# Sensitivity Analysis of PROMETHEE II for the Evaluation of Environmental Websites
## 3 Results
### Comparison with Other MCDM  
In order to check whether PROMETHEE II is effective in ranking environmental websites, a comparison with three other common MCDM models was conducted. More specifically, SAW, WPM and TOPSIS, were selected for this purpose. These three models are very popular and have been implemented in the past for the evaluation of websites and have proved to be rather effective [9; 16; 22].  
\begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline  & **SAW** & **SAW Ranking** & **WPM** & **WPM Ranking** & **PROMETHEE II** &
\begin{tabular}{l} **PROMETHEE** \\ **II Ranking** \\ \end{tabular} & **TOPSIS** & **TOPSIS Ranking** \\ \hline  
\hline \end{tabular}
\end{table}
Table 4: Values and Ranking for all websites using SAW, WPM, TOPSIS and PROMETHEE II.  
For this purpose, we use the values of criteria given by all users that are presented in Table 3. According to SAW, we calculate the multi-attribute utility function \(U\) for each one of the 23 websites as a linear combination of the values of the 10 criteria:  
\[U(A_{j})=\sum_{i=1}^{10}w_{i}x_{ij},\]  
where \(A_{j}\) is one alternative website and \(x_{ij}\) is the value of the \(i\) criterion for the \(A_{j}\) website.  
Similarly, according to WPM the utility function \(U\) is calculated for each one of the same websites.  
\[U(A_{j})=\prod_{i=1}^{10}(ux_{ij})^{w_{i}},\]  
for \(j=1,\ldots,23\).  
The term \(U(A_{j})\) denoted the total performance value of the alternative \(A_{j}\).  
The central principle in TOPSIS model is that the best alternative should have the shortest distance from the ideal solution and the farthest distance from the negative-ideal solution.  
Identify Positive-Ideal and Negative-Ideal Solutions. The positive ideal solution is the composite of all best attribute ratings attainable and is denoted:  
\[A^{*}=\{v_{1}^{*},v_{2}^{*},v_{3}^{*}\}\]  
where \(v_{i}^{*}\) is the best-weighted rating for the dimension \(i\) among all alternatives. The negative-ideal solution is the composite of all worst attribute ratings attainable, and is denoted:  
\[A^{-}=\left\{v_{1}^{-},v_{2}^{-},v_{3}^{-}\right\}\]  
where \(v_{i}^{-}\) is the worst value for the dimension \(i\) among all websites.  
Calculate the separation measure from the positive-ideal and negative-ideal alternative. The separation of each alternative from the positive-ideal solution \(A^{*}\) is given by the \(n\)-dimensional Euclidean distance:  
\[S_{j}^{*}=\sqrt{\sum_{i=1}^{3}(v_{ij}-v_{i}^{*})^{2}},\]  
where \(j\) is the index related to the alternatives and \(i\) one of the \(n\) attributes. Similarly, the separation from the negative-ideal solution \(A^{-}\) is given by  
\[S_{j}^{-}=\sqrt{\sum_{i=1}^{10}(v_{ij}-v_{i}^{-})^{2}}.\]  
Calculate Similarity Indexes. The similarity to positive-ideal solution, for alternative \(j\), is finally given by  
\[C_{j}^{*}=\frac{S_{j}^{-}}{S_{j}^{*}+S_{j}^{-}}\]  
with \(0\leq C_{j}^{*}\leq 1\). The alternatives can then be ranked according to \(C_{j}^{*}\) in descending order.  
The values of utility function \(U\) using SAW, WPM, C of TOPSIS, and the final value \(\phi(\alpha)\) of PROMETHEE II for each alternative website are used for ranking the 23 websites. The higher the value of \(U\), C or \(\phi(\alpha)\), the better the website is considered. The values of \(U\) according to SAW and WPM, the values of C according to TOPSIS and the values of \(\phi(\alpha)\) according to PROMETHEE II as well as the ranking order to the websites using the four different models are presented in Table 4.