## 4 Results  
In this section, we used BTOPSIS to rank cities based on their quality of life, using data gathered from a survey questionnaire [47]. The process begins by formulating the problem as a multi-criteria decision-making task, where the data from the questionnaire are represented in the form of belief structures (BS). The evaluation criteria are divided into four distinct grades: \(H_{1}\) (Very satisfied), \(H_{2}\) (Rather satisfied), \(H_{3}\) (Rather unsatisfied), and \(H_{4}\) (Very unsatisfied). Here, the grades \(H_{1},\ H_{2},\ H_{3}\), and \(H_{4}\) align with the ordinal measurement scale (from 1 to 4), with \(H_{k}\) representing the \(k\)-th evaluation grade, and it is assumed that \(H_{k}\) is more preferred than \(H_{k+1}\). Each object is evaluated by assigning a belief degree to each grade. Note that the degree of ignorance, \(\beta_{H}\), represents the portion of the assessment where respondents did not provide a clear answer, corresponding to the category \"Don't know/No Answer/Refused\" (see Formula (2)). Figure 2: The steps in the BTOPSIS method. For example, in the case of Zurich and the criterion related to public transport, the belief structure S = {(very satisfied, 0.7598), (rather satisfied, 0.1852), (rather unsatisfied, 0.0474), (very unsatisfied, 0.0054)} (see Table 4). The degree of ignorance is equal to \(1-(0.7598+0.1852+0.0474+0.0054)=0.0022\). Equivalently, following Formula (3), the belief structure of Zurich can be represented as the vector \(B=[0.7598,\,0.1852,\,0.0474,\,0.0054]\). Once such belief structures are created for all cities and all criteria, these evaluations are arranged into a belief decision matrix \(BM=\left[S_{ij}\right]\)\((i=1,\ldots,\,83;\,j=1,\ldots,10)\) (see Formula (9)). To ensure consistency across different criteria, the belief matrix \(BM\) is then normalized to \(BCM=\left[SC_{ij}\right]\) according to Formula (4). For example, for Zurich and satisfaction with public transport, the center of gravity for the belief structure (SC) is as follows: SC = {(very satisfied, 0.7604), (rather satisfied, 0.1858), (rather unsatisfied, 0.0479), (very unsatisfied, 0.0060)} or equivalently, the belief vector (\(BC\)) can be represented as \(BC=[0.7604,0.1858,0.0480,0.0060]\). Following this, utility values are assigned to each of the four grades to quantify their importance. For this analysis, the utility values are as follows [46]:  
\[U(H_{1})=1,\,U(H_{2})=0.7,\,U(H_{3})=0.4,U(H_{4})=0. \tag{17}\]  
These values reflect the desirability of each grade, where \"Very satisfied\" carries the highest utility and \"Very unsatisfied\" carries no utility. These utility values generate a similarity matrix to capture how closely different evaluations align. The similarity matrix is then generated using Formulas (5) and (6) and takes the following form:  
\[\overset{\sim}{S}= \begin{bmatrix}1&0.7&0.4&0\\ 0.7&1&0.7&0.3\\ 0.4&0.7&1&0.6\\ 0&0.3&0.6&1\end{bmatrix} . \tag{18}\]  
The similarity matrix \(\overset{\sim}{S}\) shows the relationships between the different satisfaction levels, where higher values indicate greater similarity between evaluations. The studies [70; 71; 73] demonstrate that equal weighting is the most commonly adopted approach in practice. They suggest that this choice may arise from a belief in the inherent importance of each criterion, the simplicity it offers in model construction or a limited understanding of causal relationships between indicators. Moreover, equal weighting is often used when there is no consensus on alternative methods, when decision-makers hold differing views, or when statistical and empirical data are insufficient to justify differentiated weights. Therefore, in our case, we assign equal weights to balance the contributions of each criterion in the overall evaluation. This approach ensures that all aspects of quality of life are considered equally important in determining the rankings of the cities. The next step in the BTOPSIS method involves defining the Positive Ideal Belief Solution and the Negative Ideal Belief Solution.