## 5 An Illustrative Example
### Conducting the Proposed Method  
The detailed process of the proposed method to solve this example is illustrated below. Each level is addressed one by one. We illustrate the quality level in detail only for economy of space. **Step 1: Get the evaluation matrix. **  
The fuzzy linguistic value evaluation matrices \(\left\{\widetilde{j}_{1},\ \widetilde{j}_{2},\ \widetilde{j}_{3},\ \widetilde{j}_{4},\ \widetilde{j}_{5},\ \widetilde{j}_{6}\right\}\) are shown as follows, in which the mapping rules of linguistic variables and fuzzy number are given in Table 4.  
\[\widetilde{J}_{1}=\left[ \begin{array}{ccccccccccccc}S&S&G&S&S&G&S&S&S&G&S&S\\ 0.98&&&&&&&&&&&&\\ G&N&G&G&N&N&G&G&S&G&G\\ S&G&S&S&S&S&N&G&G&S&G&S\\ G&G&S&G&S&G&S&S&G&S&S&S\\ \end{array} \right],\]  
\[\widetilde{J}_{2}=\left[ \begin{array}{ccccccccccccc}B&B&B&T&T&B&B&B&B&T&B&B\\ 0.83&&&&&&&&&&&&&&&&\\ N&N&S&G&N&N&N&G&N&N&G&G\\ N&G&G&N&G&S&G&G&G&N&G\\ G&G&G&S&G&G&G&S&G&N&G\\ \end{array} \right]\]  
\[\widetilde{J}_{3}=\left[ \begin{array}{ccccccccccccc}S&S&G&S&S&S&G&G&S&G&S&S\\ 0.92&&&&&&&&&&&&&&&&\\ N&N&S&G&N&N&N&G&N&B&N&N\\ N&N&S&B&N&B&B&B&N&N&N&B\\ N&N&N&B&N&N&S&G&G&N&N&N\\ \end{array} \right],\]  
Figure 4: Supplier evaluation criterion system. \[\overline{J}_{4}=\left[ \begin{array}{cccccccccccc}G&G&G&G&G&G&S&G&G&N&G\\ 0.93&&&&&&&&\\ G&G&G&N&G&G&G&G&S&G&S\\ G&G&N&N&G&G&G&G&S&G&G\\ G&N&N&G&G&G&G&N&G&G&G\\ \end{array} \right],\]  
\[\overline{J}_{5}=\left[ \begin{array}{cccccccccccc}G&S&S&S&S&S&S&G&S&S&S&S\\ 0.99&&&&&&&&\\ N&N&G&G&N&N&N&G&N&G&N&N\\ N&G&N&G&N&N&N&N&N&G&N&N\\ N&N&G&N&N&N&G&N&N&N&N\\ \end{array} \right]\]  
\[\overline{J}_{6}=\left[ \begin{array}{cccccccccccc}S&S&G&G&S&S&S&S&S&G&S&S\\ 0.95&&&&&&&&\\ G&G&G&N&G&G&N&N&G&G&G&G\\ N&N&N&N&G&G&N&N&G&N&N&G\\ N&N&N&G&N&N&N&G&N&N&N&G\\ \end{array} \right],\]  
where each matrix represents the evaluation data of an alternative, and each row in the matrix represents the evaluation data of one qualitative criterion given by 12 experts expect for the second row of each matrix that is the evaluation data of the quantitative criterion and has just one value. In fact, we can give the evaluation of qualitative criteria and quantitative criteria, respectively, but for the convenience of writing, we give them in one matrix. We transformed the fuzzy linguistic values into fuzzy numbers according to (6). Then these fuzzy numbers were aggregated according to Equation (6) and defuzzified according to Equations (8)-(13). Then the evaluation matrix \(J\) was found as follows:  
\[J=\left[ \begin{array}{cccccccc}0.8295&0.9800&0.6635&0.7839&0.7858\\ 0.2579&0.8300&0.5662&0.6635&0.7111\\ 0.8150&0.9200&0.5166&0.4172&0.5166\\ 0.6965&0.9300&0.7111&0.6800&0.6489\\ 0.8440&0.9900&0.5662&0.5497&0.5331\\ 0.8295&0.9500&0.6489&0.5662&0.5497\\ \end{array} \right],\]  
where each row represents the evaluation data of an alternative, and each column represents the evaluation data of a criterion. The 3-dimensional histogram of \(J\) is portrayed in Figure 5.  
Figure 5: The 3-dimensional histograms of \(J\). **Step 2: Standardize evaluation matrix. **  
The evaluation matrix \(J\) can be standardized as \(C\) according to Equation (14), as shown as below. \[C=\left[ \begin{array}{cccc}0.5129&0.8115&0.6977&1.3681&1.4790\\ -1.9837&-1.7970&-0.6225&0.4202&0.7949\\ 0.4494&-0.2319&-1.2968&-1.5178&-0.9849\\ -0.0679&-0.0580&1.3439&0.5502&0.2263\\ 0.5763&0.9854&-0.6225&-0.4755&-0.8334\\ 0.5129&0.2898&0.5002&-0.3452&-0.6819\\ \end{array} \right].\]  
The 3-dimensional histograms of \(C\) is depicted in Figure 6.  
**Step 3: Obtain weights of criteria. **  
**Step 3.1: Calculated the objective weights of criteria. **  
The objective weight vector which was calculated by Equation (15) on the standardized evaluation matrix was \(\omega^{1}=[0.2216,\ 0.2098,\ 0.2159,\ 0.1818,\ 0.1709]\). **Step 3.2: Obtain the subjective weights of criteria. **  
The subjective weight vector which was given by experts was \(\omega^{2}=[0.1990,\ 0.6040,\ 0.2080,\ 0.2990,\ 0.3100]\). **Step 3.3: Integrate objective and subjective weights.