## 5 The IDEMATEL Algorithm
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Criteria** & **FO** & **UA** & **SI** & **QS** & **CV** & **SC** & **RR** \\ \hline FO & 0.000 & 3.000 & 3.000 & 4.000 & 4.000 & 2.000 & 3.000 \\ UA & 2.667 & 0.000 & 2.000 & 3.000 & 2.000 & 2.000 & 2.333 \\ SI & 2.667 & 2.000 & 0.000 & 2.000 & 2.000 & 2.000 & 4.000 \\ QS & 3.333 & 3.000 & 3.000 & 0.000 & 3.000 & 2.333 & 2.667 \\ CV & 3.667 & 2.000 & 2.000 & 3.000 & 0.000 & 4.000 & 3.333 \\ SC & 1.000 & 2.000 & 2.000 & 2.000 & 2.000 & 0.000 & 3.667 \\ RR & 2.000 & 2.000 & 2.000 & 2.000 & 2.000 & 2.000 & 0.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Initial direct-relation matrix \(\mathbf{D}\) representing the influence scores among the criteria for selecting GenAI tools. fluence itself. This matrix forms the basis for further normalization and analysis in the IDEMATEL method. Note that in matrix \(\mathbf{D}\), the maximum sums of row elements (19) are equal to the maximum sums of column elements (19). In this case, we have \(p=19\). The characteristic polynomial of matrix \(\frac{1}{p}\mathbf{D}\) is  
\[f(z)=\det\left(z\mathbf{I}-\frac{1}{p}\mathbf{D}\right)=z^{7}+(7.078\times 10^ {-16})z^{6}-0.3896z^{5}-0.1674z^{4}-0.0313z^{3}-0.003z^{2}-0.00015z-2.988\times 1 0^{-6}.\]  
Since the coefficient of \(z^{6}\) is extremely small (approximately zero), the polynomial can be simplified to:  
\[f(z)=z^{7}-0.3896z^{5}-0.1674z^{4}-0.0313z^{3}-0.003z^{2}-0.00015z-2.988\times 1 0^{-6}\]  
The maximum value of the roots of the characteristic polynomial is equal to 0.8070 (approximately 0.80699641). This represents the largest eigenvalue of the matrix or the spectral radius of matrix \(\frac{1}{p}\mathbf{D}\). Since \(\rho\left(\frac{1}{p}\mathbf{D}\right)\cong 0.80699641<1\), it follows that the powers of \(\frac{1}{p}\mathbf{D}\) tend to zero. In this case, we have \(s=p=19\) and \(\overline{\mathbf{D}}=\frac{1}{p}\mathbf{D}\). The normalized initial direct-relation matrix \(\overline{\mathbf{D}}\) is displayed in Table 6. Each value of Table 6 represents the normalized direct influence of one criterion on another, scaled between 0 and 1 according to the IDEMATEL method. The normalization ensures comparability across criteria and is a key step in deriving the total influence matrix for further analysis. Since the powers of \(\overline{\mathbf{D}}\) tend to zero, it follows that the matrix \(\mathbf{T}\) can be computed. The total influence matrix \(\mathbf{T}\) is displayed in Table 7.  
This matrix reveals the overall impact pathways within the system and is essential for identifying the most influential and dependent criteria in the decision-making process. The non-zero diagonal entries in Table 7 indicate the presence of indirect self-influence resulting from the interconnected structure of the criteria, while the zero diagonal entries in Tables 5 and 6 reflect the absence of direct self-influence by definition. The sums of elements from the rows (vector \(\mathbf{a}\)) and from columns (vector \(\mathbf{b}\)) of matrix \(\mathbf{T}\) are calculated. Then, the \(\mathbf{r^{+}}\) (importance vector), \(\mathbf{r^{-}}\) (relation vector), and \(\mathbf{w}\) (criteria weights) are determined (Table 8). \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Criteria** & **FO** & **UA** & **SI** & **QS** & **CV** & **SC** & **RR** \\ \hline FO & 0.000000 & 0.157895 & 0.157895 & 0.210526 & 0.2105263 & 0.105263 & 0.157895 \\ UA & 0.140351 & 0.000000 & 0.105263 & 0.157895 & 0.105263 & 0.105263 & 0.122807 \\ SI & 0.140351 & 0.105263 & 0.000000 & 0.105263 & 0.105263 & 0.105263 & 0.210526 \\ QS & 0.175439 & 0.157895 & 0.157895 & 0.000000 & 0.157895 & 0.122807 & 0.140351 \\ CV & 0.192982 & 0.105263 & 0.105263 & 0.157895 & 0.000000 & 0.210526 & 0.175439 \\ SC & 0.052632 & 0.105263 & 0.105263 & 0.105263 & 0.105263 & 0.000000 & 0.192982 \\ RR & 0.105263 & 0.105263 & 0.105263 & 0.105263 & 0.105263 & 0.105263 & 0.000000 \\ \hline \end{tabular}
\end{table} 
Table 6: The normalized initial direct-relation matrix \(\overline{\mathbf{D}}\).