."},{"key":"ref_72","doi-asserted-by":"crossref","unstructured":"Prokopowicz, P., Czerniak, J., Miko\u0142ajewski, D., Apiecionek, \u0141., and Slezak, D. (2017). Theory and Applications of Ordered Fuzzy Number. Tribute to Professor Witold Kosi\u0144ski, Springer. Studies in Fuzziness and Soft Computing, 356.","DOI":"10.1007\/978-3-319-59614-3"},{"key":"ref_73","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_74","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1016\/S0065-2458(08)60273-9","article-title":"Developments in uncertainty-based information","volume":"36","author":"Klir","year":"1993","journal-title":"Adv. Comput."},{"key":"ref_75","unstructured":"Gupta, M.M., Ragade, R.K., and Yager, R.R. (1979). Entropy and energy measures of fuzzy sets. Advances in Fuzzy Set Theory and Applications, North-Holland Publishing Company."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/S0019-9958(72)90199-4","article-title":"A definition of a non-probabilistic entropy in the settings of fuzzy set theory","volume":"20","author":"Termini","year":"1972","journal-title":"Inf. Control"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/0020-0255(86)90006-X","article-title":"Fuzzy entropy and conditioning","volume":"40","author":"Kosko","year":"1986","journal-title":"Inf. Sci."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1080\/00207727808941724","article-title":"Operations on fuzzy numbers","volume":"9","author":"Dubois","year":"1978","journal-title":"Int. J. Syst. Sci."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/S0165-0114(96)00144-3","article-title":"On a canonical representation of fuzzy numbers","volume":"93","author":"Delgado","year":"1998","journal-title":"Fuzzy Sets Syst."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/0165-0114(79)90005-8","article-title":"Fuzzy real algebra: Some results","volume":"2","author":"Dubois","year":"1979","journal-title":"Fuzzy Sets Syst."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/0165-0114(94)90004-3","article-title":"Fuzzy system reliability analysis using fuzzy number arithmetic operations","volume":"64","year":"1994","journal-title":"Fuzzy Sets Syst."},{"key":"ref_82","doi-asserted-by":"crossref","unstructured":"Piasecki, K., and Stasiak, M.D. (2019). The Forex Trading System for Speculation with Constant Magnitude of Unit Return. Mathematics, 7.","DOI":"10.3390\/math7070623"},{"key":"ref_83","unstructured":"Nison, S. (1991). Japanese Candlestick Charting Techniques, New York Institute of Finance."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1086\/294846","article-title":"Mutual fund performance","volume":"39","author":"Sharpe","year":"1966","journal-title":"J. Bus."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1086\/295182","article-title":"Risk, the pricing of capital assets, and the evaluation of investment portfolios","volume":"42","author":"Jensen","year":"1969","journal-title":"J. Bus."},{"key":"ref_86","first-page":"63","article-title":"How to rate management of investment funds","volume":"43","author":"Treynor","year":"1963","journal-title":"Harv. Bus. Rev."},{"key":"ref_87","first-page":"59","article-title":"Performance measurement of downside risk framework","volume":"3","author":"Sortino","year":"1994","journal-title":"J. Investig. Fall"},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"45","DOI":"10.3905\/jpm.23.2.45","article-title":"Risk adjusted performance","volume":"31","author":"Modigliani","year":"1997","journal-title":"J. Portf. Manag."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"431","DOI":"10.2307\/1907413","article-title":"Safety-first and the holding of assets","volume":"20","author":"Roy","year":"1952","journal-title":"Econometrics"},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","article-title":"Intuitionistic fuzzy sets","volume":"20","author":"Atanassov","year":"1986","journal-title":"Fuzzy Sets. Syst."},{"key":"ref_91","doi-asserted-by":"crossref","unstructured":"Piasecki, K., and Stasiak, M.D. (2020). Optimization Parameters of Trading System with Constant Modulus of Unit Return. Mathematics, 8.","DOI":"10.3390\/math8081384"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/10\/1672\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:20:40Z","timestamp":1760178040000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/10\/1672"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,13]]},"references-count":91,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2020,10]]}},"alternative-id":["sym12101672"],"URL":"https:\/\/doi.org\/10.3390\/sym12101672","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,10,13]]}}}