## 4 Safety and Sustainability Assessments of U-Shaped Stair Projects: A Case Study
### Safe and Sustainable Stair-Project Selection, Applying the Novel CORST Methodology  
Using the same decision matrix presented in Table 5, and the criteria weight vector depicted in Table 9, alternatives A1-A9 were rated, using four different MCDM approaches: ARAS, SAW, TOPSIS and PROMETHEE. Alternative rating results \(\rho_{i}\) for each of the alternative \(i\) are presented in Table 10.  
The evaluation results for the alternatives, calculated using different MCDM methods, are obtained on different measurement scales; therefore, the normalisation of the values presented was carried out using Equation (8), to obtain values \(\widetilde{\rho}_{i}\) (Table 11). According to the algorithm, the final stability values \(S_{\tau}\), for each of the analysed MCDM methods, are calculated as the median of 10 attempts. The method stability coefficients \(\Phi_{\tau}\) were determined by normalizing the stability values \(S_{\tau}\) which were determined for the ARAS, SAW, TOPSIS, and PROMETHEE approaches (Equation (7)). Their values are presented in Table 12.  
The analysis of the results presented in Table 12 shows that the ARAS and SAW approaches show similar method stability coefficients (namely, \(\Phi_{\text{ARAS}}\) = 0.281, \(\Phi_{\text{SAW}}\) = 0.276, and TOPSIS \(\Phi_{\text{TOPSIS}}\) = 0.262), while PROMETHEE shows the lowest robustness toward minor changes in criteria valuations (\(\Phi_{\text{PROMETHEE}}\) = 0.181). The novel CORST methodology presented in this paper was applied to make a final decision for the most suitable stair-project selection, considering stair safety and sustainability. According to Equation (9), values \(\widetilde{\rho}_{i}\) are multiplied by the method stability coefficient \(\Phi_{\tau}\) of the specific MCDM method. The products are summed from the alternatives to give \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Method** & **A0** & **A1** & **A2** & **A3** & **A4** & **A5** & **A6** & **A7** & **A8** & **A9** \\ \hline ARAS & 0.278 & 0.008 & 0.272 & 0.010 & 0.012 & 0.270 & 0.011 & 0.008 & 0.118 & 0.012 \\ \hline SAW & 0.010 & 0.387 & 0.011 & 0.015 & 0.384 & 0.014 & 0.009 & 0.157 & 0.014 \\ \hline TOPSIS & 0.236 & 0.970 & 0.455 & 0.584 & 0.970 & 0.659 & 0.512 & 0.607 & 0.584 \\ \hline PROMETHEE & \(-0.559\) & \(1.568\) & \(-0.613\) & \(-0.511\) & \(1.542\) & \(-0.511\) & \(-0.559\) & 0.201 & \(-0.559\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Multicriteria decision-making results \(\rho_{i}\), calculated by applying ARAS, SAW, TOPSIS and PROMETHEE approaches. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Method** & \(S_{\tau}^{\mathbf{1}}\) & \(S_{\tau}^{\mathbf{2}}\) & \(S_{\tau}^{\mathbf{3}}\) & \(S_{\tau}^{\mathbf{4}}\) & \(S_{\tau}^{\mathbf{5}}\) & \(S_{\tau}^{\mathbf{6}}\) & \(S_{\tau}^{\mathbf{7}}\) & \(S_{\tau}^{\mathbf{8}}\) & \(S_{\tau}^{\mathbf{9}}\) & \(S_{\tau}^{\mathbf{10}}\) & \(S_{\tau}\) & \(\Phi_{\tau}\) \\ \hline ARAS & 77.57 & 77.32 & 77.85 & 77.44 & 77.54 & 77.61 & 77.69 & 77.37 & 77.49 & 77.46 & 77.515 & 0.281 \\ \hline SAW & 75.89 & 75.73 & 76.33 & 75.91 & 75.89 & 75.93 & 76.15 & 75.68 & 75.83 & 75.80 & 75.890 & 0.276 \\ \hline TOPSIS & 72.51 & 72.12 & 72.06 & 71.95 & 72.10 & 72.12 & 72.24 & 72.37 & 71.94 & 72.13 & 72.120 & 0.262 \\ \hline PROMETHEE & 49.95 & 49.72 & 49.94 & 49.96 & 49.90 & 49.73 & 49.87 & 49.93 & 50.12 & 49.92 & 49.925 & 0.181 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Calculation of the stability value \(S_{\tau}\) and method stability coefficients \(\Phi_{\tau}\) for the chosen MCDM methods. the combined estimate \(i_{best}(\rho)\), which is also used to identify the best alternative for the analysed problem, by applying the CORST methodology (Table 13). Data presented in Table 13 show that the best inside-stair project with respect to safety and sustainability is A2, which represents U-shaped wooden stairs (width 120 cm) located in the open living room, with natural lightning from the southwest and southeast, and with carpet coating.