## 4 The Fuzzy Grey TOPSIS Method
### The Procedure of Fuzzy Grey TOPSIS
#### 4.2.2 Defuzzify the Fuzzy Numbers
(Positive fuzzy ideal solution distance): Let positive distances of all criteria \(\{d_{ij}^{+}=\left(c_{ij}-c_{j}^{+}\right)^{2};i=1,2,\ldots,m,j=1,2,\ldots,n\}\) be reordered in an ascending order such that \(d_{i1}^{+}\geq\ldots\geq d_{ij}^{+}\geq\ldots\geq d_{in}^{+}\). The positive fuzzy ideal solution distance of the \(i\)-th object is defined as  
\[ \begin{array}{ll}D_{i}^{+}&=[d_{in}^{+}\omega_{12\ldots n}+\left(d_{in-1}^{+ }-d_{in}^{+}\right)\omega_{12\ldots(n-1)}\\ &+\ldots+\left(d_{i1}^{+}-d_{i2}^{+}\right)\omega_{1}]^{\frac{1}{2}}\end{array} . \tag{25}\]  
(Negative fuzzy ideal solution distance): Let negative distances of all criteria \(\{d_{ij}^{-}=\left(c_{ij}-c_{j}^{-}\right)^{2};i=1,2,\ldots,m,j=1,2,\ldots,n\}\) be reordered in an ascending order such that \(d_{i1}^{-}\geq\ldots\geq d_{ij}^{-}\geq\ldots\geq d_{in}^{-}\). The positive ideal solution fuzzy distance of the \(i\)-th object is defined as  
\[ \begin{array}{ll}D_{i}^{-}&=[d_{in}^{-}\omega_{12\ldots n}+\left(d_{in-1}^{- }-d_{in}^{-}\right)\omega_{12\ldots(n-1)}\\ &+\ldots+\left(d_{i1}^{-}-d_{i2}^{-}\right)\omega_{1}]^{\frac{1}{2}}\end{array} . \tag{26}\]  
**Step 7: Get grey fuzzy TOPSIS. **  
The positive grey fuzzy Choquet integral and the negative ideal solution distance depend on the close degree to the positive ideal solution of each scheme on the shape and distance, respectively. Their combination can represent the comprehensive close degree to the positive ideal solution of each scheme. Analogously, the combination of negative grey fuzzy Choquet integral and the positive ideal solution distance can characterize the close degree to the negative ideal solution of each scheme. To express the effects of \(R^{+},R^{-},D^{+},\text{ and }D^{-}\), they need to be normalized before combination. \[N_{i}^{new}=\frac{N_{i}}{\underset{1\leq i\leq m}{\max}N_{i}};i=1,2,\ldots,m, \tag{27}\]  
where \(N_{i}\) represent \(R^{+}(R^{-};D^{+};D^{-})\), respectively. For the conciseness, the normalized values of them are still denoted as \(\text{R}^{+},\text{R}^{-},\text{D}^{+},\text{D}^{-}\), respectively. The positive comprehensive proximity degree vector \(\text{S}^{+}=\left[\text{S}_{1}^{+},\text{S}_{2}^{+},\ldots,\text{S}_{m}^{+}\right]\) and the negative comprehensive proximity degree vector \(\text{S}^{-}=\left[\text{S}_{1}^{-},\text{S}_{2}^{-},\ldots,\text{S}_{m}^{-}\right]\) can be obtained by combining \(\text{R}^{+},\text{R}^{-},\text{D}^{+},\text{ and }\text{D}^{-}\) with a proper weight parameter p according to Equation (28). \[\left\{ \begin{array}{ll}\text{S}_{1}^{+}=pR_{i}^{+}+(1-p)D_{i}^{-};i=1,2, \ldots,m\\ \text{S}_{i}^{-}=pR_{i}^{-}+(1-p)D_{i}^{+};i=1,2,\ldots,m\end{array} \right., \tag{28}\]  
where parameter \(p\in(0,1)\) is determined by the decision-maker's preference. The comprehensive evaluation vector \(CS=[CS_{1},CS_{2},\ldots,CS_{m}]\) can be calculated by Equation (29). \[CS_{i}=\frac{S_{i}^{+}}{S_{i}^{-}+S_{i}^{+}};i=1,2,\ldots,m. \tag{29}\]The \(m\) objects can be evaluated reasonably according to the comprehensive evaluation vector \(CS=[CS_{1},CS_{2},\dots,CS_{m}]\). The object whose comprehensive evaluation value is big is better than the one whose comprehensive evaluation value is small. The flow chart of the proposed method is given in Figure 3.