## 3 Property Analysis  
Next, we shall analyze the computational properties of some special cases. If all of the returns are interval fuzzy variables, we have the following results. **Theorem 6**. : _Let \(r_{s}=[a_{s},b_{s}]\) be interval fuzzy variables, \(s=1,2,\ldots,S\). Then, the variance can be calculated according to the following equation._  
\[V\left[\sum_{s=1}^{S}r_{s}x_{s}\right]=\lambda\cdot\max\left\{\lambda^{2}\left( \sum_{s=1}^{S}b_{s}x_{s}-\sum_{s=1}^{S}a_{s}x_{s}\right)^{2},(1-\lambda)^{2} \left(\sum_{s=1}^{S}b_{s}x_{s}-\sum_{s=1}^{S}a_{s}x_{s}\right)^{2}\right\}\]  
**Proof of Theorem 6. ** Since the returns \(r_{s}\), \(s=1,2,\ldots,S\) are interval fuzzy variables, then the total fuzzy return is also an interval fuzzy variable for each feasible solution, given below. \[\sum_{s=1}^{S}r_{s}x_{s}=\left[\sum_{s=1}^{S}a_{s}x_{s},\sum_{s=1}^{S}b_{s}x_{s }\right].\]  
Then, by using Examples 1 and 4, we can easily prove this theorem. Typically, if all of the parameters \(r_{s}\), \(s=1,2,\ldots,S\) are interval fuzzy variables, we can use analytical methods to calculate the objective function for each feasible solution \(X\). **Theorem 7**. : _(i) Let \(r_{s}=(a_{s},b_{s},c_{s})\) be a triangular fuzzy variable, \(s=1,2,\ldots,S\). Then, the expected total return can be calculated according to the following equation._  
\[E\left[\sum_{s=1}^{S}r_{s}x_{s}\right]=\frac{(1-\lambda)\sum_{s=1}^{S}a_{s}x_{ s}+\sum_{s=1}^{S}b_{s}x_{s}+\lambda\sum_{s=1}^{S}c_{s}x_{s}}{2}.\]  
_(ii) Let \(r_{s}=(a_{s},b_{s},c_{s},d_{s})\) be a trapezoidal fuzzy variable, \(s=1,2,\ldots,S\). Then, the expected total return can be calculated according to the following equation._  
\[E\left[\sum_{s=1}^{S}r_{s}x_{s}\right]=\frac{(1-\lambda)\sum_{s=1}^{S}a_{s}x_{ s}+(1-\lambda)\sum_{s=1}^{S}b_{s}x_{s}+\lambda\sum_{s=1}^{S}c_{s}x_{s}+ \lambda\sum_{s=1}^{S}c_{s}x_{s}}{2}.\]  
By using Examples 1 and 2, the proof of this theorem is obvious. In general, for a common fuzzy variable with a complex membership function, it is difficult to calculate its expected value and variance by analytic methods. In this situation, we have to use a simulation algorithm to achieve the approximate values of these two indexes. In the following, we shall design the detailed simulation algorithm to simulate the expected value and variance of a fuzzy variable. Consider a function \(f(\mathbf{x},\mathbf{\xi})\). We next design the simulation procedure of the expected value \(E[f(\mathbf{x},\mathbf{\xi});\lambda]\). Typically, if we set \(f(\mathbf{x},\mathbf{\xi})=\zeta\), this procedure corresponds to the simulation process of the expected value of \(\zeta\); moreover, if we set \(f(\mathbf{x},\mathbf{\xi})=(\xi-E[\xi;\lambda])^{2}\), this procedure corresponds to the simulation process of variance of \(\xi\). For this purpose, we firstly need to compute the chance measure with form \(m_{\lambda}\{f(\mathbf{x},\mathbf{\xi})\leq r\}\), and chance measure \(m_{\lambda}\{f(\mathbf{x},\mathbf{\xi})\geq r\}\) can be simulated similarly. [PERSON] and [PERSON] [30] give the simulation procedure as follows. * The procedure of simulating \(m_{\lambda}\{f(\mathbf{x},\mathbf{\xi})\geq r\}\): Step 1. Randomly generate crisp vectors \(\mathbf{u}_{k}\) from the \(\varepsilon\)-level set of fuzzy vector \(\mathbf{\xi}\), respectively, where \(\varepsilon\) is a sufficiently small number and \(k=1,2,\ldots,N\); Step 2. Let \(v_{k}=\mu_{\mathbf{\xi}}(\mathbf{u}_{k})\) for \(k=1,2,\ldots,N\) where \(\mu_{\mathbf{\xi}}\) is the membership function of \(\mathbf{\xi}\); Step 3. Return the following value: \[L=\lambda\cdot\max_{1\leq k\leq N}\{v_{k}|f(\mathbf{x},\mathbf{u}_{k})\leq r\}+(1- \lambda)\cdot\min_{1\leq k\leq N}\{1-v_{k}|f(\mathbf{x},\mathbf{u}_{k})>r\}\]* The procedure of simulating \(E[f(\mathbf{x},\mathbf{\xi});\lambda]\): Step 1. Set \(e=0\); Step 2. Randomly generate crisp vectors \(\mathbf{u}_{k}\) from the \(\varepsilon\)-level set of fuzzy vector \(\mathbf{\xi}\), respectively, where \(\varepsilon\) is a sufficiently small number and \(k=1,2,\ldots,N\); Step 3.