## 2 Materials and Methods
### Methodology
#### 2.2.2 Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS)  
The TOPSIS method was created by [PERSON] and [PERSON] in 1981 [26] and introduced in 1992 by [PERSON] and [PERSON] [27]. It is one of the strongest techniques for solving multi-attribute decision making (MADM) within Discrete MCDM. The selected alternatives should have the shortest distance from the Positive Ideal Solution (PIS) and the longest distance from the Negative Ideal Solution (NIS) [28]. This method is quite useful when the decision making is faced with several qualitative and quantitative factors [29].  
Among the relevant characteristics considered for the use of TOPSIS in this investigation, the following are noteworthy:  
* TOPSIS evaluates the relative closeness of each alternative to an ideal solution, making it straightforward to understand and implement [30].
* TOPSIS is compensatory, meaning that poor performance in one criterion can be compensated by good performance in another, which is often more realistic in practical decision-making scenarios [31].
* TOPSIS requires the normalization of criteria to ensure comparability. This step helps in handling criteria measured in different units and scales, allowing a fair comparison among alternatives [32].
* Criteria weights can be assigned based on their importance, reflecting the decision-maker's priorities. This flexibility allows customization of the decision-making process according to specific needs [32].
* TOPSIS provides a clear ranking of alternatives, facilitating easy interpretation of results. The ranking helps decision-makers to understand the relative performance of each alternative quickly [33].  
The primary advantage of the TOPSIS method is that it ranks \(X\) options based on \(N\) criteria, aligning with the objective of our research. The TOPSIS method can be succinctly explained through the following phases [34; 35]:  
_Phase 1: Calculate the normalized performance ratings._  
Vector normalization is applied to obtain normalized performance ratings from Matrix \(X_{ij}\), which represents the performance ratings for each alternative against each attribute. Standardizing the collected indicator data is essential.  
Matrix \(X_{ij}\)  
\[X_{ij}=\begin{bmatrix}X_{11}&\cdots&X_{ij}\\ \vdots&X_{22}&\vdots\\ X_{i1}&\cdots&X_{ij}\end{bmatrix} \tag{2}\]  
Vector \(Y_{ij}\)  
\[Y_{ij}=\frac{X_{ij}}{\sqrt{\sum_{k=1}^{i}(X^{2})_{ij}}} \tag{3}\]  
_Phase 2: Integrate weight with ratings._  
A normalized vector \(Y_{ij}\)  
\[Y_{ij}=\begin{bmatrix}Y_{11}&\cdots&Y_{ij}\\ \vdots&Y_{22}&\vdots\\ Y_{i1}&\cdots&Y_{ij}\end{bmatrix} \tag{4}\]  
The weighted and normalized \(v_{ij}\), for the specific weights, will use the scores obtained through the PCA method, when the PCA method is used.  
\[v_{ij}=W_{j}*Y_{ij} \tag{5}\]The weighted and normalized decision Matrix \(V_{ij}\).  
\[V_{ij}=\begin{bmatrix}V_{11}&\cdots&V_{ij}\\ \vdots&V_{22}&\vdots\\ V_{i1}&\cdots&V_{ij}\end{bmatrix} \tag{6}\]  
_Stage 3: Find positive and negative ideal solutions._  
Determine the positive values \(A\)* and \(A^{-}\) from the matrix \(V_{ij}\) as per Equation (5).  
\[A^{*}=\left(V_{1}{}^{*},V_{2}{}^{*}\cdot\ldots\ V_{j}{}^{*}\right) \tag{7}\]  
\[A^{-}=\left(V_{1}{}^{-},V_{2}{}^{-}\cdot\ldots\ V_{j}{}^{-}\right) \tag{8}\]  
_Stage 4: Obtain the separation values._  
The separation measure is the distance of each alternative rating from both the positive and negative ideal solutions, obtained by applying the Euclidean Distance Theory.  
\[S_{i}{}^{*}=\sqrt{\sum_{j=1}^{j}\left(v_{ij}-v_{j}{}^{*}\right)^{2}} \tag{9}\]  
and  
\[S_{i}{}^{-}=\sqrt{\sum_{j=1}^{j}\left(v_{ij}-v_{j}{}^{-}\right)^{2}} \tag{10}\]  
_Stage 5: Calculate the overall preference score._  
The ranking is obtained from the overall uniqueness score \(V_{i}\) for each alternative \(A_{i}\).  
\[V_{i}=\frac{S_{i}{}^{-}}{S_{i}{}^{-}+S_{i}{}^{*}} \tag{11}\]