## 4 The Fuzzy Grey TOPSIS Method
### The Procedure of Fuzzy Grey TOPSIS
#### 4.2.2 Defuzzify the Fuzzy Numbers  
The fuzzy numbers need to be defuzzified into numerical values to take part in the calculation procedure later. There are many defuzzification methods, each of which has its advantages and limits. Three methods are adopted in this paper. In the case that the operation is too simple, and its effectiveness cannot be verified [83], the mean value of the results of the three methods is taken. Fuzzy numbers \(\overline{j}_{ik}=(a_{ik},b_{ik},c_{ik})\) can be defuzzicated through the three methods stated as follows:  
(1) Distance measure method [81]:  
\[M_{ik}^{1}=\frac{d_{ik}^{-}}{d_{ik}^{-}+d_{ik}^{\ast}}. \tag{8}\]  
If the optimal fuzzy evaluation value is defined as \(\overline{j}_{ik}^{\ast}=(1,1,1)\) and the worst fuzzy evaluation value is defined as \(\overline{j}_{ik}^{\ast}=(0,0,0)\), then  
\[d_{ik}^{-}=\sqrt{\frac{1}{4}\big{(}a_{ik}^{2}+2b_{ik}^{2}+c_{ik}^{2}\big{)}}, \tag{9}\]  
\[d_{ik}^{\ast}=\sqrt{\frac{1}{4}\big{[}\big{(}1-a_{ik}\big{)}^{2}+2\big{(}1-b_{ ik}\big{)}^{2}+\big{(}1-c_{ik}\big{)}^{2}\big{]}}. \tag{10}\]  
(2) Central value method [84]:  
\[M_{ik}^{2}=b_{ik}+\frac{(c_{ik}-b_{ik})-(b_{ik}-a_{ik})}{6} \tag{11}\]  
(3) Gravity method [84]:  
\[M_{ik}^{3}=\left\{ \begin{array}{ccc}a_{ik}&,&a_{ik}=b_{ik}=c_{ik}\\ \frac{(c_{ik})^{2}-(a_{ik})^{2}+b_{ik}(c_{ik}-a_{ik})}{3(c_{ik}-a_{ik})}&,& \text{otherwise}\end{array} \right.. \tag{12}\]  
The numerical value \(j_{ik}\) can be found by taking the mean value of the three results, i.e.,  
\[j_{ik}=\frac{M_{ik}^{1}+M_{ik}^{2}+M_{ik}^{3}}{3}. \tag{13}\]  
The fuzzy evaluation matrix \(\overline{J}\) can be transformed into a numerical evaluation matrix \(J^{\ast}\). \[J^{\ast}=\left[ \begin{array}{ccc}j_{1(n-i+1)}&\cdots&j_{1n}\\ \vdots&\ddots&\vdots\\ j_{m(n-i+1)}&\cdots&j_{mn}\end{array} \right].\]  
**Step1.2: Find evaluation data of quantitative criteria. **  
The evaluation data of a few numerical criteria \([j_{kl},j_{22},\ldots,j_{kl}]k=1,2,\ldots,m]\) are obtained through statistics and measurement methods. By seaming the numerical evaluation matrix of qualitative and quantitative criteria, the evaluation matrix \(J\) can be found, as shown below,  
\[J=\left[ \begin{array}{ccc}j_{11}&\ldots&j_{1n}\\ \vdots&\ddots&\vdots\\ j_{m1}&\ldots&j_{mn}\end{array} \right],\]  
where \(J_{kl}\) represents the evaluation value of the \(k\)-th object on the \(l\)-th criterion, \(k=1,\ 2,\ \ldots,\ m,\ l=1,\ 2,\ \ldots,\ n\). **Step 2: Standardize evaluation matrix. **  
To eliminate the influence of dimension and order of magnitude, the evaluation matrix \(J\) needs to be standardized according to Equation (14):  
\[c_{ik}=\frac{j_{ik}-\overline{j_{k}}}{S_{k}}, \tag{14}\]  
where \(\overline{j}_{k}\) is the mean and \(S_{k}\) is the standard deviation of the evaluation values of the \(k\)-th criterion. In that way, the standardized evaluation matrix \(C\) can be found. \[C=\left[ \begin{array}{ccc}c_{11}&\ldots&c_{1n}\\ \vdots&\ddots&\vdots\\ c_{m1}&\cdots&c_{mn}\end{array} \right],\]  
where \(J_{kl}\) represents the standardized evaluation value of the \(k\)-th object on the \(l\)-th criterion, \(k=1,\ 2,\ \ldots,\ m,\ l=1,\ 2,\ \ldots,\ n\). **Theorem 1. **_The standardization transformation satisfies the following two axioms:_  
_(1) Isotonicity:_  
\[\text{If }j_{xk}<j_{yk},\text{ then }c_{xk}<c_{yk}.\text{If }j_{xk}>j_{yk}, \text{ then }c_{xk}>c_{yk}.\]  
**Proof. ** If \(j_{xk}<j_{yk}\), then \(j_{xk}-\overline{j_{k}}<j_{yk}-\overline{j_{k}}\). Thus \(\frac{j_{ik}-\overline{j_{k}}}{S_{k}}<\frac{j_{yk}-\overline{j_{k}}}{S_{k}}\), namely, \(c_{xk}<c_{yk}\). Similarly, if \(j_{xk}>j_{yk}\), then \(c_{xk}>c_{yk}\). _(2) Difference remaining:_  
\[\forall\ x,y,p,q,\ \frac{j_{xk}-j_{yk}}{j_{pk}-j_{yk}}=\frac{c_{xk}-c_{yk}}{c_{ pk}-c_{yk}}.\]  
\(\Box\)  
**Proof.