## 5 The IDEMATEL Algorithm
\right.. \tag{8}\]  
_Step 6._ From Theorem A1 (Appendix A), it follows that the sequence of powers of the initial direct-relation matrix \(\overline{\mathbf{D}}\) converges to zero. This implies that the series that defines the total influence matrix: \(\mathbf{T}=\overline{\mathbf{D}}+\overline{\mathbf{D}}^{2}+\ldots=\sum_{m=1 }^{\infty}\overline{\mathbf{D}}^{m}\) converges. Hence, the total influence matrix \(\mathbf{T}=\left(t_{ij}\right)\) is calculated as:  
\[\mathbf{T}=\overline{\mathbf{D}}\big{(}\mathbf{I}-\overline{\mathbf{D}}\big{)} ^{-1}. \tag{9}\]  
The total influence matrix \(\mathbf{T}\) captures the complete picture of how criteria influence each other, considering both direct and indirect effects. It serves to produce the causal diagram map. _Step 7._ The row sums and column sums of the total influence matrix \(\mathbf{T}\) are calculated in the vectors \(\mathbf{a}=(a_{1},a_{2},\ldots,a_{n})\) and \(\mathbf{b}=(b_{1},b_{2},\ldots,b_{n})\). The entries of \(\mathbf{a}\) and \(\mathbf{b}\) are calculated as follows:  
\[a_{i}=\sum_{j=1}^{n}t_{ij};i=1,2,\ldots,n;\ b_{j}=\sum_{i=1}^{n}t_{ij};j=1,2, \ldots,n, \tag{10}\]  
The entries of the importance vector denoted by \(\mathbf{r}^{+}=\left(r_{1}^{+},r_{2}^{+},\ldots,r_{n}^{+}\right)\) and the entries of the relation vector denoted by \(\mathbf{r}^{-}=\left(r_{1}^{-},r_{2}^{-},\ldots,r_{n}^{-}\right)\) are calculated as follows:  
\[r_{i}^{+}=a_{i}+b_{i}\text{ and }r_{i}^{-}=a_{i}-b_{i}\,, \tag{11}\]  
The element \(r_{i}^{+}\) represents the overall influence of criterion Ci, encompassing both the influence it exerts and the influence it receives. A higher \(r_{i}^{+}\) value indicates greater importance within the system. Meanwhile, the element \(r_{i}^{-}\) determines the causal nature of the criterion. If \(r_{i}^{-}\) is positive, \(C_{i}\) has a stronger influence on other criteria than it receives, placing it in the cause group. Conversely, if \(r_{i}^{-}\) is negative, \(C_{i}\) is more influenced by other criteria, categorizing it within the effect group. By plotting the vectors \(\mathbf{r}^{+}\) and \(\mathbf{r}^{-}\), the data can be visualized in a causality diagram, which illustrates the relationships between criteria. In this diagram:  
The horizontal axis (Importance) represents \(\mathbf{r}^{+}\), indicating the overall significance of each criterion. The vertical axis (Cause/Effect Relation) corresponds to \(\mathbf{r}^{-}\), distinguishing between cause-and-effect groups. Criteria with positive \(\mathbf{r}^{-}\) values belong to the cause group, meaning they exert more influence than they receive. Conversely, criteria with negative \(\mathbf{r}^{-}\) values fall into the effect group, indicating they are more influenced by other factors. _Step 8._ The criteria weights are calculated in the vector \(\mathbf{w}=(w_{1})\). The entries of \(\mathbf{w}\) are calculated as  
\[w_{i}=r_{i}^{+}/\sum_{\mathbf{r}^{-}\mathbf{r}^{+}_{k}}^{\mathbf{r}^{-}_{k}}. \tag{12}\]  
_Step 9._ Determine a threshold value to generate the Impact Relations Map (IRM), a directed graph representing the influence between criteria. In some cases, incorporating all values from matrix \(\mathbf{T}\) into the IRM may result in an overly complex diagram, making decision-making difficult. To simplify the representation, a threshold is applied to exclude negligible influences. Only criteria with influence values in matrix \(\mathbf{T}\) exceeding the threshold are included in the IRM, ensuring a clearer and more meaningful visualization of key relationships. Two methods are considered for computing the threshold: the arithmetic mean method and the arithmetic mean--standard deviation (MSD) method. _Arithmetic Mean Method_  
In this approach, the threshold is set as the arithmetic mean \(\mu\) of matrix \(\mathbf{T}\). The entries of the matrix \(\mathbf{G}=\left(g_{ij}\right);i,\;\;j=1,2,\ldots,n\) are computed as  
\[\mu=\left(\sum_{j=1}^{n}\sum_{i=1}^{n}t_{ij}\right)/n^{2};\;g_{ij}= \begin{cases} 1\;\text{if}\;\;t_{ij}\geq\mu\\ 0\;\text{if}\;\;t_{ij}<\mu\end{cases} .