## 4 Results
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multicolumn{1}{c}{**City**} & **Country** & **City Group** & **Population** & \(\mathbf{D^{+}}\) & \(\mathbf{D^{-}}\) & **BTOPSIS** & **Range** & **QSL** & **Range** \\ \hline Vilnius & LT & Eastern MS & 3 & 0.4609 & 0.7221 & 0.6104 & 42 & 92.76\% & 17 \\ \hline Warszawa & PL & Eastern MS & 4 & 0.4943 & 0.6879 & 0.5819 & 56 & 90.43\% & 36 \\ \hline Wien & AT & Western MS & 4 & 0.3587 & 0.7841 & 0.6861 & 5 & 87.96\% & 52 \\ \hline Zagreb & HR & Eastern MS & 3 & 0.4788 & 0.6899 & 0.5903 & 54 & 86.38\% & 56 \\ \hline Zürich & CH & EFTA & 3 & 0.3091 & 0.8336 & 0.7295 & 1 & 97.08\% & 1 \\ \hline  & Source: [19; 47]. Notes: Country codes: BE: Belgium; BG: Bulgaria; CZ: Czechia; DK: Denmark; DE: Germany; EE: Estonia; IE: Ireland; EL: Greece; ES: Spain; FR: France; HR: Croatia; IT: Italy; CY: Cyprus; LV: Latvia; LT: Lithuania; LU: Luxembourg; HU: Hungary; MT: Malta; NL: Netherlands; AT: Australia; PL: Poland; PT: Portugal; RO: Romania; SI: Slovenia; SK: Slovakia; FI: Finland; SE: Sweden; ME: Montenegro; AL: Albania; CH: Switzerland; IS: Iceland; MK: North Macedonia; NO: Norway; RS: Serbia; TR: Turkey; UK: United Kingdom; MS: Member States. Population: 1. Less than 250,000 inhabitants; 2: Between 250,000 and 500,000 inhabitants; 3: Between 500,000 and 1,000,000 inhabitants; 4: Between 1,000,000 inhabitants; 5: More than 5,000,000 inhabitants. \\ \hline \end{tabular}
\end{table} 
Table 6: _Cont._\(BT(0.7;\ 0.4)\) (BTOPSIS) lie between \(BT(0.6;\ 0.3)\) and \(BT(0.8;\ 0.5)\) (Figure 3). We can also observe that the distribution trends of the values are similar across all variants of BTOPSIS. The evaluation of the quality of life in European cities, based on eight variants of BTOPSIS, is presented in Tables A1 and A2 (see Appendix A). The results show a solid Pearson correlation (greater than 0.0991) between the values of the different BTOPSIS variants, as well as a strong Spearman correlation (greater than 0.0994) between the rankings of these variants (see Appendix A, Tables A3 and A4). This indicates a high level of stability in the rankings obtained using these utility functions. Notably, for all nine variants of BTOPSIS, the top three positions in the ranking are occupied by Zurich, Groningen, and Aalborg, while the bottom three are Palermo, Skopie, and Tirana. The above nine variants of the BTOPSIS method draw our attention to changes in the evaluation and ranking of alternatives due to extreme modifications in the form of the utility functions, but they are unable to effectively determine the scale of these changes across the entire feasible range we assumed for \(H_{2}\) and \(H_{3}\). Therefore, a comprehensive sensitivity analysis would necessitate the use of a simulation technique. We used the Monte Carlo simulation approach to generate various forms of utility functions by randomly drawing the \(H_{2}\) and \(H_{3}\) values from their feasible ranges with uniform probability, using 5000 replications. For each replication with potentially different utility functions, we calculated the BTOPSIS scores for all 83 alternatives and established their rank order. Our primary interest was the stability of the rankings obtained in this way. Stability can be measured in various ways. The most straightforward approach is determining the probability distribution of each city occupying different rank positions. If utility functions had no influence on the rankings, we would expect a degenerate distribution for each city, assigning a probability of 1 to a single rank value. Given the 83 cities in our example, the most straightforward way to analyze these distributions is to visualize the cumulative distributions of rank positions for all cities and examine their shapes. A degenerate distribution would correspond to a \"single step\" in the cumulative distribution, while multiple steps indicate ambiguity in rank position.