## 2 Materials and Methods
### Description of the Proposed Decision-Support Tool
#### 2.2.2 Multi-Criteria Decision Method and Weight Factors  
A complex decision-making system involves multiple sub-factors (i.e., criteria) that must be meticulously assessed based on their significance within the system. However, assigning weighting values to these factors is a crucial topic, as it often relies on the expertise and knowledge of the experts involved. In response to this challenge, unbiased MCDMs have been developed to effectively address various interrelated objectives. Practically, MCDMs \"translate\" the qualitative preferences of designers into quantitative weight factors by utilizing specific scientific procedures that systematically decompose complex decision-making systems into sub-factors. AHP, developed by [PERSON] in the 1970s [56], provides a consistent framework for evaluating alternatives and selecting the best course of action by considering both qualitative and quantitative factors. Its flexibility and ability to handle complex decision-making systems have made AHP a well-known and highly effective MCDM method that has been extensively applied in numerous complex and multidisciplinary case studies [57]. Through this process, decision-makers are able to prioritize the criteria based on their importance, leading to a more systematic and structured approach to decision-making. After identifying the criteria for the problem, the following n \(\times\) n comparison matrix (i.e., judgment matrix), A, is constructed:  
\[[A]=[a_{ij}]= \begin{bmatrix}C_{1}&C_{2}&...&C_{n}\\ C_{2}&a_{21}&1&...&a_{2n}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ C_{n}&a_{n1}&...&...&1\\ \end{bmatrix} \quad\quad i,j=1,2,3\,...,n. \tag{1}\]  
where \(C_{i}\) (\(i=1,2,3\ldots,n\)) denotes the considered criteria along the matrix rows and columns, while n represents the number of criteria. Each cell of the judgment matrix includes the ranking score, \(\alpha_{ij}\), which has been qualitatively assigned to each criterion based on an evaluation scale ranging from 1/9 to 9. It should be noted that the values of \(\alpha_{ij}\) and its transpose \(\alpha_{ji}\) are inversely proportional, i.e., \(a_{ji}=1/a_{ij}\). A ranking score equal to 9 indicates that the criterion in the row of the matrix is much more important than the criterion in the column, whereas the reciprocal value (i.e., 1/9) implies the opposite. A value equal to 1 indicates that the two criteria are of the same importance and appear along the diagonal of the matrix. Having defined the judgment matrix, A, the qualitative factors (i.e., ranking score) are then normalized and averaged to obtain an average weight for each criterion through pairwise comparisons. More specifically, the ranking scores in every column of A are summed up, and subsequently, each cell of A is divided by the sum of the specific columns, thus forming the normalized criteria comparison matrix A as follows:  
(2)  
where, \(\Sigma_{k}=a_{1j}+a_{2j}+\ldots+a_{nj}\ k,j=1,2,3\ldots,n\) is the sum of each column of A. In the sequence, the priority vector W, which includes the weights for each criterion, is derived after calculating the average of each row according to:  
\[\{W\}=\overline{W_{i}}=\sum_{j=1}^{n}\overline{a_{ij}}\qquad i,j=1,2,3\ldots,n. \tag{3}\]  
It should be noted that W determines which of the adopted criteria is the most dominant for the design process. Numerical measures, such as the consistency index (CI) and consistency ratio (CR), have been established to evaluate the consistency of the hierarchy.