## 2 Problem Statement and Mathematical Models
### \(m_{\lambda}\) Measure and Expected Value Operator  
Fuzzy set theory was first proposed by [PERSON] in 1965 and further developed by many researchers, such as [PERSON] (2017), [PERSON] and [PERSON] (2017), etc. In this theory, the possibility measure and necessity measure are two effective tools to characterize the chance of a fuzzy event. For instance, let \(\xi\) be a fuzzy variable with membership function \(\mu_{\xi}(x)\) and \(B\) a subset of real numbers. Then, a fuzzy event can be expressed as \(\{\xi\in B\}\). Its possibility and necessity, respectively, can be calculated as follows:  
\[\text{Pos}\{\xi\in B\}=\sup_{x\in B}\mu_{\xi}(x),\ \text{Nec}\{\xi\in B\}=1- \sup_{x\
ot\in B}\mu_{\xi}(x).\]  
In general, we have the following relationship between these two measures, i.e., \(\text{Pos}\{\cdot\}\geq\text{Nec}\{\cdot\}\) for any fuzzy event \(\{\cdot\}\). Additionally, even if the possibility of a fuzzy event achieves one, it cannot necessarily guarantee the occurrence of this event; on the other hand, if the necessity of a fuzzy event is zero, it is possible that this event can occur. In this sense, in the process of optimizing the chance of fuzzy events, the possibility measure is more suitable for the optimistic decision makers, while the necessity measure is more favored by pessimistic decision makers. In order to make a trade-off between the optimism and pessimism in the decision-making process, [PERSON] and [PERSON] (2010) proposed a linear combination of this two fuzzy measures by introducing a weighted parameter \(\lambda\), called the \(m_{\lambda}\) measure, which has successfully applied to a variety of fields in handling the fuzziness of the decision-making process, for instance carbon capture, utilization and storage ([PERSON] et al. (2013)), water quality management ([PERSON] et al. (2014)), etc. In detail, the \(m_{\lambda}\) measure is defined as follows:  
\[m_{\lambda}\{\cdot\}=\lambda\text{Pos}\{\cdot\}+(1-\lambda)\text{Nec}\{\cdot\}.\]  
Theoretically, if parameter \(\lambda\) is close to one, this measure is more suitable for risk-loving decision makers; on the contrary, it is suitable for risk-averse decision makers. In particular, if \(\lambda\) is set as 0.5, this measure degenerates to the credibility measure proposed by [PERSON] and [PERSON] (2010). For this measure, [PERSON] and [PERSON] (2010) proved that for any fuzzy event \(A\), we have \(m_{\lambda}\{A\}+m_{1-\lambda}\{A^{c}\}=1\), which implies that \(m_{\lambda}\) and \(m_{1-\lambda}\) are two dual measures in scaling the chance of fuzzy events. With this property, we hereinafter define the expected value of a fuzzy variable, which has the form of a scalar integral. **Definition 1**. : _Let \(\xi\) be a fuzzy variable with the membership function \(\mu_{\xi}(x)\). Then, the \(\lambda\)-expected value of this fuzzy variable is defined by:_  
\[E_{\lambda}[\xi]=\int_{0}^{+\infty}m_{\lambda}\{\xi\geq r\}\mathrm{d}r-\int_{ -\infty}^{0}m_{1-\lambda}\{\xi\leq r\}\mathrm{d}r \tag{2}\]  
_provided that at least one integral is finite._  
Typically, this definition has a common form of the expected value operator for random variables where the probability measure is self-dual. To illustrate the calculation of expected value operator, we here give several illustrations for clarity. **Example 1**. : _Let \(\xi=(a,b,c)\) be a triangular fuzzy variable with the following membership function:_  
\[\mu_{\xi}(x)=\left\{ \begin{array}{ll}\frac{x-a}{b-a},&\text{if }a\leq x \leq b\\ \frac{x-c}{b-c},&\text{if }b\leq x\leq c\\ 0,&\text{otherwise.}\end{array} \right. \tag{3}\]  
_Then, we have:_  
\[E_{\lambda}[\xi]=\frac{(1-\lambda)a+b+\lambda c}{2}.\]Proof. : In the following, we only consider the case of \(0<a<b<c\), and the other situations can be proven similarly. In this case, it is sufficient to calculate the first integral in the definition. In fact, we have:  
\[m_{\lambda}\{\xi\geq r\}=\left\{