## 3 Materials and Methods
### Methods
#### 3.2.6 TOPSIS Procedure Based on Belief Structure Model Adapted for Survey Data  
Let us consider a scenario in which respondents evaluate a set of objects \(O=\{O_{1},O_{2},\ldots O_{m}\}\) in relation to various criteria \(C=\{C_{1},C_{2},\ldots C_{n}\}\). They provide their assessments using an ordinal scale with \(N-\) categories \(H_{1},H_{2},\ldots H_{N}\) with an additional response: \"Don't know/No Answer/Refuses\". The evaluation of the \(i\)-th object with respect to the \(j\)-th criterion can be represented by a BS model as follows:  
\[S_{ij}=\Big{\{}\Big{(}H_{k};\beta_{ijk}\Big{)};\;k=1,\ldots,N\Big{\}}, \tag{8}\]  
where \(H_{k}\) is the \(k\)-th evaluation grade and is assumed that \(H_{k}\) is ranked higher than \(H_{k+1}\). In this model \(\beta_{ijk}=\frac{n_{ijk}}{n_{ij}}\) represents the fraction of respondents who chose category \(k\) for the \(i\)-th object with respect to the \(j\)-th criterion. Here, \(n_{ijk}\) is the number of respondents who assigned category \(k\) to the \(i\)-th object for the \(j\)-th criterion, while \(n_{ij}\) is the total numberof respondents who provided an answer for the evaluated \(i\)-th object concerning the \(j\)-th criterion. It is important to note that \(\sum_{k=1}^{N}\beta_{k}\leq 1\). The term \(\beta_{H}=1-\sum_{k=1}^{N}\beta_{k}\), i.e., the degree of ignorance, indicates the proportion of respondents who selected the \"Don't know/No Answer/Refuses\" option, meaning they did not provide a clear answer. TOPSIS [24] is a widely recognized method for multi-criteria decision-making that evaluates alternatives based on their proximity to an ideal solution [23; 25; 27; 28]. In the context of evaluation questionnaires, the TOPSIS procedure will be adapted to utilize the Belief Structure model, which effectively integrates respondents' evaluations of alternatives across various criteria. When applying TOPSIS with BS, the process unfolds in several key steps:  
Step 1: Formulate the Belief MCDM Problem and Construct the Belief Decision Matrix  
Begin by formulating the Belief MCDM problem and constructing the Belief Decision Matrix as follows:  
\[BM= \begin{bmatrix}S_{ij}\end{bmatrix}  \tag{9}\]  
where each entry \(S_{ij}\) represents the belief structure values for the \(j\)-th criterion pertaining to the \(i\)-th object, \(i=1,\ldots m,\ j=1,2,\ldots,n\). Step 2: Normalize the Incomplete BS Model in the Belief Decision Matrix  
If the BS model is incomplete, meaning that the sum of the belief degrees \(\sum_{k=1}^{N}\beta_{k}<1\), normalization is necessary to account for the missing or uncertain information. The normalized belief decision matrix is calculated as follows:  
\[BCM= \begin{bmatrix}SC_{ij}\end{bmatrix}  \tag{10}\]  
where \(SC_{ij}\) represents the center of gravity applied to the BS model using Formula (4) for the \(i\)-th alternative and \(j\)-th criterion, ensuring proper normalization. By applying the center of gravity, the incomplete BS model is normalized, ensuring that the sum of the belief degrees is adjusted to account for any missing information. This step ensures that all objects are evaluated consistently, even when some data are incomplete. Step 3: Determine the Utility of Evaluation Grades and Generate the Similarity Matrix  
Determine the utility \(U(H_{k})\) for each evaluation grade \(H_{k}\), ensuring that the values satisfy the condition \(0\leq U(H_{k})\leq 1\) for \(k=1,\ldots N\). This utility function quantifies the satisfaction level associated with each evaluation grade. Next, the Similarity Matrix \(\widetilde{S}\ (i=1,\ldots m,\ j=1,2,\ldots,n)\) is computed using Formula (6). Step 4: Evaluate the Weights of the Criteria  
Assign weights to the criteria in the form of the vector:  
\[w=[w_{1},\ldots,w_{n}], \tag{11}\]  
where each weight \(w_{j}>0\) for \(j=1,\ldots,n\). It is essential that the sum of the weights satisfies the condition: \(\sum_{j=1}^{n}w_{j}=1\). These weights reflect the relative importance of each criterion in the decision-making process.