## 4 The Fuzzy Grey TOPSIS Method
### The Procedure of Fuzzy Grey TOPSIS
#### 4.2.2 Defuzzify the Fuzzy Numbers
**\(\forall\ x,y,p,q,\ \frac{c_{xk}-c_{yk}}{c_{pk}-c_{yk}}\)  
\[=\left(\frac{j_{xk}-\overline{j_{k}}}{S_{k}}-\frac{j_{yk}-\overline{j_{k}}}{S_ {k}}\right)\left(\frac{j_{pk}-\overline{j_{k}}}{S_{k}}-\frac{j_{yk}-\overline {j_{k}}}{S_{k}}\right)=\frac{j_{xk}-j_{yk}}{j_{pk}-j_{yk}}.\]  
\(\Box\)  
**Step 3: Obtain weights of criteria. **  
To aggregate the evaluation data, some series of weights that are reasonable and accurate as much as possible is certainly needed. The weights are considered in two aspects. On the one hand, the objective weights are calculated through the entropy weight method such that the influence of datacan be considered. On the other hand, the subjective weights are given by the experts to express the decision preference. Finally, the weighted average value of objective and subjective weights is taken. **Step 3.1: Calculate objective weights of criteria. **  
The objective initial weights are mainly calculated through the entropy weight method. However, some minor amendments are made to make the formula meaningful. By using the data of standardized evaluation matrix \(C=\left\{c_{ij},i=1,\;2,\ldots,m,j=1,\;2,\ldots,n\right\}\), the objective weights vector \(\omega^{1}=\left[\omega^{1}_{1},\omega^{1}_{2},\ldots,\omega^{1}_{n}\right]\) can be calculated according to Equation (15). \[\left\{ \begin{array}{l}\omega^{1}_{j}=\frac{1-H_{j}}{\sum_{i=1}^{m}(1-H_{j} )}\\ H_{j}=\frac{\sum_{i=1}^{m}\left(1+P_{ij}\right)\ln\left(1+P_{ij}\right)}{-\ln \left(m\right)}\\ P_{ij}=\frac{c_{ij}}{\sum_{i=1}^{m}\left|c_{ij}\right|}\end{array} \right.. \tag{15}\]  
**Step 3.2: Obtain subjective weights of criteria. **  
The subjective weights vector \(\omega^{2}=\left[\omega^{2}_{1},\omega^{2}_{2},\ldots,\omega^{2}_{n}\right]\) is given by experts. **Step 3.3: Integrate objective and subjective weights. **  
The comprehensive weights vector \(\omega^{0}=\left[\omega^{0}_{1},\omega^{0}_{2},\ldots,\omega^{0}_{n}\right]\) can be found according to Equation (16). \[\omega^{0}_{j}=\theta\omega^{1}_{j}+(1-\theta)\omega^{2}_{j}, \tag{16}\]  
where the parameter \(\theta\in(0,1)\) is determined by the decision maker according to the degree of trust in objective data and subjective judgment of experts. We set it as 0.5 in this paper. **Step 4: Identification of \(\lambda\)-fuzzy-measures. **  
It is generally believed that many interactions exist among criteria. If we integrate the evaluation matrix by weights only, the interactions are ignored, which will lead to an inadequate decision. Thus, the \(\lambda\)-fuzzy-measures are employed here to express the interactions. Let us regard the comprehensive weights vector \(\omega^{0}=\left[\omega^{0}_{1},\omega^{0}_{2},\ldots,\omega^{0}_{n}\right]\) as the fuzzy densities. The parameter \(\lambda\) can be obtained by Equation (17):  
\[\frac{1}{\lambda}\left|\prod_{i=1}^{n}\left(1+\lambda\omega^{0}_{\rm i}\right) -1\right|=1. \tag{17}\]  
Note that the comprehensive weights vector \(\omega^{0}=\left[\omega^{0}_{1},\omega^{0}_{2},\ldots,\omega^{0}_{n}\right]\) and the parameter \(\lambda\) are obtained. All the fuzzy-measures \(\left\{\omega_{Q};Q\in P(X)\right\}\) can be calculated according to Equation (18), where \(P(X)\) is the power set of \(X\). \[ \begin{array}{ll}\omega_{Q}&=\sum\limits_{i=1}^{p}\omega^{0}_{i}+\lambda \sum\limits_{i_{1}=1}^{p-1}\sum\limits_{i=i_{1}+1}^{p}\omega^{0}_{i_{1}}\omega^ {0}_{i_{2}}\\ &+\ldots+\lambda^{p-1}\omega^{0}_{1}\omega^{0}_{2}\ldots\omega^{0}_{p}\end{array}  \tag{18}\]  
**Step 5: Calculate the grey fuzzy correlation degree. **  
**Step 5.1: Calculate correlation coefficients. **  
To describe the difference in shapes of all objects, their positive and the negative correlation coefficients need to be calculated. First of all, the positive and the negative ideal solution schemes \(C^{+}=\left[c^{+}_{1},c^{+}_{2},\ldots,c^{+}_{n}\right]\) and \(C^{-}=\left[c^{-}_{1},c^{-}_{2},\ldots,c^{-}_{n}\right]\) should be selected.