## 5 Numerical Examples  
In this section, we aim to implement a series of numerical experiments to test the performance of the proposed approaches for the variance-constrained Model (9). All of the experiments are implemented in a personal computer with a 1.60-GHz CPU and 4.00 GB of memory.  
Assume that there are 20 securities in our decision-making process. We use the triangular fuzzy variables to denote the security returns in the model, given in Table 2. In addition, to generate a favorite investment strategy, we set the threshold for each security as 0.1 in the experiments. Thus, at most ten securities can be finally selected for investment.  
1. Steadiness of the proposed algorithm With the above-mentioned decision data, we implement this experiment by the genetic algorithm in C++ software, in which the relevant model parameters are set as \(\lambda=0.8\), \(\bar{V}=40\). The computational results are listed in Table 3. Specifically, we randomly choose the critical parameters in the algorithm, including the crossover probability, mutation probability and population size, to test the steadiness of the algorithmic implementation. The algorithm terminated after 500 cycles, i.e., \(M=500\), and a total of ten tests is finally executed. To further show the performance of the computational results, we compute the relatively errors among different results with respect to the best objective function (i.e., 80.23) according to the following equation, \[Relative\ Error=\frac{Best\ Objective\ Value-Current\ Objective\ Value}{Best\ Objective\ Value}\times 100\%.\] Typically, among all of these experiments, the relative errors are not greater than 4.00%, which implies the steadiness of the proposed algorithm with respect to these critical parameters. For the parameter setting \(P_{c}=0.6\), \(P_{m}=0.7\) and \(Pop\_size=40\), we can find the near-optimal solution with objective value 80.23, and the corresponding optimal solution is \(x_{3}=0.11\), \(x_{6}=0.13\), \(x_{8}=0.76\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Security Index** & **Fuzzy Return** & **Security Index** & **Fuzzy Return** \\ \hline
1 & (35, 45, 50) & 11 & (45, 53, 70) \\
2 & (54, 80, 90) & 12 & (44, 60, 70) \\
3 & (44, 60, 65) & 13 & (33, 45, 50) \\
4 & (35, 40, 50) & 14 & (43, 54, 70) \\
5 & (55, 70, 75) & 15 & (45, 66, 70) \\
6 & (63, 70, 80) & 16 & (44, 50, 60) \\
7 & (45, 54, 60) & 17 & (35, 50, 55) \\
8 & (72, 80, 85) & 18 & (28, 40, 50) \\
9 & (66, 73, 90) & 19 & (33, 40, 45) \\
10 & (35, 40, 50) & 20 & (45, 50, 55) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Fuzzy returns for each security in the decision-making process.  
variance can be used to denote the risk of an investment strategy, a small parameter corresponds to a risk-averse decision. In order to show the influence of this parameter on the optimal solution, we here especially implement a set of experiments to test this performance. Specifically, we take a total of ten cases for parameter \(\bar{V}\), i.e., \(\bar{V}=40,41,42,43,44,45,46,47,48,49\). In addition, we set \(\lambda=0.8\) in the \(m_{\lambda}\) measure, and \(P_{c}=0.6\), \(P_{m}=0.7\), \(Pop\_size=40\) in the genetic algorithm. The computational results are listed in Table 5. Clearly, different parameters \(\bar{V}\) might produce different optimal solutions. For instance, if we take \(\bar{V}=43\), the optimal solution turns out to be \(x_{2}=0.10\), \(x_{6}=0.42\), \(x_{8}=0.48\); on the other hand, when we set \(\bar{V}=48\), the outputted optimal solution is \(x_{3}=0.11\), \(x_{6}=0.37\), \(x_{8}=0.52\).