# GIS-Based Optimal Route Selection of Submarine Cables Considering Potential Seismic Fault Zones
## 2 Materials and Methods
### Analytical Hierarchical Process  
A complex decision-making system consists of several sub-factors (i.e., criteria) that have to be comprehensively weighted according to their importance in the system. Nevertheless, determining the weight values constitutes a challenging task, since it is usually influenced by the experience and the knowledge of the experts [46]. For this reason, unbiased multi-criteria decision methodologies have been gradually developed. The rationality of weight distribution has been improved, since the aforementioned methodologies are based on scientific procedures that efficiently decompose complex decision-making systems into sub-factors. These sub-factors are stratified by a dominance relationship, which results in the desired weight values after a pairwise comparison. The analytic hierarchy process (AHP), which was introduced by [PERSON] in the 1970s [47], constitutes a consistent multi-criteria methodology that is based on decision-making theory and has been proven very effective in a wide variety of real case studies that involve several interrelated objectives [48].  
When utilizing AHP, the criteria of the project are identified, and a \(n\)\(\times\)\(n\) criteria comparison matrix \(A\) is constructed as follows:  
\[\left[A\right]=\left[a_{ij}\right]=\begin{array}{c}C_{1}\\ C_{2}\\ \vdots\\ C_{n}\end{array}\begin{array}{c}C_{1}\\  
It is noted that \(W\) determines which of the adopted criteria is the most dominant for the design process.  
A fundamental aspect of AHP is to ensure that the assigned evaluation factors are consistent. Hence, the numerical measures, namely the consistency index (\(CI\)) and consistency ratio (\(CR\)), have been established, aiming to guarantee the existence of the transitive property of matrix \(A\), i.e., if criterion X is more important than \(\Upsilon\), and \(\Upsilon\) is more important than \(Z\), then X should be more important than \(Z\). Initially, vector \(W_{s}\) is defined as the product of \(A\) and \(W\), and subsequently, consistency vector \(C_{v}\) is derived by dividing each cell value of the matrix multiplication result vector by the corresponding priority vector cell as follows:  
\[\{W_{s}\}=[A]\{W\} \tag{4}\]  
\[\{C_{v}\}=\{W_{s}\}\bigg{\{}\frac{1}{W}\bigg{\}} \tag{5}\]  
The eigenvalue of the algebraic system \(\lambda\) is obtained from the average of the elements of consistency vector \(C_{v}\) and it is utilized to calculate the value of \(CI\). Moreover, the \(CR\) is calculated as:  
\[CR=\frac{CI}{RI} \tag{6}\]  
where \(RI\) denotes the random index, which represents the \(CR\) of a randomly generated matrix and its value depends on the number of criteria. \(RI\) can be derived from related publications [47]. Finally, if the resulted \(CR\) is lower than 0.1, the pairwise comparison matrix is consistent, which means that there are no pair combinations that would be unreasonable [51].  
The authors have taken advantage of the advanced capabilities and high functionality of GIS to develop customized tools that are suitable for the needs of the examined applications using the Python site package for analysis and data management, ArcPy. Accordingly, a Python script has been programmed in order to successfully integrate the aforementioned matrices and relationships of AHP into the developed GIS model.