On two dominances of fuzzy variables based on a parametrized fuzzy measure and application to portfolio selection with f
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On two dominances of fuzzy variables based on a parametrized fuzzy measure and application to portfolio selection with fuzzy return Justin Dzuche1 · Christian Deffo Tassak2 · Jules Sadefo Kamdem4 Louis Aimé Fono3
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Accepted: 11 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Yang and Iwamura (Appl Math Sci 46:2271–2288, 2008) introduced a new fuzzy measure as a convex linear combination of possibility and necessity measures. This measure generalizes the credibility measure and the real parameter associated to the possibility measure is considered as the decision making’s optimism level. In this paper, we introduce by means of that measure, two new dominances as binary relations on fuzzy variables. The first one generalizes the first order dominance based on credibility measure and introduced recently by Tassak et al. (J Oper Res Soc 68:1491–1502, 2017) and the second one, based on the investor’s optimism level, is more stronger than the other. Moreover, we study some properties of those dominances and characterize them on the particular family of trapezoidal fuzzy numbers. We implement the second dominance in a numerical example to illustrate the impact of the investor’s attitude through the set of best portfolios. Keywords Fuzzy measure · Fuzzy variable · Generalized first order dominance · Optimism dominance · Set of best portfolios
1 Introduction Portfolio selection theory consists on the choice of “best portfolios” among several other ones following the assumption that assets’ future returns are uncertain (Markowitz 1952; Carlsson
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Jules Sadefo Kamdem [email protected]
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Laboratoire de Mathématiques et Faculté des sciences, Université de Yaoundé I, Yaoundé, Cameroon
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Laboratoire de Mathématiques et Applications Fondamentales, UFDR MIBA, CRFD STG, Université de Yaoundé I, B.P. 812, Yaoundé, Cameroon
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Université de Douala, B.P. 24157, Douala, Cameroon
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MRE EA 7491, Université de Montpellier, Montpellier, France
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Annals of Operations Research
et al. 2002; Sharpe 1971; Ogryczak and Ruszczynski 1999; Stone 1973; Huang 2008; Li et al. 2010; Sadefo et al. 2012). Therefore, the literature presents a huge variety of methods usually used to determine efficient portfolios. More precisely, one can subdivide those methods following two approaches: the first one deals with investors’ preferences expressed through target values for expected benefit or investment risk in optimization models, whereas the second one proposes various efficient portfolios to the investor independently of target values. Indeed, two mathematical tools are used as supports to analyze those approaches. Moments and semi-moments of a random (or fuzzy) variable, defined respectively by means of crisp (or fuzzy) measure, are used through optimization models in the first approach (Markowitz 1952; Ogryczak and Ruszczynski 1999; Huang 2008; Li et al. 2010; Sadefo et al. 2012; Dzuche et al. 2017) whereas the core of portfolios and the set of best portfolios with respect to a
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