## 5 The IDEMATEL Algorithm  
The IDEMATEL method will be used in the analysis of criteria used in the selection of a GenAI tool for academic research. The algorithm of this method is presented in the following. The algorithm aims to analyze and visualize the causal relationships between a set of criteria. It combines expert opinions with mathematical calculations to create a causal diagram. _Step 1._ Form a group of experts \(E=\left\{E_{1},E_{2},\ldots,E_{p}\right\}\) who possess knowledge and experience relevant to the criteria being evaluated. _Step 2._ Define a comprehensive set of criteria (factors) \(C=\left\{C_{1},C_{2},\ldots,C_{n}\right\}\) that will be analyzed for their causal relationships. _Step 3._ Each expert \(E_{k}\) evaluates the influence of each criterion \(C_{i}\) on every other criterion \(C_{j}\) and generates the initial direct-relation matrix \(\mathbf{D}^{(k)}=\left(d_{ij}^{(k)}\right);i,j=1,2,\ldots,n;k=1,2,\ldots,p\). The entry \(d_{ij}^{(k)}\) represents expert \(k\)'s assessment of the influence of criterion \(C_{i}\) on criterion \(C_{j}\). The principal diagonal entries of matrix \(\mathbf{D}^{(k)}\) are equal to zero: \(d_{ii}^{(k)}=0\). The assessments are made using a predefined DEMATEL scale: 0: No influence, 1: Very low influence, 2: Low influence, 3: High influence and 4: Very high influence. _Step 4._ Combine the individual direct-relation matrices from all experts into an aggregate direct-relation matrix \(\mathbf{D}=\left(d_{ij}\right)\). This is achieved by averaging the corresponding elements \(\mathbf{D}^{(k)}\) across all expert matrices:  
\[d_{ij}=\frac{1}{p}\sum_{k=1}^{p}d_{ij}^{(k)}. \tag{1}\]  
_Step 5._ Normalize the aggregate direct-relation matrix \(\mathbf{D}\) to create a normalized direct-relation matrix \(\overline{\mathbf{D}}=\left(\overline{d}_{ij}\right)\). Denote \(Q=\left\{z\in\mathbb{C}:|z|=1\right\}\). Let \(p\) and \(s\) be defined as follows:  
\[p=\max(\max_{1\leq i\leq n}(\sum_{j=1}^{n}d_{ij});\max_{1\leq j\leq n}(\sum_{i =1}^{n}d_{ij})\right), \tag{2}\]  
\[s=\left\{ \begin{aligned} & p\;\;\text{if}\;\;\det\Bigl{(}z \mathbf{I}-\frac{1}{p}\mathbf{D}\Bigr{)}\
eq 0\;\;\text{for every}\;\;z\in Q\\ & p+\varepsilon\;\;\text{if}\;\;\det\Bigl{(}z\mathbf{I}-\frac{1} {p}\mathbf{D}\Bigr{)}=0\;\;\text{for some}\;\;z\in Q\end{aligned} \right. \tag{3}\]  
where \(\varepsilon\) is a small positive number and \(\mathbf{I}\) is the identity matrix. Build the \(n\times n\) normalized initial direct-relation matrix \(\overline{\mathbf{D}}=\left(\overline{d}_{ij}\right)\) as  
\[\overline{d}_{ij}=\frac{d_{ij}}{s}\;\text{for all}\;i,j \tag{4}\]  
In matrix \(\overline{\mathbf{D}}\), all principal diagonal elements are equal to zero. Some details for computing parameter \(s\) will be given in the following. Consider the characteristic polynomial of the matrix \(\frac{1}{p}\mathbf{D}\):  
\[f(z)=\det\biggl{(}z\mathbf{I}-\frac{1}{p}\mathbf{D}\biggr{)},\;z\in\mathbb{C}. \tag{5}\]Note that \(f\) is a polynomial function with real coefficients of degree \(n\), that is  
\[f(z)=\sum_{q=0}^{n}a_{q}z^{q}, \tag{6}\]  
with \(a_{q}\in\mathbb{R}\). The largest eigenvalue of the matrix \(\frac{1}{p}\mathbf{D}\) is the spectral radius of this matrix. It is equal to the greatest absolute value of the roots of polynomial \(f\). One can easily see that \(\rho\left(\frac{1}{p}\mathbf{D}\right)<1\) if and only if polynomial \(f\) has no roots on \(Q\). In order to see that \(f\) has a root in \(Q\) we shall check if the absolute value of \(f\) vanishes on \(Q\). This is equivalent to the fact that the function  
\[g(t)=\left(\sum_{q=0}^{n}a_{q}\text{cos}(qt)\right)^{2}+\left(\sum_{q=0}^{n}a_ {q}\text{sin}(qt)\right)^{2},\ t\in[0,2\pi), \tag{7}\]  
has a root in the interval \([0,2\pi)\). We shall define \(s\) as follows:  
\[s=\left\{ \begin{aligned} p&\text{ if }\ \ \text{g}(t)\
eq 0 \text{ for every }\ t\in[0,2\pi)\\ p&+\varepsilon&\text{ if }\ \ \text{g}(t)=0 \text{ for some }\ t\in[0,2\pi)\end{aligned}