## 3 Performance Attributes and Scenarios
### MCDM Approach for Evaluating Thermal Gasification Techniques for MSW
#### 3.4.3 Normalization of the Decision Matrix
As a decision-maker, it is very difficult to reach a unique preference under multiple criteria. Therefore, the closeness to an ideal solution for each alternative thermal gasification technique for a given criterion and the alternative thermal gasification technique ranking is essential. Here, in this study, five weighting scenarios are adopted (refer to Table 11 below), and the rankings are obtained. \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline  & \multicolumn{3}{c}{**E\({}_{\text{j}}\) Entropy Weight Non-Recycling Scenario**} & \multicolumn{3}{c}{**E\({}_{\text{j}}\) Entropy Weight Recycling Scenario**} \\ \hline
**Evaluation Criterion (Cj)** \(\downarrow\) & **V\({}^{\star}\)** & **Alternative \({}^{\#}\)** & **V\({}^{-}\)** & **Alternative \({}^{\star}\)** & **Alternative \({}^{\star}\)** & **Alternative \({}^{\star}\)** & **Alternative** \\ \hline C\({}_{5}\) & 0.000 & 6 & 0.096 & 1 & 0.005 & 3 and 6 & 0.068 & 1 \\ \hline C\({}_{6}\) & 0.029 & 3 & 0.059 & 1 & 0.043 & 2 and 3 & 0.053 & 1/5/6 \\ \hline C\({}_{7}\) & 0.007 & 6 & 0.041 & 1 & 0.007 & 6 & 0.041 & 1 \\ \hline C\({}_{8}\) & 0.029 & 3 & 0.056 & 6 & 0.031 & 3 & 0.062 & 6 \\ \hline C\({}_{9}\) & 0.015 & 1 & 0.077 & 6 & 0.013 & 1 & 0.064 & 6 \\ \hline C\({}_{10}\) & 0.057 & 2 & 0.000 & 6 & 0.057 & 2 & 0.000 & 6 \\ \hline C\({}_{11}\) & 0.052 & 1 & 0.009 & 6 & 0.052 & 2 & 0.009 & 6 \\ \hline C\({}_{12}\) & 0.005 & 2 & 0.065 & 3 & 0.014 & 2 & 0.058 & 1/4/5/6 \\ \hline \hline \end{tabular} \({}^{\#}\) Note: 1: Updraft; 2: Downdraft; 3: Cross-draft; 4: Bubbling bed; 5: Circulating fluidized bed; and 6: Dual fluidized bed.  
\end{table} 
Table 10: _Cont._Determine the Closeness to an Ideal Solution for Each Alternative Thermal Gasification Technique for a Given Criterion and Rank the Alternative  
For a given alternative thermal gasification technique for a given criterion, its distance from the best ideal gasification strategy is obtained using Equation (14). \[\text{D}_{\text{i}}^{+}=\sqrt{\sum_{\text{j}}\left(\text{V}_{\text{j}}^{+}-\text{ V}_{\text{ij}}\right)} \tag{14}\] \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Scenario 2: Entropy Weights Based on the Recycling Data Set**}} & \multicolumn{6}{c}{**For Recycling MSW**} \\ \cline{2-9}  & \multicolumn{2}{c}{**For Non-Recycling MSW**} & \multicolumn{6}{c}{**For Recycling MSW**} \\ \cline{2-9}  & \multicolumn{2}{c}{**D\({}_{\text{i}}^{+}\)**} & \multicolumn{2}{c}{**D\({}_{\text{i}}^{-}\)**} & \multicolumn{2}{c}{**D\({}_{\text{i}}^{-}\)**/(D\({}_{\text{i}}^{+}\) + D\({}_{\text{i}}^{-}\))**} & \multicolumn{2}{c}{**Rank C\({}_{\text{i}}\)**} & \multicolumn{2}{c}{**D\({}_{\text{i}}^{+}\)**} & \multicolumn{2}{c}{**D\({}_{\text{i}}^{-}\)**} & \multicolumn{2}{c}{**D\({}_{\text{i}}^{-}\)**/(D\({}_{\text{i}}^{+}\) + D\({}_{\text{i}}^{-}\))**} & \multicolumn{2}{c}{**Rank C\({}_{\text{i}}\)**} \\ \hline Updraft & 0.1088 & 0.1093 & 0.5010 & 5 & 0.0981 & 0.0783 & 0.4440 & 4 \\ \hline Downdraft & 0.0565 & 0.1506 & 0.7272 & 1 & 0.0485 & 0.1150 & 0.7032 & 1 \\ \hline Cross-draft & 0.1204 & 0.1083 & 0.4735 & 6 & 0.0776 & 0.0951 & 0.5508 & 2 \\ \hline Bubbling bed & 0.0855 & 0.1138 & 0.5711 & 3 & 0.0964 & 0.0614 & 0.3891 & 6 \\ \hline Circulating fluidized bed & 0.0751 & 0.1208 & 0.6167 & 2 & 0.0829 & 0.0735 & 0.4697 & 3 \\ \hline Dual fluidized bed & 0.1020 & 0.1220 & 0.5446 & 4 & 0.1077 & 0.0804 & 0.4274 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Cont. For a given thermal gasification technique, its distance from the worst ideal thermal gasification technique is obtained using Equation (15). \[\mathrm{D_{i}^{-}}=\sqrt{\sum_{\mathrm{j}}\left(\mathrm{V_{j}^{-}-V_{\mathrm{\bar{ ij}}}}\right)} \tag{15}\]  
For a given thermal gasification technique, its closeness to the ideal thermal gasification technique is obtained using Equation (16). \[\mathrm{C_{i}}=\frac{\mathrm{D_{i}^{-}}}{\mathrm{D_{i}^{+}+D_{i}^{-}}} \tag{16}\]  
In Equation (16) above, the \(\mathrm{C_{i}}\) value ranges between one and unity.