## 3 Materials and Methods
### Methods
#### 3.2.6 TOPSIS Procedure Based on Belief Structure Model Adapted for Survey Data
Numerous approaches for determining weights are available in the literature, generally grouped into three main types: subjective, objective, and integrated [66; 67; 68; 69]. Subjective weights rely on expert judgment, objective weights are derived from the statistical properties of the data, and integrated weighting systems combine both approaches to capture expert insights alongside data-driven accuracy. In socio-economic multi-criteria analysis, assigning weights to criteria often involves subjective decisions due to the absence of a universally accepted method [21; 70; 71]. Weights are typically determined based on factors such as economic relevance, data reliability, and availability, and they can significantly influence the final results. The OECD's Handbookon Constructing Composite Indicators provides recommendations on weighting indicators using both formal methods, such as factor analysis, data envelopment analysis, analytic hierarchy processes, and conjoint analysis, as well as approaches based solely on expert opinion [72]. In their review of well-being indices, Decancq and Lugo [71] classify weighting approaches into three types: data-driven, normative, and hybrid, each with distinct advantages and limitations. Step 5: Determine the Positive Ideal Belief Solution (PIBS) and Negative Ideal Belief Solution (NIBS)  
In this step, identify the Positive Ideal Belief Solution (PIBS) \(A^{+}\) and the Negative Ideal Belief Solution (NIBS) \(A^{-}\). These represent the best and worst possible alternatives based on the Belief Structure model for each criterion. The Positive Ideal Belief Solution (PIBS) \(A^{+}\) is defined as follows:  
\[A^{+}=\left\{S_{1}^{+},\ldots,S_{n}^{+}\right\} \tag{12}\]  
where \(S_{j}^{+}\) represents the best belief structure model for the \(j\)-th criterion, \(j=1,2,\ldots,n\). The Negative Ideal Belief Solution (NIBS) \(A^{-}\) is defined as follows:  
\[A^{-}=\left\{S_{1}^{-},\ldots,S_{n}^{-}\right\} \tag{13}\]  
where \(S_{j}^{-}\) represents the least favorable (worst) belief structure model for the \(j\)-th criterion. These solutions are derived by selecting the belief structure models corresponding to each criterion's most and least desirable outcomes. \(S_{j}^{+}\) maximizes the utility for the \(j\)-th criterion, while \(S_{j}^{-}\)--minimizes it. Step 6: Calculate the Separation Measures from PIBS and NIBS  
For each object \(O_{i}\), calculate the separation measures \(D_{i}^{+}\) and \(D_{i}^{-}\) which represent the distance of the object from the PIBS \(A^{+}\) and the NIBS \(A^{-}\), respectively. The separation measure from the PIBS, \(D_{i}^{+}\) is calculated as follows:  
\[D_{i}^{+}=D_{i}^{+}\left(O_{i},A^{+}\right)=\sqrt{\sum_{j=1}^{n}w_{j}d_{BS} \left(S_{ij},S_{j}^{+}\right)^{2}} \tag{14}\]  
The separation measure from the NIBS, \(D_{i}^{-}\) is calculated as follows:  
\[D_{i}^{-}=D_{i}^{-}\left(O_{i},A^{-}\right)=\sqrt{\sum_{j=1}^{n}w_{j}d_{BS} \left(S_{ij},S_{j}^{-}\right)^{2}} \tag{15}\]  
where \(w_{j}\) is the weight of criterion \(C_{j}\) and \(d_{BS}\left(S_{ij},S_{j}^{+}\right)\)\(d_{BS}\left(S_{ij},S_{j}^{-}\right)\) are belief distance measures between two \(BS\) models \(S_{ij}\) and \(S_{j}^{+}\) or \(S_{j}^{-}\), respectively. Step 7: Calculate the Relative Closeness for Each Object  
For each object \(O_{i}\), calculate the relative closeness \(R_{i}\) to the Positive Ideal Belief Solution (PIBS). This value quantifies how close each alternative is to the ideal solution relative to the worst-case solution. The formula for calculating \(R_{i}\) is as follows:  
\[R_{i}=\frac{D_{i}^{-}}{D_{i}^{-}+D_{i}^{+}} \tag{16}\]  
where \(D_{i}^{-}\) is the separation measure of option \(O_{i}\) from the NIBS, and \(D_{i}^{+}\) is the separation measure of alternative \(O_{i}\) from the PIBS, \(i=1,2,\ldots,m\). A higher value of \(R_{i}\) indicates that the alternative is closer to the ideal solution and further from the worst-case scenario. Step 8: Rank the Objects  
Based on the relative closeness values \(R_{i}\) rank the objects in descending order.