Mathematical Analyses of the Upper and Lower Possibilistic Mean – Variance Models and Their Extensions to Multiple Scenarios

Abstract

Possibility theory is the one of the most important and widely used uncertainty theories because it is closely related to the imprecise probability and expert knowledge. The possibilistic mean - variance (MV) model is the counterpart of the Markowitz’s MV model in the possibility theory. There are variants of the possibilistic MV model, which are called as the upper and lower possibilistic MV models. However, to the best of our knowledge, analytical solutions and exact efficient frontiers of these variants are not presented in the literature when the possibil-ity distributions are given with trapezoidal fuzzy numbers. In this study, under this assumption, we make mathemat-ical analyses of the upper and lower possibilistic MV models and derive their analytical solutions and exact efficient frontiers. Based on the max-min optimization framework, we also propose their extensions where there are multiple upper (lower) possibilistic mean scenarios. We show that the proposed extensions have the ease of use as the upper and lower possibilistic MV models. We also illustrate and compare the upper and lower possibilistic mean - variance models and their proposed extensions with an explanatory example. As we expect, we see that these extensions can be effectively used in portfolio selection by conservative investors.

Keywords

• Max-min optimization
• portfolio selection
• possibility theory
• scenario analysis
• trapezoidal fuzzy numbers

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