## 3 Preliminaries  
### The \(\lambda\)-Fuzzy-Measure and the Choquet Integral  
In the FGT method, the weights of all subsets of a criterion set are expressed through the \(\lambda\)-fuzzy-measure.  
Fuzzy-measures [39]: Assume that the criterion set \(X\) is nonempty and \(P(X)\) denotes its power set. If the set function \(g:P(X)\rightarrow[0,1]\) satisfies the following three axioms, then it is called a fuzzy-measure:  
(1) Boundness  
\[g(\varnothing)=0,g(X)=1.\]  
(2) Monotonicity  
\[\forall A,\ B\in P(X),\ \text{if}\ A\subseteq B,\ \text{then}\ g(A)\leq g(B).\]  
(3) Weak continuity:  
\[\text{If}\ \{A_{i}\}\subset P(X)\ \text{and}\ \{A_{i}\}\ \text{is monotonic},\ \text{then}\ \underset{i=\infty}{\lim}g(A_{i})=g\bigg{(}\underset{i=\infty}{\lim}A_{i} \bigg{)}.\]  
\(\lambda\)-fuzzy-measures [39]: If the fuzzy-measure \(g\) satisfy the following axiom, then it is called \(\lambda\)-fuzzy-measures:  
\[\text{If}\ A,B\in P(X)\ \text{and}\ A\cap B=\varnothing,\ \text{then}\ \ \exists\lambda\in[-1, \infty),\ g(A\cup B)=g(A)+g(B)+\lambda g(A)g(B).\]  
Let \(X=\{x_{1},x_{2},\ldots,x_{n}\}\) be a finite set. If the fuzzy density of \(x_{i}\) is \(g_{i}\), then \(g_{\lambda}\) can be calculated as follows:  
\[\begin{array}{rl}g_{\lambda}(\{x_{1},x_{2},\ldots,x_{k}\})&=\sum\limits_{i=1 }^{k}g_{i}+\lambda\sum\limits_{i_{1}=1}^{k-1}\sum\limits_{i_{2}=i_{1}+1}^{k}g _{i_{1}}g_{i_{2}}\\ &+\ldots+\lambda^{k-1}g_{1}g_{2}\ldots g_{k}\\ &=\frac{1}{\lambda}\bigg{|}_{i=1}^{k}(1+\lambda g_{i})-1\bigg{|}\end{array} \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Typical Reference** & **Tool Type** & **Advantage** & **Limitation** \\ \hline \multirow{3}{*}{[PERSON] et al. [71]} & Fuzzy-AHP, Choquet & \multirow{3}{*}{Realistic and reliable} & Complicated and \\  & integral and trapezoidal & & abstract models \\  & fuzzy sets & & \\ \hline \multirow{2}{*}{[PERSON] et al. [72]} & VIKOR and interval type-2 & \multirow{2}{*}{Effectively improve accuracy} & Complicated and \\  & fuzzy sets & & & abstract models \\ \hline [PERSON] et al. [73] & Fuzzy integral & \multirow{2}{*}{Convenient} & Complicated \\ \hline [PERSON] et al. [74] & Choquet fuzzy integral & Effectively improve accuracy & Complicated \\ \hline \multirow{2}{*}{[PERSON] et al. [75]} & Fuzzy Integral with Particle & \multirow{2}{*}{Solving the uncertainty problem} & Dependent on \\  & Swarm Optimization & & problem \\ \hline [PERSON] et al. [76] & Fuzzy multi-attribute & \multirow{2}{*}{Novel and convenient} & Dependent on \\  & decision making & & index selection \\ \hline \multirow{2}{*}{[PERSON] and [PERSON] [77]} & Hesitation institution & \multirow{2}{*}{More realistic and reliable} & Complicated and \\  & fuzzy number & & abstract model \\ \hline \multirow{2}{*}{[PERSON] et al. [78]} & TOPSIS and aggregation & Efficient and robust, realistic & Complicated and \\  & approach & and reasonable & abstract model \\ \hline \hline \end{tabular}
\end{table}
Table 3: Literature review of fuzzy integral and fuzzy set.  
Choquet integral [79]: Without loss of generality, assume that the evaluation values on \(n\) criteria \(\{f(x_{i})\big{|}i=1,2,\ldots,n\}\) satisfy \(f(x_{1})\geq\ldots\geq f(x_{i})\geq\ldots\geq f(x_{n})\). Then, the Choquet fuzzy integral of \(f(x)\) among \(g(x)\) on \(X\) is:  
\[\begin{array}{rl}\int fdg&=f(x_{n})g_{\lambda}(X_{n})\\ &+\Big{[}f\Big{(}x_{n-1}\Big{)}-f(x_{n})\Big{]}g_{\lambda}\big{(}X_{n-1}\Big{)} \\ &+\ldots+\Big{[}f\Big{(}x_{1}\Big{)}-f\Big{(}x_{2}\Big{)}\Big{]}g_{\lambda} \big{(}X_{1}\Big{)}\end{array} \tag{2}\]  
where \(g_{\lambda}\big{(}X_{j}\big{)}=g_{\lambda}\big{(}\big{(}x_{1},x_{2},\ldots,x_{ j}\big{)}\big{)}\) represents the fuzzy measure of the corresponding criterion set.