## 3 Methodology
### Apply TOPSIS for Final Ranking  
The TOPSIS method is a widely used MCDM approach that refines rankings by considering proximity to the ideal solution, aiming at the shortest geometric distance to the positive-ideal solution (PIS) and the longest geometric distance to the negative-ideal solution (NIS) [54]. TOPSIS is valued for its simplicity, computational efficiency, and ability to provide clear, interpretable results. It starts by normalizing the decision matrix to eliminate the effects of different measurement units, and weights are applied to the criteria to reflect their relative importance. With a maximum value for benefit criteria or a minimum value for cost criteria, as appropriate, we define an ideal solution and determine an opposite or negative-ideal solution; the maximum or minimum value to be defined depends on the specific criterion depending on whether maximization or minimization is considered. For example, if one wants to study the safety of a building for emergencies and the criterion is the number of evacuation routes, the objective for that criterion will be the maximization of the number of routes; in this case, the PIS is the maximum value of available choices, and the minimum value is the NIS. The distance of each alternative from the ideal and negative-ideal solutions is calculated using the Euclidean distance. A relative closeness coefficient is computed for each alternative to indicate its proximity to the ideal solution, and alternatives are ranked, with the highest-ranking option being the closest to the ideal solution and the farthest from the negative-ideal solution.  
Steps:  
1. Normalize the decision matrix according to (12): \[\overline{\xi_{ij}}_{TOPSIS}=\frac{\xi_{ij}}{\sqrt{\sum_{i=1}^{m}\xi_{ij}^{2 }}}\] (12)
2. Obtain the weighted normalized decision matrix (\(\
u_{ij}\)) according to (13): As discussed above, we use CRITIC to calculate the weight criteria: \[\
u_{ij}=w_{j}\times\overline{\xi_{ij}}_{TOPSIS}\] (13)
3. Identify the PIS and NIS, considering the sets \(J^{+}\) and \(J^{-}\) of criteria with a positive and negative impact, respectively: * PIS: Maximum values for benefit criteria and minimum values for cost criteria: \[PIS=\{\
u_{1}^{+},\,\ldots,\,\
u_{n}^{+}\}\;\text{where}\;\
u_{j}^{+}=\frac{ \max(\
u_{ij}),\;j\in J^{+}}{\min(\
u_{ij}),\;j\in J^{-}}\] (14) PIS (positive-ideal solution): \(\{\
u_{1}^{+},\ldots,\
u_{n}^{+}\}\), where \(\bullet\quad\
u_{j}^{+}=\max(\
u_{ij})\) for beneficial criteria (\(j\in J^{+}\)); \(\bullet\quad\
u_{j}^{+}=\min(\
u_{ij})\) for cost criteria (\(j\in J^{-}\)).
* NIS: Opposite of the PIS: \[NIS=\{\
u_{1}^{-},\,\ldots,\,\
u_{n}^{-}\}\;\text{where}\;\
u_{j}^{-}=\frac{ \min(\
u_{ij}),\;j\in J^{+}}{\max(\
u_{ij}),\;j\in J^{-}}\] (15) NIS (negative-ideal solution): \(\{\
u_{1}^{-},\ldots,\
u_{n}^{-}\}\), where \(\bullet\)\(\
u_{j}^{-}=\min\bigl{(}\
u_{ij}\bigr{)}\) for beneficial criteria (\(j\in J^{+}\)); \(\
u_{j}^{-}=\max\bigl{(}\
u_{ij}\bigr{)}\) for \(\cos\operatorname{t}\) criteria \(\bigl{(}j\in J^{-}\bigr{)}\).
4. Calculate the separation measures: \[S_{i}^{+} =\sqrt{\sum_{j=1}^{n}\left(\
u_{ij}-\
u_{j}^{+}\right)^{2}}\] (16) \[S_{i}^{-} =\sqrt{\sum_{j=1}^{n}\left(\
u_{ij}-\
u_{j}^{-}\right)^{2}}\] (17)
5. Determine the relative closeness: \[C_{i}=\frac{S_{i}^{-}}{S_{i}^{+}+S_{i}^{-}}\] (18) where \(S_{i}^{+}\) is the Euclidean distance of alternative \(i\) from the PIS: \(\sqrt{\sum_{j=1}^{n}\left(\
u_{ij}-\
u_{j}^{+}\right)^{2}}\); \(S_{i}^{-}\) is the Euclidean distance of alternative \(i\) from the NIS: \(\sqrt{\sum_{j=1}^{n}\ \left(\
u_{ij}-\
u_{j}^{-}\right)^{2}}\); \(C_{i}\) is the relative closeness to the ideal solution, defined as \(\frac{S_{i}}{S_{i}^{+}+S_{i}}\). This produces a refined ranking.