## 3 Performance Attributes and Scenarios
### MCDM Approach for Evaluating Thermal Gasification Techniques for MSW
#### 3.4.2 Estimation of Criterion Entropy Weights  
The criterion followed for evaluation of the different gasification methods is represented by the entropy weight. A more important criterion is given more weight and vice versa. Equation (8) gives the entropy weight (Eij) calculation methodology used for the jth criterion of the ith alternative. Equation (9) gives the methodology to determine the criterion entropy weight (\(W_{i}\)).  
\[E_{j}=-\frac{\sum_{i=1}^{m}[S_{ij}*ln(S_{ij})]}{\ln(m)} \tag{8}\]  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Evaluation Criterion (j)\(\rightarrow\)** & **1** & **2** & **N** \\ \multicolumn{4}{l}{**Alternative Thermal Gasification Techniques (i) \(\downarrow\)**} \\ \hline
1 & C11 & C12 & C1n \\ \hline
2 & C21 & C22 & C2n \\ \hline. &. &. &. \. &. &. &. \. &. &. &. \\ \hline M & Cm1 & Cm2 & Cmn \\ \hline Criterion Weight \(\rightarrow\) & W1 & W2 & Wn \\ \hline \hline \end{tabular}
\end{table}
Table 4: Matrix for decision making for the ith thermal gasification technique and jth criterion.  
\[\mathrm{W_{j}}=\frac{1-\mathrm{E_{j}}}{\left[1-\sum_{j=1}^{n}\mathrm{E_{j}}\right]} \tag{9}\]  
Table 7 displays the entropy weight \(\mathrm{E_{j}}\) for each of the thermal gasification method evaluation criteria.  
\begin{table}
\begin{tabular}{l l l l l l l} \hline \multicolumn{6}{c}{\(\mathbf{A_{i}}\): Alternative Thermal Gasification Techniques} \\ \hline \multicolumn{6}{c}{**Evaluation**} \\ \multicolumn{6}{c}{**Criterion (\(\mathrm{C_{j}}\)) \(\downarrow\)**} & **Updraft** & **Downdraft** & **Cross-Draft** & **Bubbling Bed** & \multicolumn{1}{c}{**Circulating Fluidized Bed**} & \multicolumn{1}{c}{**Dual Fluidized Bed**} \\ \hline \(\mathrm{C_{1}}\) & 0.671 & 0.871 & 0.414 & 1.000 & 0.812 & 0.000 \\ \hline \(\mathrm{C_{2}}\) & 0.344 & 0.000 & 0.500 & 0.500 & 1.000 & 0.813 \\ \hline \(\mathrm{C_{3}}\) & 0.083 & 0.108 & 0.000 & 0.100 & 0.175 & 1.000 \\ \hline \(\mathrm{C_{4}}\) & 0.000 & 0.627 & 0.104 & 0.851 & 0.851 & 1.000 \\ \hline \(\mathrm{C_{5}}\) & 1.000 & 0.000 & 0.009 & 0.620 & 0.375 & 0.009 \\ \hline \(\mathrm{C_{6}}\) & 0.000 & 1.000 & 1.000 & 0.000 & 0.000 & 0.000 \\ \hline \(\mathrm{C_{7}}\) & 0.000 & 1.000 & 1.000 & 0.000 & 0.000 & 0.000 \\ \hline \(\mathrm{C_{12}}\) & 0.000 & 1.000 & 0.921 & 0.000 & 0.000 & 0.000 \\ \hline \end{tabular}
\end{table}
Table 6: Standardization matrix \(\mathrm{S_{ij}^{{}^{\prime}}}\) for alternative thermal gasification techniques for the MSW recycling scenario.  
\begin{table}
\begin{tabular}{l l l l l l l} \hline \multicolumn{6}{c}{\(\mathbf{A_{i}}\): Alternative Thermal Gasification Techniques} \\ \hline \multicolumn{6}{c}{**Evaluation**} \\ \multicolumn{6}{c}{**Criterion (\(\mathrm{C_{j}}\)) \(\downarrow\)**} & **Updraft** & **Downdraft** & **Cross-Draft** & **Bubbling Bed** & \multicolumn{1}{c}{**Circulating Fluidized Bed**} & \multicolumn{1}{c}{**Dual Fluidized Bed**} \\ \hline \(\mathrm{C_{1}}\) & 0.583 & 0.708 & 0.338 & 1.000 & 0.689 & 0.000 \\ \hline \(\mathrm{C_{2}}\) & 0.305 & 0.000 & 0.666 & 0.721 & 1.000 & 0.782 \\ \hline \(\mathrm{C_{3}}\) & 0.106 & 0.064 & 0.000 & 0.077 & 0.123 & 1.000 \\ \hline \(\mathrm{C_{4}}\) & 0.000 & 0.714 & 0.143 & 0.857 & 0.857 & 1.000 \\ \hline \(\mathrm{C_{5}}\) & 1.000 & 0.015 & 0.004 & 0.138 & 0.076 & 0.000 \\ \hline \(\mathrm{C_{6}}\) & 0.000 & 1.000 & 1.000 & 0.000 & 0.000 & 0.000 \\ \hline \(\mathrm{C_{7}}\) & 0.000 & 0.400 & 0.200 & 0.596 & 0.810 & 0.000 \\ \hline \(\mathrm{C_{9}}\) & 1.000 & 0.941 & 0.941 & 0.647 & 0.765 & 0.000 \\ \hline \(\mathrm{C_{10}}\) & 0.874 & 1.000 & 0.202 & 0.093 & 0.143 & 0.000 \\ \hline \(\mathrm{C_{11}}\) & 1.000 & 0.300 & 0.100 & 0.090 & 0.400 & 0.000 \\ \hline \(\mathrm{C_{12}}\) & 0.725 & 1.000 & 0.000 & 0.725 & 0.725 & 0.725 \\ \hline \end{tabular}
\end{table}
Table 5: Standardization matrix \(\mathrm{S_{ij}^{{}^{\prime}}}\) for alternative thermal gasification techniques for the MSW non-recycling scenario.