## 2 Problem Statement and Mathematical Models
### New Reliable Models  
In this problem, if we use the least expected return to evaluate the quality of the portfolio selection strategy, we can formulate the following least expected return model for the fuzzy portfolio selection problem. \[\left\{ \begin{array}{l}\max E_{\lambda}[r_{1}x_{1}+r_{2}x_{2}+\ldots+r_{5}x _{5}]\\ s.t.\\ x_{1}+x_{2}+\ldots+x_{5}=1\\ x_{s}\geq 0,s\in\{1,2,\cdots,S\}\end{array} \right. \tag{7}\]  
In this model, the objective function aims to minimize the expected return in the decision-making process. However, we note that the optimal strategy for this model is not always favored in all situations, because the least expected return strategy, occasionally creating large, systemic and poorly understood risks, might be subject to high risks under some extreme scenarios and would be specifically undesirable to risk-averse travelers. By recognizing this critical requirement, we here particularly propose three models to reduce the the diversity of the fuzzy total return. In addition, to avoid the excessive decentralization of the final investigation, the threshold for each sector needs to be given. That is, let \(h_{s}\) be the threshold of investigation ratio \(x_{s}\), and then, we should have \(x_{s}=0\) or \(x_{s}\geq h_{s}\), \(s=1,2,\cdots,S\) in the process of the formulation. Firstly, we add the variance of the total return into the objective function, and then, an expectation-variance reliable model is formulated, given by:  
\[\left\{ \begin{array}{l}\max E_{\lambda}\left[\sum_{s=1}^{S}r_{s}x_{s}\right]- \beta\sqrt{V_{\lambda}\left[\sum_{s=1}^{S}r_{s}x_{s}\right]}\\ s.t.\\ x_{1}+x_{2}+\ldots+x_{S}=1\\ x_{s}=0\text{ or }x_{s}\geq h_{s},s\in\{1,2,\ldots,S\}\end{array} \right. \tag{8}\]  
In the literature, this type of index has been successfully applied to the transportation field (e.g., [PERSON] and [PERSON] (2006); [PERSON] et al. (2017)) as a guidance for finding a reliable path. In this model, the parameter \(\beta\) is a reliability coefficient to reflect the significance of total return variability. This reliability coefficient can also vary for different decision makers. In general, if the decision maker is risk-averse, he/she can set a relative large reliability coefficient; otherwise, a smaller coefficient can be taken for the optimistic decision makers. Typically, in the process of maximizing the objective, a small variance of the fuzzy return is desirable in evaluating the investment strategies. Secondly, to reduce the risk, the variance of total return is assumed to be less than a threshold \(\bar{V}\), which is regarded as a constraint in the formulation. This constraint can be referred to as a side constraint, which has been widely used to handle the routing optimization problems effectively (e.g., [PERSON] et al. (2018); [PERSON] et al. (2019)). By maximizing the expected return, we hereinafter reformulate the risk-guaranteed reliable model as a variance-constrained reliable model, given by:  
\[\left\{ \begin{array}{l}\max E_{\lambda}\left[\sum_{s=1}^{S}r_{s}x_{s}\right] \\ s.t.\\ x_{1}+x_{2}+\cdots+x_{S}=1\\ V_{\lambda}\left[\sum_{s=1}^{S}r_{s}x_{s}\right]\leq\bar{V}\\ x_{s}=0\text{ or }x_{s}\geq h_{s},s\in\{1,2,\ldots,S\}\end{array} \right. \tag{9}\]  
In Model (9), the parameter \(\bar{V}\) is an upper limit of the variance, which is determined by decision makers. Obviously, a risk-seeker often select a relatively large threshold \(\bar{V}\), while a smaller \(\bar{V}\) will lead to a risk-averse strategy. Thirdly, in the condition of minimizing the variance of the total return, we regard the expected return as a resource constraint, and thus, the expectation-constrained reliable model is formulated by:  
\[\left\{ \begin{array}{l}\min V_{\lambda}\left[\sum_{s=1}^{S}r_{s}x_{s}\right] \\ s.t.\\ x_{1}+x_{2}+\cdots+x_{S}=1\\ E_{\lambda}\left[\sum_{s=1}^{S}r_{s}x_{s}\right]\geq\bar{E}\\ x_{s}=0\text{ or }x_{s}\geq h_{s},s\in\{1,2,\ldots,S\}\end{array} \right.