## 2 Problem Statement and Mathematical Models
### \(m_{\lambda}\) Measure and Expected Value Operator
\begin{array}{cl}1,&\text{if }0<r\leq a \\ 1-(1-\lambda)\cdot\frac{r-a}{b-a},&\text{if }a<r\leq b\\ \lambda\cdot\frac{r-c}{b-c},&\text{if }b<r\leq c\\ 0,&\text{if }r>c\end{array} \right.\]  
Thus,  
\[E_{\lambda}[\xi] = \int_{0}^{a}\mathrm{d}r+\int_{a}^{b}\left[1-(1-\lambda)\cdot\frac {r-a}{b-a}\right]\mathrm{d}r+\int_{b}^{c}\lambda\cdot\frac{r-c}{b-c}\mathrm{d}r\] \[= a+(b-a)-(1-\lambda)\int_{a}^{b}\frac{r-a}{b-a}\mathrm{d}r+ \lambda\int_{b}^{c}\frac{r-c}{b-c}\mathrm{d}r\] \[= b-\frac{(1-\lambda)(b-a)}{2}+\int_{b}^{c}\frac{\lambda(c-b)}{2} \mathrm{d}r\] \[= \frac{(1-\lambda)a+b+\lambda c}{2}.\]  
**Remark 1**. : _If we take \(\lambda=0.5\), the expected value of fuzzy variable can be simplified as:_  
\[E_{\lambda}[\xi]=\frac{a+2b+c}{4},\]  
_which just coincides with the situation proposed by [PERSON] and [PERSON] [32]._  
**Example 2**. : _Let \(\xi=(a,b,c,d)\) be a trapezoidal fuzzy variable with the following membership function:_  
\[\mu_{\xi}(x)=\left\{ \begin{array}{cl}\frac{x-a}{b-a},&\text{if }a\leq x\leq b \\ 1,&\text{if }b\leq x\leq c\\ \frac{x-c}{b-c},&\text{if }c\leq x\leq d\\ 0,&\text{otherwise.}\end{array} \right. \tag{4}\]  
_Then, we have:_  
\[E[\xi;\lambda]=\frac{(1-\lambda)a+(1-\lambda)b+\lambda c+\lambda d}{2}.\]  
**Example 3**. : _Let \(\xi=[a,b]\) be an interval fuzzy variable with the following membership function:_  
\[\mu_{\xi}(x)=\left\{ \begin{array}{cl}1,&\text{if }a\leq x\leq b\\ 0,&\text{otherwise.}\end{array} \right. \tag{5}\]  
_Then, we have:_  
\[E_{\lambda}[\xi]=(1-\lambda)a+\lambda b.\]  
In this definition, we have the following relationship with respect to the different parameter \(\lambda\). **Theorem 1**. : _Let \(\xi\) be a fuzzy variable and \(\lambda_{1},\lambda_{2}\in[0,1]\) with \(\lambda_{1}\leq\lambda_{2}\); we have \(E_{\lambda_{1}}[\xi]\leq E_{\lambda_{2}}[\xi]\)._Proof of Theorem 1.: Since \(\lambda_{1},\lambda_{2}\in[0,1]\) with \(\lambda_{1}\leq\lambda_{2}\), we have \(m_{\lambda_{1}}\{\cdot\}\leq m_{\lambda_{2}}\{\cdot\}\) for any fuzzy event \(\{\cdot\}\) (see [PERSON] and [PERSON] [30]). Then, we have:  
\[E_{\lambda_{1}}[\xi] = \int_{0}^{+\infty}m_{\lambda_{1}}\{\xi\geq r\}\mathrm{d}r-\int_{- \infty}^{0}m_{1-\lambda_{1}}\{\xi\leq r\}\mathrm{d}r\] \[\leq \int_{0}^{+\infty}m_{\lambda_{2}}\{\xi\geq r\}\mathrm{d}r-\int_{- \infty}^{0}m_{1-\lambda_{2}}\{\xi\leq r\}\mathrm{d}r=E_{\lambda_{2}}[\xi].\]  
The proof is thus completed. To capture the variant of the expected values with respect to parameter \(\lambda\), we show the curve of the \(\lambda\)-expected value of a triangular fuzzy variable in Figure 1. Clearly, the expected value has a linear increasing relationship with respect to \(\lambda\). If \(\lambda=1\), the expected value turns out to be \(\dfrac{b+c}{2}\); if \(\lambda=0\), the expected value takes the value \(\dfrac{a+b}{2}\). Actually, if \(\xi\) denotes the fuzzy return, this expected value function is reasonable for different types of decision makers, since optimistic decision makers can usually overrate the expected return, and pessimistic decision makers prefer to underestimate the expected return. Next, we aim to investigate the linearity of the proposed \(\lambda\)-expected value operator. To this end, we firstly introduce some closely related concepts and theorems in the following discussion. Definition 2: (Liu [35]) Suppose that \(\xi\) is a fuzzy variable and \(\alpha\in(0,1]\). Then:  
\[\xi_{\sup}(\alpha)=\sup\{r|\mathrm{Pos}\{\xi\geq r\}\geq\alpha\}\]  
is called the \(\alpha\)-optimistic value to \(\xi\); and:  
\[\xi_{\inf}(\alpha)=\inf\{r|\mathrm{Pos}\{\xi\leq r\}\geq\alpha\}\]  
is called the \(\alpha\)-pessimistic value to \(\xi\). Theorem 2.1: (_Liu [35]) Suppose that \(\xi\) and \(\eta\) are two fuzzy variables. Then, for any \(\alpha\in(0,1]\), we have:_  
1. \((\xi+\eta)_{\sup}(\alpha)=\xi_{\sup}(\alpha)+\eta_{\sup}(\alpha)\)_;_
2. \((\xi+\eta)_{\inf}(\alpha)=\xi_{\inf}(\alpha)+\eta_{\inf}(\alpha)\)_;_
3.