## 3 Background
### Analytic Hierarchy Process (AHP)  
AHP is a means of decomposing a complicated problem into a hierarchy of simpler and more manageable sub-problems [45]. These sub-problems are usually called decision factors and are weighted according to their relative dominance to the problem. The bottom decision factors are actually the solution alternatives (Figure 4).  
AHP selects the solution alternative with the largest synthesized weight. AHP performs following four main steps:  
* Decomposition;
* Pairwise comparison;
* Local weight calculation;
* Weight synthesis.  
Structuring a problem as a hierarchy of multiple criteria is the first step of implementing AHP. The decision factors of the problem are identified and inserted into the hierarchy. The overall objective is placed at the topmost node of the hierarchy. The subsequent nodes present the decision factors. The solution alternatives are located at the bottom nodes. The second step is the evaluation stage, where each factor is compared to all other factors within the same parent. The comparison results within each parent are presented in a square matrix. The pairwise comparisons are finished through answering the question: \"Which of  
Figure 4: Analytic Hierarchy Process model.  
the two is more important?\" and, second: \"By how much?\". A fundamental 1 to 9 scale is used in AHP to allow the decision-maker to express the strength of preference based on his intuition, experience, and knowledge [46]. Thus, we obtain a reciprocal matrix where the entries are symmetrical with respect to the diagonal. The strategy decomposes the original problem into many smaller sub-problems and deals with the latter [47]. The notation for the AHP method is in the Table 2.  
Formally, the pairwise comparisons are collected into a pairwise comparison matrix,  
\[A=\left(a_{ij}\right)_{n\ X\ n}\text{ whit }a_{ij}>0 \tag{1}\]  
According to [PERSON]'s theory, each entry is supposed to approximate the ratio between two weights.  
\[a_{ij}\approx\ \frac{\omega_{i}}{\omega_{j}}\quad\forall i,j \tag{2}\]  
This means that, if the entries exactly represent the ratios between weights, then matrix \(A\) can be expressed in the following from:  
\[A=\left(\omega_{i}/\omega_{i}\right)_{n\ X\ n} \tag{3}\]  
For the eigenvector, we take matrix \(A\), for which entries are exactly obtained as ratios between weights, and multiply it by w, which obtains:  
\[A\mathbf{w}=\begin{bmatrix}\omega_{1}/\omega_{1}&\omega_{1}/\omega_{2}&\cdots &\omega_{1}/\omega_{n}\\ \omega_{2}/\omega_{1}&\omega_{2}/\omega_{2}&\ldots&\omega_{2}/\omega_{n}\\ \vdots&\vdots&\ddots&\vdots\\ \omega_{n}/\omega_{1}&\omega_{n}/\omega_{2}&\cdots&\omega_{n}/\omega_{n}\\ \end{bmatrix}\begin{bmatrix}\omega_{4}\\ \vdots\\ \omega_{n}\\ \end{bmatrix}=\begin{bmatrix}n\omega_{4}\\ \vdots\\ \omega_{n}\\ \end{bmatrix}=n\mathbf{w} \tag{4}\]  
We know that a formulation of the kind \(A\mathbf{w}=n\mathbf{w}\) implies that \(n\) and \(\mathbf{w}\) are an _eigenvalue_ and an eigenvector of \(A\), respectively. Moreover, by knowing that the other eigenvalue of \(A\) is 0, and it has multiplicity \(\left(n-1\right)\), then we know that \(n\) is the largest eigenvalue of \(A\). Hence, if the entries of \(A\) are ratios between weights, then the weight vector is the eigenvector of \(A\) associated with the eigenvalue \(n\).