## 2 Multiple-Criteria Decision-Making
### The TOPSIS Method  
The basis of TOPSIS is to rank the alternatives or to discover the best alternative by simultaneously minimizing the distance to the positive ideal solution and maximizing the distance from the negative ideal solution [30]. The positive ideal solution (\(a^{+}\)) is shaped as a mixture of the best performance values of any alternative for each criterion, whilst the negative ideal solution (\(a^{-}\)) corresponds to the mixture of the worst performance values. Afterwards, the procedure follows by computing the separations of each alternative \(a_{i}\) from the positive and negative ideal solutions, \(d_{i}^{+}\) and \(d_{i}^{-}\), using the \(N\)-dimensional Euclidean distance. Finally, the relative proximity to the ideal solution is computed as \(R_{i}=d_{i}^{-}/(d_{i}^{+}+d_{i}^{-})\). Note that \(R_{i}\in[0,1]\) because \(d_{i}^{+}\geq 0\) and \(d_{i}^{-}\geq 0\). Then, the alternatives can be ranked using this index in decreasing order, without the need for criterion preferences to be independent [31]. Let us assume an MCDM problem with \(M\) alternatives and \(N\) criteria represented as a decision matrix (Table 1); then, the TOPSIS method can be defined following the steps of Algorithm 1. It is worth noting that the alternatives are completely ranked based on their global utilities and, on the other hand, the criterion preferences are not required to be independent [30]. ```
1: Compute the normalized decision matrix, where the normalized value \(n_{ij}\) of the original score \(z_{ij}\) is computed as \(n_{ij}=\frac{z_{ij}}{\sqrt{\sum\limits_{i=1}^{M}z_{ij}^{2}}}\quad i=1,\ldots,M \quad j=1,\ldots,N\). 2: Compute the weighted normalized values \(v_{ij}=w_{j}z_{ij}\), where \(w_{j}\) denotes the weight of the criterion \(c_{j}\) and \(\sum_{j=1}^{N}w_{j}=1\)
3: Compute the positive and negative ideal solutions \(a^{+}=\{v_{1}^{+},\ldots,v_{N}^{+}\}=\{(\max\limits_{j}v_{ij}|i\in I),(\min \limits_{j}v_{ij}|i\in J)\}\), \(a^{-}=\{v_{1}^{-},\ldots,v_{N}^{-}\}=\{(\min\limits_{j}v_{ij}|i\in I),(\max \limits_{j}v_{ij}|i\in J)\}\), where \(I\) and \(J\) are associated with benefit and cost criteria, respectively
4: Compute the separation of each alternative from the positive and negative ideal solutions \(d_{j}^{+}=\sqrt{\sum\limits_{j=1}^{N}(v_{ij}-v_{j}^{+})^{2}}\quad\text{and} \quad d_{j}^{-}=\sqrt{\sum\limits_{j=1}^{N}(v_{ij}-v_{j}^{-})^{2}}\quad\ i=1, \ldots,M\). ```  
**Algorithm 1** TOPSIS```
5: Compute the relative proximity to the ideal solution. The relative closeness of the alternative \(a_{i}\) with respect to \(a^{+}\) is defined as \(R_{i}^{+}=\frac{d_{i}^{-}}{d_{i}^{+}+d_{i}^{-}}\) \(i=1,\ldots,M\)
6: Rank alternatives based on the decreasing order of \(R_{i}^{+}\)
```

**Algorithm 1**_Cont_. ### The PROMETHEE Method

The PROMETHEE methodology [32] intends to select the best alternatives (PROMETHEE I) or to sort the alternatives based on their values over different criteria (PROMETHEE II). As an outranking relations technique, the PROMETHEE method quantifies a ranking through the pairwise comparisons (differences) of alternatives \((a_{i},a_{j})\) to determine the preference index \(\pi(a_{i},a_{j})\in[0,1]\), which reflects how \(a_{i}\) is preferred to \(a_{j}\) on criterion \(c_{k}\). The calculation of the preference index is based on the specification of the normalized weights \(w_{k}\) and the preference functions \(P_{k}(a_{i},a_{j})\) for each criterion \(c_{k}\). The idea of this index is similar to that of the global concordance index in the ELECTRE methodology: the higher the preference index is, the higher the strength of the preference for \(a_{i}\) over \(a_{j}\). On the other hand, the PROMETHEE methodology also makes use of the concepts of positive and negative preference flows [33]: the positive preference flow \(\phi^{+}(a_{i})\) evaluates how a given alternative \(a_{i}\) outranks the remaining alternatives, and the negative preference flow \(\phi^{-}(a_{i})\) measures how an alternative \(a_{i}\) is outranked by all the other alternatives.