Recently, fuzzy set theory has been widely employed in building portfolio selection models where uncertainty plays a role. In these models, future security returns are generally taken for fuzzy variables and mathematical models are then built to maximize the investment profit according to a given risk level or to minimize a risk level based on a fixed profit level. Based on existing works, this paper proposes a portfolio selection model based on fuzzy birandom variables. Two original contributions are provided by the study: First, the concept of technical analysis is combined with fuzzy set theory to use the security returns as fuzzy birandom variables. Second, the fuzzy birandom Value-at-Risk (VaR) is used to build our model, which is called the fuzzy birandom VaR-based portfolio selection model (FBVaR-PSM). The VaR can directly reflect the largest loss of a selected case at a given confidence level and it is more sensitive than other models and more acceptable for general investors than conventional risk measurements. To solve the FBVaR-PSM, in some special cases when the security returns are taken for trapezoidal, triangular or Gaussian fuzzy birandom variables, several crisp equivalent models of the FBVaR-PSM are derived, which can be handled by any linear programming solver. In general, the fuzzy birandom simulation-based particle swarm optimization algorithm (FBS-PSO) is designed to find the approximate optimal solution. To illustrate the proposed model and the behavior of the FBS-PSO, two numerical examples are introduced based on investors' different risk attitudes. Finally, we analyze the experimental results and provide a discussion of some existing approaches.

The objective of our research is to build a statistical test that can evaluate different risks of a portfolio selection model with fuzzy data. The central points and radiuses of fuzzy numbers are used to determine the portfolio selection model, and we statistically evaluate the best return by a fuzzy statistical test. Empirical studies are presented to illustrate the risk evaluation of the portfolio selection model with interval values. We conclude that the fuzzy statistical test enables us to evaluate a stable expected return and low risk investment with different choices for k, which indicates the risk level. The results of numerical examples show that our method is suitable for short-term investments.