## 3 Methodology
### Alternatives Ranking Employing Different MCDM Approaches
\begin{table}
\begin{tabular}{c c} \hline \hline
**Method Name** & **Description and Calculation Characteristics** \\ \hline ARAS [26] & The Additive Ratio Assessment (ARAS) method is a Multiple Criteria Decision Making (MCDM) method that aims to simplify complex decision-making problems and select the “best” alternative through the relative indicator (utility degree), which can reflect the difference between the alternatives and the ideal solution, and eliminate the influence of different measurement units. ARAS uses a bottom-up approach, starting with the lowest level of the hierarchy and constructing outranking relations at the upper levels, allowing for a decomposed analysis of the results, not only at one level, but also at the intermediate levels of the hierarchy. \\ \hline SAW [32] & The Simple Additive Weighting (SAW) method is a simple, intuitive, and easy-to-apply MCDM method. It assigns weights to criteria, based on their relative importance (the weights should sum up to 1). The weighted scores for each alternative are calculated by summing the products of the criteria weights and the alternative performance values. The sum normalisation method is applied for final alternatives rating. The alternative with the highest total score is considered the best option. \\ \hline TOPSIS [32] & The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method has gained popularity in addressing multi-criteria decision-making problems, due to its ability to consider the shortest distance of each alternative to the ideal and the maximum distance to the worst solutions, thereby allowing for a more nuanced evaluation of the alternatives. This method obviates the need for transforming minimized (maximized) criteria into maximized (minimized) criteria. Moreover, the results of this method are obtained on a scale from 0 to 1, making it easy to convert into percentage values. \\ \hline PROMETHEE (Preference Ranking Organization METHod for Enrichment Evaluations) is an MCDM approach which works by calculating the difference between each alternative’s performance and the best and worst performances observed in the set of alternatives for each criterion. PROMETHEE uses pairwise comparisons to determine the relative importance of each criterion. Priority function is applied to aggregate the differences for each criterion into a single overall preference value, where 0 means that the alternative is strongly inferior to another, and 1 means that it is superior. The normalised difference is weighted according to the weight of each criterion, and summed to a total. The algorithm calculates the difference twice, reversing the order of the alternatives among themselves and recording the intermediate results as positive and negative values. The final result is calculated as the difference between the positive and negative values. It can be either positive or negative. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Review of MCDM methods used for the alternatives ranking. Step 4. The modified decision matrix D* and weight vector \(\omega^{*}\) are saved and processed by \(\tau\) different MCDM approaches. The calculated results are stored in the next empty row of the data matrices \(M_{\tau}\). Step 5. Steps 3-4 are repeated 10,000 times. Step 6. For each row \(v\) in the data matrices \(M_{\tau}\), the alternative \(i\) ranks \(r_{vi}^{\tau}\) are determined, where 1 determines the best rank in a row. Step 7. The difference of the first rank vector \(r_{0i}^{\tau}\) values and the rest of the 10,000 vectors in data matrices \(M_{\tau}\) is compared separately, capturing the frequency of the original rank repetition for each of the alternatives. The frequency values are stored in a separate matrix \(L^{\tau}=\left(l_{ki}^{\tau}\right)\), where \(k=1,\ldots,10\). Step 8. Steps 3-7 are repeated 10 times. Step 9. Calculation of the stability value \(S_{\tau}\) for each of the \(\tau\) methods.