## 4 The Fuzzy Grey TOPSIS Method
### The Procedure of Fuzzy Grey TOPSIS
#### 4.2.2 Defuzzify the Fuzzy Numbers
With the premise that \(J^{+}\) represents the set of some criteria which are the bigger the better, and \(J^{-}\) represents the set of some criteria which are the smaller the better, the vector \(C^{+}\) and \(C^{-}\) can be determined according to Equations (19) and (20). \[c_{j}^{+}=\left\{ \begin{array}{l}\underset{1\leq i\leq m}{\max}(c_{ij}),\text{ if}\,j\in J^{+}\\ \underset{1\leq i\leq m}{\min}(c_{ij}),\text{if}\,j\in J^{-}\end{array} \right., \tag{19}\]  
\[c_{j}^{-}=\left\{ \begin{array}{l}\underset{1\leq i\leq m}{\min}(c_{ij}),\text {if}\,j\in J^{+}\\ \underset{1\leq i\leq m}{\max}(c_{ij}),\text{if}\,j\in J^{-}\end{array} \right.. \tag{20}\]  
The positive correlation coefficients \(\left\{\xi_{ij}^{+},i=1,2,\ldots,m,j=1,2,\ldots,n\right\}\) and the negative correlation coefficients \(\left\{\xi_{ij}^{-},i=1,2,\ldots,m,j=1,2,\ldots,n\right\}\) of the \(j\)-th criterion can be calculated according to Equations (21) and (22):  
\[\xi_{ij}^{+}=\frac{\underset{i}{\min}\underset{k}{\min}\left|c_{ij}^{+}-c_{ ij}\right|+\rho\underset{i}{\max}\underset{k}{\max}\left|c_{ij}^{+}-c_{ij} \right|}{\left|c_{ij}^{+}-c_{ij}\right|+\rho\underset{i}{\max}\underset{k}{ \max}\left|c_{ij}^{+}-c_{ij}\right|}, \tag{21}\]  
\[\xi_{ij}^{-}=\frac{\underset{i}{\min}\underset{k}{\min}\left|c_{ij}^{-}-c_{ ij}\right|+\rho\underset{i}{\max}\underset{k}{\max}\left|c_{ij}^{-}-c_{ij} \right|}{\left|c_{ij}^{-}-c_{ij}\right|+\rho\underset{i}{\max}\underset{k}{ \max}\left|c_{ij}^{-}-c_{ij}\right|}. \tag{22}\]  
Then the positive correlation matrix \(\xi^{+}\) and the negative correlation matrix \(\xi^{-}\) can be obtained as  
\[\xi^{+}=\left[ \begin{array}{ccc}\xi_{11}^{+}&\ldots&\xi_{1n}^{+}\\ \vdots&\ddots&\vdots\\ \xi_{m1}^{+}&\ldots&\xi_{mn}^{+}\end{array} \right],\]  
\[\xi^{-}=\left[ \begin{array}{ccc}\xi_{11}^{-}&\ldots&\xi_{1n}^{-}\\ \vdots&\ddots&\vdots\\ \xi_{m1}^{-}&\ldots&\xi_{mn}^{-}\end{array} \right].\]  
**Step 5.2: Calculate the grey fuzzy Choquet integral. **  
To take the interactions among criteria into account, the Choquet fuzzy integral is used to integrate the correlation coefficients of every object. The positive grey fuzzy integral vector \(R^{+}=\left[R_{1}^{+},R_{2}^{+},\ldots,R_{m}^{+}\right]\) and the negative grey fuzzy integral vector \(R^{-}=\left[R_{1}^{-},R_{2}^{-},\ldots,R_{m}^{-}\right]\) can be calculated as follows. **Definition 1**. : _(Positive grey fuzzy Choquet integral): Let correlation coefficients \(\left\{\xi_{ij}^{+};i=1,2,\ldots,m,j=1,2,\ldots,n\right\}\) be reordered in an ascending order such that \(\xi_{i1}^{+}\geq\ldots\geq\xi_{ij}^{+}\geq\ldots\geq\xi_{in}^{+}\). The positive grey fuzzy Choquet integral of the \(i\)-th object is defined as_  
\[ \begin{array}{rl}\text{R}_{i}^{+}&=\xi_{in}^{+}\omega_{12\ldots n}+\left[\xi _{in-1}^{+}-\xi_{in}^{+}\right]\omega_{12\ldots(n-1)}\\ &+\ldots+\left[\xi_{i1}^{+}-\xi_{i2}^{+}\right]\omega_{1}\end{array} , \tag{23}\]  
_where \(\omega_{12\ldots k}\) represents the weight of the set \(\left\{x_{1},x_{2},\ldots,x_{k}\right\}\)\(\left(k\leq n\right)\) after rearrangement._  
**Definition 2**. : _(Negative grey fuzzy Choquet integral): Let correlation coefficients \(\left\{\xi_{ij}^{-};i=1,2,\ldots,m,j=1,2,\ldots,n\right\}\) be reordered in an ascending order such that \(\xi_{i1}^{-}\geq\ldots\geq\xi_{ij}^{-}\geq\ldots\geq\xi_{in}^{-}\). The negative grey fuzzy Choquet integral of the \(i\)-th object is defined as_  
\[ \begin{array}{rl}\text{R}_{i}^{-}&=\xi_{in}^{-}\omega_{12\ldots n}+\left[\xi _{in-1}^{-}-\xi_{in}^{-}\right]\omega_{12\ldots(n-1)}\\ &+\ldots+\left[\xi_{i1}^{-}-\xi_{i2}^{-}\right]\omega_{1}\end{array} . \tag{24}\]  
**Step 6: Calculate fuzzy distance to ideal solution. **  
To embody the difference the in distance of all objects, the distance between all objects to the positive and negative ideal solution schemes should be obtained. The Choquet fuzzy integral should also be employed to amend the Euclidean distance such that the interactions can be expressed in the distance.