## 2 Materials and Methods
### The PROMETHEE Method
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Generalized Criterion** & **Definition** & **Parameters to Fix** \\ \hline Type 1: usual criterion & \(P(d)=\left\{\begin{array}{ll}0&d\leq 0\\ 1&d>0\end{array}\right.\) & - \\ Type 2: U-shape criterion & \(P(d)=\left\{\begin{array}{ll}0&d\leq q\\ 1&d>q\end{array}\right.\) & \(q\) \\ Type 3: V-shape criterion & \(P(d)=\left\{\begin{array}{ll}0&d\leq p\\ \frac{d}{p}&0\leq d\leq p\end{array}\right.\) & \(p\) \\ Type 4: Level criterion & \(P(d)=\left\{\begin{array}{ll}1&d>p\\ \frac{1}{2}&q\leq d\leq p\\ 1&d>p\end{array}\right.\) & \(p,q\) \\ Type 5: V-shape with indifference criterion & \(P(d)=\left\{\begin{array}{ll}0&d\leq q\\ \frac{d-q}{p-q}&q\leq d\leq p\\ 1&d>p\end{array}\right.\) & \(p,q\) \\ Type 6: Gaussian criterion & \(P(d)=\left\{\begin{array}{ll}0&d\leq 0\\ 1-e^{-\frac{d^{2}}{2^{2}}}&d>0\end{array}\right.\) & \(s\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Types of preference function. For all the criteria, a classification is available for the various alternatives necessary to define the so-called outranking flows, which are the fundamental units for the PROMETHEE methodology. Each alternative \(a\) faces \((n-1)\) other alternatives that belong to the generic space A. The two following outranking flows are defined:  
\[\left\{ \begin{array}{l}\Theta^{+}(a)=\frac{1}{n-1}\sum\limits_{x\in A}\pi(a,x) \\ \Theta^{-}(a)=\frac{1}{n-1}\sum\limits_{x\in A}\pi(x,a)\end{array} \right., \tag{5}\]  
where \(x\) represents the deviation of the specific preference function with respect to the same function of preference for the other alternatives. \(\Theta^{+}(a)\) expresses how alternative \(a\) outranks all the others, otherwise \(\Theta^{-}(a)\) expresses how alternative \(a\) is outranked by all the others. The higher \(\Theta^{+}(a)\) (lower \(\Theta^{-}(a)\)) is, the more likely alternative \(a\) is strongest; otherwise, alternative \(a\), compared to the others, is weakest when \(\Theta^{+}(a)\) assumes small values. Once these two flows have been defined, it becomes very simple to make comparisons between alternatives and subsequently establish their order. PROMETHEE offers several ways to view the results; the main ones are illustrated below:  
* PROMETHEE I Partial Ranking: This is a partial ranking of the alternatives, based on positive and negative flows, and includes preferences, indifference, and incompatibility. This scheme allows, therefore, to compare, where possible, the alternatives and establish their partial order of preference through the indices and the related outranking flows. * PROMETHEE II Complete Ranking: This is useful when the decision maker needs a complete hierarchy among the alternatives of the problem. In this case, the alternatives will be compared in relation to their net flow \(\Theta(a)=\Theta(a)^{+}-\Theta^{-}(a)\). PROMETHEE II allows a complete classification of the alternatives; however, it is less realistic and poor in information as it eliminates any possible factor of incomparability between the different alternatives. * PROMETHEE Table: This displays the \(\Theta\), \(\Theta^{+}\), and \(\Theta^{-}\) scores. The actions are ranked according to the PROMETHEE II complete ranking. * PROMETHEe Rainbow: This is a diagram that allows one to highlight, for each alternative, the criteria that positively or negatively affect the final result. * Profile of alternatives: This is a diagram that shows, for each alternative, the net flow \(\Theta\) of each criterion.