Most of existing portfolio selection models are based on a probabilistic approach combined with optimization techniques. The uncertainty is equated with randomness. However, there are many non-probabilistic factors that affect the financial markets and a recent literature has increasingly been interested in modeling the fuzziness of portfolios returns. In this paper we consider the mean-variance portfolio optimization problem for LR-shaped fuzzy-valued returns of risky assets. Our approach is based upon Puri and Lalescu concept of fuzzy random variables and a suitable -metric on the Hilbert space of fuzzy variables whose outcomes have square-integrable support functions. The latter are defined with respect to the left and right spreads of the LR-shaped fuzzy-valued returns and can be used to obtain pessimistic and optimistic estimates of the expected returns and their corresponding covariance matrices, both with possibilistic interpretation. This allows formulating and solving three possibilistic bi-objective portfolio optimization problems, which result in pessimistic, optimistic and combined efficient possibilistic portfolios and their corresponding efficient possibilistic frontiers. A fourth problem, the classical Markowitz’s mean–variance one (defined with respect to crisp returns) is also considered as a benchmark. We employ a multi-objective metaheuristic (namely, ANSGAIII) to solve these four portfolio optimization problems for a universe of assets listed at Bucharest Stock Exchange.